GB2605973A - A method and system for reducing the amplitude of an oscillating electric field at the equilibrium of a trapped ion - Google Patents

Abstract

A method of reducing the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap, comprising: trapping at least one ion in a trapping electric field having an electric field amplitude; using an interferometry sequence comprising: applying at least three laser pulses, with changes of the trapping electric field amplitude between successive laser pulses; measuring a state of the ion; repeating the interferometry sequence in order to obtain a plurality of measurements of the state of the ion; determining a probability that the trapped ion changes state; and adjusting the trapping electric field based on the determined probability in order to reduce the magnitude of said quasi-static dipole field.

Description

A METHOD AND SYSTEM FOR REDUCING THE AMPLITUDE OF AN OSCILLATING

ELECTRIC FIELD AT THE EQUILIBRIUM POSITION OF A TRAPPED ION

Field

The present disclosure relates to a method, system and software instructions for reducing the magnitude of a quasi-stafic dipole electric field at the position of a null of an oscillating electric quadrupole field. The application also relates to an optical clock; a quantum computing system; a quantum simulator system; a trapped ion electric field sensor; a trapped ion quantum network node and a trapped ion force sensor comprising the system.

In particular, this disclosure relates to taking advantage of the electric field dependence of the equilibrium position of a trapped ion in order to identify imperfections, in the form of a dipole electric field at the null of an oscillating quadrupole field, in the trapping electric field and to use identified imperfections in order to reduce the magnitude of the dipole field at the said null.

Summary

According to a first aspect of the present disclosure, there is provided a method of reducing the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an on trap, the method comprising: trapping one or more ions in a trapping electric field, wherein the trapping electric field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude

of the oscillating electric field;

inducing a change in an equilibrium position of one of the one, or more trapped ions and measurino said change using an interterometry sequence comprising: applying at least three laser pulses to the one of the one or more trapped ions; changing the trapping electric field amplitude between application of successive laser pulses; and measuring a state of the one of the one or more trapped ions after the application of the at least three laser pulses; repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the one or more trapped ions; and determining a probability that the one or more trapped ions change state during the interferometry sequence based on the plurality of measurements of the state of the ion.

In one or more embodiments, the method may further comprise adjusting the trapping electric field based on the determined probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole s field of the ion trap. In one or more embodiments, the method may thither comprise adjusting one or more parameters of the system in which the one or more trapped ions are implemented to account for unwanted electric field effects. In either approach, the probability that the trapped ion, or the ions, changes state during the interferometry sequence is indicative of the magnitude of the quasi-static electric dipole field at the null of the oscillating electric quadrupole field of the ion trap and the effects on the system in question can be reduced by either adjusting the trapping electric field or adjusting the parameters of the system.

In one or more embodiments, determining the probability of the trapped ion being in a given state may comprise calculating the statistical likelihood of the ion moving from a first state to a second state during the interferometry sequence. In one or more embodiments, the state of the ion may refer to the electronic state in which an unpaired valence electron is situated. In one or more embodiments, the state of the on may refer to the electronic state of one or more valence electrons in an atomic ion, or a molecular orbital state of a molecular ion. In one or more embodiments, the state of the ion may refer to an atomic hyperfine state of an atomic ion, or a molecular hyperfine state of a molecular ion.

In one or more embodiments, adjusting the electric field may comprise one or more of: altering the voltage applied to one or more compensation electrodes; moving one or more electrodes configured to generate the trapping electric field; and changing the voltage on one or more electrodes configured to generate the trapping electric field.

In one or more embodiments, the trapping electric field may further comprise a static electric field and wherein the trapping electric field amplitude is additionally comprised of an electric field amplitude of the static electric field.

In one or more embodiments, repeating the interferometry sequence may be performed the plurality of times by one or a combinafion of: performing the interferometry sequence on the same trapped ion a plurauty of times; and trapping a plurality of ions in the oscillating electric field and perform ng the interferometry sequence on each of the ions.

In one or more embodiments, the first laser pulse may comprise a resonant pi/2 pulse and the final laser pulse may comprise a resonant pit2 pulse and the other laser pulses may comprise resonant pi pulses.

In one or more embodiments, the laser pulses may be coherent laser pulses and controlled phase differences between the pulses may be chosen such that the probability of the ion changing state depends on the strength of the offset electric field. This may be achieved by using a total control phase, which is a function of the controlled phase differences between the pulses, of pi/2.

In one or more embodiments, the trapping electric field amplitude may be changed between two different trapping electric field amplitude values between successive laser pulses.

In one or more embodiments, each laser pulse may be provided at least a predetermined delay after the previous laser pulse.

In one or more embodiments, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; may be performed a first plurality of times wherein, for each repeat of these steps in the first plurality of times, different controlled phase differences between the laser pulses are used; and wherein adjusting the trapping electric field may be based on the first plurality of measurements of the probability.

In one or more embodiments, the steps of.

repeating the interferometry sequence a plurality of times in order to obtain plurality of measurements of the state of the ion: and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; may be performed a second plurality of times wherein, for each repeat of these steps in the second plurality of times, a different set of trapping electric field amplitudes is used during the pulse sequence; and wherein adjusting the trapping electric field may be based on the second plurality s of measurements of the probability.

In one or more embodiments, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; may be performed a third plurality of times wherein, for each repeat of these steps in the third plurality of times, a different number of laser pulses is applied; and wherein adjusting the trapping electric field may be based on the third plurality of measurements of the probability.

In one or more embodiments, for each of the first plurality of times thesteps of repeating the interferometry sequence a plurality of times in order to obtain plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of fie in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the second plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different sets of trapping electric field amplitudes; and wherein adjusting the trapping electdc field may be based on all of the determined probabilities.

In one or more embodiments, for each of the first plurality of times the steps o repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of Vie state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain plurality of measurements of the state of the ion: and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different numbers of laser pulses; and wherein adjusting the trapping electric field may be based on all of the determined probabilities.

In one or more embodiments, for each of the second plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferornetry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferornetry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the third plurality of times such that a plurality of probabilities are obtained at different sets of trapping electric field amplitudes and different numbers of laser pulses; and wherein adjusting the trapping electric field may be based on all of the determined probabilities.

In one or more embodiments, for each of the first plurality oltirnes the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion: and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of mes in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the second plurality of times; and that for each of the second plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurahty of measurements of the state of the ion: and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferornetry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferornetry sequence based on the plurality of measurements of the state of the ion may be repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different sets of trapping electric field amplitudes and different numbers of laser pulses; and wherein adjusting the trapping electric field may be based on all of the determined probabilities.

In one or more embodiments: a first time the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are performed, the method may comprise providing the laser pulses along a first direction; and a subsequent time the steps of: repeating the interferometry sequence a plurality of times in ordiarto obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are performed, the method may comprise providing the laser pulses along a second direction, different to the first direction wherein adjusting the trapping electric field may be based on the first and subsequently determined probabilities. It will be appreciated that the plurality of repetitions of the interferometry sequence of the first time and the plurality of repetitions of the interferornetry sequence of the subsequent time may be performed either sequentially, i.e., wherein all of the repetitions of the first time are perfoimed followed by all of the repetitions of the subsequent time, or may be performed in an interleaved matter, i.e., one or more of the first repetitions of the first time may be taken followed by one or more of the repetitions of the subsequent time followed by one or more of the repetitions of the first time and so on.

In one or more embodiments, each of the first and second direction may have one of: a directional vector entirely in the plane of the oscillafino electric field; and a directional vector having a component out of the plane of the oscillating electric field. In these embodiments, the method may be performed with a system arranged as a linear Paul trap.

In one or more embodiments,the first and second directions may be relatively orthogonal directions.

