Improved Quantum Information Storage and Retrieval The research underpinning this work was supported by the Engineering and Physical Sciences Research Council (EPSRC) through the Distributed Quantum Computing and Applications grant number EP/W032643/1. The invention relates to quantum information storage and retrieval in quantum memory devices, and in particular to improvements in the efficiency of the information storage and/or retrieval. Presently, there is a huge drive to harness quantum mechanics to build new devices that can outperform current technology. One key example is the universal quantum computer, a device that can run algorithms which require significantly less computational steps to complete than their classical counterparts. Included within the umbrella of quantum computers are quantum simulators, where the goal is to efficiently simulate other quantum systems that may be difficult to study directly. This can be done for example with the use of (Gaussian) Boson sampling, or atom trapping. A further notable quantum technology is quantum communication, where quantum information is distributed over a network, such as a quantum internet, allowing for 100% secure communication, along with linking several quantum nodes together. Quantum metrology is another growing discipline, with the potential to drastically improve the sensitivity of measurement devices. For all of the technologies mentioned above, photons are ideal for quantum information processing, at both local and global scales. Photons offer large bandwidths, allowing for a high clock rate of operations, as well as resilience to noise at ambient conditions. Photons also travel at the speed of light, which together with the existing fibre optic infrastructure makes them the obvious choice for global networks. The main limitations of photons, despite significant progress over the years, are the probabilistic generation of quantum states of light and the probabilistic success of photonic entangling gates. In order to overcome the issue of requiring many probabilistic elements to succeed consecutively, a quantum memory saves and buffers the successes, thereby allowing the outputs to be temporally multiplexed. Having the ability to repeat until successful, significantly reduces the resources required to realise quantum technologies. There are several key metrics when evaluating a quantum memory. First and foremost is total efficiency, which quantifies the probability of retrieving information, such as encoded in photons, squeezed states, etc., given that an attempt was made to store the photon. This can further be partitioned into storage and retrieval efficiency, or the efficiency of writing into and subsequent reading out of the memory, respectively. Another key metric is the fidelity of the
quantum memory. This is a measure of how identical the returned photon is compared with the state that was stored. In other words: does storage and retrieval alter the state of the photon? In order to reach high fidelity it is necessary to suppress all possible noise processes that could cause the memory to add to, or alter, the photon upon retrieval. In general, quantum information processing requires incredibly high fidelities of ≃ 99%, and thus necessitates quantum memories to operate with ≥ 99% efficiency and ≥ 99% fidelity. Other important figures of merit are the bandwidth of the memory, the memory lifetime and the related time-bandwidth product. The former refers to the maximum bandwidth of an optical signal that can be stored, where higher bandwidths correspond to faster potential quantum operations. The memory lifetime measures the time taken for the stored atomic coherence to decay, and quantifies the maximum storage time after which the photon can be retrieved from the memory. The time- bandwidth product is defined as the product of the memory lifetime and the bandwidth, quantifying the maximum number of storage events within the lifetime of the memory, and is thus a useful measure of synchronisation capacity. Using cold atomic traps or cryogenic doped crystals can allow for long memory lifetimes on the order of milliseconds, and high efficiencies have been demonstrated (> 92%), albeit with sub-megahertz bandwidths and therefore low time-bandwidth products. The main limitation of these platforms is the density of the quantum systems making up the ensemble, typically referred to as optical depth. Larger optical depth allows for greater bandwidths to be stored with higher efficiencies. Alternatively, atomic vapours at temperatures around 100°C offer large optical depths, which has resulted in state-of-the-art measurements showing the storage and retrieval of light at 100MHz bandwidth with 82% efficiency. However, the differing velocities of the atoms within the warm vapour causes atoms that have contributed to the storage process to leave the interaction region, leading to a reduction in memory lifetime. In addition, to store such high bandwidths, the memory protocol is typically operated far-detuned from the atomic transition (e.g. in Raman and ORCA based systems), which requires much larger control field intensities in order to be efficient. To achieve these intensities with limited laser power, the control field is focused to a small beam waist. While this allows large efficiencies to be reached, the smaller interaction region further reduces the lifetime of the quantum memory. A further consequence of higher control field intensities is an increase in noise, along with a distortion of the output photon temporal wave packet, resulting in a reduced fidelity of the quantum memory. To summarise, we require quantum memories with near-unity efficiencies for large photon bandwidths. To achieve this, large optical depths and control field intensities are needed. This makes cold atom traps or cryogenic doped crystals, with great memory lifetime performance,
impractical due to the low optical depths achievable in these systems. With warm vapours we can achieve high optical depth, albeit at the expense of atoms leaving the interaction region during storage, since atoms have a non-zero velocity at room temperature. Moreover, the high control field intensities required for warm vapours imply narrow beam sizes, thus shrinking the interaction region (region where light interacts with atoms) and reducing the lifetime of the memory since atoms quickly leave the region. On top of this, high control field intensities lead to unwanted noise processes, reducing the fidelity of the memory. Finally, there is also a scalability argument in that for future quantum technologies many quantum memories operating in parallel will be needed, therefore it is desirable to use fewer resources. Summary of invention This invention, focuses on quantum memories in which a control field reversibly maps arbitrary photonic states to long-lived matter based coherences, such that it is possible to retrieve the photon on-demand at a later time. Examples include, e.g. Raman, Electromagneticaly Induced Transparency (EIT), Autler-Townes Splitting (ATS), Atomic Frequency Comb (AFC), and Gradient Echo Memory (GEM), among others. These types of memories utilise a memory having one or many quantum systems (e.g. and ensemble of atoms), which can absorb light under control. An laser pulse, referred to as the control pulse, mediates this mapping of the photonic field into the memory, thereby generating atomic coherence. In the case of the ensemble, this is of the form of coherence between atoms in the ground state and a small subset of the ensemble (e.g. when storing a single photon, one atom) in another state. When the light needs to be retrieved a second control pulse reverses the operation, resulting in the emission of light from the quantum system(s) (thus bringing all atoms in the same ground state). In this invention, we propose the use of a novel type of quantum interference, atom-light interference, to enhance the efficiency of quantum memories, while simultaneously requiring less optical depth and also lower control field intensities. This scheme has the potential to significantly improve all of the other metrics discussed in previous paragraphs, and we foresee that it is compatible with a multitude of quantum memory protocols, from warm vapours to cold atoms, and thus go beyond current state-of-the-art quantum memories, making the use of quantum memories in large-scale networks feasible in the near-term future. In general, the invention relates to a set of closely related solutions to the problem of improving storage and/or retrieval efficiency of quantum information using matter-based quantum memory. Each of the solutions makes use of two traversals of a signal field as it passes across the quantum memory, whereby a phase difference between information stored in the quantum
memory and the signal field on the second transition mediates an interference between the wavefunctions describing the stored information and the information encoded on the signal field. The interference between these wavefunctions leads to enhanced storage or retrieval based on the value of the phase difference. In general, the present invention results in an improvement over known methods which use a single application of a control field during a single traversal of the quantum memory device for storage and retrieval. This improvement is because the present invention requires less overall control pulse power, and that the signal field retrieved can have a higher fidelity compared with what was stored by virtue of the presently disclosed use of interference, as discussed in more detail below. In addition, the method requires a smaller quantum memory device (e.g. fewer atoms or other quantum storage elements) due to the improved efficiencies provided by the interference-based methods and apparatuses set out herein. Accordingly, disclosed herein is a method for enhanced optical quantum memory storage, the method comprising: receiving a signal field into an ingress access port of a quantum memory device including one or more matter elements having at least two internal states, the signal field having quantum information encoded thereon; applying a first mapping procedure to the quantum memory device while the signal field traverses the quantum memory device to cause: a first part of the signal field to be stored to encode the quantum information in a superposition of the two internal states of the quantum memory device; and a second part of the signal field to be transmitted and to exit an egress access port of the quantum memory device; redirecting the second part of the signal field into the ingress access port of the quantum memory device to cause the second part of the signal field to traverse the quantum memory device for a second time; and applying a second mapping procedure to the quantum memory device while the second part of the signal field traverses the quantum memory device to cause either constructive quantum interference between waveforms corresponding to the second part of the signal field and the first part of the signal field encoded in the quantum memory device to enhance the storage efficiency of the quantum information in the quantum memory device; or destructive quantum interference between waveforms corresponding to the second part of the signal field and the first part of the signal field encoded in the quantum memory device to enhance a retrieval efficiency of mapping the quantum information stored in the quantum memory device onto the signal field; and wherein the quantum interference is mediated by a phase difference, Θ, between the first and second parts of the signal field as the second part of the signal field traverses the quantum memory device for the second time, wherein the value of Θ is based at least in part on a phase ^^^ accumulated by the second part of the signal field as it is redirected.
Here the use of a second traversal of the quantum memory device with a carefully controlled phase difference in the two transitions allows for the leveraging of interference to enhance the storage efficiency of the information. The mapping procedures may be the same or a different procedure (or the same process with different parameters). The procedures themselves may be provided by lasers, magnetic fields and so forth, depending on the properties of the quantum memory device, discussed in more detail below. Pump lasers may be used in some examples to shape the absorption spectrum of the quantum memory device. The path length taken by the signal beam between the first and second transition of the signal beam through the quantum memory device introduces a phase shift between the first and second parts of the signal field, The interference of the control and signal fields with the stored matter field can be controlled by means of this phase shift. A suitable adjustment causes the signal field to be stored into the memory with higher efficiency than if there were no excitation in the memory. A different setting can cause the stored signal to be read out more efficiently than if there were no signal field present. The constructive and destructive interference of the quantum fields in these two setting enhances both the storage and readout efficiency of the signal light. The incoming signal field may be generated locally or received from further away. In some examples it may include single photons having quantum information encoded thereon, but may also include e.g. coherent states of light as well. Note that depending on the temporal shape, frequency and polarisation of the control field, it is possible to change the spatial distribution of the stored coherence and which part of the temporal envelope of the signal field is transmitted (also which frequency or polarisation components). When optimising this interference method, the control field may be varied to obtain the best spatial distribution of the stored coherence and best temporal envelope of the transmitted signal to maximise the desired interference. In this way, the first and second parts should be thought of as the storage process being only partially effective, leading to partial storage and partial transmission in a controllable manner. A transmitted portion is redirected to interact with the quantum memory device (and the information already partially stored therein) a second time, whereby the interference is carefully controlled to provide constructive interference and enhance the storage efficiency. Note that the phase difference mediating the interference may have optimal values which lead to the maximum constructive interference, but there is a wide range of such values which lead to constructive interference to some degree, and which therefore provide a benefit in terms of increased storage efficiency.
