Quantum quench for the biaxial spin system

Dynamical non-equilibrium effects after a quantum quench in a quantum spin system, which permits exact analytical solution in both closed and open cases, are studied. The exact analytic results are obtained for the biaxial paramagnet, both for the “easy axis”- and the “easy-plane”-like situations,...

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Дата:	2018
Автор:	Zvyagin, A.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2018
Назва видання:	Физика низких температур
Теми:	Низькотемпературний магнетизм
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/176477
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Цитувати:	Quantum quench for the biaxial spin system / A.A. Zvyagin // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1501-1509. — Бібліогр.: 21 назв. — англ.

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spelling	irk-123456789-1764772021-02-05T01:28:52Z Quantum quench for the biaxial spin system Zvyagin, A.A. Низькотемпературний магнетизм Dynamical non-equilibrium effects after a quantum quench in a quantum spin system, which permits exact analytical solution in both closed and open cases, are studied. The exact analytic results are obtained for the biaxial paramagnet, both for the “easy axis”- and the “easy-plane”-like situations, and for the field directed along both principal axes of the system. Quantum quench of the external magnetic field produces a nonlinear response. For the closed system the average magnetic moment oscillates with time and with the final value of the external field. Such oscillations exist also for the open system, connected to the bath, in the dynamical regime. For the steady-state regime in the open case the oscillations are damped. Non-equilibrium effects yield specific hysteresis phenomena in the considered single spin system. Досліджено динамічні нерівноважні ефекти після квантового гартування в квантових спінових системах, які допускають точні аналітичні рішення як в закритому, так і у відкритому випадках. Точні аналітичні рішення отримані для двовісного парамагнетика як в легкоплощинних, так і легковісній ситуаціях і для поля, прикладеного вздовж двох головних осей системи. Квантове гартування зовнішнім магнітним полем призводить до нелінійного відгуку. Для закритої системи середній магнітний момент осцилює із часом та з фінальним значенням зовнішнього поля. Такі осциляції існують також і у відкритій системі, що контактує з термостатом, в динамічному режимі. Для сталого режиму у відкритій системі осциляції пригнічені. Нерівноважні ефекти створюють специфічні гістерезисні явища в даній моноспіновій системі. Исследованы динамические неравновесные эффекты после квантовой закалки в квантовых спиновых системах, которые допускают точные аналитические решения как в закрытом, так и в открытом случаях. Точные аналитические решения получены для двуосного парамагнетика как в легкоплоскостной, так и легкоосной ситуациях и для поля, приложенного вдоль двух главных осей системы. Квантовая закалка внешним магнитным полем приводит к нелинейному отклику. Для закрытой системы средний магнитный момент осциллирует со временем и с финальным значением внешнего поля. Такие осцилляции существуют также и в открытой системе, контактирующей с термостатом, в динамическом режиме. Для установившегося режима в открытой системе осцилляции подавлены. Неравновесные эффекты создают специфические гистерезисные явления в рассматриваемой моноспиновой системе. 2018 Article Quantum quench for the biaxial spin system / A.A. Zvyagin // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1501-1509. — Бібліогр.: 21 назв. — англ. 0132-6414 http://dspace.nbuv.gov.ua/handle/123456789/176477 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	Низькотемпературний магнетизм Низькотемпературний магнетизм
spellingShingle	Низькотемпературний магнетизм Низькотемпературний магнетизм Zvyagin, A.A. Quantum quench for the biaxial spin system Физика низких температур
description	Dynamical non-equilibrium effects after a quantum quench in a quantum spin system, which permits exact analytical solution in both closed and open cases, are studied. The exact analytic results are obtained for the biaxial paramagnet, both for the “easy axis”- and the “easy-plane”-like situations, and for the field directed along both principal axes of the system. Quantum quench of the external magnetic field produces a nonlinear response. For the closed system the average magnetic moment oscillates with time and with the final value of the external field. Such oscillations exist also for the open system, connected to the bath, in the dynamical regime. For the steady-state regime in the open case the oscillations are damped. Non-equilibrium effects yield specific hysteresis phenomena in the considered single spin system.
