EP1368673A1 - Procede de calcul de modulation de frequence et d'amplitude pour des impulsions electromagnetiques adiabatiques - Google Patents

Procede de calcul de modulation de frequence et d'amplitude pour des impulsions electromagnetiques adiabatiques

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Publication number
EP1368673A1
EP1368673A1 EP02750521A EP02750521A EP1368673A1 EP 1368673 A1 EP1368673 A1 EP 1368673A1 EP 02750521 A EP02750521 A EP 02750521A EP 02750521 A EP02750521 A EP 02750521A EP 1368673 A1 EP1368673 A1 EP 1368673A1
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EP
European Patent Office
Prior art keywords
coordinate system
hamilton operator
adiabatic
parameters
hamilton
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EP02750521A
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German (de)
English (en)
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Lorenz Mitschang
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Br Deutschland Vertrd Bundesministerium F Wi
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Bundesrepublik Deutschland Vertrdurch Den Bundesminister fur Wirtschvertr Durch Den Praes Der Phys-Techn Bundesanstalt
Bundesrepublik Deutschland
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Publication of EP1368673A1 publication Critical patent/EP1368673A1/fr
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R33/00—Arrangements or instruments for measuring magnetic variables
    • G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
    • G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
    • G01R33/446—Multifrequency selective RF pulses, e.g. multinuclear acquisition mode

Definitions

  • the invention relates to a method for calculating parameters for frequency and / or amplitude-modulated electromagnetic pulses for adiabatic transfer processes in a quantum mechanical system, the dynamic processes of the quantum mechanical system being described with a Hamilton operator H (t) in a current coordinate system ,
  • the magnetism of matter is based on magnetic moments m of elementary particles from which matter is composed, the magnetic moments being able to interact with an external magnetic field.
  • the magnetic moments perform a precession movement in a static external magnetic field.
  • the precession frequency of the magnetic moments depends on the magnetic field strength. If, in addition to the static magnetic field, an alternating magnetic field is applied with a frequency that is matched to the precession frequency of the magnetic moments, then resonance absorption of energy from the alternating magnetic field takes place. All phenomena that are subsumed under the term "magnetic resonance" are based on this principle.
  • the cause of the magnetic moments of the elementary particles is their total angular momentum, in which the spin is contained as a quantum mechanical quantity.
  • the theoretical treatment of magnetic resonances is therefore based on quantum mechanics and relativity.
  • the dynamics of the quantum mechanical system are described by the so-called Hamilton operator H, which is contained in the so-called Schrödinger equation.
  • the Schrödinger equation is an equation of motion for the state vector describing the state of a quantum mechanical system. To define the Schrödinger equation and the Hamilton Operators are referred to Lexikon der Physik, Vol. 3 and 5, Spektrum Akademischer Verlag, Heidelberg, 2000.
  • Frequency and amplitude modulated electromagnetic pulses are used for adiabatic polarization and coherence transfer in atomic and molecular spectroscopy (laser optics, magnetic resonance) and in imaging processes (magnetic resonance imaging).
  • laser optics, magnetic resonance laser optics, magnetic resonance
  • imaging processes magnetic resonance imaging
  • adiabatic techniques are used, which are particularly advantageous if a spectrocopical parameter is broadband (variation of the strength of a molecular coupling, distribution of transition frequencies generated by field gradients), or if a difficult adjustment is to be avoided (external homogeneity fields).
  • the pulse shapes and parameters of the frequency- and / or amplitude-modulated electromagnetic pulses are conventionally determined empirically.
  • the control criterion here is that a successful pulse shape, the so-called adiabatic condition, is fulfilled.
  • This adiabatic condition is a criterion in the form of an inequality for the so-called adiabatic approximation and can be formulated in various ways. If, for example, the square of the quotient of the maximum angular velocity of an eigenstate (energy level) and minimum transition frequency to an adjacent eigenstate (Ener ⁇ max 2nd energy level) is much smaller than 1 ( ⁇ m, n «1), the adiabatic becomes
  • adiabatic i.e. dynamic in the ne adiabatic approximation
  • Ehrenfest in such a way that the change in the system during the action of the electromagnetic field is so slow that the change in energy levels is negligibly small.