In cases where the second traversal includes destructive interference, the invention allows for the use of the quantum memory device to provide a temporal delay instead of using a delay line. Optionally, the value of Θ is further based on ^^^ combined with at least one of a phase difference ^^ଶ between the first and second procedures; and a phase ^^ଷ induced in
the first part of the signal field after the of the signal field has exited the egress access port and before the second part of the signal field enters the ingress access port of the quantum memory device. The total phase difference is then given by a linear sum of the phase differences from each of the sources ^^^ ; ^^ = {1,2,3}, as well as any other sources of phase difference in the system (such as due to dispersion, or AC stark shift due to intense optical fields, etc.). The interference is mediated by the control field, allowing the phase difference to be altered via control field parameters (as well as other parameters such as path length, etc.), but also for example any phase difference between the two control fields, as well as any phase accumulated by the matter in the quantum memory device, such as due to an applied magnetic field. This allows the maximum of constructive interference to be adjusted by iteratively changing the phase of the stored quantum information, which provides another parameter to be used to adjust the net phase Θ difference to achieve the desired result. In some examples, this means that the additional phase shifts ^^^ ; ^^ = {2,3} represent corrections to the phase necessarily gained by traversing the redirected path, to bring the total phase shift into maximum constructive interference. The phase difference, Θ, measured in radians may be approximately 0 modulo 2^^ in cases where the quantum interference is constructive. This is the phase difference which theoretically leads to maximum constructive interference of the stored coherence and destructive interference of the retrieved optical signal. However, since absolute phase is not directly measurable, in practice this total phase difference is detected by adjusting the components which make up the total phase difference until maximum constructive interference is obtained. In practice, then, the method may in fact (rather than aiming for a specific value for Θ) include calibrating the method (or equivalently, the apparatus as discussed below) using a test signal and measuring the storage efficiency, while adjusting the phase shifts until a maximum efficiency is achieved. This allows a proxy measurement of the total phase shift, but
more importantly allows the calibrated method/apparatus to be used to efficiently store quantum information after the calibration has been performed. Alternatively, where the quantum interference is destructive, the phase difference, Θ, measured in radians may be approximately ^^ modulo 2^^. This is the phase difference which theoretically leads to maximum destructive interference of the stored coherence and constructive interference of the retrieved optical signal. However, since absolute phase is not directly measurable, in practice this total phase difference is detected by adjusting the components which make up the total phase difference until maximum constructive interference is obtained. In practice, then, the method may in fact (rather than aiming for a specific value for Θ) include calibrating the method (or equivalently, the apparatus as discussed below) using a test signal and measuring the storage efficiency, while adjusting the phase shifts until a maximum efficiency is achieved. This allows a proxy measurement of the total phase shift, but more importantly allows the calibrated method/apparatus to be used to efficiently preserve quantum information after the calibration has been performed, for later retrieval and transmission. Where the mapping procedure uses optical fields, then a phase difference between the two instances of the mapping procedure being applied can be used to adjust the total phase difference. Where the mapping field uses a field (e.g. a magnetic field gradient or strain gradient), then if a static field is used (e.g. a static magnetic field or static strain gradient field), then a phase difference results from the signal field travelling the loop, or through varying the phase accumulated for the stored coherence through the application of a magnetic field for instance. These fields (magnetic field and strain gradient) are static during mapping procedure, thus do not allow to change the phase with these fields. The field must therefore be dynamic to allow for phase shifts to be adjusted via the field itself. Also disclosed herein is a method for enhanced optical quantum memory retrieval, the method comprising: applying a mapping procedure to a quantum memory device including one or more matter elements having at least two internal states and having quantum information encoded in a superposition of the two internal states of the quantum memory device, to cause: a first part of the quantum information to be mapped onto a signal field which traverses at least part of the quantum memory device a first time before exiting the quantum memory device at an egress access port; and a second part of the quantum information to remain encoded in the two internal states of the quantum memory device; redirecting the signal field into an ingress access port of the quantum memory device, wherein the redirecting causes the signal field to gain a phase, ^^^; applying a second mapping procedure to the quantum memory device while
the signal field traverses the quantum memory device for a second time to cause destructive quantum interference between waveforms corresponding to the first and second parts of the quantum information to enhance the retrieval efficiency of mapping quantum information from the quantum memory onto the signal field; and allowing the signal field to exit the quantum memory device for a second time; wherein the destructive interference is mediated by a phase difference, Φ, between the signal field as it traverses the quantum memory device for a second time and the second part of the quantum information, wherein the value of Φ is based at least in part on ^^^. Here the use of a second traversal of the quantum memory device with a carefully controlled phase difference in the two transitions allows for the leveraging of interference to enhance the retrieval efficiency of the information. The mapping procedures may be the same or a different procedure (or the same process with different parameters). The procedures themselves may be provided by lasers, magnetic fields and so forth, depending on the properties of the quantum memory device, discussed in more detail below. Pump lasers may be used in some examples to shape the absorption spectrum of the quantum memory device. The path length taken by the signal beam between the first and second transition of the signal beam through the quantum memory device introduces a phase shift between the first and second parts of the quantum information, which can be tuned to bring the stored and transmitted quantum information into maximum destructive interference, thereby providing enhanced retrieval efficiency of the quantum information in the quantum memory device (i.e. minimising storage efficiency of the quantum information in the quantum memory device). This storage example makes use of destructive interference of the stored quantum information to enhance the retrieval efficiency, which corresponds to constructive interference of the leaked signal field. This can be thought of as destructive interference of the probability amplitude associated with the event of memory storage, which results in an improved retrieval efficiency by suppressing the undesirable outcome (further storage of the information stored in the memory). Note that depending on the temporal shape, frequency and polarisation of the control field, it is possible to change the spatial distribution of the stored coherence and which part of the temporal envelope of the signal field is retrieved (also which frequency or polarisation components). When optimising this interference method, the control field may be varied to obtain the best spatial distribution of the stored coherence and best temporal envelope of the transmitted signal to maximise the desired interference. In this way, the first and second parts should be thought of as the retrieval process being only partially effective, leading to partial
retrieval and partial retention of the information in the quantum memory device in a controllable manner. The transmitted portion is redirected to interact with the quantum memory device (and the information remaining partially stored therein) a second time, whereby the interference is carefully controlled to provide destructive interference and enhance the retrieval efficiency. Note that the phase difference mediating the interference may have optimal values which lead to the maximum destructive interference, but there is a wide range of such values which lead to destructive interference to some degree, and which therefore provide a benefit in terms of increased retrieval efficiency. Optionally the value of Φ is further based on ^^^ combined with at least one of: a phase difference ^^ଶ first and second procedures; and a phase ^^ induced in
ଷ
the stored part of the signal field after the has exited the quantum memory device at the egress access port and before the signal field enters the quantum memory device at the ingress access port. The total phase difference is then given by a linear sum of the phase differences from each of the sources ^^^ ; ^^ = {1,2,3}, as well as any other sources of phase difference in the system (such as due to dispersion, or AC stark shift due to intense optical fields, etc.). The interference is mediated by this phase difference, but also for example any phase difference between the two control fields, as well as any phase accumulated by the matter in the quantum memory device, such as due to an applied magnetic field. This allows the maximum of destructive interference to be adjusted by iteratively changing the phase of the stored quantum information, which typically allows finer control of the net phase difference Φ than can be achieved by adjusting the path length of the signal field as it traverses the exiting
and entering the quantum memory device. In some examples, this means that the additional phase shifts ^^^ ; ^^ = {2,3} represent corrections to the phase necessarily gained by traversing the redirected path, to bring the total phase shift into maximum destructive interference. The phase difference, Φ, measured in radians may be approximately ^^ modulo 2^^. This is the phase difference which theoretically leads to maximum destructive interference of the stored coherence and constructive interference of the retrieved optical signal. However, since absolute phase is not directly measurable, in practice this total phase difference is detected by adjusting the components which make up the total phase difference until maximum destructive interference is obtained. In practice, then, the method may in fact (rather than aiming for a specific value for Φ) include calibrating the method (or equivalently, the apparatus as discussed below) using a test signal and measuring the storage efficiency, while adjusting
the phase shifts until a maximum efficiency is achieved. This allows a proxy measurement of the total phase shift, but more importantly allows the calibrated method/apparatus to be used to efficiently retrieve quantum information after the calibration has been performed. Where the mapping procedure uses optical fields, then a phase difference between the two instances of the mapping procedure being applied can be used to adjust the total phase difference. Where the mapping field uses a field (e.g. a magnetic field gradient or strain gradient), then if a static field is used (e.g. a static magnetic field or static strain gradient field), then a phase difference results from the signal field travelling the loop, or through varying the phase accumulated for the stored coherence through the application of a magnetic field for instance. These fields (magnetic field and strain gradient) are static during mapping procedure, thus do not allow to change the phase with these fields. The field must therefore be dynamic to allow for phase shifts to be adjusted via the field itself. Also disclosed herein is a method for enhanced optical quantum memory storage and retrieval, the method comprising: applying any of the methods discussed above for enhanced storage of quantum information in a quantum memory device; allowing a storage time, ^^, to elapse; and applying any of the methods above for retrieving the information to map the stored quantum information onto a signal field. This arrangement demonstrates that the method (and correspondingly the apparatus discussed below) can be used to store information for a time ^^, before retrieving and using that information in a further process such as a quantum computation or a measurement of the information. This allows for the information from one quantum computation to be temporarily stored while quantum information processes are carried out to enact a second quantum computation, and finally for the output of the second quantum computation to be combined with the output of the first quantum computation by retrieving the output of the first quantum computation and enacting a third quantum computation on both outputs. Optionally the signal field is further directed to a detector or to a quantum computing circuit after exiting the quantum memory device for a second time as part of a retrieval process. This allows information retrieved in the retrieval process to be used in calculations, measured as part of an output, and so forth. Optionally the signal field is a low intensity laser pulse or a quantum state of light containing one or more photons, having information encoded thereon. The signal field can be any coherent state, for example a squeezed state. The information may be encoded, for example, on a polarisation state of a photon (or frequency or time bin of a photon) . In yet further
examples, the information may be encoded in a photon-number superposition of 0 +1 for example? In other examples, which the information can be encoded in other degrees of freedom, such as frequency or time bin. Optionally the or each mapping procedure includes applying a control laser pulse, optionally where the signal field is a laser pulse or a quantum state of light containing one or several (e.g. fewer than 1000, fewer than 100, or even 10 or fewer) photons, and wherein the control laser pulse is more intense than the signal field pulse. By “more intense” here, a factor of 105 or 106 is generally considered as the ratio of the mapping procedure pulse to the signal field. However, in some cases, where a very high quality factor optical cavity is used to house the quantum memory device, the pulse used for the mapping procedure may in fact be approximately equal in intensity relative to the signal field. Note that the ratio between the intensity of the signal field and the mapping procedure pulse (along with the optical properties of the quantum memory device) is more important than the absolute magnitude of the intensity of either the signal field or the mapping procedure. The frequencies of the control pulse(s) and the signal field may be selected based on an energy gap between electronic energy states of the quantum memory device. This allows the interactions of the mapping procedure and the signal field to be resonant with certain electronic transitions in the device. For example, the frequencies may be selected to be on resonance with the quantum memory device. For example, the quantum memory device may include individual atoms of a single element (e.g. alkali earth metals); an ensemble of alkali metal atoms, optionally with one or more noble gases; a crystal doped with one or more rare-earth ions; a crystal doped with one or more molecules; a bulk material having one or more defects; a quantum dot substrate; or one or more 2D materials. In each case, an electronic excitation is available in the material, for example: • Transitions between two or more of: a ground state and one or more excited electronic states of atoms, where the memory is based on ensembles of a single element. • Transitions between ground and/or excited electronic states in a quantum dot. • Transitions between electronic states in crystals having molecules or defects. • Transitions between energy levels of different electronic states in 2D materials. By tuning the frequency to correspond to the energy gaps between the energy levels of interest, the information can be efficiently encoded in a resonance between the two states.