format	Article
author	Zvyagin, A.A.
author_facet	Zvyagin, A.A.
author_sort	Zvyagin, A.A.
title	Quantum quench for the biaxial spin system
title_short	Quantum quench for the biaxial spin system
title_full	Quantum quench for the biaxial spin system
title_fullStr	Quantum quench for the biaxial spin system
title_full_unstemmed	Quantum quench for the biaxial spin system
title_sort	quantum quench for the biaxial spin system
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2018
topic_facet	Низькотемпературний магнетизм
url	http://dspace.nbuv.gov.ua/handle/123456789/176477
citation_txt	Quantum quench for the biaxial spin system / A.A. Zvyagin // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1501-1509. — Бібліогр.: 21 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT zvyaginaa quantumquenchforthebiaxialspinsystem
first_indexed	2025-07-15T14:15:01Z
last_indexed	2025-07-15T14:15:01Z
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fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11, pp. 1501–1509 Quantum quench for the biaxial spin system A.A. Zvyagin B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Nauky Ave., Kharkiv 61103, Ukraine V.N. Karazin Kharkiv National University, 4 Svobody sq., Kharkiv 61002, Ukraine Max-Planck-Institut für Physik komplexer Systeme, 38 Nöthnitzer Str., D-01187, Dresden, Germany E-mail: zvyagin@ilt.kharkov.ua Received May 14, 2018, published online September 26, 2018 Dynamical non-equilibrium effects after a quantum quench in a quantum spin system, which permits exact analytical solution in both closed and open cases, are studied. The exact analytic results are obtained for the bi- axial paramagnet, both for the “easy axis”- and the “easy-plane”-like situations, and for the field directed along both principal axes of the system. Quantum quench of the external magnetic field produces a nonlinear response. For the closed system the average magnetic moment oscillates with time and with the final value of the external field. Such oscillations exist also for the open system, connected to the bath, in the dynamical regime. For the steady-state regime in the open case the oscillations are damped. Non-equilibrium effects yield specific hystere- sis phenomena in the considered single spin system. Keywords: quantum quench, biaxial magnetic anisotropy, dynamical hysteresis. 1. Introduction Quantum systems out of equilibrium, e.g., after abrupt changes of their parameters, are basically not susceptible to general principles of equilibrium systems [1]. This is why, studies of non-equilibrium dynamics of quantum models are necessary for the fundamental understanding of how me- chanics emerges under the unitary time evolution. The time evolution of quantum averages depends on the initial state through the values of a large number of parameters of the quantum system. It disagrees with the standard ensem- bles of statistical mechanics which use few conserved val- ues of the dynamical system and usually describe the be- havior after relaxation. Theoretical studies of dynamical characteristics of many-body quantum systems are more difficult than of their static counterparts, because all eigen- states contribute to dynamics, and there is no possibility to limit the consideration by the low-energy eigenstates, as, e.g., in low-temperature thermodynamics. Since the dynamics of a quantum system typically involve many excited eigen- states, with a non-thermal distribution, the time evolution of such a system provides an unique way for investigation of non-equilibrium quantum statistical mechanics. Last de- cade such new subjects like quantum quenches, thermal- ization, pre-thermalization, equilibration, generalized Gibbs ensemble, etc. are among the most attractive topics of in- vestigation in modern quantum physics. Abrupt changes of some parameters, i.e., quantum quenches, in which the sys- tem is prepared in an eigenstate of the initial Hamiltonian and its time evolution driven by the final Hamiltonian, lead to such a unitary time evolution, and the final (long time) state strongly depends on the type of the system. Their studies can provide the information