In one or more embodiments, the method may further comprise: measuring a detuninu of a laser from a transition resonance frequency using interferometry by: applying a first laser pulse to the trapped ion when the electric field amplitude comprises a fixed electric field amplitude and the first laser pulse has a first phase; applying a second laser pulse to the trapped ion when the electric field amplitude comprises the fixed electric field amplitude and the second laser pulse has a second phase different to the first phase; and measuring a state of the ion after the application of e first and second laser pulses; repeating the process of measuring the deturung of the laser a plurality times in order to obtain a plurality of measurements of the state of the ion; determining a fixed electric field amplitude probability of the trapped ion being in the given state based on the plurality of measurements of the state of the ion; wherein detuning of the laser may be accounted for based on the fixed electricfield amplitude probability.

In one or more embodiments, the method may comprise alternating between determining the state of the trapped on at electric field amplitudes which change between application of the laser pulses and determining the state of the trapped ion at the fixed electric field amplitude. In one or more embodiments, the same predetermined delays may be used whether the electric field amplitude is varying or fixed.

In one or more embodiments, the average of the square of the amplitude of the oscillating electric field of the on trap over the course of the pulse sequence, comprising any initialisation step; the coherent pulses and the measurement, may be equal to the square of the amplitude of the oscillating electric field of the ion trap during an operational mode.

s In one or more embodiments, the state of the on may be measured by fluorescence detection, in one or more embodiments, the fluorescence detection may be preceded by a quantum logic transfer step of quantum logic spectroscopy experiments. In one or more embodiments, the ion trap comprises a linear Paul trap or a ring Paul trap.

According to a second aspect of the present disclosure, there is provided a system configured to reduce the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap comprising: a plurality of electrodes configured to generate a trapping electric field for trapping one or more ions wherein the trapping electric field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field; a first laser beam configured to apply laser pulses to the trapped ion and a detector; the system configured to induce a change in equilibrium position of one of the one or more trapped ions and use interferometry to measure said change using an interferometry sequence, by controlling: the laser to apply laser pulses to the one of the one or more trapped ions and to change the trapping electric field amplitude between application of successive laser pulses; and the detector to measure the state of the one of the one or more trapped ions after the application of the laser pulses; wherein the system is further configured to repeat the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the one or more trapped ions; determine a probability that the one or more trapped ions change state during the interferometry sequence based on the plurality of measurements of the state of the ion; and adjust the trapping electric fields based on the probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating

electric quadrupole field.

According to a third aspect, there is disclosed a computer readable medium having stored thereon software instructions that, when executed by a processor, cause the processor to generate control signals to cause a system of the second aspect to perform the method of the first aspect.

According to a fourth aspect of the present disclosure, there is provided an optical clock comprising the system of the second aspect.

According to a fifth aspect of the present disclosure, there is provided a quantum computing system comprising the system of the second aspect.

According to a sixth aspect of the present disclosure, there is provided a qu ntun simulator system comprising the system of the second aspect.

According to a seventh aspect of the present disclosure, there is provided a trapped ion electric field sensor comprising the system of the second aspect.

According to an eighth aspect of the present disclosure, there is provided a trapped ion force sensor comprising the system of the second aspect.

Brief Descriøtion of the Drawings One or more embodiments will now be described by way of example only wth reference to the accompanying drawings in which: Figures 1A -13 show an example of the electric quadrupole arrangement configured to generate the oscillating electric field of an ion trap; Figure 2 shows an example of the time averaged effect of the osculating electric field of Figures 1A and 13 of the ion trap; Figure 3 shows an example embodiment of a method for reducing the magnitude of a quasi-static electric dipole field at the null position of the oscillating electric quadrupole field of an ion trap; Figure 4 shows a second example embodiment of a method for reducing the magnitude of a quasi-static electric dipole field at the null position of the oscillating electric

ss quadrupole field of an on trap

Figure 5 shows an example system comprising a linear ion trap; Figure 6 shows the example system comprising an ion trap in a view looking along the axial direction; Figure 7 shows an example pulse sequence of laser pulses and the corresponding electric field amplitude during said pulse sequence; Figure 8 shows an example pulse sequence of laser pulses and the corresponding electnc field amplitude during said pulse sequence; Figure 9 shows experimental results of changing an electric field offset on pp and on the phase (c.1)1 -4)2): and Figure 10 shows experimental results representing the degree to which the lo magnitude of the offset field along two directions can be reduced depending on the interrogation time used.

Detailed Description

Ions can be trapped in a variety of configurations of ion traps by using arrangements of electric fields. These arrangements can comprise at least one oscillating electric field S (such as a radio frequency, RF, quadrupole field) and, in some examples, may also include one static electric field. Figures 1A ---15 show an example of how an ion is trapped in an oscillating electric field. In this example, at a first time instant, diagonally opposing electrodes are charged with the same polarities. As shown in Figure 1A, a first pair of electrodes 101A; 1015 are positively charged and a second pair of electrodes 1010, 1010 are negatively charged. Assuming a positively charged ion 102, an ion 102 near the centre of the trap is attracted towards the negatively charged electrodes 1010; 101D and it is repulsed from the positively charged electrodes 101A, 1015. As shown in Figure 15, at a second time instant, the charges of the electrodes 101A, 1015, 1010, 1010 may be inverted such that the first pair of electrodes 101A, 101B are negatively charged and the second pair of electrodes 1010, 1010 are positively charged. This causes the ion to then be attracted to electrodes 101A, 1015 and repulsed from electrodes 101C, 1010. By alternating between the configuration of Figure 1A and the configuration of Figure 1B, the time-averaged force acting on the ion may be towards the centre of the trap such that the ion is dynamically trapped, as is simplistically shown in Figure 2.

It will be appreciated that the examples shown in Figures 1A-1B show a simplified two-dimensional example of an ion trap. In order to provide for trapping in the third dimension; out of the plane in the examples of Figures 1A -15, several different options can be used. In the arrangement of a linear ion trap, a static (DC) electric field may be generated by electrodes having a like-charge arranged on either side of the ion in the third dimension.

The time-averaged position of a trapped ion shown in Figure 2 is referred to as the trapped ion's equilibrium position. At a trapped ion's equilibrium position the time-averaged electric field is zero. Usually ion traps are configured with the aim of having the equilibrium position of a trapped ion coincide with a null of the oscillating trapping electric field. Trap imperfections and external field sources may give rise to a slowly varying (quasi-static) unwanted dipolar electric field near the centre of an ion trap. A quasi-static offset field at the null position of the oscillating field causes the equilibrium position of a trapped ion to be shifted from the null of the oscillating field by

-

Where orthogonal directions defined by the ion's secular motion are indexed by is the displacement of the ion equilibrium position from the null position of the oscillating field in the I direction, q is the ion charge, K1 is the component of the quasi-static offset field in the direction, in is the ion mass, wi is the frequency of the ion's secular motion in the I direction. As a result, a trapped ion will experience an oscillating electric field at its equilibrium position. This unwanted oscillating field will cause the ion to exhibit additional motion at the frequency of the oscillating field, called excess micromotion. This unwanted field will also exacerbate the Stark effect on the energy levels of the ion.

If effects of the unwanted offset field on the ion can be accurately measured, then information about the unwanted offset electric fields may be determined and, therefrom, it may be possible to make changes to the system in order to account for the unwanted offset electric field. It may also be possible that if effects of the unwanted offset field on the ion can be accurately measured, changes may be made to the system to reduce these effects, and this may include reduction of the magnitude of the unwanted offset field.