Optionally, the frequency of a first one of the control pulse and the signal field is selected based on the energy gap between an excited state energy and a first one of the hyperfine split ground state energies and the frequency of a second, different, one of the control pulse and the signal field is selected based on the energy gap between the excited state and a second, different, one of the hyperfine split ground state energies, or a difference between magnetic sublevels in the same hyperfine ground state. This allows the methods and apparatuses set out herein to operate with known atom-light interaction schemes, such as the Raman, Electromagnetically Induced Transparency (EIT), and Autler-Townes Splitting (ATS) protocols. In some cases, the selection of frequency is based on the energy gap in the sense that the photon energy is approximately equal to the energy gap. This tunes the interactions to the energy gaps, thereby placing us in the EIT or ATS regime. In other cases the frequencies are detuned from the excited state by an amount larger than an inhomogeneous linewidth of the matter in the quantum memory device, as in e.g. a Raman process. This may include Doppler dephasing, for example, which dominates in warm atomic systems, or in cold ensembles of matter other sources of broadening of the energy levels may be present, such as spacing of the hyperfine splitting of the excited states. In other cases, such as where the electronic energy levels are due to dopants or defects in a crystal, broadening of the energy levels is dominated by the interaction of lattice vibrations (phonons) in the crystal with the defect or dopant. Where detuning is present, it should be much larger than the broadening of the excited state, ideally such that there is no overlap between the bandwidth of the signal field and the intermediate state broadening. The larger the overlap in this way, the more the protocol begins to depart from the Raman regime and moves toward the EIT or ATS regime, although there is no exact boundary between these regimes, more of a gradual crossover. An approximate rule of thumb for Raman systems is that the energy (e.g. frequency) of the signal field or mapping procedure is shifted from the exact energy gap by an energy shift (i.e. a frequency shift corresponding to such an energy) which is at least as large as twice the signal bandwidth plus the broadening width of the excited state (e.g. due to Doppler effects, etc. as discussed above). In such a case, a 1GHz signal field and a Doppler width of 500MHz would give a detuning threshold of 500MHz + (2 x 1GHz) = 2.5GHz. Note that the Doppler width depends on the atomic mass, temperature and frequency of the transition, since the first two of these relate to the velocity distribution of the atoms, while the third relates to the baseline value which is scaled by the red-/blue-shift caused by the motion of the atoms –
larger baseline values lead to shifts by larger absolute amounts when a constant shifting factor is applied. I With a simple three level model which is usually appropriate, blue/red detuning and which ground state the signal field and mapping procedure are interacting with doesn’t matter – provided that the mapping procedure is not near resonant with the signal transition. For example, in 87Rb the ground hyperfine splitting is 6.8GHz. If the mapping procedure transition involved the lower hyperfine ground state, and was red detuned from the intermediate excited state by 6.8GHz, then the control would be resonant with the other transition (intermediate excited state to upper hyperfine ground state), which would have a much stronger interaction. The larger the detuning, the weaker the interaction. For more complex realistic systems, certain detuning values can be beneficial for reducing the noise. By detuning Raman systems in this way, we do not need to consider the hyperfine states of the intermediate (i.e. non-ground) states, which can lead to interference and a corresponding reduction in encoding and retrieval efficiency. Optionally, the frequency of a first one of the control laser pulse and the signal laser pulse is selected based on the energy gap between a first excited state energy and a ground state energy and the frequency of a second, different, one of the control laser pulse and the signal laser pulse is selected based on the energy gap between a second excited state energy and the first excited state energy; wherein the first excited state energy is lower than the second excited state energy. This allows the methods and apparatuses set out herein to operate with known atom-light interaction schemes, such as the Off-Resonance Cascaded Absorption (ORCA) and Fast Ladder Memory (FLAME). In some cases, the selection of frequency is based on the energy gap in the sense that the photon energy is approximately equal to the energy gap. This tunes the interactions to the energy gaps, thereby placing us in the FLAME regime. In other cases the frequencies are detuned from the excited state by an amount larger than an inhomogeneous linewidth of the matter in the quantum memory device, as in e.g. an ORCA process. This may include Doppler dephasing, for example, which dominates in warm atomic systems, or in cold ensembles of matter other sources of broadening of the energy levels may be present, such as spacing of the hyperfine splitting of the excited states. In other cases, such as where the electronic energy levels are due to dopants or defects in a crystal,
broadening of the energy levels is dominated by the interaction of lattice vibrations (phonons) in the crystal with the defect or dopant. Similarly, the frequencies may be detuned from the first excited state by an amount larger than the inhomogeneous linewidth of the matter in the quantum memory device. This detuning has the same general effect as discussed above in respect of Raman and EIT/ATS, where small or no detuning leads to the FLAME protocol, and large dephasing (larger than the Doppler width or other broadening plus twice the signal bandwidth, for example) leads to ORCA. Optionally, the laser supplying the control pulse and the laser supplying the signal laser pulse are aligned prior to ingress into the quantum memory device using a partial beam splitter and/or wherein the laser supplying the control pulse and the laser supplying the signal laser pulse are separated after egress from the quantum memory device using a partial beam splitter. This can be achieved by using a polarisation or frequency based filter, for example. This allows the beams to be absolutely aligned to better overlap their cross section on the same parts of the quantum memory device, thereby improving storage and/or retrieval efficiency. As used herein, “aligned” in this context can mean that either beams are copropagating or counter propagating. That is to say that the beams may have vectors which are parallel or antiparallel (i.e. propagating in an aligned manner, but in directly opposite directions). Optionally, each mapping procedure is supplied by the same mapping procedure source. This can be used to make the mapping procedure more efficient by not requiring additional equipment to enact the process. For example the same control pulse could be used for both mapping procedures. In other examples, different sources may be used for the first and second mapping procedures, leading to benefits in improved timing of the two applications of the mapping procedure, or allowing for greater overall power, independently of the required timing since the second mapping procedure would have a tuneable timing with respect to the first mapping procedure, without sacrificing overall power. Optionally, the ingress access port and the egress access port are different access ports arranged on opposed sides of the quantum memory device. This allows the signal field (and e.g. the control laser pulse) to enter and exit the quantum memory device along a single axis, thereby simplifying the optical pathways in the system. In other examples, the access ports may be located in other places, which can allow the signal field to take a longer path through the quantum memory device and thereby increase the optical depth. In other examples the ingress and egress ports are the same access port.
Optionally the value of ^^^ and/or ^^^ is determined by a length of a path taken as the signal field is redirected by optical devices provided along the path. Optionally, the value of
^^^ and/or ^^^ is altering the path length and/or the operational parameters of optical provided along the path. For example, certain materials can be used to introduce a and controlled additional phase, in addition to the phase shift necessarily incurred in travelling a given distance. This can help to control the total phase shift. Optionally, the method further comprises applying a phase shift to the stored quantum information. This can be used to control the relative phase shift. For example, if the phase shift is applied to the stored information between the first and second transitions of the quantum memory by the signal field, then the total phase difference can be calibrated to maximise the constructive or destructive interference (as needed for the storage or retrieval respectively) via the additional phase shift applied to the stored information. Optionally, a path taken during redirection of the signal beam is intrinsically phase-stable. This means that while the phase acquired by the signal field as it traverses the setup can change over time, due to small fluctuations in path length, there is no effect on the interference. This allows for other parameters to be set so as to provide optimal interference (i.e. maximally constructive or maximally destructive, as required), secure in the knowledge that the phase shift due to redirecting along the path will have the same effect each time. To achieve this, the control field could enter at the egress port of the quantum memory device, and travel the same path as the redirected signal to the ingress port of the quantum memory device. In this way, any phase accumulated by the signal field (which could vary each time the memory device is used) is also accumulated by the control field, such that there is no net effect of the phase accumulated while traversing the loop. The phase may then be adjusted for example, through varying the input phase of the control field in order to provide optimal interference. Also disclosed herein is an apparatus for enhanced optical quantum memory storage and/or retrieval, comprising: a quantum memory device including one or more matter elements having at least two internal states; a mapping procedure source arranged to apply a mapping procedure to the quantum memory device; and optics for redirecting a signal field exiting the quantum memory device at an egress access port around a path and into an ingress access port; wherein the optics introduces a phase shift ^^^ as the signal field traverses the path. It will be appreciated that such a system allows for the enactment of the methods discussed above. For example, where information from an incoming signal beam is to be encoded on
the quantum memory device using the above enhanced encoding process, then a first transition is possible before the signal beam exits the quantum memory device and is redirected around the path to re-enter the quantum memory device having picked up a phase ^^^ (here equivalent to ^^^ as used above), which can be tuned to give constructive interference
As a specific example, the apparatus may be adapted for enhanced optical quantum memory storage of quantum information encoded on a signal field by: applying a first mapping procedure to the quantum memory device as the signal field traverses the quantum memory device a first time, causing a first part of the signal field to be encoded in a superposition of the two internal states of the quantum memory device and resulting in a partial storage of the quantum information in the quantum memory device; allowing a second part of the signal field to exit the quantum memory device at the egress access port and using the optics to redirect the second part of the signal field into the ingress access port; and applying a second mapping procedure to the quantum memory device as the second part of the signal field traverses the quantum memory device a second time to cause constructive quantum interference between waveforms corresponding to the second part of the signal field and the first part of the signal field encoded in the quantum memory device, resulting in enhanced storage of the quantum information in the quantum memory device; wherein the constructive interference is mediated by a phase difference, X, between the first and second parts of the signal field as the second part of the signal field traverses the quantum memory device for the second time, wherein the value of X is based at least in part on the phase ^^^ accumulated by the second part of the signal field as it is redirected. Another specific example is where the apparatus is adapted for enhanced optical quantum memory storage of quantum information encoded on a signal field by: applying a first mapping procedure to the quantum memory device as the signal field traverses the quantum memory device a first time, causing a first part of the signal field to be encoded in a superposition of the two internal states of the quantum memory device and resulting in a partial storage of the quantum information in the quantum memory device; allowing a second part of the signal field to exit the quantum memory device at the egress access port and using the optics to redirect the second part of the signal field into the ingress access port; and applying a second mapping procedure to the quantum memory device as the second part of the signal field traverses the quantum memory device a second time to cause destructive quantum interference between waveforms corresponding to the second part of the signal field and the first part of the signal field encoded in the quantum memory device, resulting in enhanced retrieval in mapping the quantum information in the quantum memory device onto the signal field; wherein the
destructive interference is mediated by a phase difference, X, between the first and second parts of the signal field as the second part of the signal field traverses the quantum memory device for the second time, wherein the value of X is based at least in part on the phase ^^^ accumulated by the second part of the signal field as it is redirected. This allows the memory to be used analogously to a stable delay line by storing information in the quantum memory and then allowing retrieval of the information after part of the signal field has traverse a path during redirection. This is more stable than a delay line as it uses the general principals of interference set out herein to introduce a time delay into a signal field. Equally where there is initially information stored in the quantum memory, it is evident that this arrangement is suitable for enhanced retrieval of information from the memory by using the above procedure for leveraging destructive interference and enhancing retrieval of information stored in the quantum memory device. In such cases, the phase shift ^^^ picked up in the redirecting is equivalent to ^^^ as used above. As a specific example, the apparatus may be adapted for enhanced retrieval of quantum information encoded in an optical quantum memory storage by: applying a first mapping procedure to the quantum memory device as the signal field traverses the quantum memory device a first time, causing a retrieved part of the quantum information to be mapped onto a signal field which traverses at least part of the quantum memory device a first time before exiting the quantum memory device at an egress access port; allowing a residual part of the quantum information to remain encoded in the superposition of the two internal states of the quantum memory device; using the optics to redirect the signal field into the ingress access port, thereby introducing a phase shift ^^^between the residual part of the quantum information and the signal field; and applying a second mapping procedure to the quantum memory device as the signal field traverses the quantum memory device a second time to cause destructive quantum interference between waveforms corresponding to the signal field and the residual part of the quantum information which has remained encoded in the quantum memory device, resulting in enhanced retrieval of the quantum information from the quantum memory device; wherein the destructive interference is mediated by a phase difference, X, between the retrieved and residual parts of the quantum information as the signal field traverses the quantum memory device for the second time, wherein the value of X is based at least in part on the phase ^^^ accumulated by the second part of the signal field as it is redirected. It will be appreciated that the apparatus for storage and retrieval makes use of the same elements, and therefore can be thought of as the same apparatus used in different “modes”