  • the magnetization follows the direction of the magnetic field if the field direction changes sufficiently slowly.
  • the adiabatic approximation corresponds to a simplification in the description of the dynamics of the system, in that a coupling between one caused by the time-variable carrier frequency and amplitude
  • the adiabatic condition takes the form of a
  • pulse forms are developed for the two-level system (spin-1/2), in which there are no internal interactions and the dynamics are determined solely by the imprinted radiation and are therefore fully controllable.
  • diabatic losses occur as a result of an imperfect adiabatic transfer. They are caused by the neglected coupling, since the actual dynamics of the system deviate from the theoretical dynamics assumed within the framework of the adiabatic approximation.
  • the size of the diabatic losses is determined by the quality of the adiabatic condition. The diabatic losses increase for faster transfer processes.
  • WO 99/355 09 describes a method for calculating parameters for frequency and / or amplitude modulated pulses.
  • the principle is to find a suitable speed profile for a given trajectory so that the corresponding pulse induces adiabatic dynamics in a coordinate system other than the rotating one.
  • Known trajectories are copied from a rotating to the other coordinate system in question and the speed profiles are numerically optimized.
  • the two variables sought to define a pulse namely frequency and amplitude modulation as a function of time, can finally be clearly calculated from the variables thus determined, the trajectory and the speed profile.
  • the disadvantage is that empirically new pulse shapes have to be adopted and examined.
  • the trajectory and speed profile reflect local (infinitesemale) properties of the resulting pulse.
  • Both variables are also used to fulfill an adiabatic condition in the sense of the adiabatic approximation - i.e. neglecting interactions - in a coordinate system other than the rotating one.
  • the object of the invention was to provide a method for calculating parameters for frequency and / or amplitude-modulated electromagnetic pulses for adiabatic transfer processes in a quantum mechanical system, with which optimized pulses are obtained which enable fast adiabatic transfer without significant diabatic losses.
  • the method is based on an exact description of the system dynamics under frequency and amplitude-modulated radiation, so that no approximation in the sense of the adiabatic approximation is required.
  • the pulse shapes obtained can even violate the conventional adiabatic approximation, but can be used by the underlying method. This means that adiabatic transfer processes can be achieved in a shorter time than is conventionally possible.
  • the pulses obtained can thus also be used for systems that relax relatively quickly.
  • l z , l x and l y are the components of the magnetic angular moment of spin-1/2 in a defined Z, X, and Y axis of the rotating coordinate system (rotating wave approximation with linearly polarized irradiation ) are.
  • Her Hamilton operator H (t) is thus understood as a vector (in the Liouvill space).
  • the resonance offset ⁇ 0 (t) defines the internal energy of the system in the rotating system and the field amplitude ⁇ 0 (t) the coupling strength of the two energy levels.
  • An arbitrary initial phase of the radiation is set to zero.
  • Ho (t) can also be effectively used by means of polar coordinates
  • the time dependency of the change in direction of the Hamilton operator Ho (t) can be described by transformation from the current, the rotating coordinate system into a following coordinate system, the Z axis of which always points along the effective field in the rotating coordinate system and the Y axis with that of the rotating coordinate system matches.
  • the X, Y and Z axes form an orthogonal system.
  • Steps a) and b) are preferably carried out iteratively, the transformation from the current coordinate system into a subsequent coordinate system by the angle of rotation ⁇ , which describes the change in direction of the Hamilton operator H (t), that is to say between the Z axis of the current coordinate system and the Hamilton operator H (t) is spanned.
  • i is the integer consecutive number of the corresponding coordinate system
  • t is the time
  • t ' is a time different from t with t and t'> 0.
  • ) a M (t) with ⁇ j constant can be met at (t) and ⁇ i (t) vary, but their ratio tan ⁇ j and thus the angle ⁇ ⁇ is time invariant.
