Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field

A closed system of the equations for the local Bloch vectors and spin correlation functions is obtained by decomplexification of the Liouville-von Neumann equation for 4 magnetic particles with the exchange interaction that takes place in an arbitrary time-dependent external magnetic field. The analytical and numerical analysis of the quantum thermodynamic variables is carried out depending on separable mixed initial state and the magnetic field modulation. Under unitary evolution, non-equilibrium reversible dynamics of power production in the finite environment is investigated.


I. INTRODUCTION
The classical thermodynamic heat engine converts heat energy in a mechanical work with the help classical mechanical system in which gas extends and pushes the piston in the cylinder. Such heat engine receives energy from the high-temperature reservoir. The energy part from this reservoir is converted in mechanical work, and the part of it is transferred in the low-temperature reservoir. The classical heat engine reaches its peak efficiency if it is reversible. Because of the impossibility to construct the heat engine which is ideally reversible, in 1824 Carnot [1] has offered the mathematical model of the ideal heat engine which is not only reversible, but is also cyclic. In the last decades great efforts has been devoted to the investigation of the quantum properties of the working substance, the search and practical implementation of the quantum analogue of the Carnot cycle in microsystems.
The operation of quantum heat engines that employ as working agents multi-level systems, for example harmonic oscillators, free particles in a box, three-level atoms or electrons submitted to magnetic fields, were introduced in [2], [3], [4]. The two-level quantum systems similar to particles with spin 1/2 are the essential ingredients for quantum computation, but the coupled spin systems can also be used as quantum thermodynamic engines [5-8, 10, 11]. The quantum analogue of the Carnot cycle requires a dynamical description of the working medium, the power output and the heat transport mechanism. The spin system " working gas " in an external field has its own physical properties. These properties at weak interaction with the environment (heat baths) can, as a rule, slightly be deformed. The purpose of this paper is to predict these properties to know what we can expect at the contact with the environment which is usually considered in the Markovian approximation for a derivation of the equation for the reduced matrix.
The paper is organized as follows. In Sec. II we introduce the model Hamiltonian. In Sec. III the Liouville-von Neumann equation for a density matrix for four particles with spin 1/2 with exchange interaction in variable magnetic field we write down in the Bloch representation in terms of the local Bloch vectors and spin correlation functions. We describe the conservation laws which effectively supervise the numerical calculations. In Sec. IV we describe the local quantum thermodynamic parameters of the spin subsystems. Our numerical results are detailed in Sec. V. For the separable mixed initial state we numerically investigate the quantum thermodynamics of a particle in the environment of three others depending on modulation of an operating field and initial conditions. Our results are summarized and discussed in Sec. VI. It will be numerically found, that under unitary dynamics the work production of one part of system is compensated by absorption of work produced by the other part. The work production by the subsystem is accompanied the entropy growth and vice-versa, the entropy becomes less with work absorption by the subsystem. Some necessary auxiliary details for numerical results are represented in the Appendix.

II. MODEL HAMILTONIAN
The Hamiltonian of four coupled particles e, p, n, u with spin 1/2 in the external ac magnetic field where h e i , h p i , h n i , h u i are the Cartesian components of the external magnetic field in the energy units, operating on corresponding particle; (1); the Pauli matrices are equal to ⊗ is the symbol of direct product [12]; J ep , J en , J pn , J eu , J pu , J nu are the constants of isotropic exchange interaction between spins; the summation over e, p, n, u is absent.