of how fast correla- tions spread in quantum systems, whether averages can decay to some time-independent values, and which param- eters can govern those processes. The study of the non-equi- librium dynamics of quantum coherence is very important for the modern theory of quantum computation, where na- mely abrupt changes (gates) are used to govern the beha- vior of ensembles of qubits [2]. On the other hand, the study of sudden changes is very important in the context of experiments on ultracold gases [3], ultrafast (e.g., THz) pulses [4] realized in solids [5], or high magnetic field ex- periments in pulse fields [6,7]. For ultracold gases, for in- stance, the coherence is maintained for much longer times than for usual condensed matter, and the time evolution of a quantum system after abrupt changes has become a real- istic concept. The analysis of nonlinear quantum dynamics of isolated spins or small particles in the mean field ap- proximation) was performed, e.g., in [8]. Nonlinear quan- © A.A. Zvyagin, 2018 A.A. Zvyagin tum dynamics of spins = 1S under action of short laser pulses has been studied in [9], see also [10] (it was shown there that such a dynamics can be totally longitudinal, i.e., with the evolution of the average value of the spin, and average values of quadrupole variables). The new field of technology, molecular spintronics, com- bines the approaches and the advantages of spintronics and molecular electronics. The main issue of the molecular spin- tronics is the creation of small devices using one or several magnetic molecules. Single molecule magnets or single atom magnets can be used there. In such systems the mag- netic relaxation time is very long at low temperatures [11]. Their single- or few-particle nature yields quantum effects of their static and dynamic magnetic properties [12]. The interest to single molecular magnets is caused by a small number of degrees of freedom (due to absent exchange between spins). Contrary, the spin-orbit coupling for a sin- gle spin together with the crystalline electric field of non- magnetic ligands governs the magnetic properties of the system, yielding local spin symmetries. In this study we consider the non-equilibrium dynamics of a simple quantum mechanical system, the paramagnetic quantum spin, which has the biaxial magnetic anisotropy in the external magnetic field. The advantage of the consid- ered model is its solvability: The characteristics of the mo- del after the quantum quench is written explicitly, in the closed form. The results are obtained for the “easy axis”- like and for the “easy-plane”-like main magnetic anisotro- py with the weak biaxial anisotropy, for the field directed along the axis of the largest and the weakest magnetic ani- sotropy. We show that for the closed system the quantum quench produces oscillations of the average magnetic mo- ment. Those oscillations persist with time and with the value of the magnitude of the quantum quench with respect to the value, determined by the parameters of the system and the values of the initial and the final values of the field. For the open system, which exchanges the energy with the bath, such oscillations persist in the dynamical regime, for small enough time values. For large values of time, in the steady-state regime, the relaxation “smears out” the oscil- lations. We show that the dependence of the steady-state average magnetic moment on the values of the initial value of the field and the final one are very different from the field dependence of the same system in the stationary re- gime. Also, we show that the behavior of the system for switching on and off the field is also very different. 2. The Hamiltonian Consider the Hamiltonian of the spin S with the biaxial magnetic anisotropy in the external magnetic field 2 2= z z xHS DS ES− + + , (1) where ,x zS are the operators of projections of the spin, D and E are the magnetic anisotropy parameters, and H is the external magnetic field (we use units in which the Bohr magneton and the effective g -factors are equal to unity). In the representation with the diagonal z -component for = 1S we can write the Hamiltonian as 1 0 2 2 = 0 2 2 0 0 E EH D E EH D E  − + +     + + =          2 0 1ˆ= 2 0 , 2 2 0 0 2 H E ED I E H D E −     + +      − +  (2) where Î is the unity matrix. We see that the magnetic field affects only the 2 2× subspace (with = 1zS ± ). In that sub- space (i.e., in the effective 2 2× matrix representation) the effective Hamiltonian can be written as = . 