Spin relaxation in a complex environment

Spin relaxation in a complex environment

Massimiliano Esposito and Pierre Gaspard

Center for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium

共Received 6 August 2003; published 24 December 2003兲

We report the study of a model of a two-level system interacting in a nondiagonal way with a complex environment described by Gaussian orthogonal random matrices共GORM兲. The effect of the interaction on the total spectrum and its consequences on the dynamics of the two-level system is analyzed. We show the existence of a critical value of the interaction, depending on the mean level spacing of the environment, above which the dynamics is self-averaging and closely obey a master equation for the time evolution of the observ-ables of the two-level system. Analytic results are also obtained in the strong coupling regimes. We finally study the equilibrium values of the two-level system population and show under which condition it thermalizes to the environment temperature.

DOI: 10.1103/PhysRevE.68.066113 PACS number共s兲: 05.50.⫹q, 03.65.Yz I. INTRODUCTION

The nonequilibrium statistical mechanics of small quan-tum systems has become a topic of fundamental importance for nanosciences. It is indeed very important to understand what the minimum sizes and conditions are under which a quantum system can display a relaxation to an equilibrium state. Isolated and finite quantum systems have a discrete energy spectrum, which has for consequence that all the ob-servables present almost periodic recurrences on long time scales. Nevertheless, their early time evolution may still present relaxation types of behavior that is important to study. Although tools have been developed in nonequilibrium statistical mechanics to describe such relaxations to a state of equilibrium by master and kinetic equations, the conditions of validity of these equations remain little known.

It is the purpose of this paper to contribute to the clarifi-cation of these questions of validity of the kinetic description by studying a simple model of a two-level system or spin coupled to a complex environment described by random ma-trices. Indeed, work done during the last decade has shown that the Hamiltonian of typical quantum systems presents properties of random matrices on their small energy scales. Here, we consider the Hamiltonian of the environment as well as the operator of coupling between the spin and the environment to be given by random matrices taken in a sta-tistical ensemble of Gaussian orthogonal random matrices. This defines a model in which many results can be obtained analytically.

Similar models using Gaussian orthogonal random matri-ces关1,2兴 or using banded random matrices 关3–7兴 have been studied. In关5兴, it has been shown that random matrices used as environment coupling operators have a universal feature.

The model we here consider differs from the spin-boson model by the density of states of the environment. Instead of monotonously increasing with energy as in the spin-boson model, the density of states of the environment obeys Wigner’s semicircular law in our model and is thus limited to an interval of energy with a maximum density in between. We notice that such densities of states appear in systems where a spin is coupled to other 共possibly dissimilar兲 spins

as, for instance, in NMR, in which case the density of states of the other spins forming the environment also present a maximum instead of a monotonous increase with energy. Our model may therefore constitute a simplification of such kinds of interacting spin systems. Our main purpose is to understand the conditions under which a kinetic description can be used in order to understand the relaxation of the spin under the effect of the coupling with the rest of the system, which we refer to as a complex environment.

The plan of the paper is the following. The model is pre-sented in Sec. II. The properties of the spectrum are de-scribed in Sec. III. The relaxation in the time evolution of the spin is studied in Sec. IV. The very long time behavior and the approach to the equilibrium is discussed in Sec. V. Con-clusions are drawn in Sec. VI.

II. THE MODEL

We are interested in the study of a total system composed of a simple system 共with a few discrete levels兲 interacting with a complex environment 共with many levels兲. We con-sider a two-level system as a prototype for the simple sys-tem.

For this kind of total system, the time-dependent Schro¨-dinger equation is of the following type:

iបd兩⌿共 t˜兲

典

d t˜ ⫽H ˜ˆ tot兩⌿共 t˜兲

典

⫽

冉

⌬˜ 2␴ˆz⫹H˜ˆB⫹␭˜␴ˆxB˜ˆ

冊

兩⌿共 t˜兲

典

, 共1兲 where␴ˆ_x, ␴ˆ_y, and␴ˆ_z are the 2⫻2 Pauli matrices, (⌬˜/2)␴ˆ_z is the Hamiltonian of the two-level system, ⌬˜ is the energy spacing between the two levels of the system, H˜ˆB is the Hamiltonian of the environment,␴ˆxis the coupling operator of the system, B˜ˆ is the coupling operator of the environment, and␭˜ is the coupling parameter between the system and the environment.

harmonic oscillator lattice and B˜ˆ is linear in the degree of freedom of the environment. Here, we want to define a new model, the spin-GORM model, also described by the Hamil-tonian 共1兲 and for which H˜ˆB and B˜ˆ are Gaussian orthogonal random matrices 共GORM兲 共see Appendix A for some basic property on GORM兲.

Let us discuss now the spin-GORM model in more detail. As we said, we want to model a two-level system that inter-acts with an environment that has a complex dynamics. Here, complex is used in a generic way. The complexity can come, for example, from the fact that the corresponding classical system is chaotic like in a quantum billiard or for the hydro-gen atom in a strong magnetic field关11,12兴. It can also come from large coupling in an interacting many-body system as in nuclear physics 关11兴 or in interacting fermion systems such as quantum computers关11,13兴. Wigner in 1960 关14–16兴 was the first to develop random-matrix theory for the pur-pose of modeling spectral fluctuations of complex quantum systems containing many states interacting with each other. This tool has now become very common in many fields from nuclear physics to quantum chaos. This is the reason why we consider random matrices to characterize the complexity of the environment operators.