  • the system thus remains in its own state relative to the i-th coordinate system over time if it was initially in its own state of Hi (0). There is therefore no change in the system over time in the i-th coordinate system (stationary state).
  • the system is “spin-locked” relative to the i-th coordinate system.
  • spin-locking is used to stabilize the deflected nuclear magnetization by continuous exposure to an electromagnetic field understand.
  • a secondary condition can be, for example, a defined path of the tip of the Hamilton operator H (t) in a fixed coordinate system, the path being, for example, circular, elliptical or rectangular.
  • constraints can e.g. be holonomic, i.e. as equations in the searched system and time variables. However, they can also be non-holonomic and, like the commutator condition, represent a system of differential equations.
  • the type of coordinates used can be selected depending on the constraints for the system under consideration. For example, polar coordinates, elliptical coordinates or Cartesian coordinates can be used.
  • the differential equation system obtained can be solved as an initial value problem or as a boundary value problem in order to calculate the parameters.
  • the Hamilton operator Hj_ ⁇ (t) is transformed from the current, the (i-1) -th coordinate system into the following, the i-th coordinate system, a fixed offset angle ⁇ , - occurs.
  • the Hamilton operator HM (t) is aligned in a defined Z axis of the (i-1) th coordinate system.
  • the Z axis of the following i-th coordinate system is assumed after rotation through the angle ⁇ M in the direction of the vector a ⁇ .
  • the rotation takes place in the plane spanned by the Z and X or Y axis of the (i-1) th coordinate system, so that the remaining Y or X axis for the (i-1) th and i-th coordinate system is identical.
  • the vector of the angular velocity ⁇ j- ⁇ points in this direction.
  • the resulting Hamilton operator H (t) points into the space of the i-th coordinate system and is offset relative to the vector a ⁇ i (t) in the Z-axis of the i-th coordinate system by the angle ⁇ .
  • the method can also advantageously be used to calculate the parameters of the Zeeman frequency or the parameters of an activated linear field gradient to determine the Zeeman frequency.
  • the method can also be used to calculate the parameters for frequency and / or amplitude-modulated electromagnetic pulses from a plurality of radiation sources.
  • the method can also be used to calculate the parameters of a mechanical rotation of a sample under consideration, such as the orientation of the rotation and the speed of rotation.
  • Figure 1 - schematic diagram of a resonance spectrometer for adiabatic transfer processes
  • FIG. 2 - vector diagram of a rotating around the Hamilton operator H
  • FIG. 3 - vector diagram of a device connected to the Hamilton operator H.
  • FIG. 4 - vector diagram of the Hamilton operator traveling through the angle ⁇ o;
  • FIG. 5 diagram of the Hamilton operator in the 0th coordinate system
  • FIG. 6 diagram of the transformed Hamilton operator in the 1 st coordinate system
  • Figure 7 Diagram of the transformed Hamilton operator in the 2nd coordinate system
  • Figure 12 Curve of the determined frequency and amplitude modulation functions of the 1st to 3rd order over time.
  • A is defined as a vector perpendicular to the area spanned by the circular current I.
  • Energy expenditure is required to rotate the dipole moment by an angle ⁇ against the magnetic induction B. According to the laws of quantum mechanics, it must be taken into account that only discrete energy states (orientations) are possible.
  • the orbiting charged particle (electron, proton, neutron) is considered as a circular electric current I.
  • the orbital angular momentum can only take discrete values. This also applies to the particle's own angular momentum or spins.
  • the orbital angular momentum quantum number of a particle occurs only number on.
  • the spin quantum number can take half-numbered values. Electron, proton and neutron are spin 1/2 particles. Therefore, only two spin settings are possible with respect to the Z direction, the Z axis being defined as the axis around which the self-rotation takes place.
  • the self-rotating pulse thus precesses around the Z axis.
  • a magnetic dipole moment ⁇ is connected to the angular momentum, which is also precessed and quantized about this Z axis.