It is possible to measure excess micromotion using techniques such as the measurement of modulation of ion fluorescence as a result of micromotion, and measuring the strength of resonance sidebands in transition spectra as a result of micromotion. The techniques mentioned above, however, suffer from the disadvantage of lower resolution compared to that of the systems and methods that are described below. One of the disadvantages of the ion fluorescence measurement is that the measurement results are sensitive to the radiation pressure of the laser field used. One of the disadvantages of the sideband method is that a measurement result in one direction gives information about the magnitude of a component of the offset field, but not about the sign of the component of the offset field. These disadvantages do not apply to the technique disclosed below.

The technique disclosed below may provide for determination of the unwanted offset electric field at a higher accuracy and in a shorter interrogation time than achieved using prior techniques.

Another effect sensitive to the unwanted offs.et field is the change of the trapped ion equilibrium position when the amplitude of the trapping fields is changed. A change of the amplitude of the trapping fields causes the ion's secular frequencies to change (on ohi and the ion's equilibrium position to change: a-u1 = cp5i ( m.w2 This change in equilibrium position Au can be detected using an imaging system. The resolution of such a technique in the object plane is limited by the resolution with which the position of the trapped ion can be determined, where that resolution is the diffraction limit. Further; this technique is less sensitive to movement of the ion out of the object plane, which gives rise to defocussing. This technique is less sensitive to a change of ion equilibrium position Au, and to the offset electric field E that causes it, than the technique presented herein.

Figure 3 shows a method 300 of reducing the magnitude of an offset field at the null position of an oscillating electric field used for trapping an ion.

The oscillating electric field may oscillate at RF frequencies. However, it will be appreciated that the oscillating electric field may oscillate at any frequency suitable to maintain the trapping of the trapped ion. For example, the oscillating electric field may oscillate at frequencies between 10kFiz and lOGHz. In some examples, the oscillating electric field may oscillate at frequencies between 1 and 100 MHz. in some examples, the oscillating electric field may oscillate at frequencies between 5 and 20 MHz.

The oscillating quadrupole electric field may be generated by four electrodes, as described with reference to Figure 1, where a first pair of diametrically opposed electrodes of the four electrodes are each configured to have a first voltage and a second pair of diametrically opposed electrodes of the four electrodes are each configured to have a second voltage different from the first voltage. The electrodes are configured such that the polarity of the first electrode pair is opposite of that of the second electrode pair, i.e., when the first pair of electrodes have a positive charge thereon, the second pair of electrodes have a negative charge thereon and vice versa. The magnitude of the potential at the electrodes may be varied during operation, as is described further below. In one or more examples the voltage on the first pair of electrodes may be set to a fixed value while an RE: voltage is applied to the second pair of electrodes.

It will be appreciated that; while quadrupole is often used herein to refer to the structure of the type of electric field, other arrangements of electric field may be implemented. For example, an octupole electric field, or even higher order, may be implemented instead. In yet other examples, combinations of quadrupole, octupole or higher ogler electric fields may be implemented. Further, any reference to quadrupole, octupole or other order of electric field arrangement does not preclude the use of one or more compensation electrodes configured to contribute to the electric field and, thereby, reduce the dipole electric field magnitude at the position of the null of the oscillating electric field.

s By reducing the magnitude of the dipole offset electric field at the position of the null of the oscillating electric field, the dependence of the ion's equilibrium position on the electric field amplitude is reduced. Reducing the dependence of the equilibrium position on the electric field amplitudes advantageously reduces the excess micromotion of the ion and also reduces the undesired Stark effect on the states of the ion.

Herein, on may refer to any of an atomic ion or a molecular ion where, in either case, the ion may comprise a single valence electron or a plurality of valence electrons. The on may have a hyperfine structure.

The method may comprise trapping 301 at least one on in a trapping electric field wherein the trapping electric field comprises an oscillating electric quadrupole field and may further comprise a static quadrupole electric field. The trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field and the electric field amplitude of the static electric field. As such, it will be appreciated that references to changing the trapping electric field amplitude may refer to changing one or both of the oscillating electric field amplitude and the static electric field amplitude. The oscillating and static electric field amplitudes may be changed by changing the amplitude of the voltages applied to the electrodes configured to generate those fields. A change of the trapping electric field amplitude would cause the ion's secular frequencies to change --->w,1. As such, with reference to Equation 2, this would cause the ion equilibrium position to change by iv.

The method further comprises inducing a change in an equilibrium position of the at least one trapped ion and measuring said change using an interferometry sequence. As can be seen from Equation 2, changing the ion's secular frequencies, cuff -> W21, results in a shill Au in the equilibrium position of the trapped ion if there is an offset electric field E. The secular frequencies are changed by changing the amplitude of the voltages applied to the electrodes to generate the trapping fields.

Performing the interferometry sequence comprises applying 303 a first pi/2 laser pulse to the trapped ion and subsequently applying 305 M-1 pi laser pulses to the trapped ion and subsequently applying 307 a second pi/2 laser pulse to the trapped ion. The index j describes the order in which the pulses are applied. The first p1/2 pulse has index = 1 and the final p1/2 pulse has index j = M + 1. The application of a laser pulse causes a change in the state of the ion. More particutarly, two different states of the ion may be considered where the laser field resonantly couples the two states. in other words, the laser field is resonant to the transition between the two states. In some examples, the ion might be prepared in a plurality of initial states and the laser might drive transitions between Me plurality of initial states to a corresponding plurality of final states.

The method further comprises measuring 309 a state of the ion after the application of the 10M+1 laser pulses. It will be appreciated herein that measuring the state is performed after the application of all of the laser pulses (i.e., after the whole pulse sequence has completed), and not individually after each laser pulse such that multiple measurements are obtained. The final state of the ion after interferornetry can be measured using any suitable technique. In one or more embodiments, the final state of the ion may be measured using a fluorescence measurement. The final state of the trapped ion may also be measured using a technique used in quantum logic spectroscopy, whereby the state of the on is coupled to the state of a second ion, a subsequent fluorescence measurement of the state of the second ion reveals the state of the ion. The phase differences of the laser fields during the laser pulses experienced by the on determines the ion's final state with: Pe,s Or) Eq. 3 where pc is the probability of finding the ion in an excited state, e; and the total phase satisfies: y + Y712,(-1 ' 20; + Ofri+j + it M is even if M is odd Ea 4 = 01+ ,7 * 26), ant,: where Of is the phase of the laser field experienced by the ion during the ph laser pulse. It will be appreciated that this describes the idealised relationship but that experimental imperfections may include errors in pulse areas and decoherence, as such, the probability variation may differ from the presented equation in true experimental conditions. The phase Oj has the components: Eq. 5 Where the component S1 can be controlled using, for example, an accusto-optic modulator, while the component & depends on the position of the ion when the jth pulse is applied.

And thus Eq. 6 = [1 + cos(0,± y5T)11 2 Where or = 01 +y (-0 0 0141-1 al YE-1Y.28 + 6" if M is even Eq. 7 + \-;(-P+1 * 20; /.2 OT = + VA-111+ 29i -Om 4-, 1.2 if M is odd Because efyj depends on the ion position, epT contains information about the ion position or about changes of the ion position. We refer to OT as the total control phase. It will be appreciated that because the phase contributions B, are controlled, the total control phase Br can be accounted for.

During the I" pulse, when the ion is at the position indexed by p, position up, the phase (pi takes the value tp. The difference between the position--dependent phases experienced at two positions up and up satisfies cf), -(Pp = Eq. 8 where k? is the component of the laser field wavevector k in the i direction u is the component of the position up in the i direction. and upi is the component of the position up in the i direction.