The environment operators of the spin-GORM model, H˜ˆB and B˜ˆ , are defined by

H ˜ˆ_B⫽␴ ND H ˜ˆ B_Xˆ B ˜ˆ_⫽␴ ND B ˜ˆ Xˆ

⬘

, 共2兲

where Xˆ and Xˆ

⬘

are two different (N/2)⫻(N/2) Gaussian orthogonal random matrices with mean zero. Their nondi-agonal共diagonal兲 elements have standard deviation ␴_NDX˜ˆ ⫽1 (␴_DX˜ˆ⫽

冑

2). Xˆ and Xˆ

⬘

are two different realizations of the same random matrix ensemble and have therefore the same statistical properties. ␴_NDH˜ˆB _and ␴

ND B ˜ˆ

are the standard devia-tions of the nondiagonal elements of H˜ˆBand B˜ˆ , respectively. For these random matrices, the width of their averaged smoothed density of state is given by

DH˜ B⫽␴ND H˜ˆ_B

冑

8N, DB˜⫽␴ND B ˜ˆ

冑

8N 共3兲

共see Appendix A兲.

It is interesting to define the model in such a way that, when N is increased, the averaged smoothed density of state of the environment increases without changing its width

DH˜

B. The width can be fixed to unity. This is equivalent to fixing the characteristic time scale of the environment. For doing this, it is necessary to rescale the parameters as fol-lows: ␣⫽␴ND H˜ˆ_B

冑

8N, t⫽␣t˜, ⌬⫽ ⌬˜ ␴ND H ˜ˆ B

冑

_8N , ␭⫽␭˜␴ND B ˜ˆ ␴ND H˜ˆ_B , N⫽N.

The time-dependent Schro¨dinger equation of the spin-GORM model becomes

iបd兩⌿共t兲

典

dt ⫽Hˆtot兩⌿共t兲

典

, 共4兲 with the rescaled total Hamiltonian

Hˆtot⫽HˆS⫹HˆB⫹␭␴ˆxBˆ⫽ ⌬ 2␴ˆz⫹ 1

冑

8NX ˆ_⫹␭␴ˆx 1

冑

8NX ˆ

_⬘

_. 共5兲 As announced, we have nowDHB⫽DB⫽1.

In the following, without loss of generality,␣will always be taken equal to unity. Notice that, to model an environment with a quasicontinuous spectrum, the random matrix must be very large (N→⬁).

In order to get ensemble averaged results, one has to per-form averages over the different results obtained for each realization of Eq.共5兲. When we use finite ensemble averages, the number of members of the ensemble average will be denoted by ␹.

We see that the Hamiltonian共5兲 is characterized by three different parameters:⌬, ␭, and N. We define three different parameter domains in the reduced parameter space corre-sponding to a fixed N in order to facilitate the following discussion. These three regimes are represented in Fig. 1: domain A with 1⬎␭,⌬; domain B with ⌬⬎1,␭; domain C with␭⬎1,⌬.

III. THE SPECTRUM

In this section we study the spectrum of the complete system for the different values of the parameters. This study is important in order to understand the different dynamical behaviors that we encounter in the model.

Let us begin defining the notations in the simple case where there is no coupling between the two parts of the total system共␭→0兲. The isolated system has two levels separated by the energy⌬:

HˆS兩s

典

⫽s ⌬

2兩s

典

, 共6兲

HˆB兩b

典

⫽Eb

B_兩b

_典

_{, 共7兲}

where b⫽1,2, . . . ,N/2. The Hamiltonian of the total system without interaction between the system and the environment is thus

Hˆ0⫽HˆS⫹HˆB, 共8兲 and the spectrum is therefore given by

Hˆ0兩n

典

⫽En 0_兩n

_典

, 共9兲 with n⫽1,2, . . . ,N and En 0_⫽s⌬ 2 ⫹Eb B . 共10兲

The eigenvectors are tensorial products of both the system and environment eigenvectors:

兩n

典

⫽兩s

典

丢兩b

典

. 共11兲 Let us now define the notations in the opposite simple situation where the coupling term is so large that the Hamil-tonian of the system and of the environment can both be neglected共␭→⬁兲. Using the unitary matrix Uˆ acting only on the system degrees of freedom

Uˆ⫽

冋

1

冑

2 1

冑

2 1

冑

2 ⫺ 1

冑

2

册

,

the total Hamiltonian becomes Hˆ ˜

0⫽␭␴ˆzBˆ . 共12兲

E_␬␩ and兩␬␩典⫽兩␬典丢兩␩典 are, respectively, the eigenvalues and eigenvectors of the Hamiltonian

H˜ˆ0兩␬␩

典

⫽␭E␬␩兩␬␩

典

⫽␭␬E␩兩␬␩

典

, 共13兲 where ␩⫽1, . . . ,N/2 and ␬⫽⫾1. After having defined the notation in the two extreme cases ␭→0 and ␭→⬁, we will start the study of the spectrum with interaction␭⫽0.

The total spectrum is given by the eigenvalues 兵E_␣其 which are solutions of the eigenvalue problem

Hˆtot兩␣

典

⫽E␣兩␣典, 共14兲 where␣⫽1,2, . . . ,N. It is very difficult to obtain analytical results for this problem. We will therefore study the total spectra using a method of numerical diagonalization of the total Hamiltonian.

A. Smoothed density of states

In order to have a quantitative understanding of the global aspect of the spectrum共on large energy scales兲, we will study the total perturbed averaged smoothed density of states.

