Rabi oscillations and quantum beats in a qubit in distorted magnetic field

In a two-level system the time-periodic modulation of the magnetic field stabilizing the magnetic resonance position has been investigated. It was shown that the fundamental resonance is stable with respect to consistent variation of the longitudinal and transverse magnetic fields. The time-depend...

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Datum:	2007
Hauptverfasser:	Ivanchenko, E.A., Tolstoluzhsky, A.P.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Schriftenreihe:	Физика низких температур
Schlagworte:	Низкотемпеpатуpный магнетизм
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spelling	irk-123456789-1209242017-06-14T03:06:32Z Rabi oscillations and quantum beats in a qubit in distorted magnetic field Ivanchenko, E.A. Tolstoluzhsky, A.P. Низкотемпеpатуpный магнетизм In a two-level system the time-periodic modulation of the magnetic field stabilizing the magnetic resonance position has been investigated. It was shown that the fundamental resonance is stable with respect to consistent variation of the longitudinal and transverse magnetic fields. The time-dependency of the Rabi oscillations and quantum beats of the spin flip probability was numerically researched in different parameter regimes taking into account dissipation and decoherence in the Lindblad form. The present study may be useful in the analysis of interference experiments and for manipulation of quantum bits. 2007 Article Rabi oscillations and quantum beats in a qubit in distorted magnetic field / E.A. Ivanchenko, A.P. Tolstoluzhsky // Физика низких температур. — 2007. — Т. 33, № 08. — С. 902–906. — Бібліогр.: 12 назв. — англ. 0132-6414 PACS: 33.35.+r, 02.30.Hq, 85.35.Gv, 03.65.Vf http://dspace.nbuv.gov.ua/handle/123456789/120924 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм
spellingShingle	Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Ivanchenko, E.A. Tolstoluzhsky, A.P. Rabi oscillations and quantum beats in a qubit in distorted magnetic field Физика низких температур
description	In a two-level system the time-periodic modulation of the magnetic field stabilizing the magnetic resonance position has been investigated. It was shown that the fundamental resonance is stable with respect to consistent variation of the longitudinal and transverse magnetic fields. The time-dependency of the Rabi oscillations and quantum beats of the spin flip probability was numerically researched in different parameter regimes taking into account dissipation and decoherence in the Lindblad form. The present study may be useful in the analysis of interference experiments and for manipulation of quantum bits.
format	Article
author	Ivanchenko, E.A. Tolstoluzhsky, A.P.
author_facet	Ivanchenko, E.A. Tolstoluzhsky, A.P.
author_sort	Ivanchenko, E.A.
title	Rabi oscillations and quantum beats in a qubit in distorted magnetic field
title_short	Rabi oscillations and quantum beats in a qubit in distorted magnetic field
title_full	Rabi oscillations and quantum beats in a qubit in distorted magnetic field
title_fullStr	Rabi oscillations and quantum beats in a qubit in distorted magnetic field
title_full_unstemmed	Rabi oscillations and quantum beats in a qubit in distorted magnetic field
title_sort	rabi oscillations and quantum beats in a qubit in distorted magnetic field
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2007
topic_facet	Низкотемпеpатуpный магнетизм
url	http://dspace.nbuv.gov.ua/handle/123456789/120924
citation_txt	Rabi oscillations and quantum beats in a qubit in distorted magnetic field / E.A. Ivanchenko, A.P. Tolstoluzhsky // Физика низких температур. — 2007. — Т. 33, № 08. — С. 902–906. — Бібліогр.: 12 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT ivanchenkoea rabioscillationsandquantumbeatsinaqubitindistortedmagneticfield AT tolstoluzhskyap rabioscillationsandquantumbeatsinaqubitindistortedmagneticfield
first_indexed	2025-07-08T18:52:34Z
last_indexed	2025-07-08T18:52:34Z