to store and/or retrieve information as desired by a used simply by adjusting the operational parameters. As used above, “optics” relates to any apparatus which is capable of performing the redirecting of the signal field in the manner required. This may include lenses, mirrors, beam splitters, waveplates, and so forth. In addition, chip-based devices, such as optical waveguides or optical fibres may form part of the optics as used herein. The optics may include one or more mirrors arranged to direct the signal beam around the path. The spatial location and/or orientation of at least one mirror is adjustable to change the path length and thereby adjust the phase shift ^^. The mirror position and/or orientation may be controlled by a piezoelectric actuator to provide precise control of the phase shift introduced in traversing the path during a redirection. The apparatus may further be adapted to perform any of the methods discussed above, with a corresponding advantage as set out above. The apparatus may further comprise a signal field source arranged to direct a signal field into the quantum memory device. The apparatus may be configured to introduce a further phase difference between quantum information stored in the quantum memory device and the signal beam; optionally wherein the apparatus includes: a magnetic field source for applying a magnetic field to the quantum memory device; or a mapping procedure source having a variable phase. As noted above, these can be used to induce a phase shift in stored information, which provides another parameter to be used to adjust the net phase Θ difference to achieve the desired result. The quantum memory device may include: atoms of a single element selected from the alkali earth metals; an ensemble of alkali metal atoms, optionally provided with one or more noble gases; a crystal doped with one or more rare-earth ions; a crystal doped with one or more molecules; a bulk material having one or more defects; a quantum dot substrate; or one or more 2D materials. As noted above, each of these is a suitable candidate for a quantum memory device because they each have stable electronic excitations which can be used to encode quantum information in a superposition of (at least) two of the internal states. Brief Description of the Figures The invention will now be described with reference to the Figures in which:
Figure 1 illustrates a schematic of the concepts underpinning the present invention, by analogy with a Mach-Zehnder interferometer; Figure 2A illustrates an apparatus for implementing the principles described herein; Figure 2B illustrates timing signals for enacting the processes described herein; Figure 2C illustrates various electronic transition schemes for use in encoding, storing, and retrieving quantum information in/from matter-based quantum memory devices; Figure 3A is a flow chart indicating a method for encoding quantum information carried by a signal field onto a quantum memory device, in accordance with the present disclosure; Figure 3B is a flow chart indicating a method for retrieving quantum information stored in a quantum memory device by mapping it onto a signal field, in accordance with the present disclosure; Figure 4A(a) illustrates the differences between known storage protocols and the enhanced storage protocols disclosed herein; Figure 4A(b) is a plot of the variation in total memory efficiency as phase is varied for known storage protocols and the enhanced storage protocols disclosed herein; Figure 4B(a) illustrates the differences between known retrieval protocols and the enhanced retrieval protocols disclosed herein; Figure 4B(b) is a plot of the variation in a product of storage and retrieval efficiencies as phase is varied for known retrieval protocols and the enhanced retrieval protocols disclosed herein, where here the retrieval efficiency shows a large gain which dominates this plot; Figure 4C(a) illustrates the differences between known write-read protocols and the enhanced write-read protocols disclosed herein; Figure 4C(b) is a plot of the variation in total write-read efficiency as phase is varied for known write-read protocols and the enhanced write-read protocols disclosed herein; Figure 4D is a plot illustrating the theoretical efficiency of known methods compared with the presently disclosed procedures as loop transmission is varied; Figure 5 illustrates a simulation of population density in a warm ensemble of atoms compared with measured experimental data; Figure 6A(a) illustrates a simulated total efficiency for the enhanced storage protocol; Figure 6A(b) illustrates an experimentally measured total efficiency for the enhanced storage protocol; Figure 6A(c) illustrates experimentally measured total efficiency for the enhanced storage protocol as a function of phase; Figure 6B(a) illustrates a simulated total efficiency for the enhanced retrieval protocol; Figure 6B(b) illustrates an experimentally measured total efficiency for the enhanced retrieval protocol;
Figure 6B(c) illustrates experimentally measured total efficiency for the enhanced retrieval protocol as a function of phase; Figure 6C(a) illustrates experimentally measured total efficiency for the enhanced combined (write-read) storage and retrieval protocol; Figure 6C(b) illustrates an experimentally measured total efficiency for the enhanced combined (write-read) storage and retrieval protocol; Figure 6C(c) illustrates experimentally measured total efficiency of enhanced combined (write-read) storage and retrieval efficiency as a function of phase; Figure 7 illustrates a numerical simulation solving for optimising pulse shapes to improve efficiency of the protocol; Figures 8a to 8d provide another illustration of the EEVI memory concept using the beam-splitter analogy; Figures 9a and 9b show an example of an experimental setup for implementing EEVI; Figures 10a to 10c show results and simulations for the experimental setup in Figures 9a and 9b for storage processes; Figures 11a to 11c show results and simulations for the experimental setup in Figures 9a and 9b for retrieval processes; Figures 12a to 12c show results and simulations for the experimental setup in Figures 9a and 9b for Raman processes; and Figures 13 a to 13d show optimised control pulses for Raman and EEVI-Raman processes. Detailed Description Turning now to Figure 1, an illustration of the concepts underpinning the general ideas set out herein is shown. In particular, a beam splitter-like interaction of a quantum memory is used to show how atom-light interference can result in an enhancement in both stored light into the spin wave and retrieved light, thus enhancing the total efficiency of a quantum memory protocol. A standard memory protocol is analogous to a temporal beam splitter between an optical mode ^^ (presented as ^^ in Figure 1) and a spin wave ^^ mode (atomic coherence between the two atomic ground states). Following the beam splitter model, the full memory interaction, mediated by an intense control field, may be written as: ൬^^^௨௧(^^) ^^ (^^) ^^ (^^) ^^ (^^ ^^ (^^)^ = ൬ ^ ଶ ^^ (^^) ^^ ( ^ ൬ ^^ ) ^ ^௨௧ ^ ଶ ^^) ^^^^(^^) where ^^^௨௧(^^), ^^^௨௧(^^), the output photonic field, the
output spin wave, the wave wave and ^^ refers to the frequency
component of the coherences. For the storage interaction, illustrated in Figure 1(a) an input photon is stored into the spin wave with a storage efficiency: ^^^ = ^^^^^|^^^(^^)|ଶ corresponding to reflection in the The non-stored component of the
photon is transmitted through the memory a of: 1 − ^^^ = ^^^^^|^^^(^^)|ଶ The coefficients ^^^(^^) and ^^^ corresponding to a change in
amplitude and phase. an wave mode, the retrieval efficiency into ^^^௨௧(^^) is given by: ^^^ = ^^^^^|^^ଶ(^^)|ଶ
The analogy is illustrated in Figure . ^^^^ (also referred to as a signal field) is partially stored in an atomic ensemble, mediated by a strong control field (also referred to as a mapping procedure), resulting in two output modes: a spin wave ^^^ and a transmitted (un-stored) signal ^^^. The form of ^^^,ଶ and ^^^,ଶ is determined by the temporal shape and phase of the control field (not shown in Fig.1), along with dispersion due to the atoms. Other parameters which affect the value of r and t include the temporal shape of the signal field, the frequency of both the signal and control fields, for ensemble based systems the number density of the matter systems being used to store the light, in an optical cavity the quality factor will contribute to the r and t, the spatial overlap of the control and signal fields Previous approaches to maximise efficiency have relied on a combination of increasing the optical depth of the ensemble, as well as increasing the intensity and temporally shaping the control field or placing the matter element(s) in a cavity. In the protocol disclosed herein, instead the photonic output of the first memory interaction (i.e. the light that was not successfully stored in the memory and was thus transmitted) is looped back into the memory, where a second memory interaction is performed, such that the spin wave and looped around photonic field interfere. An example of the present protocol for memory storage is shown in Figure 1(b), where the non-stored portion of the input photon ^^^(^^) acquires a round-trip phase ^^^ఏ as it is looped back into the front of the memory. In the lower portion of Figure 1(b), we show the full interaction using the beam splitter model, where
it is clear that the protocol achieves something analogous to Mach-Zehnder interference between the stored atomic spin wave ^^^ and the non-stored optical field ^^^. The equivalent beam splitter model is shown where ^^^^ is split into two output modes and where the reflectivity of the beam splitter captures the storage efficiency. The phase acquired by the two modes upon reflection (storage) or transmission (non-storage) is determined by the atoms and the control field. In the upper portion of Figure 1(b) the present protocol applied to a quantum memory is shown. Firstly, ^^^^ is partially stored and we obtain the output modes ^^^ and ^^^. The mode ^^^ is then looped back to the input to the memory, where a second storage interaction takes place. This second storage interaction can be thought of as another beam splitter, (analogously with the lower part of Fig.1(b)), where the modes ^^^ and ^^^ are interfered, similar to a Mach-Zehnder interferometer. Unlike an optical Mach-Zehnder interferometer with two interfering optical modes, here the interference is between atomic and light modes. Destructive interference into one of the final output ports, such as ^^^௨௧, may be realised by introducing a phase to ^^^ or ^^^, or through adjusting the amplitude and phase of the control field. This results in constructive interference in the quantum memory and therefore an enhancement in the storage efficiency. A similar approach is taken for retrieval, except that the interference is destructive, thereby enhancing retrieval. The full interaction may be written as: ^^(ଶ)(^^) ^^(ଶ)(^^) ^^(^)(^^) ^^(^)(^^) ൬^^^௨௧(^^) ^ = ൭ ^ ଶ ^൭ ^ ଶ ^^^^(^^) (ଶ) (ଶ) (^) (^ ^൬ ^ ^^ ^^ ^^ ) ^^^^ where respectively.
For no input wave = ^^^௨௧(^^) = ^^(ଶ) ^ (^^)^^(^) ^ (^^)^^^^(^^)^^^ఏ + ^^(ଶ) ଶ (^^)^^(^) ^ (^^)^^^^(^^)
Setting ^^(^,ଶ) ( ) ^,ଶ (^^) = ^1/2 and ^^^ ^,ଶ ,ଶ (^^) = ^^^/2 (i.e. constant with respect to ^^) recovers the beam splitter:
(^^)^^^ఏ − ^^ ( ) ௧( ) ^^ ^^ ^^^௨ ^^ = 2 ^ which for ^^ = 2^^^^ for interference into ^^^௨௧ and
destructive into ^^^௨௧, i.e. the spin wave (storage or read
in). Conversely, where ^^ = 2^^^^ + ^^ for integer values of ^^ leads to destructive interference into ^^^௨௧ and constructive into ^^^௨௧, i.e. mapping from the spin wave onto the optical channel (retrieval or read out). For a Raman memory interaction (discussed below), the coefficients ^^ (^,ଶ) ^,ଶ ( ^^ ) and ^ (^,ଶ) ^^ ,ଶ ( ^^ ) are in general not constant with respect to ^^, and depend on both the control field and the signal field. Nevertheless, through varying the phase acquired by the looped non-stored signal field, or through modification of the two control fields used for each of the beam-splitter interactions, we can obtain constructive atom-light interference into ^^^௨௧, thereby enhancing the memory storage efficiency. Similarly for retrieval, as shown in Figure 1(c) we can use atom-light interference to improve the retrieval efficiency. In this instance, the input to the memory interferometer is a previously stored spin wave, with ^^^^ = 0, and the constructive interference is realised for the ^^^௨௧ port. Again, the lower part of Figure 1(c) illustrates the interferometer analogy, while the upper part shows the process as applied in the present invention. Note that it is also possible for the spin wave ^^^ to acquire a relative phase, through the application of a magnetic field or by varying the two-photon detuning for example. Additionally, although we present in this document the enhanced protocol enhancing a Raman memory, this can generally be applied to any memory protocol, with further specific, illustrative examples listed below. While this basic model suggests combining these two protocols should lead to an improved efficiency from 25% to 100%, this single-mode model is an oversimplification. In reality, the spin-wave ^^(^^) is a collective spatial excitation across the medium and the interaction with the photon temporal wavefunction is a multimode process that involves the temporal mode of the control field. These spatial-temporal dynamics can prevent obtaining perfect visibilities. Some effects include: the leakage wavefunction of the photon that was unsuccessfully stored, when looped around, can be a distorted temporal shape, which means it will not necessarily interact in the same way during the second memory process; mismatch between spin waves generated from the first and second passes; and time varying phase accumulation on the leakage photon due to the AC Stark shift. To better understand these dynamics, full numerical simulations taking the temporal and spatial dynamics into account have been performed and are presented below.
Consider now Figure 2A, which illustrates an experimental setup suitable for performing the above methods. The memory operation sub-image illustrates the main features in which a quantum memory device (“memory”) is adapted to receive a signal field (“signal”), which has quantum information encoded thereon, and a mapping procedure (illustrated here as “control” – a pulse entering the memory simultaneously with the signal, although other mapping procedures can be used) to encode the signal onto the quantum memory device. The signal is partially stored on a first transition through the memory, leaving part of the signal on the signal field. This remaining part of the signal is redirected around a loop to re-enter the quantum memory device. A second mapping procedure (the same or different from the first one) is used to enact a second encoding event. Given that the redirected signal has gained a phase by traversing the loop, there is an interference mediated by the phase difference gained by the redirected signal. Additional phase shifts may be included by e.g. altering the phase of the mapping procedure or including additional phase shifts in the redirected path. This allows for constructive interference (in some cases maximal constructive interference) to be provided, and thereby to enhance the storage efficiency. An alternative use can be considered, in which the memory begins with quantum information encoded thereon. A mapping procedure is applied to the memory, which leads to a partial mapping of the stored information onto a signal field. This signal field then traverses at least part of the memory and exits the memory. The signal field is then redirected around the loop to re-enter the quantum memory device as in the storage case above. A second mapping procedure (the same or different from the first one) is used to enact a second retrieval event. Given that the redirected signal has gained a phase by traversing the loop, there is an interference mediated by the phase difference gained by the redirected signal. Additional phase shifts may be included by e.g. altering the phase of the mapping procedure or including additional phase shifts in the redirected path. This allows for destructive interference (in some cases maximal destructive interference) to be provided, and thereby to enhance the retrieval efficiency. In a little more detail, the setup includes a single pulse carving system to construct different temporal shapes for both the signal and control, shown in the “pulse creation” inset. Amplified voltage signals are sent to the “pulse carver,” which carves pulses out of a continuous wave (CW) light field. This may be performed with an arbitrary Waveform Generator AWG (Tektronix® 7000A), electrical amplifiers, and a Fiber-Integrated Electro-Optical Mach- Zehnder Interferometer.