The environment-averaged smoothed density of states obeys the semicircular Wigner law 关see Eq. 共A4兲 in Appen-dix A兴 nw_共⑀兲⫽

再

4N ␲

冑

冉

1 2

冊

2 ⫺⑀2 _{if兩⑀兩⬍}1 2, 0 if兩⑀兩⭓1 2, 共15兲

where⑀is the continuous variable corresponding to the en-vironment energy Eb

B .

Therefore, when ␭⫽0, the total averaged smoothed den-sity of states is the sum of the two environment semicircular densities of states which correspond to both states of the two-level system 关see Eq. 共9兲兴:

n共␧兲⫽nw

冉

␧⫺⌬ 2

冊

⫹n w

冉

_␧⫹⌬ 2

冊

⫽

冦

4N ␲

冑

冉

1 2

冊

2 ⫺

冉

␧⫺⌬ 2

冊

2 ⫹4N ␲

冑

冉

1 2

冊

2 ⫺

冉

␧⫹⌬ 2

冊

2 if

冉

1 2⫺ ⌬ 2

冊

⬍兩␧兩⬍

冉

1 2⫹ ⌬ 2

冊

, 0 elsewhere, 共16兲

where␧ is the continuous variable corresponding to the total energy E_n0. The semicircular densities nw关␧⫺(⌬/2)兴 and nw_{关␧⫹(⌬/2)兴 are schematically depicted in Fig. 2 for} differ-ent values of ⌬. The numerical density of states of the total system corresponding to␭⫽0 is depicted in Fig. 3.

When ␭→⬁ 共meaning that the coupling term becomes dominant in the Hamiltonian兲, the averaged smoothed den-sity of states of the total system 关see Eq. 共13兲兴 is given by

n共␧兲⫽nw

冉

␧ ␭

冊

⫹n w

冉

_⫺␧ ␭

冊

⫽

冦

8N ␭␲

冑

冉

␭ 2

冊

2 ⫺共␧兲2 _{if 兩␧兩⬍}␭ 2, 0 if 兩␧兩⭓␭ 2, 共17兲

where␧ is the continuous variable corresponding to the total energy E_␬␩. This result can be observed in Fig. 3共a兲 for ␭⫽10 共because ⌬,1Ⰶ␭兲.

When␭⫽0, the total averaged smoothed density of states is also plotted in Fig. 3. The main observation is that there is a broadening of the complete spectrum when one increases ␭. In Fig. 3共a兲 we see that the averaged smoothed density of states changes in a smooth way from共16兲 to 共17兲. But in Fig. 3共b兲 and much more in Figs. 3共c兲 and 3共d兲, the two semicir-cular densities nw_{关␧⫺(⌬/2)兴 and n}w_{关␧⫹(⌬/2)兴 seem to} re-pel each other as␭ increases. This is due to the fact that the levels of a given semicircular density do not interact with each other but only interact with the levels of the other semi-circular density. This is a consequence of the nondiagonal form of the coupling. Therefore, having in mind the pertur-bative expression of the energies关see Eq. 共C6兲 of Appendix C兴, one understands that when ⌬ is nonzero, the eigenvalues that are repelling each other with the most efficiency are the ones closest to the center of the total spectrum.

B. Eigenvalue diagrams

The global effect of the increase of ␭ on the eigenvalues has been studied with the average smoothed density of states. But in order to have an idea of what happens on a finer

FIG. 2. Smoothed densities of states nw关␧⫺(⌬/2)兴 and nw关␧

⫹(⌬/2)兴 for different values of ⌬. The total smoothed averaged

density of states of the nonperturbed spectrum is obtained by the sum of them关see Eq. 共16兲兴.

FIG. 3. Total smoothed aver-aged density of states obtained nu-merically for different values of⌬ and␭. 共a兲 corresponds to ⌬⫽0.01,

共b兲 to ⌬⫽0.5, and 共c兲 and 共d兲 to ⌬⫽5. For all of them N⫽500 and ␹⫽50. Notice that the ␭⫽0 and

energy scale inside the spectrum, it is interesting to individu-ally follow each eigenvalue E_␣ as a function of ␭ on an eigenvalue diagram.

The first thing to note共see Fig. 4兲 is that the increase of the coupling induces a repulsion between the eigenvalues. This has as a result the broadening of the total spectrum as we already noticed on the smoothed averaged density of states. If one looks closer inside the fine structure of the eigenvalue spectrum, we see that there is no crossing be-tween the eigenvalues. This is a consequence of the fact that there is no symmetry in the total system. Therefore, the non-crossing rule is always working. Each time two eigenvalues come close to each other, they repel each other and create an avoided crossing. One can notice that there is a large number of avoided crossings inside the region where the two semi-circular densities nw关␧⫺(⌬/2)兴 and nw关␧⫹(⌬/2)兴 overlap 关see Fig. 2 in order to visualize the overlapping zone that extends from共⌬/2兲⫺1 to 1⫺共⌬/2兲兴. But outside this overlap-ping zone, there are very weak avoided crossings 共and of course no crossing兲 and all the eigenvalues appear to move in a regular and smooth way. The regions seen in Figs. 4共a兲, 4共b兲, and 4共d兲 are inside the overlapping zone and the one in Fig. 4共e兲 is outside. In Fig. 4共c兲 the lower part of the diagram is inside the overlapping zone and the upper part is outside.