Changing the trap stiffness from al"i alqi when there is an offset electric field E affecting the trapped ion system causes the ion position to change up -+ up. By combining Equation 2 and Equation 8 we see the phase difference ch,,, D. is related to the unwanted electric

field E by

q 1 Eq. 9 9 La in vopi-The phase difference tp ch" is sensitive to E along the dirertion d, where Eq. 10 d= k, f 2:)-q-z7)1 If the trap stiffness is changed cop; cogi between application of the laser pulses of an interferometry sequence, the phase difference <hp -4), will affect the probability Pe of measuring the ion in the state e, as is described by Equation 6. Because (1--)., ch, depends on E, the probability Pe of measuring the ion in the state e will depend on the unwanted electric field E. Each time the interferometry sequence is performed, the state of the ion in that instance will be determined. Repeating the process of performing the interferometry sequence with the at least one trapped ion and measuring the final state of the at least one trapped ion a plurality of times will allow a probability of the trapped ion being in a given state to be calculated. Repeating the process of performing the interferometry sequence with the at least one trapped ion and measuring the final state of the at least one trapped ion a plurality of times may be represented as a loop repeated N times of the steps: initializing 302 the state of the ion; applying 303 the first pi/2 laser pulse; changing 304 the trapping electric IS field amplitude between a first trapping electric field amplitude and a second trapping electric field amplitude; applying a sub-loop M-1 times involving applying 305 a pi laser pulse and then changing 308 the trapping electric field amplitude between a first trapping electric field amplitude and a second trapping electric field amplitude: applying 307 the second pi/2 laser pulse; setting 308 the trapping electric field amplitude to the first trapping electric field amplitude; and measuring 309 the state of the ion. It will be appreciated that when M is an even number step 308 will have no effect.

The method 300 also includes determining 310A the probability of the trapped ion being in a given state. This may, for example, comprise calculating the fraction of the plurality of measurements in which the state of the ion is e.

It will be appreciated that, generally, one may discuss exciting the ion into a higher state from a lower state. However, the ion may equally be driven into a lower state from a higher state. Additionally, regardless of whether the ion is driven into a higher or lower state, the probability of the ion being in either of those states may be determined at step 310A.

Because the final state of the on after the interferometry sequence has concluded depends on the change of the ion position Au, interferometry can be used to measure the unwanted offset electric field E which causes excess rnicrornotion and exacerbates Stark shifts. Between the laser pulses in the pulse sequence the trap stiffness is alternated between a first trap stiffness value w1i and a second trap stiffness value (1/21. This causes the on position to change between position u1 when the pulses with odd-valued indices are applied, and position u2 when pulses with even-valued positions are applied. Then * = 42. when j is even (Pi And using Equation 7 and Equation 8 and Equation 2 (pj- 4 --,2) -a2i) = 'Cl LA VII.

ePi. when j is odd Eq. 11 Eq. 12 And using Equation 6 -2- cs101, + Al Ohl (Ii2)11 --2 1 COS ± 1 wei2) 1 ( _ cikiEi It / 1 Information about the unwanted offset electric field E may be determined by: applying the laser pulses 303, 305, 307; inducing the change in the equilibrium position Au of the at least one trapped ion 304, 306, between application of the laser pulses; measuring 309 said change a plurality of times; and determining 310A the probability of the ion being in a given state. Using Equation 13 and taking into account the other variables and constants of the equation, which are either known or may be independently determined; the component of the unwanted offset electric field E in the direction d, described by Equation 10, can be determined.

Having calculated the probability of the ion being in a given state, the method can calculate one or more pieces of information such as (1)7. or the component of E in a direction d. The method may comprise adjusting 311 the trapping electric field in order to reduce the magnitude of E in the direction d, as discussed in detail below, based on the probability of the ion being in the given state and/or one or more pieces of the calculated information.

This may thereby reduce the magnitude of the unwanted offset electric field E at the position of the null of the oscillating trapping electric field.

Eq. 13 Pe It will be appreciated that one may implement interferometry sequences to gain information about E, which comprise more than two trap stiffness settings, and thus involve more than two equilibrium positions of a trapped ion.

Adjusting 311 the trapping electric field may comprise one or more of: - applying or changing a static voltage at one or more electrodes configured to

generate the trapping electric field;

moving one or more of the electrodes configured to generate the trapping electric field relative to the trapped ion; adjusting the trapping electric field by applying or changing a voltage at one or more compensation electrodes; and - moving one or more of the compensation electrodes relative to the trapped ion.

Taking any of the above actions will change the local electric field around the ion and; if done based on the probability of the ion being in a given state as described, will advantageously reduce the magnitude of the unwanted offset electric field Eat the position of the null of the oscillating trapping electric field.

Figure 4 shows a simpler version of the sequence presented in Figure 3, In Figure 4 the initialisation step 302 is not repeated in each iteration, because the state of the ion before application of the first laser pulse will be known from the previous measurement step. This would mean that instead of determining the probability of the ion being excited to state le> (step 308A) that instead the probability of the state changing would be determined (step 308E). It will be appreciated that, in the first measurement, determining the probability of the ion being in a given state 308A may be directly equivalent to determining a probability of the ion changing its state when 308A is performed after an initialisation step.

Another simplification that may be introduced is the trapping electric field amplitude is not set to the first trapping electric field amplitude after applying the laser pulses (step 308 is removed). In this case, if M is odd, the iterations may alternate between iterations in which the first trapping field amplitude is used during the odd numbered laser pulses and the second trapping field amplitude is used during the even numbered laser pulses (and in step 304 the ion is displaced by Au) and iterations in which the second trapping field amplitude is used during the odd numbered laser pulses and the first trapping field amplitude is used during the even numbered laser pulses (and in step 304 the ion is displaced by -AO. The interferometry measurements in each case would be sensitive to the phase offsets (1); = MOD" ft)2) and eT = P4(4).2 respectively. This can be accounted for during step 309.

Figures 5 and 6 show an example system 500 comprising a linear ion trap and laser geometry configured to carry out the above-described method. The example on trap represented in Figures 5 and 6 comprise a linear on trap 501 using a quadrupole electric field arrangement. It will be appreciated, however, that any suitable ion trap arrangement may be used such as, but not limited to, a hyperbolic ion trap, linear ion trap, a ring ion trap, a cylindrical ion trap, a planar on trap, a wheel ion trap, or a sandwich ion trap. In this example, a 88Sr+ ion 502 is confined in the linear ion trap 501 and the subsequently described experimental details are suitable for trapping a Sr'88 ion. It will be appreciated that other ions may be used in this arrangement; and the same or different experimental parameters may be used to trap the ion such as different electrode configurations, different laser arrangements and different laser frequencies.

In this example, voltages are applied to four gold-coated blade electrodes 503 to confine the ion in the x and y radial directions. The voltages on these four blade electrodes 503 are configured to generate the oscillating electric field in the x, y plane. Static voltages are applied to two gold-coated endcap electrodes 504 to confine the ion in the, z, axial direction. Any other suitable electrode design may be used. The co-ordinate axes defined in the figures are defined by the ion's secular motion and the electrode 503, 504 geometry. As such, the electrodes 503 are configured to generate the oscillating electric field in the x, y plane. The electrodes 503 may also generate a static electric field which produces the non-degeneracy of the radial modes, cox and ay. The electrodes 504 configured to generate the static electric field which provides confinement along the z direction are arranged in diametric opposition along the z direction. It will be appreciated that, while the electrodes 504 configured to generate the static electric field may start in a diametrically opposed arrangement, the exact relative position of these electrodes 504 may be adjusted in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field. The relative position of the electrodes 503 may also be adjusted in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field.