  • the magnetic dipole moment is in a magnetic field with a magnetic induction B z , then a preferred direction is defined. Relative to the magnetic induction B z , the magnetic dipole moment can only occur in certain values, since only discrete energy states (orientations) are possible.
  • a magnetic alternating field with a resonance frequency is now radiated perpendicular to the induced magnetic field, there is a transition between the energy levels of the elementary particles and a flipping of the magnetic moment.
  • microwaves in the gigahertz range are used for electron spin resonance and radio waves in the range from 60 MHz to currently 1 GHz for nuclear magnetic resonance. It is not only the frequency that is decisive for the respective experiment, but also the pulse shape and, in the case of multiple irradiation, the relative phase of the pulses is also important.
  • FIG. 1 The basic structure of a resonance spectrometer is shown in FIG.
  • a magnet arrangement 1 a homogeneous and constant magnetic field B z is applied, so that a preferred direction is determined.
  • a sample 2 is introduced into the magnetic field.
  • frequency and / or amplitude-modulated electromagnetic pulses for excitation of the sample 2 are introduced with radio frequency coils 3 and a transition between the energy levels of the elementary particles under consideration is forced.
  • the resonance signals are registered in an evaluation unit 4 and possibly reconstructed with the aid of a Fourier transformation.
  • a location-dependent resonance frequency linear field gradient generated by gradient coils
  • a location-dependent measurement can be carried out by frequency or phase coding. This is used primarily for medical magnetic resonance imaging.
  • the applied pulses are generated in a pulse generator 5 and are dimensioned such that diabatic losses are avoided as much as possible.
  • the transfer processes should therefore proceed adiabatically.
  • the adiabatic hypothesis in quantum theory states that a system that is initially in a certain stationary quantum mechanical state remains in this state if the change in the parameters of the system, such as the external field strengths, is sufficiently slow.
  • the adiabatic hypothesis thus makes a statement about the time dependence of the Hamilton operator H (t) for the limit case of large times. To avoid diabatic losses, it is therefore desirable for the change in direction (rotation) of the Hamilton operator H (t) to be as slow as possible by the angle ⁇ . That is, the rotational angular velocity ⁇ approaches zero.
  • H 0 (t) ⁇ 0 (t) l z + ⁇ 0 (t) l x ,
  • H 0 (t) is in a rotating coordinate system that is oriented around the direction of the applied carrier frequency of the radiation outer constant Zeeman field, in which the Z axis of the coordinate system is located, rotates.
  • the resonance offset ⁇ 0 (t) defines the internal energy of the system and the
  • Eating is when the angle between the magnetic moment m and the Hamilton operator H 0 (t) is as small as possible, that is, when the vector
  • the Hamilton operator can also be defined as
  • H 0 (t) a 0 (t) ⁇ cos ⁇ 0 (t) l z + sin ⁇ 0 (t) l x ⁇ .
  • variable ao (t) is the length or amplitude of the Hamilton operator Ho (t) and is defined as
  • the angle of rotation ⁇ o (t) is the tilt angle of the Hamilton operator relative to the Z axis of the rotating coordinate system and is defined as
  • ⁇ 0 (t) arctan ⁇ ° (t)
  • FIG. 5 shows the Hamilton operator H 0 (t) with the amplitude a 0 in the rotating coordinate system.
  • the amplitude ao is determined here from the resonance offset ⁇ 0 and the field amplitude ⁇ o.
  • the length and direction (ao (t) and ⁇ o (t)) of Ho (t) change.
  • the latter can be described by a transformation into a first coordinate system, the Z axis of which is always along H 0 (t).
  • the first and the rotating coordinate system have a common Y axis.
  • the Hamilton operator in the first coordinate system is
  • Torque which generates the change in direction of H 0 (t) in the (original) rotating coordinate system.
  • the adiabatic factor Q (t) is a measure of the pulse used (in
  • the invention is based on the finding that the modulation functions sought must keep the system in its own state (nestle against the Hamilton operator), although the Hamilton operator changes its direction in the rotating coordinate system. This can be achieved if the Hamilton operator H ⁇ (t) does not act as a disturbance at any time in the first coordinate system (adiabatic transfer process), that is, the self-commutation condition of the Hamilton operator
  • ⁇ 1 is the angle of the Hamilton operator relative to the Z axis of the first coordinate system.