This phenomenon is due to the fact that the eigenvalues of a given semicircular density do not interact with the eigenval-ues of their own semicircular density but only with those of the other semicircular density. It is the consequence of the nondiagonal nature of the coupling in the spin degrees of freedom. When the coupling becomes large enough 共␭⬎1兲, the avoided crossings also disappear inside the overlapping zone. Around␭⫽1 and inside the overlapping zone, there is a smooth transition from an avoided-crossing regime that gives rise to a turbulent and complex␭ evolution to another regime without much interaction between the eigenvalues that gives rise to a smooth␭ evolution.

C. Spacing distribution

The eigenvalue diagrams only give us a qualitative under-standing of the fine energy structure of the spectrum. For a more quantitative study, it is interesting to look at the spac-ing distribution of the spectrum.

When␭⫽0, each semicircular distribution, corresponding to a different system level, has a Wignerian level spacing distribution,

Pw_共s兲⫽␲ 2se

⫺(␲/4)s2

. 共18兲

FIG. 4. Different parts of the eigenvalue diagrams with N

⫽500, corresponding to 共a兲 and 共b兲 ⌬⫽0.01, 共c兲 and 共d兲 ⌬⫽0.5,

But the total spectrum is made by the superposition, with a shift ⌬, of two of such semicircular distributions. The shift creates a Poissonian component to the total spacing distribu-tion in the overlapping zone关共⌬/2兲⫺1 to ⫺共⌬/2兲⫹1兴. A Pois-sonian distribution is given by

Pp共s兲⫽e⫺s. 共19兲

Therefore, we choose to fit the total spacing distribution by the mixture

Pf it共s兲⫽C1Pw共s兲⫹共1⫺C1兲Pp共s兲. 共20兲 This choice of the form of the fit is empirical but reasonable because the correlation coefficient of the fit is always close to one共between 0.963 and 0.982兲.

We computed the spacing distribution and made the fit 共20兲 in order to compute the mixing coefficient C1 for dif-ferent values of ␭ in the case where the overlapping zone covers almost the whole spectrum 共⌬Ⰶ1兲. The results are plotted in Fig. 5. We see that there is a specific region of␭ values where the total spacing distribution is close to a pure Wignerian one. This region corresponds to the situation where ␭2_{N⫽O(1), i.e., when the typical intensity of the}

interaction between the nonperturbed levels is O(␭2) 共since the first nonzero correction in perturbation theory is of sec-ond order due to the nsec-ondiagonal coupling in our model兲, becomes of the order of the mean level spacing O(1/N) in the total system. In this region the level repulsion is maximal and effective among almost all the states.

In the limit␭→⬁ the spectrum is again the superposition of two semicircular distributions, one for the E1␩’s and one for the E_⫺1␩’s according to Eq.共13兲. As a consequence, one gets a Poissonian type of spacing distribution.

The other⌬ cases will have a Wignerian spacing distribu-tion outside the overlapping zone and a mixed one 共like for the case ⌬Ⰶ1兲 inside. This can be visually seen on the ei-genvalue diagram that we studied before.

D. The shape of the eigenstates„SOE…

We want now to have some information about the eigen-states of the total system inside the overlapping zone of the two semicircular densities. Following 关6兴, we define the quantity

␰共␧,␧0_兲⫽

兺

␣,n 円

具␣

兩n

典

円

2_␦共E

␣⫺␧兲␦共En⫺␧0兲. 共21兲 If we fix␧0 and study␰(␧,␧0) as a function of␧, we will call it the local density of states 共LDOS兲. If we fix ␧ and study ␰(␧,␧0) as a function of␧0, we will call it the SOE 共shape of the eigenstates兲.

We here focus on the SOE. The SOE tells us how close a perturbed eigenstate 共␭⫽0兲 at energy ␧ is from the nonper-turbed eigenstate 共␭⫽0兲 at energy ␧0. If the SOE is a very narrow function centered around ␧⫽␧0_{, the concept of a} nonperturbed eigenstate is still useful. This regime corre-sponds to very small coupling, for which the interaction in-tensity is lower than the mean level spacing 0⬍␭2Nⱗ1, and will be called to the localized regime. In the limit N→⬁ this regime disappears. If one increases the coupling, the interac-tion intensity between the nonperturbed states begins to be larger than the mean level spacing between the states: ␭2N ⬎1. The nonperturbed levels start to be ‘‘mixed’’ by the interaction and the SOE starts then to have a Lorentzian shape with a finite width ⌫, centered around ␧⫽␧0. This regime is called the Lorentzian regime. If one further in-creases the coupling parameter, the SOE begins to spread over almost the whole spectrum. This regime is called the delocalized regime.

In the banded random matrix model of 关6兴, the regimes are classified according to a different terminology and there is an additional regime corresponding to a spreading that goes beyond the energy range where the coupling acts 共due

FIG. 5. 共a兲 Spacing distribution for different values of the cou-pling parameter␭. 共b兲 Fitted coefficient C1of Eq.共20兲. The closer

is C1from unity, the closer is the spacing distribution to the Wigner

spacing distribution. In the two figures, ⌬⫽0.01, N⫽500, and

␹⫽50.

to the finite coupling range of the banded matrices兲. The motivation for our change of terminology will become clear in the study of the dynamics.

The Lorentzian regime can be separated into two parts. For small coupling, the width of the Lorentzian⌫ is smaller or of the order of magnitude of the typical energy scale of variation of the averaged smoothed density of state of the environment ␦⑀: n(⑀⫹␦⑀)⬇n(⑀). For larger coupling, the Lorentzian width extends on an energy scale larger than the typical energy scale of variation of the density of states of the environment. Therefore, one has⌫ⱗ␦⑀in the former case and⌫⬎␦⑀in the latter case. The different regimes are repre-sented in Figs. 6 and 7.