2z x EHσ − σ + σ (3) The density matrix = exp ( / ) / Tr [exp ( / )]T Tρ −  , where T is the temperature (we use units in which the Boltzmann constant is unity), in this representation is 12= 2sinh exp 2 D E T T − σ ε − ρ + ×   ˆ cosh sinhzI T T′σ ε ε ′× + σ    , (4) where Îσ is the unity matrix in the 2 2× subspace, and = ( / ) ( / 2 )z z xH E′σ ε σ + ε σ , with 2 24= . 2 H E+ ε (5) For the spin = 3 / 2S case we can write the expression for the Hamiltonian in the diagonal in zS representation 3/2 3 1ˆ= 4 2 E D I+ + × 3 4 3 0 0 3 0 0 0 0 3 0 0 3 3 4 H D E E H E H E E E H D  − +    + ×   − +   +  . (6) One can see that the magnetic field acts independently in two 2×2 subspaces (for = 3 / 2, 1/ 2zS − and = 3 / 2, 1/ 2zS − , respectively). In the effective 2 2 2 2× ⊗ × (let us denote it as 1σ – 2σ ) representation for those two subspaces the effec- tive Hamiltonians can be written as 1502 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 Quantum quench for the biaxial spin system 1,2 1,2 1,21,2 1 ˆ= (2 2 ) 3 . 2 z xI H D E H Eσ  ± σ − + σ   (7) The density matrix in this representation can be written as 1 1 2 ,1 2 = 2exp cosh 2exp cosh 2 2 H H T T T T − σ σ ε ε− ρ + ×   1 1 1 1 ˆ exp cosh exp sinh 2 2z H HI T T T T ε ε ′× + σ + 2 2 2 2 ˆ exp cosh exp sinh , 2 2z H HI T T T T ε ε− − ′+ + σ  (8) where 1,2Î are the unity matrices in those two 2 2× subspaces, and 1,2 1,2 1,2 1,2 1,2= [( / 2) / ] ( 3 / 2 )z z xH D E E′σ ± ε σ + ε σ with 2 2 1,2 3 (2 2 ) = . 2 E D E H+ − ε  (9) 3. Static characteristics Then the quantum mechanical average value of the pro- jection of the magnetic moment along z direction is calcu- lated as = Tr ( )z zM S ρ . We can calculate that value using the effective 2 2× representations written above. For the spin = 1S we obtain (using = σρ ρ ) 0 2 sinh ( / ) . 2 cosh ( / ) exp [(2 ) / 2 ]z H TM T D E T ε = ε ε + − (10) It can be compared with the approximate expression [13] 2 2 ,HS H K 〈 〉 ≈ + (11) valid at low temperatures (here K is the anisotropy con- stant in the basis plane). One can see that for the “easy axis”-like case for = 1S the approximate expression is reminiscent to the exact one. On the other hand, for the “easy-plane” case the situation is different. For the spin = 3 / 2S we get (using , 21 = σ σρ ρ ) [0 1= exp ( / 2 ) cosh ( / )zM H T Tε + ] 1 2exp ( / 2 ) cosh ( / 2 )H T T −+ − ε × 1 1 ( / 2)exp ( / 2 ) sinh ( / )H D EH T T   − + × ε +  ε 1 2 cosh ( / 2 ) ( / 2)exp ( / 2 ) 2 T H D EH T ε + −+ + − × ε  2 2 cosh ( / 2 ) sinh ( / ) . 2 TT ε × ε −   (12) We can also calculate the projection of the average magnetic moment along x direction. For this purpose we replace x z↔ in the Hamiltonian . Hence, the answers for the quantum mechanical averaged value of the x- projection of the magnetic moment xM can be obtained by the formal replacement D E↔ in Eqs. (10) and (12). Figures 1 and 2 manifest the magnetic field dependenc- es of the z - and x-projections of the magnetic moments for the “easy axis” = 1D − and “easy-plane” paramagnet with Fig. 1. (Color online) Calculated magnetic field dependences of the projections of the magnetic moment 0M for = 1S at = 0.1T . For the “easy axis” case = 1D , with the small biaxial anizotropy = 0.1E − the solid black line shows the field is directed along z axis; the dashed blue line shows the field is directed along x axis. For the “easy-plane” case = 1D − , with the small in-plane an- izotropy = 0.1E − the dashed-dotted orange line shows the field is directed along z axis; the dotted red line shows the field is di- rected along x axis. Fig. 2. (Color online) The same as in Fig. 1 but for = 3 / 2S . The parameters and the notations are the same as in Fig. 1. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1503 A.A. Zvyagin the weak biaxial magnetic anisotropy = 0.1E − at low tem- perature = 0.1T for = 1S and 3/2, respectively. We see that for different directions of the field the magnetic field behavior manifests different features. We point out that the change of the sign of the weak biaxial anisotropy does not produce essential changes in the magnetic field behavior. At higher temperatures (of order of the maximal value of the magnetic anisotropy) all those features are “smeared out”. 4. Dynamics after the quantum quench Now consider the following situation. Suppose at = 0t we change the value of the magnetic field from iH (valid at 0t ≤ ) to fH (valid for > 0t ), known as the quantum quench. Dynamics of any quantum system can be describe- ed in two ways. In the first way one considers the time evolution of the considered operator (using the Heisenberg equations), and then average the obtained time-dependent value of the operator with respect to the wave function (for the pure state), or the density matrix (for the mixed state). The other way, is to find the time evolution of the wave function (using the Schrödinger equation) or the density matrix (using the Liouville equation), and then average the considered operator with the obtained time-dependent wave function or the density matrix. In the case of exact calcula- tions both ways yield the same answer. To describe dynamics of the studied spin system under the action of the linearly polarized ac magnetic field let us use, for instance, the first approach. The Liouville equation for density matrix ρ has the form = [ , ]iρ ρ  , where [.,.] denotes the commutator. Such a behavior is characteristic for a closed system. However, as a rule, the spin system is not isolated. For example, there are processes, which take the energy from the system, i.e., relaxation processes. The relaxation can be considered in a number of ways. The reason for the relaxation of the density matrix is the interaction of the considered system with some environ- ment; such an interaction takes the energy from the system, i.e., our considered system is the open one. For example, for the studied quantum spin system the lattice (i.e., the elastic subsystem of the crystal) can serve as such an envi- ronment. Dynamics of the density matrix of our open system for general Markovian processes is described by the Lindblad master equation [14] (here we write it in the diagonal form) 2 1 † † =1 1= [ , ] { , } 2 N j j jj j j i i −  ρ ρ + γ ρ + ρ    ∑      , (13) where N is the dimension of the system, {.,.} denotes the anticommutator, and the orthonormal and traceless opera- tors j are the Lindblad (jump) operators. For = 0jγ the Lindblad equation is, obviously, the Liouville equation. In the model of random collisions [15] one can write the Lindblad operators as 0= ( ) \| \|j jj j j′ρ 〉〈 , and suppose that all jγ are equal, which yields 0= [ , ] ( ) .i iρ ρ + γ ρ −ρ  (14) This form of the master equation was first suggested by Karplus and Schwinger [16]. It was used to describe the relaxation processes of quantum systems under the action of the ac electromagnetic field. It describes the interaction of the considered system with the bath, with the relaxation of the density matrix to 0ρ in the steady state. It is useful to substitute = exp ( )t′ρ γ ρ; we obtain 0= [ , ] exp ( )i t′ ′ρ ρ + γ γ ρ  . The used approximation implies equal relaxation times for all eigenmodes of the system. It is equivalent to the Bloch form of relaxation in the theory of the nuclear magnetic resonance [17]. Two relaxation times as in the Bloch ap- proach can be easily introduced in the above scheme by using different relaxation rates for diagonal and non-dia- gonal components of the density matrix. One can also gen- eralize the approach using, e.g., Torrey’s phenomenological theory [18], which adds diffusion processes to the Bloch equations. It is possible to show that the effect of the linear relaxation in the Bloch form is similar to the effect of the relaxation in the Landau–Lifshitz form for magnetic sys- tems [19]. Here we are interested mostly in the homogene- ous response, and can neglect the spatial dependence of relaxation. 