In the following section, directional vectors will be referred to in the format (x, y, z). In this example; three 674 nrn laser beams 505 are provided to illuminate the ion with unnormalized propagation directions of (0, 0, 1); (-1, 1, 42) and (-1, -1, 0). These independently controllable laser beams may be provided by a single laser or may be provided by a plurality of lasers. In one or more embodiments, the axial (0, 0, 1) laser beam 505 may be configured to propagate through one or more holes in the endcap electrodes A first laser beam, such as the laser beam arranged along (-1" 0) may be configured to provide the laser pulses with a wavevector k in the plane of the oscillating electric field. In one or more embodiment, a further laser beam may be provided along another direction k in the plane, or having components in the plane, of the oscillating electric field which is also configured to provide the laser pulses of the interferometry sequence. As previously described, the probability of the ion 502 being in a given state can be calculated when a laser field wavevector ks used and, therefrom, information about the electric field E in the d direction may be obtained. The system may be configured such that each laser beam is configured to provide the one or more laser pulse sequences separately in order to is determine the probability of the ion being in a given state when laser pulses are provided along different k directions so information about the electric field in different d directions may be determined, that a 2D or 3D measurement of offset electric field E can be determined, it will be appreciated that, where a measurement is sensitive to the offset field E in a direction d which has components dl! in the plane of the oscillating electric field and d'E, out of the plane of the oscillating electric field and a value of El! is being sought which is in the plane of the oscillating electric field, it will be necessary to resolve the components of the electric field in the direction (ill. The axial laser beam may also be used to measure unwanted electric fields in the z direction by using the method 300 of Figures 3 or 4. It will be appreciated that in an ideal linear Paul trap there is no oscillating field along the axial direction, but that imperfections, such as machining imperfections, may give rise to such an oscillating field along the axial direction. This unwanted field may be probed using the technique disclosed herein, using a laser beam which has a wave vector with an axial component. It will be appreciated that, where measurements of the unwanted electric field are sought in the x, y and z directions, it may be particularly beneficial for the laser beams that are configured to provide the laser pulses and for the changes of the trapping field amplitudes to be arranged such that the different directions d are orthogonal to each other. It may also be beneficial for the lasers that are configured to provide both the one or more laser pulse sequences and the changes of the trapping field amplitudes, to be arranged such that the different directions d are aligned with the x, y and z axes which are defined by the ion's secular motion.

A Doppler cooling laser may be provided, such as the laser in the (1, -1, A/2) direction 505 in Figure 5. The Doppler cooling laser may be configured to provide a laser configured to slow the movement of the ion and, thereby, increase the degree to which the interferometry results are described by Equation 3. An additional Doppler cooling laser beam may be employed when radial secular frequencies of the system are degenerate, such as the laser in the (1, 1, 0) direction 505 in Figure 6.

One of the lasers 505 of the system 500 may comprise a repump laser. The repump laser may be configured to counter optical pumping to an unwanted state driven by the Doppler cooling laser beam. One of the lasers of the system 500 may comprise a quench laser.

The quench laser 505 may be configured to transfer an on from an excited state to the ground state. The Doppler cooling, repump and quench laser beams 505 may be configured to copropagate. Due to experimental set-up restrictions, optical access to the ion may be restricted. Providing for copropaoating laser beams 505 may allow the beams to be focussed together and thereby make efficient use of available set-up space.

By way of specific example for illustrative purposes, a strontium 88 ion may be initialised in a particular sublevel of its ground state 5S112 by optical pumping. A pulse of 674nrn laser light may transfer the ion from state 581/2 rm=1/2 to 405/2 rm=-3/2. A pulse of the 1033nm laser light may transfer the ion from 4D5,--2 to 5P312, from which it may decay to either sublevel of 551;2. The process does not affect the ion if it was initially in state 55112 m- /2. By repeating this process (typically 10 times) if the ion was initially in state 55112 nb=1/2, it will likely finish in 551;2 m-l/2. The 1092nm laser field is turned on during this process to prevent optical pumping to state 4D3/2.

The system 500 may also comprise one or more compensation electrodes. The compensation electrodes 506 may comprise additional electrodes to which a different voltage can be applied in order to cause a change in the overall trapping electric field of the system 500. Alternatively, the compensation electrodes 506 may be moved in order to cause a change in the overall trapping electric field of the system 500. Each of the one or more compensation electrodes 506 may comprise a pair of rods to which a voltage is applied. Each of the rods of a compensation electrode 506 may be arranged adjacent to one of the electrodes configured to generate the oscillating electric field. In some embodiments, the first rod in a pair of rods of a compensation electrode may be arranged adjacent to a first oscillating electric field generating electrode and the second rod in the pair of rods of the compensation electrode may be arranged adjacent to a second oscillating electric field generating electrode. it will be appreciated that a first pair of oscillating electric field generating electrodes may be configured to have a fixed voltage applied thereto and a second pair of oscillating electric field generating electrodes may be configured to have a time-varying voltage applied thereto. The first oscillating electric field generating electrode may be one of those configured to have a fixed voltage applied s thereto and the second oscillating electric field generating electrode may be one of those configured to have a time-varying voltage applied thereto.

The system 500 may further comprise a photon-collection device (not shown) configured to provide a measure indicative of the number of photons emitted by the on 502 during the measurement step 309 of the interferometry sequence. In one or more embodiments, the photon-collection device may comprise a photomultiplier tube (PMT) or it may comprise another photon-collection device.

As has been described already, the interferometry sequence is repeated a plurality of in order to obtain a plurality of measurements of the state of the ion. Then a probability of the trapped ion being in a given state can be determined. Determining the state of the same ion a plurality of times may comprise repeatedly performing the interferometry sequence on a single trapped ion in an ion trap. Repeatedly performing the interferometry sequence on a single trapped ion may comprise performing the interferometry sequence on the same trapped ion a plurality of times or may comprise performing the interferometry sequence on a first trapped on in the trap, removing that trapped on from the trap and trapping a new ion in the same trap and then repeating the interferometry sequence. A combination of these approaches may be taken. Alternatively, where a plurality of ions are trapped in the ion trap, performing the interferometry sequence a plurality of times may be achieved by performing the method on each, or a subset, of the plurality of trapped ions to simultaneously obtain a plurality of measurements. It will further be appreciated that the method may be repeated a plurality of times on a plurality of trapped ions, thereby utilising a combination of said techniques, In a two-level system {1g), le)}, initialized in state 1g), application of a laser field resonant to the I g) +-> e) transition couples the states, and causes the state of the system to oscillate between la), superposition states of 1g) and le), and state le). A pi/2 pulse causes state 1g) or state le) to evolve to a superposition state with equal la) and le) components. Application of a pi pulse to a superposition state which has equal 1 g) and le) components may cause the system to evolve to a different superposition state which has equal 1g) and le) components or it may remain in the same superposition state, whichever of these outcomes that occurs depends on the phase relation between the superposition state and the laser field. Application of a p1t2 pulse to a superposition state which has equal 1,g) and le) components may cause the system to evolve to state 1,a), or to state le), or to a different superposition state of I g) and le), or it may even remain in the same superposition state, whichever of these outcomes that occurs depends on the phase relation between the superposition state and the laser field.

Figure 7 shows a simpl Bed example pulse sequence 701 for the process OT inducing a change in the equilibrium position of the at least one trapped ion and measuring said change using interferometry. In a first time period 702; the system is initialised into a starting state while the trapping electric field amplitude is at a first electric field amplitude 709. In a second time period 703, a pi/2 pulse is applied to the on when the trapping electric field is at the-first electric field amplitude 709. In a third time period 704, a pi pulse is applied to the ion when the trapping electric field is at a second electric field amplitude 710, different to the first electric field amplitude. In a fourth time period 705, a pi pulse is applied to the ion when the trapping electric field is at the first electric field amplitude 709. The steps 704 and 705 are repeated M12-1 times. In a fifth tirne period 706, a pi pulse is applied to the ion when the trapping electric field is at the second electric field amplitude 710. In a sixth time period 707, a pi/2 pulse is applied to the ion when the trapping electric field is at the first electric field amplitude 709. In a final time period 703; the state of the ion is detected while the electric field amplitude is at the first electric field amplitude 709.