III. DECOMPLEXIFICATION OF THE LIOUVILLE-VON NEUMANN EQUATION
The Liouville-von Neumann equation for the density matrix ρ, describing the dynamics of four-spin system, looks like Let us present the solution of the equation (2) as Hereinafter summation is taken over the repeating Greek indices from zero up to four, and over Latin indices from one up to three. The four coherence vectors (the Bloch vectors) widely used in the magnetic resonance theory, are written as These vectors characterize the local properties of individual spins, whereas the other tensors describe the spin correlations. All correlation functions are in the limits As i∂ t ρ n = [Ĥ, ρ n ] (n = 1, 2, 3, . . . ) at unitary evolution there is the numerable number of conservation laws Tr ρ = C 1 = 1, Tr ρ 2 = C 2 , . . . , where C n are the constants of motion, from which only the first C 2 , C 3 , ..., C 16 are algebraically independent [13]. From the conservation of purity, for which (ρ 2 ) ik def ≡ (ρ) ik , the polynomial (square-law) invariants are obtained. The square polynomials also control the signs R αβγδ .
The length of the generalized Bloch vector b epnu is conserved under unitary evolution: The Liouville-von Neumann equation accepts the real form in terms of the functions R αβγδ as closed system of 255 differential equations for the local Bloch vectors and spin correlation functions The derivation algorithm of the system equation (7) has been presented in [14]. The set of equations (7) with the initial conditions has wide applications, since the magnetic field enters in the form of arbitrary functions. It allows to make numerical calculations for: 1. continuous (paramagnetic resonance in a continuous mode), as well as for pulse modes (nuclear magnetic resonance); 2. by means of this system it is possible to investigate the entanglement dynamics of qubits in a magnetic field [15] as the entanglement measures are expressed in terms of the reduced density matrices or of populations; 3. the important application of the system (7) is quantum approach to the Carnot cycle [1], [2], [3], [4], [5], [6], [7], when a working body is a finite spin chain. In case of equivalent particles at h e = h p = h n = h u from the equations (7) it follows, that the square length of the total magnetization (R q000 + R 0q00 + R 00q0 + R 000q ) 2 is conserved.
In the external field the energy of system is defined by formula The change of the total energy expectation value is equal to For the external dc field the hamiltonian is independent of time, hence ∂ t E = 0, that is the energy of system is the constant in time.
The change of work W can be associated with the term where only spectrum changes The last integral is equal zero due to the equation of motion. Therefore the work production system is given by expression of the form The change of the full system energy in time t which consists of the performed work and heat energy, in our problem it is equal to the work production, as in the closed system heat energy is not produced.
In system (7), assuming, for example, J eu = J pu = J nu = 0, we get the closed system of equations for the description of three-qubit dynamics [16]. Assuming J en = J pn = J eu = J pu = 0, we get the closed system of equations for the description of two-qubit dynamics where R q0 = Tr ρσ q ⊗ σ 0 , R 0q = Tr ρσ 0 ⊗ σ q , R kq = Tr ρσ k ⊗ σ q .