  • the parameters are not calculated on the basis of boundary conditions, which are not defined by the adiabatic inequality, but on the basis of the consideration of the complete physical system.
  • the dynamics of the system are taken into account by transforming the Hamilton operator from a current to a subsequent coordinate system. This describes the time dependency of the change in direction of the Hamilton operator H (t).
  • H 0 (t) ⁇ (t) l z + ⁇ 0 (t) l x .
  • the rotation by the angle ⁇ 0 can be described as a rotation of the coordinate system or as a transformation of the Hamilton operator from the zero coordinate system into the following coordinate system, as outlined in FIG. 6.
  • the new Z axis of the following first coordinate system extends in the direction of the amplitude a 0 .
  • the transformed Hamilton operator H ⁇ (t) is then by the equation
  • the speed of rotation ⁇ 0 is a vector along the Y axis because the rotation takes place in the ZX plane of the original coordinate system.
  • the result is an offset angle ⁇ 1 of the resulting Hamilton operator H- ⁇ (t) relative to the Z axis Zi of the first coordinate system.
  • the angle of rotation ⁇ 1 realigns faster, which leads to shorter pulse lengths.
  • the angle of the precession cone is disadvantageously
  • the first coordinate system also aligned exactly along the magnetization at the beginning of the pulse. If a time-dependent Hamilton function can be determined such that the increase in the angle of rotation ⁇ ., (T) is adiabatic, large resulting angles of rotation ⁇ ., ( ⁇ ) can be realized without noticeable diabatic losses.
  • T angle of rotation
  • ⁇ large resulting angles of rotation
  • ⁇ can be realized without noticeable diabatic losses.
  • the movement of the Hamilton operator or effective field in the first coordinate system can be described by transformation into a second coordinate system analogously to the transformation carried out previously. Again, the exact adiabatic condition must apply, which is expressed by the self-commutation of the Hamilton operator H 2 (t) in the second coordinate system.
  • Hamilton operator H 2 (t) a., (T) l z - ⁇ (t) l x with suitable modulation functions ⁇ 2 (t) and ⁇ 2 (t) implies a fixed angle of rotation ⁇ 2 between the effective field and the Z axis of the second coordinate system.
  • the assumption of ⁇ 2 ⁇ ⁇ is always fulfilled because the effective field in the er ⁇
  • H n (t) a. ⁇ (T) l z + ⁇ __., (T) l xy
  • the differential equations of spin-locking represent a system of differential equations, with the help of which the parameters of the optimized modulation signal can be calculated. They can be solved as an initial value or boundary value problem depending on the system under consideration and the secondary condition.
  • the following differential equation system for determining the nth order pulse can be used to determine the Modulations ⁇ n (t) and ⁇ n (t) (the index n refers to the solution of the commutator in the nth coordinate system) are set up as parameters:
  • ⁇ 0 (t) tan ⁇ 1 (t) ⁇ a 0 (t)
  • the path described by the tip of the Hamilton operator in the zero coordinate system can also be used as a secondary condition.
  • This curve can be semicircular, elliptical or rectangular, for example.
  • the two unknown parameters ⁇ n (t) and ⁇ n (t) can be determined from the quantities ⁇ 0 (t) and a 0 (t) determined by the relationships already mentioned
  • the key element of the method according to the invention is the self-commutating property of the Hamilton operator in the nth coordinate system. Self-commutation is therefore a main condition.
  • the secondary conditions can be chosen relatively freely in a known manner according to the system under consideration.
  • the initial state of the system is thermal equilibrium (magnetization along the Z axis of the 0th coordinate system).
  • the corresponding differential equation system for the first, second and third order modulation functions are numerically integrated with a Runge-Kutta algorithm according to FIG.
  • the reorientation of the corresponding Hamiltonian in time and the trajectory of the state are shown on the unit sphere of the rotating or 0th coordinate system.