The sequence in Figure 7 works with even values of M. Figure 8 shows a simplified example pulse sequence 701 for the process of inducing a change in the equilibrium position of the at least one trapped ion and measuring said change using interferometry. The sequence in Figure 8 works with odd values of M. In the sequence in Figure 3, steps 704 and 705 are repeated (M-1)12 times, and step 706 is omitted.

A predetermined delay may be provided between the laser pulses. The predetermined delay may be a sufficient time for the trapping electric field to be changed from the first electric field amplitude to the second electric field amplitude. In one or more embodiments, the predetermined time delay may be between lps and 20ms. The predetermined time delay may between 5 ps and 20 pa. The predetermined time delay may be 10 pa. In one or more embodiments, the predetermined delay may be equal between each of the laser pulses. In other embodiments, the one or more of the predetermined delays may be different between the laser pulses.

In one or more embodiments; the laser pulses may be coherent laser pulses. The contribution from the controlled phase shifts (which is referred to herein as the total control phase) Br may satisfy OT = g/2. it will be appreciated that eT = rr/2 may be achieved using different combinations of controlled phase shifts 9. It \AM be appreciated that other values of OT may be used, for instance, OT = $7/2 may be used and OT = 0 may be used.

More specifically, in one or more embodiments, the first laser pulse may comprise a resonant pi,12 pulse and the final laser pulse may also comprise a resonant pit2 pulse.

Experiments involving two pi/2 pulses may be referred to as Ramsey interferometry experiments. When M x It au; = P.// (1)1 -(1)21 = I ehi « 2P and when control phases OT = 7E/2 or 07 = 37/2 are used, the probability of the on being in the given state may respond most strongly to changes in (PT because, at this point, the magnitude of the rate of change of the probability pa with respect to OT is greatest. Absent of an offset electric field E. a probability of 0.5 may be expected when a phase difference 0.1-of pi/2 or 3ti/2 is used. In some embodiments, q5T and therefrom information about the unwanted electric field E. may be estimated based on measurements of the probability of the ion being in the given state when the phase combination OT is pii2 and 30/2. Where the probability of the ion being in the given state when 0T is pi/2 is denoted as pc and the probability of the ion being in the given state when OT is 3*pii2 is denoted as o"', the phase OT can be estimated from: 4 /1):2 chi -= pe Re; Estimation of P T using two sets of measurements in this fashion may have the advantage of robustness to errors in the pulse strength and to errors caused by decoherence. If N12 repetitions are conducted to determine pc and P1/2 repetitions are conducted to determine po', when IcPri <.< 27 and ik pe" 0.5, then the uncertainties ark ape. Pz 11(2/41) (due to quantum projection noise, using the normal approximation) and statistical uncertainty aCIS It will be appreciated that, OT and therefrom information about the unwanted electric field E. may be estimated based on measurements of the probability of the ion being in the given state when other phase combinations are used. Where the probability of the ion being in the given state when OT is 0 is denoted as Pe and the probability of the ion being in the given state when Or is 3pil2 is denoted as pec the phase 457, can be estimated from: I)n arctan2pi --" 2 Eq. 14 Eq. 15 0 The estimate using the approach described by Equation 15 may be more precise than the estimate using Eq. 14 if the condition ç5T <.< 2 IT is not satisfied.

It will be appreciated that herein, where a phase difference is referred to, the unit of that phase difference will be radians. The omission of "radians" herein is provided for ease of readability and in line with the practice of those skilled in the art and is not intended to imply that the phase differences referred to herein are measured in any other units.

Figures 9A-C shows example datasets which represent the results of how the probability of exciting to the excited state p" and the phase difference ti,421 -(132) change with changes to the voltage applied to a compensation electrode, and thus changes to the offset electric field E. Figures 9A and 9B each show the sinusoidal response of pe to changes to the voltage applied to the compensation electrode, and thus to changes of the offset electric field E. This is consistent with Equation 13. Using another technique, namely the resolved sideband technique, we independently checked that the magnitude of the offset electric field Ewes minimised when the voltage applied to the compensation electrode was around is 19.5V, which is indicated by the vertical dashed line in the figures. At this voltage the response of p40 voltage changes was high when the controlled phases satisfied OT = ±:4, as shown in Figure 10A, and low when the controlled phases satisfied 81 = oor OT = 21, as shown in Figure 10B. This is also consistent with Equation 13. We conducted the measurements with interferometry sequences with different numbers of pulses; i.e., with four different values of M. Values M=1, M=2, M=4 and M=8 are used. It can be seen that the sensitivity of pe to the offset electric field E can be enhanced by increasing the value of M, as expected from Equation 13. Using the probabilities pen Figures 9A and 9B, we estimated the values of (,(P, -4'2), using Equation 15. The estimated values are shown in Figure 9C. When the voltage applied to the compensation electrode was near 19.5V, where E=0, the determined values of -eibi were close to 0, as expected from Equation 9.

The solid lines throughout represent fits to the experimental data The probability p, varies sinusoidally with the voltage applied to the compensation electrode, since oe depends sinusoidally on the component of E in the direction d, and because the components of E depend linearly on the voltage applied to the compensation electrode. The phase (EDI. -02) varies linearly with the voltage applied to the compensation electrode, since (4), 4,;) depends linearly on the component of the offset field E in the direction d, and because the components of E depend linearly on the voltage applied to the compensation electrode.

For the smallest value of M used here. M=1, a wide range of voltages can be scanned in order to determine the voltage to apply to the compensation electrode to achieve a zero or near-zero phase (hi -(1)2) and zero or near-zero offset electric field component along the direction d along which the measurement is sensitive. Using larger values of M may allow for an enhancement in sensitivity and thereby assist in identifying a more optimal compensation electrode voltage which provides for a reduction in the magnitude of the offset electric field E and the amplitude of the oscillating electric field at the ion equilibrium position. Since Equation 13 is cyclic, phase differences of leL (DJ > n/M cannot be measured. Thus, in some examples. an enhancement in sensitivity provided by obtaining a plurality of probabilities of the trapped ion being in the given state [which is a function of (cbi -02)] using a sequence with a large value of M may come at the expense of reducing the total range of offset electric field values which may be interrogated. it is possible, if using a sequence with a value of M that is too large, to fall into a situation in which P2) = 0 while the magnitude of the offset electric field along the direction d along which the measurement is sensitive is far from zero, such as when M=4 is used around 21.0 V or 18.0 V in Figure 9C instead of the actual optimal trapping electric field using a compensation electrode voltage of approximately 19.5 V. This can be checked by changing the value of M. If the point is reached where the component of E along the direction d is 0, then etti --4)23 = 0 for all values of M. Alternatively this can be checked by changing the difference between the first trapping electric field amplitude and the second trapping electric field amplitude, since this affects the dependence of (c131 -4)2) on E. as can be appreciated from Equation 9.

Datasets with different values of M allow 4.iT, and therefrom E, to be estimated with different accuracies and from within different ranges. it may be advantageous to combine to use results of measurements using different values of M, to achieve a high accuracy from within a broad range Such an approach may be similar to methods used in quantum phase estimation techniques.

Thus, in some embodiments the method may comprise repeating the interferometry sequence and determining a probability that the trapped ion changes state during the interferometry sequence a first plurality of times. For each repeat of this group of steps in the first plurality of times, a different set of control phases ei may be used. This information may be used to obtain data such as that shown in Figure 9. Adjusting the trapping electric field may be based on the first plurality of measurements of the probability of the trapped ion being in the given state.

Further, in some embodiments the method may comprise repeating the interferometry sequence, and determining a probability that the trapped on changes state during the interferometry sequence a second plurality of times. For each repeat of this group of steps in the second plurality of times, a different set of trapping electric field amplitudes during s the pulse sequence may be used. Adjusting the trapping electric field may be based on the second plurality of measurements of the probability of the trapped ion being in the given state.