To represent the SOE of the spin-GORM model, we dis-cretize the energy axis in small cells of the order of the mean level spacing 1/N and we average the SOE over ␹ realiza-tions of the random matrix ensemble. We see in Fig. 8共a兲 the

typical shape of the SOE going from the perturbative regime 共␭⫽0.01,0.05兲 to the beginning of the Lorentzian one 共␭ ⫽0.1兲. In Fig. 8共b兲 we see the SOE across the Lorentzian regime 共0.2⭐␭⭐0.8兲. We also see the delocalized regime, when␭→⬁ and the SOE gets completely flat 共␭⫽10兲. Figure 8共c兲 shows the width of the Lorentzian from a fit made on the SOE curve. The correlation coefficient of the fit helps us to determine the region of the Lorentzian regime where the SOE is very well fitted by a Lorentzian. It has been verified that the width of the Lorentzian is independent of N in the Lorentzian regime. Figure 8共d兲 共log-log兲 shows that the width of the Lorentzian has a power-law dependence in the coupling parameter close to two in the Lorentzian regime.

E. Asymptotic transition probability kernel„ATPK… An interesting quantity, which is close to the SOE, but which has a nice physical interpretation, is the asymptotic transition probability kernel共ATPK兲. The transition probabil-ity kernel共TPK兲 gives the probability at time t to be in the level兩n

典

if starting from␳ˆ (0). It is defined as

⌸t„n円␳共0兲…⫽

具

n兩e⫺iHˆtot

t_{␳ˆ共0兲e}iHˆ_tott_兩n

_典

_{. 共22兲} The ATPK is the time average of the TPK

⌸⬁„n円␳共0兲…⫽ lim T→⬁ 1 T

冕

0 T dt⌸t„n兩␳共0兲… ⫽

兺

␣ 円

具␣

兩n

典

円 2

_具␣

_{兩␳ˆ共0兲兩␣}

_典

_{. 共23兲} The distribution of the ATPK in energy is given by

FIG. 7. Schematic representation of the different regimes in the plane of the reduced parameter ␭ and N. The Lorentzian 1 regime corresponds to⌫ⱗ␦⑀ and the Lorentzian 2 regime to ⌫⬎␦⑀.

FIG. 8. 共a兲 SOE in the pertur-bative regime. 共b兲 SOE in the Lorentzian regime.共c兲 The width and the correlation coefficient of the fit of the SOE by a Lorentzian.

共d兲 Power-law dependence of the

⌸_⬁„␧0_{円␳共0兲…⫽}

兺

n ⌸⬁„n円␳共0兲…␦共En⫺␧

0_{兲. 共24兲} The ATPK has an intuitive physical interpretation. It repre-sents the probability after a very long time to end up in a nonperturbed state兩n

典

, having started from the initial condi-tion ␳ˆ (0). The ATPK is the convolution of the SOE

⌸⬁共␧0兩␧0⬘兲⫽

兺

n,m

兺

␣ 円

具␣

兩n

典

円

2_円

_具␣

_兩m

_典

_円2

⫻␦共En⫺␧0兲␦共Em⫺␧0⬘兲, 共25兲 with␳(0)⫽兺m兩m

典具

m兩␦(Em⫺␧0⬘).

Because we are interested in a random matrix model and for the purpose of studying the dynamics, we will perform averages of two different kinds. The first kind of average is a microcanonical average over states belonging to the same given energy shell of width␦␧ for a given realization of the random matrix ensemble used in our total Hamiltonian. The second kind of average is an ensemble average over the ␹ different realizations of the random matrices ensemble. For the microcanonical average, a choice of the width of the energy shell␦␧ has to be done in such a way that it is large enough to contains many levels共to get a good statistics兲 and small enough to be smaller than or equal to the typical en-ergy scale of variation of the averaged smoothed density of states of the environment␦⑀. Therefore, the adequate choice of the width of the energy shell corresponds to 1/N⬍␦␧ ⬍␦⑀.

Different ATPK are depicted in Fig. 9 where there is no random matrix ensemble average ␹⫽1. In Fig. 9共a兲 the ATPK is a microcanonical average inside the energy shell at energy␧0⬘and of width␦␧0⬘. ‘‘Pin’’ denotes the total prob-ability of staying inside the energy shell after a very long time. N␭2⫽1 is on the border between the localized and the Lorentzian regime and␭⫽1,10 in the delocalized regime. We see that, until N␭2⫽1, the main probability stays in the ini-tial energy shell. But when the Lorentzian regime starts, the probability spreads over energies larger than ␦⑀ 共here ␦⑀ ⬇␦␧0⬘_{). In Figs. 9共b兲 and 9共c兲 no average has been done.} The initial condition is a nonperturbed pure state 共corre-sponding to an energy close to zero兲 and one can see the individual probability of being on another nonperturbed state for the different regimes. We can see that in the localized regime the probability of staying on the initial state is much more important than the probability of leaving it. We also see that the Lorentzian regime starts when the initial state loses its privileged position containing the main probability and, therefore, when the neighboring levels begin to have an im-portant fraction of the total probability.