  • large deviations of the state trajectory from the path of the Hamilton operator occur. This shows a non-adiabatic dynamic.
  • the angle of rotation of the first coordinate system can change over time.
  • FIG. 12 shows the modulation functions obtained in the abovementioned examples, the curves dropping from the 25 kHz value representing the resonance offset, that is to say the actual frequency modulation function, and the curves rising from 0 kHz representing the amplitude modulation over time.
  • the fine dashed curves represent a first order modulation function.
  • the steep rise in the resonance offset function can be interpreted as a large opening angle for the rotating magnetic moment vector.
  • the long dashed curves are second order modulation functions. Optimal results can already be achieved with a third-order modulation function as outlined as a solid line.
  • This theoretically described method can be used to determine pulse shapes in two-level or multi-level systems and requires a coherent radiation source with variable amplitude and carrier frequency, such as radio frequency waves or lasers. This opens up a wide range of applications in magnetic resonance and coherent optics. The method can therefore preferably be used in medical technology and the optical industry.
  • An additional condition can be introduced into the method for the respective application, so that the specific pulse shapes ensure an improved or even optimal execution.
  • criteria can be used that have already proven beneficial in the past and have been published in various ways.
  • One field of application of the invention is magnetic resonance imaging. It is based on the so-called “slice selection", that is, the inversion of population in two-level systems for a given band of crossover frequencies (broadband inversion). This is the classic application of adiabatic techniques. In magnetic resonance imaging, it is particularly important to generate an inversion profile with a sharp transition from the inverted and non-inverted area.
  • Various criteria for this are described in the literature, which are incorporated into the process as secondary conditions. For example, D. Rosenfeld and Y. Zur, Magnetic Resonance in Medicine 36, 124 (1996) describe that the effective field should follow a rectangular path with rounded corners. In contrast to this, the case of an elliptical path was discussed.
  • the method according to the invention enables a qualitatively better “slice selection” and the expansion of the imaging to tissue that relaxes more quickly.
  • pulse forms for optimized inversions of populations over a given frequency band can be determined with the method using a minimal frequency exposure (heating) of the sample or by specifying a maximum permissible amplitude of the irradiated field.
  • Another area of application for the method according to the invention is
  • adiabatic techniques are used to detect the inversion of population or the excitation of magnetization in a two-level system in the presence of inhomogeneities in the amplitude of the irradiated field, that is, a location-dependent amplitude over the sample volume, since notoriously inhomogeneous surface coils are used.
  • the systematic method also allows the determination of pulse shapes for optimized inversion and excitation under various constraints, such as minimal sample heating or rapid execution (the field amplitude virtually replaces the transition frequency when used in magnetic resonance imaging as a broadband parameter).
  • the modulation functions determined by the method according to the invention can be built into the "plan rotation pulses" scheme.
  • adiabatic inversion has hitherto required considerably longer (10 to 100 times) than conventional pulse techniques (non-selective pulses), which precludes their use on rapidly relaxing systems, such as in experiments on biomolecules. This is where the systematic procedure can help with fast adiabatic transfers.
  • Polarization transfer from sensitive to less sensitive nuclei is a basic technique of solid-state resonance. It was shown in S. Heediger, BH Meier, ND Kurur, G. Bodenhausen and RR Ernst, Chemical Physics Letters, 223, 283 (1994) that this polarization transfer formally corresponds to an inversion of population in the two-level system and can be successfully performed adiabatically. Optimizing an adiabatic polarization transfer with the aid of the method according to the invention can both improve the signal-to-noise ratio in a large number of solid-state resonance experiments and can also enable polarization transfer in rapidly relaxing systems.
  • the method according to the invention allows the determination of pulse shapes for optimized polarization and coherence transfer processes in static and, in particular, rotating solid bodies ("magic angel spinning") with inherently time-dependent interactions.
  • the direct execution of two inversions in a row restores the original population in the two-level system.