Further, in some embodiments the method may comprise repeating the interferometry sequence, and determining a probability that the trapped on changes state during the interferometry sequence a third plurality of times. For each repeat of this group of steps in the third plurality of times, a different value of M may be used. Adjusting the trapping electric field may be based on the third plurality of measurements of the probability of the trapped ion being in the given state.

It will be appreciated that varying the control phases 01 and varying the trapping electric field amplitudes and varying Al are described as first and second and third pluralities of times or measurements respectively, however, this nomenclature is provided for convenience of description. The nomenclature does not require that measurements of the plurality of control phases must be performed before the measurements of the plurality of trap stiffness changes, or that the measurements of the plurality of trap stiffness changes must be performed before the measurements of the plurality of M values or even that one set of measurements must be provided at all in order to perform, and obtain information from, another set of measurements.

In some embodiments, for each of the first plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the second plurality of times. In this way, a plurality of probabilities of the ion being in the given state are obtained at different combinations of control phases and different sets of trapping electric field amplitudes.

Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities.

In some embodiments, for each of the first plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the third plurality of times. in this way, a plurality of probabilities of the ion being in the given state are obtained at different combinations of control phases and different M values. Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities. Obtaining this data may allow for the collection and use of data such as that shown in Figure 9, In some embodiments, for each of the second plurality of times the steps of performing the interferornetry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the third plurality of times. In this way, a plurality of probabilities of the ion being in the given state are obtained at different sets of trapping electric field amplitudes and different M values. Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities.

In some embodiments, for each of the first plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped on being in a given state are performed, the same steps are repeated the second plurality of times, and that for each of the second plurality of times the steps of performing the interferornetry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the third plurality of times. In this way, a plurality of probabilities of the ion being in the given state are obtained at different combinations of control phases and different sets of trapping electric field amplitudes and different Ael values. Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities.

It has been described and shown in the above equations that inducing a change in an equilibrium position au of the at least one trapped ion and measuring said change using interferometry to determine a probability of the trapped ion being in a given state provides information about the offset electric field, E, along a direction d. Because of this, in order to provide enhanced reduction of the magnitude of the offset electric field E, in one or more embodiments, the method may comprise performing the method to determine said probability sensitive to E along at least each of a first direction d, and a second direction d2 different from the first direction. Adjusting the trapping electric field may then be based on the measurement of the probability along the first direction, pl, and the measurement of the probability along the second direction, p2. The method may equally be implemented along a third direction (13 in order to determine a probability along a third direction, p3 to provide for information in three dimensions.

The first direction and second directions may comprise a wave vector k which is entirely, in the case of a linear Paul trap, in the plane of the oscillating electric field, such as along the (-1, -1, 0) direction or may comprise a wave vector k having a component out of the plane of the oscillating electric field, such as the direction (-1, 1, 42). Where the direction s of a laser pulse comprises a wave vector k having a component out of the plane of the oscillating electric field, it may comprise a projection onto the plane of the oscillating electric field. In other examples, the directional vector may comprise no projection onto the plane of the oscillating electric field. While the relative angle between the wave vectors k of the first and second directions may be any relative angle, in some embodiments the angle between the two vectors may be orthogonal.

Figure 10 shows the reduction of the magnitude of the residual unwanted field E in two dimensions by conducting measurements of ("chi --41/2) along two directions d that may be achieved using the method disclosed herein, With increasing interrogation time, the magnitude of the residual unwanted electric field decreases before it levels out and increases. The levelling out and increase may be due to time-variation of the background electric fields or to noise in the voltage supplies used to apply voltages to the trap electrodes. Using the method disclosed herein, the offset field E is reduced to a lower value in two dimensions than has been achieved using the prior art. The magnitude of the offset field E in two dimensions is reduced also at a higher rate than has been achieved using the prior art. The second y-axis shows the amplitude of the residual oscillating field at the ion equilibrium position that arises due to the residual quasi-static offset field at the position of the null of the oscillating field (shown on the first y-axis).

It will be appreciated that such measurements may also be made using a probe laser along the z-direction in order to obtain three-dimensional information on E. It will be appreciated, that the control phases Of may be chosen to achieve an estimate of (4)1 -(P2) which is robust against different types of error, caused by, for example, detuning of the laser field from the lg) 10. resonance, or heating of the ion motion caused by changes of the amplitude of the trapping electric field.

By measuring the offset field Eat different times the drift of E over time can be measured. By taking into account this drift, the magnitude of the offset field E may be further reduced.

In some embodiments, where a plurality of probability measurements are being made, such as where a plurality of different probabilities are being measured using different control phases ej or with different trap stiffness changes, the interferometry sequence may also be conducted using a fixed electric field amplitude. Probabilities derived from such measurements may be used to correct for the time-varying detuning of the laser field from the transition resonance frequency. This will allow systematic offsets in the estimate of ch7-to be reduced or corrected.

In one or more embodiments, the average of the square of the amplitude of the oscillating electric field of the ion trap over the pulse sequence, comprising any initialisation step; the coherent pulses and the measurement, may be equal to the square of the amplitude of the oscillating electric field of the ion trap during an operational mode. An operational mode may be a mode of operation that the system is designed for; such as an optical clock or a quantum computing system. It will be appreciated that the method defined here is provided to reduce the magnitude of the offset electric field E and consequently the magnitude of the oscillating electric field at the ion's equilibrium position in order to reduce unwanted effects of excess micromotion and to reduce Stark shifts of energy levels. The trapped ion may then be implemented in any of a wide variety of applications. Such applications will operate under a standard trapping electric field amplitude. It will be understood that the square of the amplitude of the oscillating electric field is proportional to the power dissipated in the system. With greater power dissipation comes greater heating of the ion trap and changes in the temperature of the ion trap impact the trapping of the ion. As such, it may be beneficial to mitigate changes to the temperature when seeking to reduce the magnitude of the offset electric field E by having the average power dissipated in the system over the pulse sequence be equal to the average power dissipated in the system during its normal mode of operation.

There is also provided a computer readable medium having stored thereon software instructions that, when executed by a processor, cause the processor to generate control signals to cause a system such as that shown in Figures 5 and 6 to perform the above-described method.

The method and system presented herein may be applicable to a wide variety of (stem such as any system where the accurate trapping of an ion in an electric field is required. Such applications may include but are not limited to: an optical clock; a quantum computing system; a quantum simulator system; a trapped ion electric field sensor; and a trapped ion force sensor.

In a trapped on system, an offset electric field that varies in time causes the trapped ion to experience an amount of excess rnicromotion that varies in time. This in turn causes the Doppler shift on a transition (due to excess micrornotion) to vary in time. A varying offset electric field also causes the Stark shift to vary in time. By applying the described method s from time to time in a system which utilises one or more trapped ions, the varying offset electric field can be kept under control, and the transition frequencies can be kept stable. Further, if the amount of excess rnicromotion changes, then the strengths with which transitions can be driven using a laser field change in time.

Trapped ion optical clocks require stable transition frequencies, and also accurate knowledge of resonance shifts. Thus, the presently disclosed method and apparatus may provide for improved trapped on optical clocks.

In trapped on quantum computers, simulators; network nodes and force sensors, different transitions need to be driven with high fidelity. If the strength of transitions changes in time, then the fidelity of operations decreases. Also, if the Doppler shifts and Stark shifts change in time, this causes resonances to shift and this also decreases the fidelity of operations. This is detrimental to their operation. Thus, by way of the advantages described herein, the presently disclosed method and apparatus may provide for improved trapped ion quantum computers, simulators, network nodes and force sensors.

In some examples, it may not be necessary to adjust the trapping electric field based on the determined probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field of the ion trap.