The memory can be implemented with a range of matter-based systems, so long as a superposition between internal states can be targeted. In examples herein, this is implemented as a Raman memory using warm caesium vapour. To form a Λ configuration, we use the 6^^^/ଶ (F = 3) and (F = 4) hyperfine states as ground states used for storage, which are separated by 9.192GHz. A strong control field off-resonantly couples the 6 S1/2(F = 3) → 6P1/2 transition, while the signal field couples the 6^^^/ଶ → 6^^^/ଶ transition. Both the signal field and the control field are red detuned by Δ = 18.4 GHz and are orthogonally polarized, which is specifically chosen to have an absorption resonance in one of the paths that mediate the FWM noise. As shown in the setup, once pulses have been generated, the light first travels through a “switching station” using a single Pockels cell (“switch”), then enters the memory part of the system, and after polarization filtering, then re-enters the switching station from another port with an orthogonal polarization. To complete a single loop, the PC is temporarily switched on, and the pulsed signal field is re-routed back into the memory a second time. Filtering and Detection), that light is sent a frequency filtering station to fully eliminate any control field photons before measuring with single photon detectors. The signal and control fields are orthogonally polarized and focused to a waist radius of ≈ 125^^m at the centre of the memory. The memory itself is heated with quad-twisted cryogenic wire and is surrounded by three layers of ^^-metal magnetic shield to reduce magnetic dephasing and to provide sufficient temperature isolation across the cell’s length. As illustrated in Figure 2A, the example uses a single CW laser, operating at the control field frequency, which acts the source for both the signal and control fields. After pulse-carving, the pulses are amplified with an optical amplifier (Pulse Intensity amplification ≈ 5), the amplified spontaneous emission (ASE) is filtered out with a diffraction grating, and the light is split in two spatial directions. One direction is sent as the control field to the memory experiment, and the remaining direction is used as the signal field – the frequency shift of which is obtained by applying a 9.2 GHz RF signal to a free-space Electro-Optic Modulator (EOM), and filtering out a single sideband with a single pass of an angle-tuned, monolithic Fabry-Perot etalon. A delay is added for the signal field and it is sent to the Pockels cell “switching station” to be sent into the memory experiment. A single pulse carving system is used to construct different temporal shapes for both the signal and control. Amplified voltage signals are sent to the“pulse carver,”which carves pulses out of the CW light field. A delay line is used to temporally multiplex the pulse sequence, in order
to use a single pulse-carver to carve temporally-overlapped pulses for both the signal and control field. First, pulses intended to represent the “signal field” are carved. Then after a waiting approximately 215ns (associated with the optical delay of a fibre delay line), the AWG/EOM carves a second sequence of pulses to represent the “control field. Therefore, the previous set of pulses, associated with the “signal field”, after traveling optically through fibre delay line are delayed by 215ns and therefore will be synchronized in time with pulses to represent the control field. Note that this method produces extra unneeded pulses for both frequencies. That is, there are extra, unnecessary pulses in the frequency of the signal field that have the desired temporal shapes as the control field – and likewise for the signal field. The extra unnecessary pulses in the frequency of the control field travel through the memory experiment earlier in time. Consequently, these pulses should not cause a change in the pulse sequence as a result, as they would only assist in preparing the S1/2(F = 4) ground state. Additionally, the unnecessary pulses in the frequency of the signal field lag in time behind the pulse sequence for storage, and do consequently do not affect the data. These extra pulses are in fact useful as they represent unstored reference pulses, and are used to determine the effective efficiency of the memory. Each of the three experiments (“Store”, “Retrieve”, and “Combined”) have their own pulse sequence protocols. Additionally each experiment has its own equivalent reference case, which simply represents normal storage for the same parameters without any interference. A full illustration of the six different pulse sequences for the three different protocols are illustrated later. The loop is illustrated in Figure 2A. To achieve a moderate loop transmission efficiency, we use an AR-coated caesium vapour cell and single-mode fibres that are AR coated on one of the two ends. As shown in the setup, light first travels through a “switching station” using a single Pockel cell (PC), then enters the memory experiment, and after polarization filtering, then re-enters the switching station from another port with an orthogonal polarization. To complete a single loop, the PC is temporarily switched on for 24 ns, and the pulsed signal field is re-routed back into the memory experiment a second time. Finally, that light is sent a frequency filtering station to fully eliminate any control field photons before measuring with single photon detectors. The loop transmission of the memory is estimated to be approximately 50%, with the largest amount of losses being attributed to the two fibre-coupling efficiencies (≈ 20% each), two lossy beamsplitters (≈ 4% each) and a lossy PC (≈ 5%).
Note that a free-space implementation or better mode matching could be implemented to achieve higher transmission efficiencies. Additionally, before conducting the experiment, CW light was used to confirm a high visibility for the interferometer, > 95%. Additionally, we note a slight fibre-coupling inefficiency (< ±7%) as a function of phase can be observed, a small degree of which can be observed due to piezo motion slightly changing the beam alignment entering a fibre. This effect is reduced by minimizing the distance from the piezo-modulated mirror and the relevant fibre coupler, which is ≈ 4cm. Additionally, care is made before the experiment to align the beam-path to the centre of the fibre to reduce this effect as much as possible. Additionally, before each pulse sequence of the experiment, a separate counter-propagating CW laser prepares the system by optically pumping the memory into the S1/2(F = 4) ground state. This CW field is switched off by a separate EOM with an extinction ratio of < 1 : 250. The CW pump field, without switching, at the memory is approximately 2 mW. For each pulsed experiment, the pump field is switched on for ≈ 2μs and is off for ≈ 1μs. To obtain both a high optical depth and preparation of light in only one of the ground states, a buffer gas of 5 Torr of N2 is mixed with the Caesium vapour. As shown in Figure 5, a nonlinear fit and simulation is used to estimate the temperature and ground state population distribution and compared to experimental results. Specifically, this plot shows the transmission spectrum of light as its frequency is scanned across some of the relevant atomic transitions. From these plots it is possible determine the atomic density and which atomic states are populated, and populated by what fraction. From this it is estimated that the pumping efficiency is ≈ 99.85% and a temperature of (approx. ≈ 73◦C). In Figure 5(a), a weak field is scanned across both hyperfine transitions. Reference spectroscopy when the pump is turned off to estimate the effective temperature of the cell. In Figure 5(b). The same measurement as in Figure 5(a) is performed but with the pump on. A fit is then performed to estimate the relative populations The pump used for the experiment was measured to have a pumping efficiency of 99.85% and a temperature of (approximately 73◦ C). These pumping values were estimated using spectroscopy, in which a weak field is scanned across both hyperfine transitions, while the pump is on. The retrieved signal field light, after traveling through the loop, is then spectrally filtered before it is measured, in order to filter out the control field light. First it is filtered by > 40 dB via polarization using a calcite beam displacer. Afterwards it is sent through 6 separate Fabry-
Perot etalon passes. The first four passes are performed with monolithic etalons with an FSRs of 18.4 GHz, followed by 2 passes via monolithic etalons with FSRs of 103 GHz. The total transmission through the etalon system, including couplings is ≈ 15%, with a total filtration of ≈ 110 dB of filtering. After frequency filtering, the signal is sent into a fibre-coupled single-photon avalanche photo diode and a time-to-digital converter. In each of the three schemes (store/encode, read/retrieve, and combined/both) a single experimental “cycle” consists of the following: the co-propagating pump is turned on to prepare the cell in a single ground state, then the pump field is switched off and a single sequence of pulses are sent into the experiment for the given scheme. This cycle is then repeated continuously as the phase of the loop is scanned by changing a mirror’s length along the loop path via a piezo. The fluctuations of the interferometer setup is estimated to be ≈ 5Hz. Therefore, we scan the phase at 100Hz, a rate faster than the interferometer fluctuations, but significantly slower than an individual experiment’s pulse sequence. The entire experimental pulse sequence is triggered by a master trigger created by a delay generator. The master trigger is used to create a mask in post processing which extracts out clicks associated with each window of each pulse. Additionally, a second trigger, associated with the piezo-controlled phase of the interferometer is used to assign a phase to each individual pulse-sequence experiment. As the piezo scans, raw time tags are collected. Pulses are extracted using the trigger timings, and each are assigned an independent time, which is associated with a piezo phase. These are binned with a histogram, representing counts vs piezo phase. These photon numbers are then corrected for detector saturation (see next section), and then relevant parameters are estimated. Illustrations of the pulse sequence data, and additional data, such as mean photon number in each pulse, for each of the three experiments (and 6 pulse sequences), are discussed in the examples below. We now briefly outline how we correct for small errors due to detector saturation of the single photon detectors. In order to see clear interference fringes in the experiment, we need to accumulate a significant amount of data within the time-window smaller than any phase drift of the experiment. Therefore, to get as much data as possible, we attenuate our light just enough so that our single-photon detector is not saturated.
In some of the measurements, the detector sees as much as 0.4 clicks per pulse. This is large enough such that a small amount of detection saturation occurs, in which two photon events and higher cannot be distinguished from single photon events. Here we outline a simple procedure that we perform to correct for this error. One reliable piece of information that can be inferred from clicks per pulse of a detector are the non-detection events. For example, a measurement of 0.4 clicks per pulse indicates that 60% of the time a photon was not measured. The probability of detecting 0 photons in a coherent state pulse is ^^(0) = ^^ି^ത. We therefore can equate the measured relative frequency of measuring no counts to ^^ି^ത and solve for ^ത^. Therefore to estimate the photon number we simply need to do the operation ^ത^ = −ln(1 − CPP), where CPP represents the “clicks per pulse” of the single-photon detector. The data are now updated in the following way: As the piezo scans, raw time tags are collected. Pulses are extracted using the trigger timings, and each are assigned an independent time, which is associated with a piezo phase. These are binned with a histogram, representing counts vs piezo phase. This is converted into average photon number by taking the counts and dividing by the amount of piezo triggers are found in each phase bin. Finally, this is converted from “clicks per pulse (CPP)” to true average photon number using the above formula. In addition to the experimental data discussed in detail below, the inventors have performed three kinds of numerical simulations: 1. Simulation of experimental results. That is, matching the data to what is expected theoretically, using estimates of the current experimental parameters. (Seen in the shaded regions of Figures 4A to 4C). 2. Extrapolation of experimental results for improved experimental parameters. (See Figures 2A and 4A to 4C). 3. Optimized pulse-shaping search for ultrahigh efficiencies (Figure 7). To achieve the first two, the inventors use techniques, in which the Maxwell-Bloch equations are numerically solved for a three level system, in which the standard adiabatic approximation is applied such that the coherences are solved for. Additionally, to simulate the system at room- temperature, the numerical simulation integrates all of the velocity classes associated with temperature.
For the third simulation, an optimization procedure is used that searches for both optimal tuneable parameters of the experiment (such as control field strength and detuning) and temporal shape of the control pulse, including phase information across the pulse. The optimization procedure uses the Krotov method to iteratively search for optimal parameters and shapes, while iterating on the temporal shape by estimating the gradient. Figure 7 illustrates the beneficial outputs of this process. By pulse shaping the control field’s phase (dashed line) and amplitude (black), the protocol can achieve > 99% total efficiency. The upper row shows a comparison between ordinary Raman storage (left) and the enhanced storage and retrieval protocols (right) of the present invention for a moderate Optical Depth (OD = 950). In this case of moderate optical depth, the ordinary Raman with optimized pulses achieves an efficiency of 54% (in light grey – see pulse at approximately 8ns), while the enhanced protocols discussed herein with optimized pulses achieves an efficiency of 84%. In the lower row, a comparison between ordinary Raman storage (left) and the enhanced store and retrieve protocols of the present invention (right) is shown for a larger optical depth (OD = 5690). In this case of large optical depth, the ordinary Raman with optimized pulses achieves an efficiency of 58%, while the enhanced protocol discussed herein with optimized pulses achieves an efficiency of 99% Figure 2B shows in a little more detail examples of the specific pulses and timings which can be used to store and retrieve information in/from the quantum memory. Inset (b) illustrates the pulse sequences needed to operate the memory under standard conditions (i.e. single storage and retrieval, no interference). For read in, the control (top) and signal (middle) pulses temporally overlap at the memory, resulting in partial storage of signal light (bottom) into the memory, which is visualised as the reduction of light in the input signal pulse and the simultaneous emergence of a spin wave in the memory. Some time later (labelled as storage time), a second control pulse (top) is sent to the memory, where it interacts with the spin wave and retrieves the stored light (small central pulse). For the proof-of-principle experiments presented herein, the (simplified) experimental setup shown in Figure 2B(a) is used, where the memory is operated in the Raman configuration (see inset), although other memory configurations are possible as shown in Figure 2C. To generate the pulses necessary for the operation of the memory, control and signal pulse sequences are carved from a continuous wave laser operating at 351.7122THz. The signal field is detuned from the control field by 9.2GHz, corresponding to the hyperfine splitting between the ground states ( | ^^ ^ and | ^^ ^ ) of the atomic ensemble.