IV. TIME EVOLUTION

We now want to understand the time evolution of our model and more specifically the population dynamics of the system. The exact evolution of the total system is described by the von Neumann equation

␳ˆ˙共t兲⫽⫺i关Hˆtot,␳ˆ共t兲兴, 共26兲 where Hˆtotis given by Eq. 共5兲. The system dynamics is ob-tained from Eq. 共26兲 using the reduced density matrix

␳ˆS(t)⫽TrB␳ˆ (t). The total system has a finite and constant energy. At initial time, the environment has a given fixed energy ⑀corresponding to a microcanonical distribution in-side a energy shell centered at ⑀ and of width ␦⑀. ␦␧ is chosen in such a way that it is large enough to contains many levels 共to get a good statistics兲 and small enough to be smaller than or equal to the typical energy scale of variation of the averaged smoothed density of state of the environment

␦⑀. Therefore, the adequate choice for the width of the en-ergy shell corresponds to 1/N⬍␦␧⬍␦⑀. The dynamics is then averaged over the ␹ realizations of the random matrix ensemble. In the following, we consider in detail the two different extreme cases of weak and strong couplings.

A. The weak coupling regime„␭™1…

We derived in Ref. 关17兴 a perturbative equation for the description of the evolution of a system 共with a discrete spectrum兲 weakly interacting with its environment 共with a quasicontinuous spectrum兲. This equation has been shown to be equivalent to the well-known Redfield equation关10,18,19兴 when the typical energy scale of the system 共typical energy spacing between the system levels兲 can be considered small compared to the typical environment energy scale␦⑀共typical energy scales on which the smoothed density of states of the environment varies兲. In the spin-GORM model, this condi-tion means that n(⑀⫹⌬)⬇n(⑀).

We will not perform in this paper the detailed derivation of this equation and its application to the spin-GOE model. This has been done in Ref. 关17兴. We will simply recall the main ideas and results of this paper and apply them to the study of the dynamics of our model.

The main idea is to suppose that the total density matrix can be described at all time by a density matrix of the fol-lowing form:

␳ˆ共t兲⫽ 1 n共HˆB兲

兺

s,s⬘

兩s

典具

s

⬘

兩P_ss⬘共Hˆ_B;t兲, 共27兲 where the environment density of states is

n共⑀兲⫽TrB␦共⑀⫺HˆB兲. 共28兲 Doing this, we neglect the contribution to the dynamics com-ing from the coherence of the environment but we keep those of the system. Therefore, the total density matrix is diagonal in the environment degrees of freedom but not in the system ones. For comparison, let us recall that in the derivation of the well known Pauli equation, both types of coherences are neglected and the total density matrix is completely diagonal 关20–22兴.

The reduced density matrix of the system becomes ␳ˆS共t兲⫽TrB␳ˆ共t兲 ⫽

冕

d⑀TrB␦共⑀⫺HˆB兲␳ˆ共t兲 ⫽

兺

s,s⬘ 兩s

典具

s

⬘

兩

冕

d⑀P_ss⬘共⑀;t兲. 共30兲

We see that each element of the reduced density matrix of the system depends on the environment energy. This is fun-damental in order to take into account the finite energy ef-fects of the total system.

Using Eq. 共27兲 and performing a perturbative expansion up to the second order in␭ on Eq. 共26兲 共see Ref. 关17兴兲, one gets for the population dynamics the equation

P˙ss共⑀;t兲⫽⫺2␭2

兺

s ¯,s¯⬘

冋

⫹

具

s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s¯

典

P¯ss共⑀;t兲 ⫻

冕

d⑀⬘F共⑀,⑀⬘兲n共⑀⬘兲sin共E¯s⫺E¯s⬘⫹⑀⫺⑀⬘兲␶

E¯s⫺E¯s⬘⫹⑀⫺⑀⬘ ⫺

具

s兩Sˆ兩s¯

典具

¯s

⬘

兩Sˆ兩s

典

n共⑀兲

冕

d⑀⬘F共⑀,⑀⬘兲 ⫻P¯ss¯⬘共⑀⬘;t兲

sin共E_s⫺E_¯s⬘⫹⑀⫺⑀⬘兲␶ Es⫺E¯s⬘⫹⑀⫺⑀⬘

册

, 共31兲

where F(⑀,⑀⬘)⫽‘‘円

具⑀

兩Bˆ兩⑀⬘典円2,’’ where the quotes denote a smoothening over a dense spectrum of eigenvalues around⑀ and⑀⬘. We see that the probability P˙¯ss¯(⑀;t), if initially con-centrated in a given energy shell, can spread in energy under the dynamics. This is a typical non-Markovian effect happen-ing on a short-time scale. It is due to the presence in the equation of the energy integral and of the sin共␰␶兲/␰function that has a finite width in energy at short time. This equation is non-Markovian in the sense that the coefficients of the differential evolution equation are time dependent and that this time dependence can be neglected on long-time scales 共performing the Markovian approximation兲 when the envi-ronment has a faster dynamics共i.e., typically a larger energy scale兲 than the system.

The Markovian approximation consists of taking the infinite-time limit of the time-dependent coefficients using the property lim_␶→⬁关sin(␰␶)/␰兴⫽␲␦(␰). This approximation is justified if the contribution of the ␰⫽0 value 共if it exists兲 has the main and almost unique contribution to the energy integral. If one further neglects the contributions of the co-herence to the population evolution共this is automatically sat-isfied for the spin-GORM model because of the nondiagonal coupling兲, one gets a Pauli-type equation 关20–22兴

FIG. 9.共a兲 ATPK from the localized regime, through the Lorent-zian regime, to the delocalized regime. The ATPK is microcanoni-cally averaged over the different initial conditions correspond-ing to the levels inside the energy shell centered at ␧0⬘_⫽0

with width ␦␧0⬘_{⫽0.05. ‘‘Pin’’ denotes the proportion of the}

ATPK that stays inside the initial energy shell after an infinite time.