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fulltext	Fizika Nizkikh Temperatur, 2007, v. 33, No. 8, p. 902–906 Rabi oscillations and quantum beats in a qubit in distorted magnetic field E.A. Ivanchenko and A.P. Tolstoluzhsky National Science Center «Kharkov Institute of Physics and Technology», 1 Akademicheskaya Str., Kharkov 61108, Ukraine E-mail: yevgeny@kipt.kharkov.ua Received December 4, 2006 In a two-level system the time-periodic modulation of the magnetic field stabilizing the magnetic reso- nance position has been investigated. It was shown that the fundamental resonance is stable with respect to consistent variation of the longitudinal and transverse magnetic fields. The time-dependency of the Rabi os- cillations and quantum beats of the spin flip probability was numerically researched in different parameter regimes taking into account dissipation and decoherence in the Lindblad form. The present study may be useful in the analysis of interference experiments and for manipulation of quantum bits. PACS: 33.35.+r Electron resonance and relaxation; 02.30.Hq Ordinary differential equations; 85.35.Gv Single electron devices; 03.65.Vf Phases: geometric; dynamic or topological. Keywords: NMR, Rabi oscillations, quantum beats. 1. Introduction Now world science centers dealt with a problem of a construction of quantum computers in which the algo- rithms of calculations based on the coherent mechanism of quantum processes are used. The idea of a quantum computer will consist in the construction of computer on the basis of quantum bits, instead of classical elementary cells. The laws of quantum mechanics determining the be- havior of quantum bits, provide huge advantages in the speed of calculations of a quantum computer in the com- parison with a classical computer. In an NMR computer the manipulation of quantum bits (nuclear spins) is extremely important. In the standard implementation of the magnetic re- sonance a constant magnetic field is perpendicular to linearly polarized, variable in time t monochromatic magnetic field. The shift of resonance frequency (the Bloch–Ziegert shift) appears [1]. The goal of the paper consists in the proposal and research of such magnetic field configuration, at which one the position of a main resonance is determined only by the Larmor frequency at arbitrary parameters of a system. This can be reached by generalizing the Rabi model [2]. Rabi studied the tempo- ral dynamics of the particle with dipole magnetic moment and spin 1 2/ in a constant magnetic field H 0, directed along the z axis, and an alternating magnetic field perpen- dicular to it, rotating uniformly with a frequency � �/ 2 : H h tx � 0 cos � , H h ty � 0 sin � , h0 is the transverse field amplitude. There are several methods for modulating magnetic fields while studying the phenomenon of magnetic reso- nance, depending on the goal of research (see [3] and ref- erences therein). In Refs. 4–6, without taking dissipation into account, it is investigated the temporal evolution of particle with dipole magnetic moment and spin 1 2/ in a distorted magnetic field H( ) ( ( , ), ( , ), ( , ))t h t k h t k H t k� 0 0 0cn sn dn� � � , (1) where one cn, sn, dn are the Jacobi elliptic functions [7]. Such a modulation of the field upon variation of the modulus k of elliptic functions from zero to unity describes a class of field shapes from trigonometric [ ( , ) cos , ( , ) sin , ( , ) ]cn sn dn� � � � �t t t t t0 0 0 1� � � [2] to [ ( , ) / , ( , ) , ( , ) / ]cn ch sn th dn ch� � � � � �t t t t t t1 1 1 1 1� � � pulsed exponential [7]. The elliptic functions cn ( , )�t k , sn ( , )�t k have a real period of 4K / �, while dn ( , )�t k has a real period half as long. Here K is the complete elliptic integral of the first kind [7]. In other words, though the field is periodic with total real period of 4K / �, it is seen © E.A. Ivanchenko and A.P. Tolstoluzhsky, 2007 that the frequency of amplitude modulation of the longi- tudinal field is twice that of the amplitude modulation of the transverse field. Such a field is called harmonized. In the paper [5] it was predicted that the position of the mag- netic resonance would be stabilized in the field (1). In the work [6] the influence of a dissipation and decoherence on a stabilization has been studied. In the present work we study the transition of the Rabi oscillations into beats de- pending on the initial conditions and parameters describ- ing the variable magnetic field (1). 2. Model The dynamics of a spin 1 2/ particle (qubit) with mag- netic moment in an ac magnetic field H( )t will be de- scribed in the formalism of the density matrix � with a dissipative environment taken into account with the help of the Liouville–von Neumann–Lindblad (LvNL) equa- tion [8] (we set � �1) i H i L L L L L Li i i i i i i � � � � �� [ � , ] ( ) 2 2 1 3 (2) with the Hermitian Hamiltonian � ( , ) ( ( , ) ( , )) ( ( , ) H k k i k k i � � � � � � � � � 0 1 1 2 2 2 dn cn sn cn sn dn( , )) ( , ) , � � k k� � � � �� 0 2 (3) in which dimensionless independent variable �� t, � �0 0 0� g H is the Larmor frequency, � �1 0 0� g h is the amplitude of transverse field in terms of angular fre- quency, g