When using the pulse sequence shown in Figure 2B(b), the memory can be operated under standard Raman conditions (i.e. no interference). Therefore first the signal field (attenuated to the single-photon level) and control field are sent towards the memory, where they are temporally overlapped. The presence of the strong control field allows the signal field to interact with the atoms (here caesium-133) inside the memory, and mediates the transition of an atom from one of the atomic ground states (|^^^) to the other ground state (|^^^), called storage state. As a result, part of the signal light is mapped onto the spin wave (coherence between the two atomic ground states) of the atomic ensemble, and thus stored. This is called Read in. The non-stored light exits the memory together with the control field where they are separated using polarisation at a polarising beam splitter (PBS3). Some time later, corresponding to the storage time, a second strong control field is sent towards the memory cell where it now mediates the reverse operation: it maps the spin wave onto the optical signal field, resulting in the read out of light. Ideally, the control field would mediate the read in and read out of all available signal field. However, due to physical limitations of the system, this is challenging and we often are left with some light that has not been read in or read out -- indicated by the remaining light pulse in the Read in bin in Figure 2B(b) and the left over spin wave in the Read out bin (middle-right of inset (b)). In inset (c) the enhanced memory protocol pulses are shown for three different scenarios: Read in (left), Read out (middle), and Combined (right). To increase the efficiency of the storage and retrieval of light, the present invention uses quantum interference between light and matter, i.e. between the signal field (here classical) and spin wave (here a quantum state). There are three scenarios we consider: enhancing the storage efficiency, enhancing the retrieval efficiency, and enhancing the total efficiency, corresponding to the pulse sequences shown in Figure 2B(c), respectively. We first consider enhancing the read in (storage) efficiency. After initial storage (corresponding to the first control pulse and signal pulse in Figure 2B(c)), the control field is removed from the non-stored signal field and the latter is sent back towards the front of the memory where a half wave plate (HWP) rotates its polarisation to correspond to the reflection axis of PBS1. A subsequent switch changes the polarisation back to the transmission axis of PBS2 and the light is sent to the memory where a second control field induces read in of the initially non- stored signal field. However, before this second storage attempt, the signal field traveling the loop has acquired an optical phase ^^. This phase determines whether the initially stored spin wave and the looped non-stored light interfere constructively or destructively, resulting in an enhanced or decreased storage of light with respect to a normal storage event. Of course, the
path length and other circuit parameters can be adjusted to increase the constructive interference toward (or even to) maximum constructive interference. Finally, as with normal read out, a third control field may sent to the memory to retrieve the stored light. This retrieved light is looped back towards the memory, however, this time the switch is turned off, and thus the retrieved signal field is sent to the detector. The application of this process is discussed generally in Figure 3A, which is a flow chart 300 illustrating the steps of an enhanced storage procedure. At step 302, a signal field is received which has quantum information encoded thereon. The signal field is received into an ingress access port of the quantum memory device. At step 304 a first mapping procedure is applied, while the signal field traverses the quantum memory device. The mapping procedure causes part of the quantum information to be stored in the quantum memory, for example as a superposition of two states in the quantum memory. A part of the information remains encoded on the signal field due to limitations of physical memory devices and associated encoding procedures. At step 306, the signal field (with the non-stored portion of the quantum information) is allowed to exit the quantum memory device and is redirected along a path to re-enter the quantum memory device at the ingress access port. The path travelled by the redirected signal field leads to an additional phase change ^^^ being gained by the quantum information which has remained encoded on the signal field. At step 308, a second mapping procedure is applied while the signal field traverses the quantum memory device for a second time, causing interference between a wavefunction corresponding to the portion of the quantum information stored in the quantum memory device and a waveform corresponding to the quantum information which has remained encoded on the signal field. Since the quantum information which has remained encoded on the signal field has gained a phase ^^^, the waveforms interact and an interference effect is seen. By controlling the relative (via ^^^ and other phase shifts applied as discussed elsewhere),
the interference can be to be constructive (in some cases optimally or maximally
constructive), thereby the efficiency of storage of the quantum information in the quantum memory device. In the second scenario, where we enhance the efficiency of the read out process through light- matter interference, we initially assume that light has encoded information in the memory
(whether by the “normal'” process of inset (b) as indicated in this Figure, or the enhanced method using constructive interference – not shown here, but see the final case below). For the read out, we have a second control pulse to perform partial retrieval of the stored signal field (which is as effective as the normal retrieval in inset (b), leaving a portion of information still encoded in the memory). This retrieved light is sent back to the memory using the loop and the switch. Using a third, and final, control pulse, the light retrieved interferes with the remaining spin wave either constructively or destructively, depending on the relative phase ^^. As a result, the second read out event can achieve an enhanced retrieval of light which is sent to the detector. The application of this process is discussed generally in Figure 3B, which is a flow chart 310 setting out the steps of an enhanced retrieval procedure. The method starts with the assumption that a quantum memory device has quantum information encoded thereon, for example as a superposition of two states in the quantum memory. This may have been encoded by the method of flow chart 300, or by any other suitable method, such as e.g. that shown in Figure 2B(b). At step 312, a first mapping procedure is applied to the quantum memory device, which causes a portion of the stored quantum information to be mapped onto a signal field. A portion of the stored information remains on the quantum memory device due to limitations of physical memory devices and associated retrieval methods. The signal field traverses a part of the quantum memory device for a first time. At step 314 the signal field is allowed to exit the quantum memory device at an egress access port. At step 316, the signal field (with the retrieved portion of the quantum information) is redirected along a path to re-enter the quantum memory device at the ingress access port. The path travelled by the redirected signal field leads to an additional phase change ^^^ being gained by the quantum information which has been encoded on the signal field.
At step 318, a second mapping procedure is applied while the signal field traverses the quantum memory device for a second time, causing interference between a wavefunction corresponding to the portion of the quantum information remaining stored in the quantum memory device and a waveform corresponding to the quantum information which has been encoded on the signal field. Since the quantum information which has been encoded on the
signal field has gained a phase ^^^, the waveforms interact and an interference effect is seen. By controlling the relative phase (via ^^^ and other phase shifts applied as discussed elsewhere), the interference can be selected to be destructive (in some cases optimally or maximally destructive), thereby the efficiency of retrieval of the quantum information from the quantum memory device. Finally, we expressly consider the case where both the storage and retrieval are enhanced to enhance the total efficiency of the memory protocol, see the rightmost combination of pulse sequences in Figure 2B(c). This can be thought of as enacting the enhanced storage protocol, and subsequently the enhanced retrieval protocol. That is, following the steps of flowchart 300 in Figure 3A, followed by the steps of flowchart 310 in Figure 3B. Note that in all experiments a mirror mounted on a piezo is used to scan the relative phase between the signal field and the spin wave (scanning at 100Hz). Each piezo scan thus scans the full phase space from 0 to 2^^, and by repeating the pulse sequence multiple times (and thus the protocol) we acquire phase-resolved statistics on the storage and retrieval of light. We now consider Figure 2C, in which different possible electronic transitions which can be used as the basis for the memory are shown. While the experimental results shown herein use the Raman memory protocol, it should be noted that it is possible to use any other form of ensemble-based quantum memory protocol, examples of which are shown in Figure 2C. For example, the Raman-based protocol is shown in Figure2C(a) with the signal and control field tuned to the |^^^-|^^^ and |^^^-|^^^ transition, respectively, where both are red-detuned from the excited state transition by a frequency far larger than the Doppler dephasing. Another (equivalent) possible configuration is shown on the right-hand side where the control and signal field are blue-detuned from the excited state by the same frequency. A similar configuration is shown in Figure 2C(b) where the signal and control field are now approximately on-resonance with the |^^^-|^^^ and |^^^-|^^^ transitions. These protocols correspond to the electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS) protocol. In all four scenarios described the light is stored into the spin wave (coherence between the hyperfine ground states |^^^ and |^^^).
Instead of mapping light onto the coherence between the hyperfine ground states of an atomic ensemble, one can also store light into an orbital wave (i.e. coherence between a ground state and a doubly excited state), see Figure 2C(c) and Figure 2C(d). Let us first consider the off- resonance cascaded absorption (ORCA) memory protocol, Figure 2C(c). There are again two possible configurations: one where the signal is tuned to the | ^^ ^-| ^^ ^ transition and the control to the | ^^ ^-| ^^ ^ transition, where | ^^ ^ is the doubly excited state of the atoms, or vice versa. In both cases, the signal and control field are detuned from the excited state by a frequency larger than the Doppler dephasing frequency. When both control and signal field arrive simultaneously at the memory, the control field thus maps the light onto coherence between the ground state | ^^ ^ and the doubly excited state | ^^ ^ . A similar protocol is the fast ladder memory (FLAME) protocol shown in Figure 2C(d). However, unlike ORCA, here the optical fields are close to resonance. The enhanced protocols described in this document are compatible with any of these memory protocols. Moreover, we expect that these protocols are not limited to the examples shown here, but could also be used to enhance the memory efficiency in protocols based on engineered absorption rather than optically-controlled memories, for example the gradient echo memory (GEM) or atomic frequency comb (AFC) memory protocols. Before moving on to demonstrations of the experimental applications and results, we briefly discuss some mathematical formalisms and other supplemental details. A Raman quantum memory can, in a single mode approximation, be represented as a beam splitter model. Here we briefly show this via simplification of the Green's function solution to a Raman memory. A simplified system of equations for the dynamics of the atom-spin wave system can be represented as: ^^௭^^(^^, ^^) = −√^^^^(^^, ^^) ^^)
In which ^^(^^, ^^) represents the quantum electric field operator for the slowly-varying envelope of the signal field, ^^(^^, ^^) represents the collective spin-wave of the ground state coherence, and ^^(^^, ^^) represents the polarizability in the frequency of the signal field, which is
proportional to the coherence between the ground and excited states driving the signal field interaction. The medium is assumed to be 1D, and the dimension ^^ is across the medium, while ^^ represents the retarded time, ^^ = ^^ − ^^/^^. Additionally, Γ represents the decay rate, and Ω(^^) represents the control field, with Ω∗(^^) being the complex conjugate of Ω(^^).