IV. QUANTUM THERMODYNAMIC VARIABLES
The reduced density matrices describe dynamics of subsystems (4 matrices of individual particles, 6 matrices of two particles, 4 matrices of three particles) and, for example, for a particle e, for the coupled particles ep, epn can be written as The matrices (11) are determined by the system solution (7), as the equations for the reduced matrices are not closed.
From the system (7) it follows, that probability of spin flip e from initial state is equal to and for the spin p It is known [9] that in weakly coupled systems in a longitudinal field h e = (0, 0, h e 3 ) the transverse coherences R 1000 , R 2000 are equal to zero for the initial state (A.1), that is the reduced matrix ρ e is diagonal in the course of unitary evolution. In this case it is possible to define the local dynamic temperature correctly. The local or dynamic temperature for two-level system can be defined according to the basic meaning of a thermal state [10], [17]: where Ω e (t) is the transition frequency in the e two-level system, equal to h e 3 . Further we set the Bolzmann constant k B equal to 1, therefore all temperatures in energy units. The von Neumann entropy after multiplication on Bolzmann constant can then be taken as the thermodynamic entropy. The e-spin entropy is equal to where the local populations are equal to The time derivative of the entropy for e-spin is defined by the formula The stated concerns each particle. The calculation of the work which are carried out by subsystems, by means of ST diagrames is applied in the the papers [10] it is shown numerically, that the entropy S and temperature T are dependent thermodynamic variables. For the closed trajectory in ST plane the change of full energy ∆W sys is equal to zero and consequently the area captured by the closed curve in ST plane, determines the work during a reversible cycle where t c is the duration of a cycle, and the sign is defined according to the rule, if at path tracing clockwise the area is situated on the right it obtains a minus sign (heat pump). The spin system is isolated and consequently, the Carnot cycles can only refer to sub-systems of the 4-spin system. We use (17) for e, p, n, u particles. The energy of the coupled particle e in a magnetic field h e 3 in an environment of three others is equal to 1 2 h e 3 R 3000 . We use the formula ∂ t ( 1 2 h e 3 R 3000 ) = 1 2 (∂ t h e 3 )R 3000 + 1 2 h e 3 ∂ t R 3000 . The work production (the heat production) by spin e during cycle is equal to During cycle the energy change of the particle e tc Having inserted the equations (13), (16) in (17) we conclude that Thus the definitions of temperature (13), entropy (14) and work (17) are coordinated with the work / heat production for parameters, for which the ST plots are closed. It concerns also p, n, u particles. The Klein-von Neumann inequality looks like where − m i=1 ρ ii ln ρ ii is the diagonal entropy d, m is the number of system states. For the initial diagonal state the diagonal entropy possesses property [18].
The dynamics of a purity measure P = Tr ρ 2 is connected with the dynamics of entropy S = −Tr ρ ln ρ as follows. If the entropy is equal to zero, the system is in a pure state. At the maximum entropy the system is in the maximally mixed state. The purity P has the maximum value 1 for a pure state and the minimum value in the mixed state, equal to 1 m , where m is the number of accessible states. The subsystem purity is expressed in terms of the square length of the local or generalized Bloch vector (6): the purity for the subsystems pn, epnu is equal to is the length of the generalized Bloch vector of the pn system. Let's define the entanglement measure p and n spins according to [19] on system solutions, having entered the two-particle entanglement tensor: The tensor m 0ij0 is equal to zero, when the two-particle correlation function R 0ij0 is factorized in terms of the local Bloch vectors (4b), (4c) and thus the matrix will be separable, i.e. ρ pn = ρ p ⊗ ρ p . By means of tensor we shall define a measure of the two-particle entanglement in the pn subsystem This measure is equal to zero for separable state and it is equal to 1 for the Greenberger-Horne-Zeilinger maximally entangled state. This measure is applicable both for the pure and mixed states (in all 6 two-particle measures).