is the factor Lande, � 0 is the Bohr magneton. The operators Li are chosen in the form Li i i� � � 2 [9], where the �i are phenomenological constants, which take into account the decoherence and dissipation in the system, and the � are the Pauli matrixes. We make the substitution � � �� 1r with the matrix � � � � � � � � f f 0 0 * , where the function f is equal to f i� �cn sn . (4) The equation for the transformed matrix r takes the form i r r rx r z� � � � � � � � � �1 2 2 [ , ] [ , ]dn � � �� i L L r r L L L r Li i i i i i i 2 21 1 1 1 1 3 � � � � � � � � �( ) (5) in which detuning � r is equal to � � �r � �0 . (6) As it is seen from Eq. (5) the detuning appears explicitly, i.e., the position of the principal resonance does not suffer a shift when the parameters of model are changed. In the case of a sharp resonance, when � �0 � , in neglect of damping, the transition probability P r1 2 1 2 2 10 2� � � �( , ) sin � � � (7) does not contain modulus k, i.e., it is does not depend of the harmonized field deformation [4,5]. 3. Decomplexification of LvNL equation In the general case for an arbitrary detuning � r (6) we write the matrix r in the form of a decomposition in the complete set of Pauli matrices: r � � 1 2 1( )�R , r r� � , Sp r �1, (8) for all . We substitute the expression for r (8) in Eq. (5). As a result, we obtain the system of three first-order dif- ferential equations with periodic coefficients, with re- spect to unknown real functions R R Rx y z, , : � � � � � � � � � � � R R R k R k k x r y z x x x dn sn cn sn � � ( ( , ) ( , ) ( ,2 ) ) ( ( , ) ( , ) ( , ) ) , R k R k k R y y x y � � � � � cn cn sn2 (9) � � � � � � � � � � � � � R R R R k R k y r x z z y x y dn cn cn 1 2 � � ( ( , ) ( , ) sn cn sn sn ( , ) ) ( ( , ) ( , ) ( , ) ) , � k R k k R k R x y x y � � � � 2 (10) � � � � � � � � � � � � � � R R Rz y x y z 1 � � . (11) Now in terms R R Rx y z, , the density matrix � becomes � � � � � � � � � � � � 1 2 1 1 2 2 R f R iR f R iR R z x y x y z ( ) ( )* . (12) The transition probability with spin flip is equal to the matrix element �22 that is P k Rr z1 2 1 2 1 2 1 � � � �( , , ) ( ) � . (13) Rabi oscillations and quantum beats in a qubit in distorted magnetic field Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 903 Using expression (13) and set of Eqs. (9)–(11), it is easy to obtain a differential equation of the third order for the transition probability, which one we do not make out here. By the selection of decay constants � � � �x y z� � � the system (9)–(11) becomes � � � � � � �R R Rx r y xdn , (14) � � � � � � � � �R R R Ry r x z ydn 1 , (15) � � � � � �R R Rz y z 1 , (16) where � �� 2� / . In some special cases exact solutions of set (14)–(16) are presented in the work [6]. 4. Numerical results Let us consider behavior of transition probability de- pending on the pure initial conditions R Rx y( ) ( )0 0 0� � , R z ( )0 1� , (17) Rx ( )0 1� , R y ( )0 0� , R z ( )0 0� , (18) R R Rx y z( ) ( ) ( ) /0 0 0 1 3� � � , (19) and the mixed initial condition R R Rx y z( ) ( ) ( ) . /0 0 0 0 25 3� � � . (20) The probability of a spin-flip transition is determined by the expression (13). To perform the numerical simulation, we have chosen the parameters in units of 2�100 MHz: �0=1 corresponds to the longitudinal field 2.3487 T for the proton resonance of a qubit. Without taking into account dissipative de- coherence and at a resonance the formula (13) accepts an obvious form (7). When the detuning increases and k � 0, the frequency of oscillations increases, and the amplitude decreases, i.e., in the Rabi–Lindblad model damping os- cillations are observed [6]. The spin-flip probabilities are presented at the initial condition (17) in Figs. 1 and 2. As it is seen from Fig. 1 the decoherence and dissipation reduce the amplitude of beats and their appreciable number. The period of beating and «the small period of oscillations» depend on initial conditions and the modulus k (Fig. 2). At frequency of an ac magnetic field � much greater of the Larmor frequency �0 and the initial condition (17) the transition probability is closely to zero and equal 1/2 for the condition (18) (Fig. 3). At all parameters and any initial conditions at the Rabi frequency � R r� � �� 2 1 2 only damping oscillations are observed. The evolution of initial condition (18) is more sensitive to the field deformation. Already for small k the beats arise (Fig. 4). In Fig. 5 we present the dynamics of imaginary part of density matrix �12 ( y component of the polarization vector) reduced by the factor 0.5 [6]. The real part �12 904 Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 E.A. Ivanchenko and A.P. Tolstoluzhsky 0 0 50 50 100 100 150 150 200 200 250 250 300 300 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 �0 =1, �0 =1, � = 0.5 � = 0.5 �1 = 0.8 �1 = 0.8 k = 0.5 k = 0.5 � = 0.001 � = 0.005 P P Fig. 1. Transition probabilities versus at � = 0.005 (top) and at � = 0.001 (bottom). 