After an adiabatic approximation and a transformation, these rate equations can have the form: ^^௭^^ = −^^^^ ^^௭^^ = ^^^^ in which we use the following change in variables: ^^(^^, ^^) ^^(^^,^^)^^ି(^ఠାௗ௭)/^ = √^^ ^^(^^) where ^^ and ^^ can be
^^ = ^ |^^(^^)|ଶ^^^^ This system has a Green's
^^^௨௧(^^) = ^ ^^(^^,^^ᇱ)^^^^(^^′)^^^^′ − ^ ^^(^^, ^^)^^^^(^^)^^^^ ^ ^ where ^^^௨௧(^^) = ^^(^^,^^) and
^^(^^,^^) are as: ( ) = ^^( 1 ^^ ^^,^^ ^^ − ^^) − ^^Θ(^^ − ^^) × ^^ ^2^^ ^^ − ^^൧ ^^ ^ ^ ^ − ^^
Additionally the as a model notion has been solved for the more general case. This has been shown by representing the Green's functions as singular value decompositions of the form: ^^(^^,^^) = ^ ^^^(^^)^^^ ^^^(1 − ^^) ^ in which ^^^ଶ + ^^ ଶ ^ = 1 and ^^^ the equation:
^ ^ ^^^^^^2^^^^^^^൧ ^^^(^^)^^^^ = ^^^^^^(^^) ^
From this, each individual solution can be represented individually as: ^^^ = ^^^^^^ − ^^^^^^ ^^^ = ^^^^^^ + ^^^^^^ From this we immediately see that individual isolated modes of the signal field and spin wave, interfere linearly as a beamsplitter in line with the conceptual discussion of this invention, likening the setup to a beamsplitter. We now turn to the experimental results to illustrate the power of the methods discussed herein. In general, Figure 6A shows results for the enhanced read in (storage) protocol, Figure 6B shows results for the enhanced read out (retrieval) protocol, and Figure 6C shows results for the combined (store-retrieve) protocol. The histograms are each accumulated over the entire duration of the measurement, where each bin corresponds to a particular position of the piezo scan, and thus can be related to a relative phase ^^,^^, or ^^ as used variously herein. In each plot, black histogram entries show the distribution in total efficiency for a standard Raman pulse sequence, where the variation is due to a combination of shot noise, along with fluctuations in the control pulses and loop efficiency. In light grey. the distribution of the total efficiency for each enhanced protocol (Read in, Read out, and Combined) is shown. Overlap is shown in mid grey. Each of Figures 6A to 6C shows three plots – (a) a simulation of the process; (b) experimental data of the process; and (c) experimentally measured total efficiency as a function of phase. To determine the total efficiency, counts were accumulated from a particular point in the piezo scan, for five scans. For the simulations, the histograms are obtained from a numerical simulation of the system, with Poissonian shot noise included. In the experimentally measured total efficiency as a function of phase plots, Figures 6A-C(c), the total efficiency was determined by the scanning piezo position. Here counts were accumulated over 50 piezo scans. Here we clearly see a wider distribution of efficiencies with respect to the standard Raman protocol. Particularly for the Read in protocol, we can distinguish two peaks at both extremes of the histogram, a feature associated with uniform sampling of the phase space. For Read out and Combined the histograms bias towards zero which is due to the small average photon-
number used in the experiments (0.55 photons per pulse). Despite the low photon-number, for all three protocols we witness maximum efficiencies exceeding the standard Raman efficiencies, thus showing the enhancement of the total efficiency when using the interference- based enhancement methods discussed herein. As expected, the biggest improvement is seen when combining enhanced Read in and enhanced Read out where we now reach 30% total efficiency (Figure 6C(b)), which is a two-fold increase with respect to normal Raman. Figures 6A-C(a) show the numerical simulations corresponding to the data presented in Figures 6A-C(b), which agree well. The simulations predict slightly higher maximum efficiencies obtainable in all three protocols, which we do not reach in the experiment. We believe this is due to phase noise during the measurement which creates a phase averaging effect and decreases the maximum efficiency. A potential solution to this phase averaging could be active stabilisation of the phase over the interferometer loop in our setup using a third (continuous wave) laser field. In Figures 6A-C(c) a snap shot of a piezo phase scan (averaged over 50 piezo scans) for each protocol is shown, where we plot an oscillating line the numerical simulation (from numerical simulations of a warm caesium vapour, with a 35% loop efficiency) of the total efficiency in enhanced configurations and using a straight horizontal line the numerical simulation (parameters as for the enhanced simulation) of total efficiency in the standard Raman configuration. Additionally, in all of the experiments, we perform an additional measurement in which the signal field is blocked to measure the noise due to the control field. No observable change in noise was observed by the implementation of the enhanced protocols as compared to ordinary storage, which can be attributed to already working in a low-noise setup. We see a clear sinusoidal variation in the efficiency as the phase of the signal input to the second memory interaction is varied. Also plotted are the experimental results, which can be seen to match their simulated values well. Again we witness the effect of phase averaging on the maximum achievable total efficiency, albeit the experimental results show lower efficiencies than expected from the simulations. This discrepancy can mostly be attributed to the non-trival spatial-temporal multimode dynamics, which we have modelled numerically. What is clear is the strong dependence of the efficiency on the phase in the enhanced protocols, as expected. Even still, in the experiment there is a lower-than-expected efficiency, in slight disagreement with the full multimode numerical simulation. This can be explained by experimental phase- noise of the amplifier used to create strong control field pulses, which can slightly scramble
the visibility of interferometer, and is in principle possible to avoid by using a system that requires less pulse amplification or by using a less noisy optical amplifier. According to optimization search of our numerical simulations, with a combination of improved loop efficiency using free space, pulse shape optimization, and reduced phase noise of the amplifier, it should be possible to achieve 99.7% total efficiency in such a system. In addition to being able to potentially achieve ultra-high total efficiencies, this protocol requires significantly less resources and less noise. In conclusion, the experimental data presented above demonstrates the viability of the enhanced protocols discussed herein to improve the efficiency of a quantum memory, without undue increase in resource usage, observing strong improvements. Additionally, we note that, unlike many other quantum optics experiments, single-photons and coherent states have the same behaviour for these systems, and we theoretically expect identical behaviour to occur in our system for storage of pure single photon Fock states. Further examples The inventors have also implemented the above methods in a Raman quantum memory in warm Cesium vapor, and achieve a more than three-fold improvement in total efficiency reaching (34.3±8.4)%, while retaining GHz-bandwidth operation and low noise levels, with a view to approaching the near-unity efficiency and fidelity desired for quantum memory applications. Numerical simulations predict that this approach can boost efficiencies in systems limited by atomic density, such as cold atomic ensembles, from 65% to beyond 96%, while in warm atomic vapours it could reduce the laser intensity to reach a given efficiency by over an order-of-magnitude, and exceed 95% total efficiency. Furthermore, the inventors have found that the method preserves the single-mode nature of the memory at significantly higher efficiencies. The protocol is applicable to various memory architectures, paving the way toward scalable, efficient, low-noise, and high-bandwidth quantum memories. Optical quantum memory protocols fundamentally rely on light-matter interactions to coherently map an incoming optical field to a spatially-extended excitation across an ensemble of absorbers such as atoms. Often such a quantum memory is operated in a lambda configuration (Fig.8a), whereby an input photonic signal^ ℇ^^(^^) is temporally overlapped with a strong control field Ω^(^^) at the input of the memory, and subsequently stored as a coherence between the two ground states |^^^ and |^^^, referred to as a spin-wave.
As noted above, the memory interaction can be viewed as a beam-splitter, where the input field is partially mapped onto an output spin-wave mode and partially transmitted as an output optical field, see in Figs 8a and 8b. An ideal quantum memory would have 100% reflectivity for both the write and read interactions, i.e. perfect storage and retrieval. However, achieving high reflectivities (>90%) for high-bandwidth input signals requires large optical depths and control field powers, as well as pulse shaping of the control pulses. These requirements are often difficult to attain, lead to additional noise, nonlinear effects, and sacrifice the single-mode nature of the memory. Notably, increasing the reflectivity becomes increasingly challenging the closer it is to unity and, for example, increasing from 90% to 99% can require orders of magnitude higher control intensities. The EEVI memory concept using the beam-splitter analogy. Is shown in detail in Figures 8a to 8d. Figure 8a shows the storage process of light using an ensemble-based Raman quantum memory. A strong write field Ω^ is temporally overlapped with an input signal field^ ℇ^^ at the memory, mediating the two-photon transition between hyperfine ground states |^^^ and |^^^ and generating a collective coherence called a spin-wave ^^^^௨௧. The non-stored light is transmitted through the memory into output mode ℇ^ ^௨௧. In the single-mode regime, this interaction can be viewed as a beam-splitter interaction where the input light couples to the modes ^ ^ ^^௨௧ and ^ ℇ^௨௧ and the reflectivity (transmissivity) is determined by the control write field amplitude (shown in the inset). Fig.8b illustrates that, after a certain storage time, a strong control read field is sent into the memory, reversing the process and thereby retrieving the stored light into output mode^ ℇ^௨௧. In the beam-splitter analogy (shown in the inset), this interaction couples an input spin-wave ^^^^^ to output modes ^^^^௨௧ and ^ ℇ^௨௧. As noted above, the new method to enhance the storage and retrieval efficiency of quantum memories uses light-matter interference, and is shown generally in Figs 8c and 8d, respectively, and can be conceptualized as a Mach-Zehnder (MZ) interferometer applied to either the storage or retrieval processes of the memory, or both. To apply EEVI to the storage interaction, a first write control pulse ^^^ is chosen such that it couples 50% of the input field^ ℇ^^ to the spin-wave and 50% of the signal is transmitted as a photonic field, analogous to a 50:50 beam splitter. The non-stored signal is looped back into the memory where a second write field ^^ଶ (also with 50% coupling) mediates interference between the spin-wave and the looped photonic field. For perfect mode overlap between the
spin-wave and photonic field this can lead to complete constructive interference, resulting in perfect storage efficiency ^^^,ாா^ூ=1. This is equivalent to an all-optical MZ interferometer with two 50:50 beam-splitters achieving full constructive interference into one output port In Figure 8c, the EEVI procedure is applied to the storage process. After initial storage with control write field ^^^, the non-stored light is looped back into the memory input, overlapped with a second control write field ^^ଶ and subsequently interferes with the previously created spin-wave. In the beam-splitter analogy (inset), this full process can be viewed as a Mach- Zehnder interferometer where an input signal field results in the interference of a spin-wave and the non-stored light at the second beam-splitter. The relative phase ∆^^ between the two interfering fields dictates the storage efficiency ^^^,ாா^ூ, and for full constructive interference it can reach 100% even when the memory is operating at 50% efficiency. Figure 8d shows EEVI applied to the retrieval procedure. The light initially retrieved with control read field ^^^ is sent back into the memory input, overlapped with a second control read field ^^ଶ, and interferes with the remaining spin-wave, resulting in an enhanced retrieval efficiency ^^^,ாா^ூ for certain relative phases ∆^^. In this case, the input to the Mach-Zehnder is a spin- wave. When considering loop loss, the storage efficiency after write field ^^ଶ is given by: ^^ ଶ ^௨௧ ଶ ∆^^ (1 − ^^^)ଶ ^^^,^^^୍ = ฬ ฬ = ^^^^^^^^^^ ൬ ^ + ^^ 2 4 where ^^^ is the loop between the looped
and the spin-wave. We can also apply EEVI to the retrieval process, and can find an equivalent expression for the retrieval efficiency ^^^,ாா^ூ. Here, the input to the light-matter interferometer is a spin-wave ^^^^^, see Fig.8d. Upon applying a read field ^^^, 50% of the spin-wave is mapped onto the signal field, and thus retrieved. The retrieved light is looped back to the input of the memory where another read field ^^ଶ (50% mapping) enables light-matter interference. For full constructive interference, 100% of the light can be retrieved from the memory. Finally, we can combine EEVI storage and retrieval, yielding a total efficiency proportional to ^^ ^௧,ாா^ூ ∝ ^^^ ସ ∆ఏ ௧ ^^^ ^ ଶ ^. When considering 50% storage and 50% retrieval efficiency, this full EEVI enhance the total memory efficiency from 25% to 100%. However,
reaching overlap to maximize the interference between the photonic and spin-