5. Closed system The response to the quantum quench is strictly nonline- ar for the studied model. After some algebra similar to the one, developed in Ref. 20 we obtain for = 0γ for the closed system for = 1S , 2sinh ( / )( ) = 2cosh ( / ) exp [(2 ) / 2 ]z z i TM t M T D E T ε + × ε + − 2 2 2 sin ( ) , 2 f f i f E H t× ε ε ε (15) where we use units in which = 1 , and , ,= ( = )i f i fH Hε ε . We see that the average magnetic moment oscillates with time and with the magnitude fH of the quantum quench. For instance, Fig. 3 shows the oscillations of the average value of the magnetic moment along z direction for = 1S “easy axis”-like case with = 1D − and = 0.1E − at = 0.1T for = 0iH . For = 3 / 2S the calculated dependence is [, 1( ) exp ( / 2 )cosh ( / )z z i i iM t M H T T= + ε + ] 2 1 2 3 exp ( / 2 )cosh ( / 2 ) 2 f i i H E H T T −+ − ε × 1504 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 Quantum quench for the biaxial spin system 2 1 12 1 1 1exp ( / 2 ) sin ( ) sinh ( / )i f i i f H T t T  × ε ε × ε ε  1 1 ( / 2) cosh ( / 2 ) 2 i i i H D E T − + ε × + −ε  2 2 22 2 2 1exp ( / 2 ) sin ( ) sinh ( / )i f i i f H T t T− − ε ε ×ε ε 2 2 ( / 2) cosh ( / 2 ) 2 i i i H D E T + − ε × −  ε   , (16) where 1,2 , 1,2 ,= ( = )i f i fH Hε ε . Notice that for = 3 / 2S we observe the interference of oscillations with two frequences, 1,2 fε , unlike the case for = 1S , where the magnetic moment oscillates with only one frequency fε . Figure 4 shows the example of oscillations of the average value of the magnetic moment along x direction for = 3 / 2S “easy-plane”-like case with = 1E and = 0.1D − at = 0.1T for = 0iH . We see that the magnitude of the oscillations of the x-pro- jection is larger than the one for the z -projection of the magnetic moment. Notice that for large values of fH the magnitude of oscillations decay, see Figs. 3 and 4. It is possible to calculate the average in time value, about which the quantum mechanical magnetic moment oscillates after the quantum quench. For = 1S we get 2 , 2 2sinh ( / )= 2cosh( / ) exp [(2 ) / 2 ] 4 f z z i i f E HTM M T D E T ε 〈 〉 + ε + − ε ε (17) and for = 3 / 2S we obtain [, 1= exp ( / 2 )cosh ( / )z z i i iM M H T T〈 〉 + ε + ] 2 1 2 3 exp ( / 2 )cosh ( / 2 ) 4 f i i H E H T T −+ − ε × 12 1 1 1exp ( / 2 ) sinh ( / )i i i f H T T  × ε × ε ε  1 1 ( / 2) cosh ( / 2 ) 2 i i i H D E T− + ε × + −ε  22 2 2 1exp ( / 2 ) sinh ( / )i i i f H T T  − − ε × ε ε  2 2 ( / 2) cosh ( / 2 ) . 2 i i i H D E T + − ε × −  ε   (18) Equations (15)–(18) describe totally dynamical behavior for the closed system. Figures 5 and 6 show the dependences of the average in time values, about which magnetic moments oscillate on the values of the initial iH and final fH values of the magnetic field at = 0.1T . In Fig. 5 the = 1S case is shown for the “easy axis”-like case ( = 1D − and = 0.1E − ) for the x-projection. In Fig. 6 the = 3 / 2S case is shown for the “easy- plane”-like case ( = 1D and = 0.1E − ) for the z -projection. These results show that the average value of the mag- netic moment is mostly determined by the initial value of the magnetic field iH , and the final value fH plays an essential role if it has the sign, different from the sign of iH , and for = 0iH . Fig. 3. (Color online) Calculated magnetic field dependence of the z-projection of the magnetic moment M on the magnitude of the quantum quench fH and time t at = 0iH for the closed = 1S “easy axis”-like system ( = 1D − , = 0.1E − ) at = 0.1T . Fig. 4. (Color online) Calculated magnetic field dependence of the x -projection of the magnetic moment M on the magnitude of the quantum quench fH and time t at = 0iH for the closed = 3 / 2S “easy-plane”-like system ( = 0.1D − , = 1E ) at = 0.1T . Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1505 A.A. Zvyagin 6. Open system If the considered system is the part of the larger subsys- tem (the bath), i.e., we deal with the open system case, we need to take into account the relaxation processes due to the exchange of the energy between the bath and our sub- system, see above. For example, we can use the Karplus– Schwinger form of the Lindblad master equation for the density matrix, where the linear relaxation γ is introduced. The result is equivalent to the standard Bloch approach to the equations of motion for spin projections. Notice that the inclusion of the linear relaxation implies multiplication of the time-dependent parts of Eqs. (15) and (16) by the multiplier exp ( )t−γ , and those equations are valid for 1<t −γ in the dynamical regime. In the steady-state regime 1t −γ after some calculations we obtain for spin = 1S st , 2sinh( / )= 2cosh( / ) exp [(2 ) / 2 ]z z i TM M T D E T ε + × ε + − 2 2 2 . 2 ( ) f i f E H × ε ε + γ (19) Here we have supposed that the system decays to the state with = iH H . The dependence of the steady-state magnetic moment M on the magnitude of the quantum quench fH for = 0iH (i.e., the switching on the field) for = 1S for two directions of the magnetic field and for the “easy axis” and the “easy-plane” cases (with the weak biaxial magnetic anisotropy) is shown in Fig. 7. Fig. 5. (Color online) Calculated dependences of the x -projection of the average in time value of the magnetic moment M on the magnitude of the quantum quench fH and the initial value of the field iH for the closed = 1S “easy axis”-like system ( = 1D − , = 0.1E − ) at = 0.1T . Fig. 6. (Color online) Calculated dependences of the z-projection of the average in time value of the magnetic moment M on the magnitude of the quantum quench fH and the initial value of the field iH for the closed = 3 / 2S “easy-plane”-like system ( = 1D , = 0.1E − ) at = 0.1T . Fig. 7. (Color online) Calculated magnetic field dependences of the projections of the magnetic moment M on the magnitude of the quantum quench fH at = 0iH (switching on the field) in the steady-state regime for = 1S at = 0.1T with = 0.01γ . For the “easy axis” case = 1D , with the small biaxial anizotropy = 0.1E − the solid black line shows the field is directed along z axis; the dashed blue line shows the field is directed along x axis. For the “easy-plane” case = 1D , with the small in-plane anizotropy = 0.1E − the dashed-dotted orange line shows the field is directed along z axis; the dotted red line shows the field is directed along x axis. 1506 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 Quantum quench for the biaxial spin system We see that the average magnetic moment decays in the steady-state regime with the magnitude of the quantum quench. The oscillations, characteristic for the dynamical regime and the closed system, are exhausted. For the case = 3 / 2S we get st , 1= exp ( / 2 ) cosh ( / )z z i f fM M H T T+ ε + 2 1 2 3 exp ( / 2 )cosh ( / 2 ) 2 f f f H E H T T − + − ε × 12 2 1 1 1exp ( / 2 ) sinh ( / ) ( ) f f i f H T T  × ε × ε ε + γ  1 1 ( / 2) cosh ( / 2 ) 2 f f f H D E T − + ε × + − ε  22 2 2 2 1exp ( / 2 ) sinh ( / ) ( ) f f i f H T T  − − ε × ε ε + γ  2 2 ( / 2) cosh ( / 2 ) . 2 f f f H D E T + − ε × −  ε   (20) Figure 8 shows the dependence on the magnitude of the quantum quench fH of the steady-state value of the aver- age magnetic moment for = 3 / 2S after switching the magnetic field from = 0iH for the “easy axis” and the “easy-plane” anisotropy (with the weak biaxial anisotropy) for z and x directions of the magnetic field. The parame- ters and the notations are the same as in Fig. 7. The aver- age magnetic moment decays in the steady-state regime with the magnitude of the quantum quench as for = 1S . However, the value of the average magnetic moment in the steady-state regime for = 3 / 2S can become negative. On the other hand, for switching off the field from = 5iH the dependence of the steady-state average magnet- ic moment on the magnitude of the quantum quench fH is shown in Fig. 9 for = 3 / 2S . The parameters and the nota- tions are the same as in Fig. 7. It turns out that the average magnetic moment goes to zero not at =f iH H− as it is naively expected; zero value of the average magnetic mo- ment is determined by the values of the magnetic anisotro- py constants. Then, at large negative values of fH , the Fig. 9. (Color online) Calculated magnetic field dependences of the projections of the magnetic moment M on the magnitude of the quantum quench fH starting from = 5iH (switching off the field) in the steady-state regime for = 3 / 2S at = 0.1T with = 0.01γ . The parameters and the notations are the same as in Fig. 7. Fig. 8. (Color online) The same as in Fig. 7 but for = 3 / 2S . The parameters and the notations are the same as in Fig. 7. Fig. 10. (Color online) The same as in Fig. 9 but for = 1S . The parameters and the notations are the same as in Fig. 7. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1507 A.A. Zvyagin average magnetic moment gets the value determined by = iH H . For = 1S the dependences for switching off the field are shown in Fig. 10. Notice that the dependences are simi- lar for = 1D ± . It turns out that for = 1S the steady-state magnetic moment for =f iH H− when switching off the field becomes negative (unlike the case = 3 / 2S ). Also, unlike = 3 / 2S the steady-state magnetic moment is mini- mal at =f iH H− . Then, for large negative values of fH , it gets the value determined by = iH H , as for = 3 / 2S . Those effects are the manifestation of the specific hys- teresis phenomenon, which is known to be totally dynam- ical in single molecular magnets [21]. 7. Summary In summary, we have studied dynamical non-equilibrium effects in a quantum spin system, which permits exact ana- lytical solution in both closed and open cases, i.e., if the system is isolated, or it is connected to the bath. The exact analytic results are obtained for the bi-axial paramagnet, both for the “easy axis”- and the “easy-plane”-like situa- tions, and for the field directed along both principal axes of the system. We have shown that quantum quench of the external magnetic field produces a nonlinear response. Namely, for the closed system the average magnetic moment oscillates with time and with the final value of the external field. The value of the magnetic moment, around which oscillations persist, is mostly determined by the initial value of the field. 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Для сталого режиму у відкритій системі осциляції пригнічені. Нерівноважні ефекти створю- ють специфічні гістерезисні явища в даній моноспіновій сис- темі. Ключові слова: квантове гартування, двовісна магнітна ані- зотропія, динамічний гістерезис. Квантовая закалка двуосных квантовых спиновых систем А.А. Звягин Исследованы динамические неравновесные эффекты по- сле квантовой закалки в квантовых спиновых системах, ко- торые допускают точные аналитические решения как в за- крытом, так и в открытом случаях. Точные аналитические решения получены для двуосного парамагнетика как в лег- коплоскостной, так и легкоосной ситуациях и для поля, при- ложенного вдоль двух главных осей системы. Квантовая закалка внешним магнитным полем приводит к нелинейному отклику. Для закрытой системы средний магнитный момент осциллирует со временем и с финальным значением внешне- го поля. Такие осцилляции существуют также и в открытой системе, контактирующей с термостатом, в динамическом режиме. Для установившегося режима в открытой системе осцилляции подавлены. Неравновесные эффекты создают специфические гистерезисные явления в рассматриваемой моноспиновой системе. Ключевые слова: квантовая закалка, двуосная магнитная анизотропия, динамический гистерезис. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1509 https://doi.org/10.1021/jacs.6b02638 https://doi.org/10.1038/s41467-017-00616-2 https://doi.org/10.1038/s41467-017-00616-2 https://doi.org/10.1007/BF01608499 https://doi.org/10.1063/1.522979 https://doi.org/10.1063/1.522979 https://doi.org/10.1103/PhysRev.73.1020 https://doi.org/10.1103/PhysRev.70.460 https://doi.org/10.1103/PhysRev.104.563 https://doi.org/10.1063/1.4931784 https://doi.org/10.1063/1.4931784 https://doi.org/10.1103/PhysRevB.92.184507 https://doi.org/10.1016/S0304-8853(99)00408-4 https://doi.org/10.1016/S0375-9601(98)00021-8 https://doi.org/10.1016/j.poly.2009.02.032 1. Introduction 2. The Hamiltonian 3. Static characteristics 4. Dynamics after the quantum quench 5. Closed system 6. Open system 7. Summary

Quantum quench for the biaxial spin system

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