20 40 60 80 1000 0.2 0.4 0.6 0.8 1.0 P k = 0.4 k = 0.7 �0 =1, � = 0.5, �1 = 0.8 � = 0.001 Fig. 2. Beats for different modulus k. (x component of the polarization vector) has the similar behavior. In Fig. 6 we record the time evolution of transition probabilities for pure (19) (dot line) and mixed (20) (solid line) initial conditions. We see that the amplitude of oscil- lations for mixed conditions less then for pure conditions, but the period of beats do not changes. We also record the time evolution of the entropy S � � � � �Sp ( ln ) ln ln� � � � � �1 1 2 2, where � �1 2, are the eigenvalues of the density matrix �. The entropy in- creases monotonically from 0 to its asymptotic limit of ln 2. At resonance (solid line) the entropy keeps the mix- ing longer (Fig. 7). The deformation of a field can be considered as the in- fluence of an environment on the qubit dynamics [10]. It is necessary to note that at � � �x y z� � the beats are kept and the beat amplitudes change only insignifi- cantly. Rabi oscillations and quantum beats in a qubit in distorted magnetic field Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 905 0 50 100 150 200 250 300 0.2 0.4 0.6 P �0 = 1, � = 10.0 �1 = 1.0 k = 0.14 � = 0.001 Fig. 3. Rabi oscillations at big detuning. The top plot corre- sponds to initial condition (18), bottom (17). 0 50 100 150 200 250 300 0.2 0.4 0.6 0.8 1.0 P �0 =1 � = 0.5 �1 = 0.8 k = 0.143 � = 0.001 Fig. 4. Transition probability at the initial condition (18). 0 50 100 150 200 250 300 – 0.6 – 0.3 0 0.3 0.6 Im � 1 2 �0 =1 � = 0.5 �1 = 0.8 k = 0.143 � = 0.001 Fig. 5. Imaginary part of density matrix �12 versus at the ini- tial condition (18). 150 300 450 6000 0.2 0.4 0.6 0.8 �0 =1, � = 0.485 �1 = 0.8 k = 0.143 � = 0.001 P Fig. 6. Transition probabilities versus at the initial condi- tions: (19) (dot line), (20) (solid line). 0 100 200 300 0 2. 0 4. 0.6 � = 0.5 � = 1 S �0 =1 �1 = 0.8 k = 0.143 � = 0.001 Fig. 7. Entropy versus at the initial condition (18). Conclusion Eventually the presence of dissipative decoherence levels the population of top and bottom levels. Depending on the field frequency, own frequency and a kind of initial conditions there are extremely a plenty of oscillations. It would be desirable to do an experiment to check the theoretical predictions as to the stability of the magnetic resonance positions for different model parameters. Such an experiment would be an extension of the experimental situation in the circular polarized field. Since the para- metric resonances in a harmonized magnetic field have a appreciable widths, it may be preferable to investigate magnetic resonance at parametric frequencies. This re- search reported here may find application in the analysis of interference experiments and for manipulation of qubits [11,12]. 1. M. Grifoni and P. Hanggi, Phys. Rev. 304, 229 (1998); I.I. Rabi, Phys. Rev. 51, 652 (1937). 2. J. Schwinger, Phys. Rev. 51, 648 (1937). 3. M. K�lin, I. Gromov, and A. Schweiger, J. Magn. Res. 160, 166 (2003); M. Fedin, I. Gromov, and A. Schweiger, J. Magn. Res. 171, 80 (2004). 4. E.A. Ivanchenko, Physica B358, 308 (2005); ArXiv: quant- ph/0404114 (2004), 11 p. 5. E.A. Ivanchenko, Fiz. Nizk. Temp. 31, 761 (2005) [Low Temp. Phys. 31, 577 (2005)]. 6. E.A. Ivanchenko and A.P. Tolstoluzhsky, Fiz. Nizk. Temp. 32, 103 (2006) [Low Temp. Phys. 32, 77 (2006)]. 7. Handbook of Mathematical Functions, M. Abramovitz and I.A. Stegun (eds.), New York: Dover (1968). 8. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 9. A.R.P. Rau and R.A. Wendell, Phys. Rev. Lett. 89, 220405 (2002). 10. Y.M. Galperin, D.V. Shantsev, J. Bergli, and L. Altshuler, ArXiv: cond-mat 0501455 (2005). 11. R.W. Simmonds, K.M. Lang, D.A. Hite, S. Nam, D.P. Pap- pas, and J.M. Martinis, Phys. Rev. Lett. 93, 077003 (2004). 12. K. Cooper, M. Steffen, R. Dermontt, R. Simmonds, S. Oh, D.A. Hite, D. Pappas, and J.M. Martinis, ArXiv: cond-mat 0405710 (2004); M. Nakahara, J. Vartiainen, Y. Kondo, S. Tanimura, and K. Hata, ArXiv: quant-ph/0411153 (2004). 906 Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 E.A. Ivanchenko and A.P. Tolstoluzhsky

Rabi oscillations and quantum beats in a qubit in distorted magnetic field

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