  • Such "identity manipulations” form the basis of complex spectroscopic techniques for spin coupling and so-called isotropic mixing, a specific type of coherence transfer.
  • Spin coupling is an integral part of many experiments in nuclear magnetic resonance on liquids, solids and in vivo, since spectral simplification and a higher signal-to-noise ratio and a higher resolution result.
  • the laser as a light source enables coherent spectroscopy in the optical wavelength range.
  • the frequency and amplitude of laser light in continuous and pulsed operation can be modulated almost arbitrarily, which opens up a wide range of adiabatic techniques in the field of coherent optics.
  • Adiabatic polarization and coherence transfers based on magnetic resonance, such as broadband inversion and excitation are possible.
  • the "Stimmulated Raman Adiabatic Passage” (STIRAP) is mentioned, which is a technique for adiabatic coherence transfer in the three-level system and by K. Bergmann, H. Theuer and BW Shore, Review of Modern Physics 70, 1003 (1998).
  • STIRAP Stimmulated Raman Adiabatic Passage
  • the STIRAP technique allows the population to be reversed between two long-lived energy levels whose electrical dipole transition is prohibited, including a third short-lived excited state without excessive losses due to spontaneous emission. The losses that still occur due to spontaneous emission are caused by imperfections of the adiabatic transfer process and thus correspond to the diabatic losses.

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  • Physics & Mathematics (AREA)
  • High Energy & Nuclear Physics (AREA)
  • Condensed Matter Physics & Semiconductors (AREA)
  • General Physics & Mathematics (AREA)
  • Magnetic Resonance Imaging Apparatus (AREA)

Abstract

L'invention concerne un procédé de calcul de paramètres pour des impulsions électromagnétiques modulées en fréquence et/ou en amplitude, destinés à des processus de transferts adiabatiques dans un système de mécanique quantique, les processus dynamiques du système de mécanique quantique étant décrits au moyen d'un opérateur de Hamilton (H(t)) dans un système de coordonnées courant. Ledit procédé consiste a) à transformer l'opérateur de Hamilton (H(t)) d'un système de coordonnées courant vers un système de coordonnées consécutif pour la description de la dépendance par rapport au temps du changement de direction de l'opérateur de Hamilton (H(t)) ; b) à déterminer l'autocommutation de l'opérateur de Hamilton (H(t)) au temps t et de l'opérateur de Hamilton (H(t')) au temps t' dans le système de coordonnées respectif, un système d'équations différentielles étant ainsi obtenu ; et, c) à calculer les paramètres à partir du système d'équations différentielles obtenu.
EP02750521A 2001-03-14 2002-03-14 Procede de calcul de modulation de frequence et d'amplitude pour des impulsions electromagnetiques adiabatiques Ceased EP1368673A1 (fr)

Applications Claiming Priority (3)

Application Number Priority Date Filing Date Title
DE10112625A DE10112625B4 (de) 2001-03-14 2001-03-14 Verfahren zur Bildung frequenz- und/oder amplitudenmodulierter elektromagnetischer Pulse zur Induzierung adiabatischer Transfervorgänge in quantenmechanischen Systemen
DE10112625 2001-03-14
PCT/DE2002/000911 WO2002073229A1 (fr) 2001-03-14 2002-03-14 Procede de calcul de modulation de frequence et d'amplitude pour des impulsions electromagnetiques adiabatiques

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CN111464154B (zh) * 2019-01-22 2022-04-22 华为技术有限公司 一种控制脉冲的计算方法及装置
CN112906927B (zh) * 2019-12-03 2024-04-19 新奥数能科技有限公司 一种热电联产机组收益优化方法及装置
CN111693556B (zh) * 2020-07-22 2022-09-27 中国工程物理研究院核物理与化学研究所 一种用于自旋回波小角中子散射谱仪的中子极化方向翻转装置
CN113507775B (zh) * 2021-06-07 2023-11-21 中国工程物理研究院激光聚变研究中心 适用大型激光装置的多用途光学汤姆逊散射光谱测量系统

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US20040167751A1 (en) 2004-08-26

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