wave mode is nontrivial, but as shown later, numerical modelling demonstrates that with pulse shaping techniques a total memory efficiency approaching 100% is possible. The experimental demonstration of EEVI applied to a GHz-bandwidth Raman memory in a warm Cesium ensemble is illustrated in Figure 9. We explore three scenarios: EEVI-storage, EEVI-retrieval, and EEVI-Raman (i.e. combining EEVI-storage and -retrieval). The setup is shown alongside the employed Raman configuration where the signal and control field have a 330ps duration, and are red-detuned by 18.4GHz from their respective transition to eliminate four-wave mixing. A free-space loop (^^^ = (63.5 ± 2.5)%), containing a piezo and Pockels cell PC, is used to switch light back into the memory and scan the relative phase ∆^^ between the light and the spin-wave (see Methods for details). More specifically, in Figure 9a, an input signal field^ ℇ is sent into the setup ①and temporally overlapped at the memory with a strong control field Ω^ to map the signal field onto a spin- wave. The non-stored signal field is looped back towards the input of the memory following path ② using polarization optics and a Pockels cell whereby the latter is triggered such that it either sends the light back to the memory ③ or to the detectors ④. A piezo mounted to the back of a mirror inside the loop scans the optical phase, resulting in constructive or destructive interference between the spin-wave and non-stored light when it is sent back into the memory ③. The same procedure can be applied to the retrieval process (see main). PBS: polarizing beam-splitter; HWP: half-wave plate; APD: avalanche photodiode. In Figure 9b, the ensemble-based memory is operated in a Raman configuration and consists of Cesium-133 atoms initially optically pumped into the |^^ = 4^ hyperfine ground state. A strong control field drives the two-photon transition to the hyperfine ground state |^^ = 3^, creating a macroscopic coherence (spin-wave) across the atomic ensemble between the two ground states. The signal and control field are both red-detuned from their respective transitions by 18.4GHz, which is twice the hyperfine ground state splitting ∆^^. For EEVI-storage, the pulse sequence and PC timings are shown in Fig.10a together with the equivalent normal Raman memory which uses a single write (^^^) and read (^^^) pulse, and all control pulse energies are set to be the same: 400pJ. The storage time, the time between the final write and first read control pulse, is 12.48ns in both cases. With these parameters, the normal Raman memory reaches an average storage efficiency of (42.1 ± 5.0)% and total internal memory efficiency of (10.8 ± 2.1)%. For EEVI-storage, the PC switches the non-stored light after the first write process (bin ^^^) back into the memory for a second write attempt (^^ଶ),
and 12.48ns later the spin-wave is retrieved by a single read pulse ^^^ (i.e. standard retrieval). The pulse sequences for the write (W) and read (R) control Ω^ and signal ℇ^ ^ fields used in normal Raman memory procedure (upper panel) and when applying EEVI to the storage process (lower panel). In the latter case a Pockels cell (PC) switches the non-stored light in bin ^^^ back into the memory in bin ^^ଶ, where a second control write field enables interference between the looped signal field and previously stored spin-wave. Fig.10b shows the storage efficiency of the EEVI-storage protocol as we scan ∆^^, as well as the average storage efficiency for the standard Raman memory (solid line). We observe a clear sinusoidal variation in storage efficiency when interfering the non-stored light with the spin-wave. For constructive interference this leads to a two-fold improvement and a maximum storage efficiency of (72.3 ± 8.3)%, the above calculations for ^^^=(63.5 ± 2.5)%. The dashed lines correspond to numerical simulations using experimentally measured parameters. This storage efficiency is comparable to state-of-the-art Raman memories, but at six times larger detuning and 30-fold increase in bandwidth without any increase in control pulse energy. Figure 10c shows a corresponding plot, but for total efficiency. The more important metric, however, is the total efficiency. Indeed, an improvement in storage efficiency does not always correspond to an improvement in total efficiency, due to spin-wave amplitude accumulation (bunching) at the input to the vapor cell, leading to re-absorption when retrieving in the forwards direction. This effect is expected to be stronger in EEVI-storage, since any spin-wave bunching after initial storage can lead to constructive interference concentrated at the input with the second write process. Regardless, we observe a sinusoidal trend in the total efficiency for EEVI-storage, demonstrating again a two-fold increase in total efficiency to (17.1 ± 3.6)%. For storage efficiencies exceeding 72%, however, a lower total efficiency was observed partly due to spin-wave bunching, but also due to nonlinear effects such as self-defocusing and self-phase modulation caused by high control field intensities. As we show later, these effects can be mitigated with EEVI through control pulse shaping. Next, we apply EEVI to the retrieval procedure, but applying standard storage, using the timings shown in Fig.11a. Here the interference is between the spin-wave and the initially retrieved signal (^^^) switched into bin ^^ଶ using a Pockels Cell (PC). We set the control pulse energy to 800pJ, resulting in a retrieval efficiency of (25.3 ± 5.0)% and total efficiency of (12.6 ± 2.1)% for standard Raman. For EEVI-retrieval, we again observe light-matter interference, but initially the observed maximum efficiency was reduced compared to EEVI-storage. This was primarily due to suboptimal overlap between the retrieved light in bin ^^^ and remaining
spin-wave. We find that higher overlap is achieved when reducing the energy of the ^^^ control pulse to 460pJ. This leads to a greater interference visibility in the retrieval (Fig.11b) and total efficiencies (Fig.11c) as we scan ∆^^. With these pulse energies, we observe a more than two- fold increase in both efficiencies with respect to standard Raman, reaching a maximum retrieval efficiency of (74.3 ± 14.0)% and total efficiency of (30.0 ± 2.3)%. The amplitudes of the control field pulses used in EEVI-retrieval are set to ^^^/^^ଶ = 0.6 to optimize efficiency. In Figures 11b and 11c, the retrieval and total efficiencies respectively are shown as a function of the optical phase ∆^^ for the EEVI-retrieval protocol. The shaded regions indicate the standard deviation. For EEVI-Raman we use the PC to switch the non-stored light in ^^^ and the retrieved light in ^^^ into the later time bins ^^^ and the retrieved light in ^^ଶ and ^^ଶ, respectively, see Fig.12a. Using a control pulse energy of 500pJ, we measure a total efficiency for the normal memory procedure of (10.4 ± 2.3)%. We optimize the total efficiency of EEVI-Raman by reducing the control pulse energy in bin ^^ଶ and ^^^ to 300pJ. As shown in Fig. 11b, this combined procedure improves the efficiency over three-fold, reaching a total efficiency of (34.3 ± 8.4)%. In this case, the relative phase ∆^^ refers to both the write and read process. Conversely, the highest total efficiency we measure for the standard Raman memory is (13.7 ± 0.4)%, obtained using control pulse energies of 1nJ, highlighting the increase in efficiency while reducing resources. Finally, we explore the noise performance of the EEVI-Raman memory. Noise in Raman memories can originate from four-wave mixing (FWM), fluorescence, spontaneous Raman scattering due to imperfect optical pumping, and insufficient filtering of the strong control field after the memory. In our demonstration we operate the memory at a specific detuning from resonance that offers built-in noise suppression by strongly absorbing the anti-Stokes field, thereby fully suppressing FWM noise. As such, we expect the noise in bin ^^^ or ^^ଶ to predominantly originate from the other three incoherent processes, which scale linearly with control pulse energy and cannot build up or interfere between subsequent applications of the control pulse. To quantify the remaining sources of noise, we perform the same measurements but with the signal field blocked. We observe no phase-dependence in the number of noise photons generated by the memory, and measure an average noise floor of (2.2 ± 1.1) x 10-2 photons
per pulse for the EEVI-Raman protocol compared to (3.9 ± 0.2) x 10-2 for the equivalent normal Raman memory, demonstrating that the EEVI process does not increase the noise. For our demonstration using weak coherent states with an average input photon-number of ^ത^ ~ 17, this results in a signal-to-noise ratio (SNR) on retrieval of 47 ± 20 for standard Raman and (187 ± 104) for EEVI-Raman at ∆^^ corresponding to maximum constructive interference. The EEVI process thus increases the memory efficiency without increasing the noise, and therefore could enable higher fidelity storage and retrieval of quantum states. Since resonant memory protocols such as EIT and ATS are also dominated by linear processes such as fluorescence noise, we expect similar improvements in SNR when applying EEVI. The experimental results of EEVI applied to a Raman memory show a clear improvement in memory efficiency, and the numerical simulations match the data well. Further improvements in efficiency can be made by optimizing the loop transmission and temporal shaping of the control field. The loop efficiency can be readily improved through the use of low-loss optics -- a similar experimental setup with a free-space loop and Pockels cell has demonstrated a loop transmission of 98.8%. Our numerical model predicts that such an improvement to the loop transmission would result in a maximum total efficiency of 55% with the same experimental parameters. Temporal shaping of the control pulses can be implemented by using electro-optic modulators and optical amplifiers. Using our numerical model, and assuming 100% loop transmission, we calculate the total efficiency for forward retrieval after a local optimization of the control field temporal profile. We note that the efficiencies obtained here are not the theoretical maximum and that in principle higher efficiencies could be reached through further optimization. We explore the performance of EEVI in a Raman configuration in different parameter regimes, while keeping the signal bandwidth fixed at 1GHz. In Fig.13a we investigate systems that are limited by optical depth such as cold atomic systems in magneto-optical traps (MOT) or micro- vapor cells. For instance, optical depths of 1000 have been demonstrated for MOTs, yet even after optimization the total efficiency in the forwards direction, the standard Raman protocol barely surpasses 60%. However, applying EEVI to the storage and retrieval processes (EEVI- Raman) improves the total efficiency to >96%, without an increase in required control field intensity. Fig. 13b shows the optimized control pulse sequences in blue for Raman (top) and EEVI- Raman (bottom) at an optical depth of 2000, where applying EEVI boosts the total efficiency from 75% to >98% without increasing the maximum Rabi frequency or changing the phase
chirp significantly. The same optimization procedure applied to resonant memories predicts similar improvements with EEVI. In Figure 13c, total efficiency for forward retrieval as a function of maximum peak control field Rabi frequency, for standard Raman memory (diamonds) and EEVI-Raman (circles) is shown. Also shown on the top x-axis is the equivalent peak power, using the same parameters as the simulation of the experiment, with a focused beam waist of w0=90µm. In Figure 13d, Schmidt number as a function of storage efficiency for standard Raman memory (diamonds) and EEVI-Raman (circles) is shown, where a Schmidt number of one indicates a fully single-mode memory. The inset shows a zoom in of the Schmidt number for EEVI-Raman as a function of storage efficiency. For warm atomic vapours, the main limitation is often posed by the available control field intensity. For a Cesium ensemble kept at 80°C and restricting the maximum Rabi frequency, we show in Fig.13c the total efficiency obtained with optimized control pulses. Indeed, EEVI significantly reduces the control field Rabi frequencies needed and the improvements become more prominent at higher efficiencies where the total efficiency begins to plateau. For example, for a total efficiency of 80% EEVI reduces the required peak Rabi frequency by more than four times, corresponding to a >16-fold reduction in peak intensity. We also highlight that a Rabi frequency of 1.4GHz is sufficient to reach >95% total efficiency with EEVI, which for a beam waist of 90µm as used in our experimental demonstration, corresponds to a peak power of 800mW. This reduction in required control intensity not only lowers the technical requirements for efficient memories, but also reduces noise contributions and unwanted nonlinear effects. For memories where FWM is present, we expect EEVI to show a significant reduction in noise since FWM scales quadratically with control field energy. EEVI thus enables a route towards efficient high-fidelity storage and retrieval of quantum states. Lastly, we investigate the single-mode nature of EEVI applied to the storage process and compare it to the equivalent standard Raman memory. The Raman memory has been shown to be single-mode in the low coupling regime, which means it can act as a coherent temporal mode filter, but there is a trade-off between single-mode capacity and efficiency. Indeed, for a normal Raman memory the single-mode capacity of the storage process drastically decreases as a function of optimized storage efficiency, as shown in Fig.13d where we plot the simulated Schmidt number quantifying the modal capacity. We can see that for all storage efficiencies EEVI outperforms the standard Raman memory. This is expected since the EEVI protocol uses two subsequent memory interactions with a lower coupling strength and therefore each
individual interaction remains more single mode, akin to the Ramsey interferometer for the quantum pulse gate. The most significant improvement is seen close to 100% storage efficiency, where the Schmidt number is reduced from 1.33 to 1.01. EEVI could thus play a crucial role in technologies such as quantum parameter estimation and high-dimensional encoding, which require single-mode capacity and high efficiency. In conclusion, we have demonstrated a new method to enhance the efficiency of existing optically-controlled quantum memory protocols, using quantum interference between an optical field and a stored material excitation. Experiments employing a high-bandwidth Raman memory show that applying this method results in a greater than three-fold improvement in memory efficiency without additional noise. Numerical modelling shows close agreement with our experimental results and predicts that with pulse shaping and improved setup transmission, near-unity efficiencies are realistically achievable, while retaining single-mode capacity and reducing requirements on the control field intensity and ensemble optical depth. These improvements to quantum memories are important in applications such as quantum networks, distributed quantum computing and sensing, and reducing the resource requirements is critical for real-world deployment. Besides the advantage of reducing technical overhead, we expect the EEVI method to reduce common noise processes such as four-wave mixing, spontaneous Raman scattering and control field leakage. The EEVI method thus provides a potential route towards scalable, efficient, low noise, high-bandwidth quantum memories.