Instead; the method may comprise determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, wherein the probability is indicative of the strength of the quasi-static electric dipole field Eat a null position of the oscillating electric quadrupole field of the ion trap and, based on the probability, parameters of a system in which the trapped ion is implemented may be varied to account for unwanted electric field effects. A system in which the trapped ion is implemented may be any relevant system, such as those discussed above including a trapped ion optical clock, a trapped ion quantum computer, simulator, network nodes or force sensor, for example. For example, in the case of a trapped ion optical clock, instead of adjusting the trapping electric field, the frequencies of the laser fields and the frequency of the optical clock may be adjusted. Similarly, in the case of a trapped ion quantum computer or simulator, instead of adjusting the trapping electric field, the frequencies of the lasers may be adjusted and the pulse areas may be adjusted. Thus, a will be appreciated that the probabilifies determined in the plurality of interferometry sequences may be used to correct for effects that result from the offset field E, such as Doppler shifts of the frequencies of transitions, rather than being used to reduce the magnitude of E.

Claims (1)

1. CLAIMS1. A method of reducing the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an on trap, the method comprising: trapping one or more ions in a trapping electric field, wherein the trapping electric s field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field; inducing a change in an equilibrium position of one of the one or more trapped ions and measuring said change using an interferometry sequence comprising: successively applying three or more laser pulses to the one of the one or more trapped ions and changing the trapping electric field amplitude between application of successive laser pulses; and measuring a state of the one of the one or more trapoed ions after the application of the three or more laser pulses; repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the one or more trapped ions; determining a probability that the one or more trapped ions change state during the interferometry sequence based on the plurality of measurements of the state of the one or more trapped ions; and adjusting the trapping electric field based on the determined probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field of the ion trap.

   9. The method of claim 1 wherein the trapping electric field further comprises a static electric field and wherein the trapping electric field amplitude is additionally comprised of an electric field amplitude of the static electric field.

   3. The method of claim 1 wherein repeating the interferometry sequence is performed the plurality of times by one or a combination of: performing interferomeny sequence on a single trapped ion a plurality of times; and trapping a plurality of ions in the oscillating electric field and performing the interferometry sequence on each of the ions.

   4. The method of claim 1 or claim 3 wherein the first laser pulse comprises a resonant p112 pulse and the final laser pulse comprises a resonant p1/2 pulse and any other laser pulse comprises a resonant pi pulse.

   5. The method of claim 4 wherein the at least three laser pulses are coherent laser pulses; controlled phase differences are introduced between the laser pulses; and a total control phase is pi/2, where the total control phase is a function of the controlled phase differences between the laser pulses.

   6. The method of any preceding claim in which the trapping electric field amplitude is changed between two different trapping electric field amplitude values.

   7. The method of any preceding claim wherein a predetermined delay is provided between the at least three laser pulses.

   8. The method of any of claims 1 -3 or 7 wherein the steps of: repeating interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion: are performed a first plurality of times wherein, for each repeat of these steps in the first plurality of times, a different combination of controlled phase differences is used; 20 and wherein adjusting the trapping electric field is based on the first plurality of measurements of the probability.

   9. The method of any preceding claim wherein the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped on changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; are performed a second plurality of times wherein, for each repeat of these steps in the second plurality of times, a different set of electric field amplitudes is used during pulse sequence wherein adjusting the trapping electric field is based on the second plurality of measurements of the probability.

   10. The method of any preceding claim wherein the steps of: repeating the interferometry sequence a plurality of imes in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped on changes state dunng the interferometry sequence based on the plurality of measurements of the state of the ion; are performed a third plurality of times wherein, foreach repeat of these steps in the third plurality of times, a different number of laser pulses is used during the pulse sequence; and wherein adjusting the trapping electric field is based on the third plurality of measurements of the probability.

   11. The method of claims 8 and 9 wherein for each of the first plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferornetry sequence a plurality of mes in order to obtain plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the second plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different trap stiffness changes; and wherein adjusting the trapping electric field is based on all of the determined probabilities.

   12. The method of claims 8 and 10 wherein for each of the first plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different numbers of laser pulses; and wherein adjusting the rapping electric field is based on all of the determined s probabilities.

   13. The method of claims 9 and 10 wherein for each of the second plurality of times the steps of repeating the interferometry sequence a plurality of mes in order to obtain a plurality of measurements of the state of the ion, and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during th.e interferometry sequence based on the plurality of measurements of the state of the ion are repeated the third plurality of times such that a plurality of probabilities are obtained at combinations of different trap stiffness changes and different numbers of laser pulses; and wherein adjusting the trapping electric field is based on all of the determined probabilities.

   14. The method of claims 8 to 10 wherein for each of the first plurality of times the steps of: repeating the interferometry sequence a plurality of mes in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the second plurality of times; and: for each of the second plurality of times the steps of: repeating the interferometry sequence a plurality of mes in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of order to obtain s plurality of measurements of the state of the ion: and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences, different trap stiffness changes and different numbers of laser pulses, and wherein adjusting the trapping electric field is based on all of the determined probabilities.

   15. The method of any preceding claim wherein: a first time the steps of: repeating the interferometry sequence a plurality ot times n order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state donna the interferornetry sequence based on the plurality of measurements of the state of the ion are performed, the method comprises providing the three or more laser pulses along a first direction; and a subsequent time the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are performed, the method comprises providing the three or more laser pulses along a second direction, different to the first direction, wherein adjusting the trapping electric field is based on the first and subsequently determined probabilities.

   16. The method of claim 15 wherein the first and second directions are relatively orthogonal directions.

   17. The method of any preceding claim wherein the method further comprises: measuring a detuning of a laser from a transition resonance frequency using interferometry by: applying a first laser pulse to the trapped ion when the electric field amplitude comprises a fixed electric field amplitude; applying a second laser pulse to the trapped on when the electric field amplitude comprises the fixed electric field amplitude and the second laser pulse has a second phase different to the first phase; and measuring a state of the on after the applicafion of the first and second laser pulses; repeating the process of measuring the detuning of the laser a plurality of times in order to obtain a plurality of measurements of the state of the ion determining a fixed electric field amplitude probability of the trapped on being in the given state based on the plurality of measurements of the state of the ion; wherein detuning of the laser is accounted for based on the fixed electric field amplitude probability.

   18. The method of any preceding claim wherein the average of the square of the amplitude of the oscillating electric field during application of the extended pulse sequence, comprising any initialisation step; the coherent pulses and the measurement, is equal to the square of the amplitude of the oscillating field of the ion trap during an operational mode.

   19. A system configured to reduce the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap comprising: a plurality of electrodes configured to generate a trapping electric field for trapping at least one ion wherein the trapping electric field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field; a first laser configured to apply laser pulses to the trapped ion and a detector; the system configured to induce a change in equilibrium position of at least one trapped ion and use interferometry to measure said change using an interferometry sequence, by controlling: the laser to apply at least three laser pulses to the trapped ion, and for the electric field amplitude to alternate between a first electric field amplitude and a second electric field amplitude different from the first electric field amplitude between application of successive laser pulses; and the detector to measure the state of the ion alter he application of the at least three laser pulses; wherein the system is further configured to: repeat the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; determine a probability that the trapped ion changes state during the interferometry s sequence based on the plurality of measurements of the state of the ion; and adjust the trapping electric fields based on the probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field.

   20. A computer readable medium having stored thereon software instrucflons that, when executed by a processor, cause the processor to generate control signals to cause a system of claim 19 to perform the method of any of claims Ito 18.

   21. An optical clock comprising the system of claim 19.

   22. A quantum computing system comprising the system of claim 19.A quantum simulat m comprising the system of claim 19 24.. A trapped ion electric field sensor comprising the system of claim 19.25. A trapped ion force sensor comprising the system of claim 19.