V. NUMERICAL RESULTS
The quantum thermodynamic devices are subdivided into heat pumps and heat engines depending on functional purpose. In our model absorbed or made by system work arises due to displacement power levels by a magnetic field [20], [21], [22], and it also depends on an initial state of the system.
We will consider the influence of the variable magnetic field  (17) is equal to 0.1331514. e, p, u spins perform the work equal to 3×(−0.0443838) = −0.1331514. Thereby the full energy change of 4 spin system ∆W sys (9) is equal to zero in full compliance with the general results for the isolated system [23][24][25][26][27]. It confirms the use of temperature (13) and entropy (14) as the effective thermodynamic characteristics. The feature of this initial state and modulation magnetic field is that these cycles are repeated without any deformation. (It is known the system equation (7) with the periodic coefficients according to the Floquet theory has periodic or quasi-periodic solution i.e. the Floquet theory does not exclude also the periodic solutions. It depends on a set of the coefficients. We have presented this set.) In other words each cycle comes to the end returning to the same initial state. The closure of the ST plots does not depend of the amplitudes of the driving field (the form and the area vary only), but it critically depends on frequency ω [10] and the module of exchange constants, that does not depend of ferromagnetic or antiferromagnetic working gas [28]. After replacement of a frequency sign ω the circulation direction becomes opposite for all particles. If the value of parameter is T n (0) > 0.2, then the particle n makes work, and the spins e, p, u absorb work. The purity of all system decreases with increase of T n (0), but cycle-after-cycle and periodicity remain as it is described, and the areas characterizing work, increase with preservation of the algebraic sum which is equal to zero. For T n (0) < 0.2 the spins e, p, u produce the work , and the spin n absorbs the work. If T n (0) approaches to 0.2 the absorbed and produced works approach to zero, as at the initial moment there is no temperature gradient in the system (passive state [29]). The numerical calculation also shows, that for the parameters of the initial state T e (0) = T p (0) = 0.2; T n (0) = T u (0) = 0.25 (other parameters as in Fig. 1) particles e, p absorb work equal to 2(-0.002265), and particles n, u produce the same work 2(+0.002265).
We would like to indicate, that in a vicinity t ≈ 117.8 the local temperature of all particles goes to zero as the frequencies Ω i (t) for all particles go to zero. It is necessary to notice, that for the minimum local temperatures the eigenvalues of the Hamiltonian (1) come nearer to zero, and with the growth of temperatures the eigenvalues become bigger. The transition probability of each particle (12) is close to 1 and makes one oscillation per cycle. The population ρ 16 16 is approximately equal to 0.925 during a cycle. For the opposite sign h e 3 (0) = h p 3 (0) = h n 3 (0) = h u 3 (0) = −1.5, ρ 1 1 ≈ 0.925 the thermodynamic characteristics do not change. In Fig. 1 it is clearly visible, that if the initial state of a subsystem is more disordered, a subsystem absorbs/produces more (on the module) work [29]. The cycle of each particle is determined by the direct coupling of the given particle with the others and does not depend of the coupling constants between other particles. The calculations confirm, that the dependence between the energy of particles p, n E pn = 1 2 (h p i R 0i00 + h n i R 00i0 ) + 2J pn R 0ii0 and the entropy S pn is also cyclic. This property is carried out for any particle pair.
The additional quantum thermodynamic characteristics of eu and pn subsystems are resulted in Fig. 2 and Fig. 3. Under unitary evolution the global purity P epnu does not depend of time. The purity P pn (19) has maximum at an entropy minimum. The work production of eu subsystem is accompanied by the entropy increase and the purity reduction and the inverse process occurs at work absorption. It is visible, that the local entanglement m pn (21) has maximum for a minimum entropy S pn = −Tr ρ pn ln ρ pn . The entanglement (21) between the particles is noticeable in the middle of cycle and it is bigger between particles with opposite signs in work, and, as the calculations confirm, grows with the disorder increase in system. This entropy is always less or equal to the sum entropies of the individual spins in pn subsystem. Also the inequality Klein-von Neumann is carried out (18), i. e. the diagonal entropy is bigger or equal to pn subsystem entropy. For the cases presented in Fig. 1 the diagonal entropy d pn (d eu ) coincides with the sum entropies of the individual spins S p + S n (S e + S u ).
The control of calculations was carried out with the help invariants of motion, described in section III, and all the correlation functions have been in the limits −1 ≤ R αβγδ ≤ 1.

VI. CONCLUSION
The closed system equations for the local Bloch vectors and spin correlation functions of four two-level systems with the exchange interaction, being in the time-dependent external magnetic field is derived. The invariants of motion have been found , which are necessary for the control of computing. The numerical analysis of thermodynamic behaviour in 4 spin system depending on the parameters characterizing an initial state and modulation of the driving field was performed. It was numerically found, that under unitary dynamics the work production of one part of system is compensated by absorption of work produced by the other part. The work production by the subsystem is accompanied the entropy growth and vice-versa, the entropy becomes less with work absorption by the subsystem. It was revealed, that the ST cycle of each particle is determined by the direct coupling of the given particle with the others and weakly depends on coupling constants between the other spins. It was numerically shown that the formulas for the temperature (13) and the entropy (14) are the effective thermodynamic characteristics for work calculation in 4 spin system in the course of unitary evolution. The research of the "spin gas" properties is necessary for the implementation of quantum thermodynamic cycles in spin systems [30,31]. Taking into account an environment zero balance will be broken and the system will become heat engine or heat pump depending on temperatures of heat baths.  Fig. 1