共b兲 and 共c兲 ATPK for a single level as an initial condition, without

P˙_ss共⑀;t兲⫽⫺2␲␭2

兺

s⬘⫽s 円

具

s兩Sˆ兩s

⬘典

円2_F共⑀,E s⫺Es⬘⫹⑀兲 ⫻n共Es⫺Es⬘⫹⑀兲Pss共⑀;t兲 ⫹2␲␭2

兺

s⬘_⫽s 円

具

s兩Sˆ兩s

⬘典

円2F共⑀,Es⫺Es⬘⫹⑀兲 ⫻n共⑀兲Ps⬘s⬘共Es⫺Es⬘⫹⑀;t兲. 共32兲 We see that the Markovian approximation strictly keeps the dynamics of the total system inside an energy shell. Starting with the probability located on a given energy shell, the dy-namics will preserve the probability inside this shell. But of course, the probability of the different states inside the shell are varying. The dynamics described by this equation can be seen as a random walk between nonperturbed states of the total system belonging to the same energy shell with transi-tion rates between these states given by the Fermi golden rule.

We now apply our equation to the spin-GORM model in order to study of the population evolution through ␴ˆ_z 共the difference between the probability of being in the upper state of the system minus the probability of being in the lower one兲. Doing this, we suppose that the environment is quasi-continuous N→⬁ and that the random matrix ensemble av-erage has been performed ␹→⬁. For the non-Markovian equation共31兲, one gets

具␴

ˆz

典

N M共t兲⫽

冕

d⑀⬘关P⫹⫹共⑀⬘;t兲⫺P⫺⫺共⑀⬘⫹⌬;t兲兴, 共33兲 where P˙_⫾⫾共⑀;t兲⫽␭ 2 ␲

冕

⫺1/2 ⫹1/2 d⑀⬘sin共⫾⌬⫹⑀⫺⑀⬘兲t 共⫾⌬⫹⑀⫺⑀⬘兲 ⫻

冋

P_⫿⫿共⑀⬘;t兲

冑

1 4⫺⑀ 2 ⫺P⫾⫾共⑀;t兲

冑

1 4⫺⑀⬘ 2

册

_{. 共34兲} In the Markovian limit, Eq.共33兲 becomes

具

␴ˆ_z

典

M共t兲⫽P_⫹⫹共⑀;t兲⫺P_⫺⫺共⑀⫹⌬;t兲, 共35兲 which obeys a Pauli-type equation so that we get

具␴

ˆz

典

M共t兲⫽

具␴

ˆz

典

_⬁M⫹关

具␴

ˆz

典

M共0兲⫺

具␴

ˆz

典

_⬁M兴e⫺␥t, 共36兲 where the equilibrium value of the populations is given by

具␴

ˆz

典

_⬁ M_⫽

冑

1 4⫺共⑀兲 2_⫺

冑

1 4⫺共⑀⫹⌬兲 2

冑

1 4⫺共⑀兲 2_⫹

冑

1 4⫺共⑀⫹⌬兲 2 共37兲

and the relaxation rate by

␥⫽␭2

冉

冑

1 4⫺共⑀兲

2_⫹

冑

1

4⫺共⑀⫹⌬兲

2

冊

_{, 共38兲} where we adopted the convection that

冑

x⫽0 if x⬍0.

In the case of the spin-GORM model, the transition prob-ability between states belonging to the same total energy shell only depends on the smoothed density of states. This is due to the fact that the environment coupling elements be-tween the environment nonperturbed states are randomly dis-tributed because the environment coupling operator is a ran-dom matrix. This has the consequence that the equilibrium value of the populations共37兲 corresponds to a microcanoni-cal distribution probability of the states belonging to the total energy shell independently of the initial distribution of these states inside this energy shell.

We notice that, in the general case, the Markovian as-sumption made on our equation does not directly give a Pauli-type equation which is an equation for the populations only. To get a Pauli equation, the further approximation, which consists of neglecting the contribution of the coher-ences to the population dynamics, has to be done. For the particular case of the spin-GORM model, this further ap-proximation is not necessary because it is automatically sat-isfied.

B. The strong coupling regime„␭š1…

We are now interested in describing the dynamical regime where the coupling parameter␭ is very large and, therefore, dominant in front of 1 and⌬. We will again suppose that the environment is continuous N→⬁ and that the random matrix ensemble average has been performed␹→⬁.

The population dynamics of the system is given by

具

␴ˆz

典

共t兲⫽Tr␳ˆ共t兲␴ˆz⫽Tr ei[(⌬/2)␴ˆz⫹HˆB⫹␭␴ˆxBˆ ]t␳ˆ共0兲 ⫻e⫺i[(⌬/2)␴ˆz⫹HˆB⫹␭␴ˆxBˆ ]t␴_ˆ

z. 共39兲

Using the following perturbative expansion of the evolution operator to order zero in 1/␭:

e⫺i␭[(1/␭)(⌬/2)␴ˆx⫹(1/␭)HˆB⫹␴ˆzBˆ ]t ⫽ 1/␭→0 e⫺i␭t␴ˆzBˆ⫹O

冉

1 ␭

冊

, 共42兲 we get

具␴

ˆz

典

共t兲 ⫽ 1/␭→0 Tr ei␭t␴ˆzBˆ_Uˆ†␳_ˆ

S共0兲Uˆ␳ˆB共0兲e⫺i␭t␴ˆzB ˆ

⫹O

冉

1_␭

冊

. 共43兲 Using the following notation:

␴ˆzBˆ兩␬␩典⫽␴ˆz兩␬典丢Bˆ兩␩典⫽␬E␩兩␬␩

典

共44兲 and HˆB兩b

典

⫽Eb B_兩b

_典

, 共45兲 we find that

具␴

ˆz

典

共t兲 ⫽ 1/␭→0

兺

_␬,␩ ei␭t␬E␩

具␩

_{兩␳ˆB共0兲兩}␩

典具␬

_兩Uˆ†_␳ˆ S共0兲Uˆ兩⫺␬

典

⫻ei␭t␬E_␩⫹O

冉

1

␭

冊

. 共46兲

Because␳ˆBis diagonal in the basis that diagonalizes HˆB, we have

具␩

兩␳ˆB共0兲兩␩典⫽

兺

b 円

具␩

兩b

典

円

2

_具

_b兩␳ˆ

B兩b

典

. 共47兲 If we perform an ensemble average over different realiza-tions of HˆBand use the random matrix eigenvectors statistics 关14–16兴, we find

具␩

兩␳ˆ_B共0兲兩␩典(HˆB)⫽

兺

b 円

具␩

兩b

典

円 2(Hˆ_B)

_具

_b兩␳ˆ B兩b

典

(Hˆ_B) ⫽

兺

b 2 N

具

b兩␳ˆB兩b

典

(Hˆ_B) ⫽_N2. 共48兲

Equation共46兲 therefore becomes

具␴

ˆz

典

共t兲(H ˆ B) ⫽ 1/␭→0₂ N

兺

_␬,␩ e 2i␭t␬E_␩

_具␬

_兩Uˆ†_␳ˆ S共0兲Uˆ兩⫺␬典 ⫽ 2 N

冋

具

1兩Uˆ †_␳ˆ S共0兲Uˆ兩⫺1

典

兺

␩ e 2i␭tE_␩ ⫹

具

⫺1兩Uˆ†_␳ˆ S共0兲Uˆ兩1

典

兺

␩ e ⫺2i␭tE␩

册

⫹O

冉

1 ␭

冊

. 共49兲 Performing now the following ensemble average over differ-ent realizations of Bˆ :

兺

_␩ e2i␭tE␩(Bˆ )_⫽

冕

⫺1/2 ⫹1/2 d⑀n共⑀兲e2i␭t⑀ ⫽4N_␲

冕

⫺1/2 ⫹1/2 d⑀

冑

1 4⫺⑀ 2_e2i␭t⑀ ⫽NJ1共␭t兲 ␭t , 共50兲 we finally get

具␴

ˆz

典

共t兲(H ˆ B,Bˆ ) ⫽ 1/␭→0 2J1共␭t兲 ␭t 关

具

1兩Uˆ†␳ˆS共0兲Uˆ兩⫺1

典

⫹

具

⫺1兩Uˆ†_␳ˆ S共0兲Uˆ兩1

典

兴⫹O

冉

1 ␭2

冊

. 共51兲 It is easy to show that the term of order 1/␭ is zero. This explains the fact that O(1/␭) has been replaced by O(1/␭2).

Choosing as an initial condition

␳ˆ_S共0兲⫽

冉

1 0 0 0

冊

, 共52兲 we get

具␴

ˆz

典

共t兲(H ˆ B,Bˆ ) ⫽ 1/_␭→0 2J1共␭t兲 ␭t ⫹O

冉

1 ␭2

冊

. 共53兲 We have found a well-defined behavior of the system dy-namics when the coupling parameter ␭ is so large that the coupling term can be considered to contribute alone to the dynamics.

C. Numerical results

We will now numerically study the validity of the ap-proximated equation that we just derived in order to under-stand the dynamical evolution of our model.

In our numerical simulations, the initial condition of the system is always the upper state 共52兲:

具␴

ˆz

典

(0)⫽1. The dif-ferent parameter domains represented in Fig. 1 will play a fundamental role in our discussion of the dynamics.

equation and the results of the Markovian version共36兲–共38兲 共of Pauli type兲 in order to understand better the consequences of the Markovian approximation. Figures 10共a兲 and 10共b兲 show the time evolution of

具

␴ˆz

典

for both equations at differ-ent values of the coupling parameter. The time axis has been scaled by the coupling parameter (␭2t). This scaling is char-acteristic of the Lorentzian SOE regime. The values of ␭ have of course to be reasonably small in order to remain consistent with the fact that these equations are obtained per-turbatively. The time scale that we are observing is the global one: from the initial condition to the equilibrium. The

char-acteristic energy of the system (⌬) is different in Figs. 10共a兲 and 10共b兲. But we are in both cases in domain A of the reduced parameter phase space 共see Fig. 1兲. For such small values of⌬, we see almost no difference between Figs. 10共a兲 and 10共b兲. The characteristic time scale of the environment is of order 2␲, and is therefore in both cases much shorter than the system one. We are in situations where the Markovian approximation makes sense on times longer than 2␲. Using the␭2t scaling, the Markovian equation is independent of␭. This is not the case for the non-Markovian equation. We see that the stronger␭ is, the larger the deviation is between both

FIG. 10. Time evolution of the z component of the spin: compari-son between the Markovian 共M兲 and non-Markovian 共NM兲 ver-sions of the perturbative equation for different values of the cou-pling parameter. In all the cases

⌬⫽0.01 and⑀⫽0.

FIG. 11. Time evolution of the z component of the spin.共a兲, 共b兲, and 共c兲 Comparison between the exact von Neumann equation and the non-Markovian共NM兲 version of the perturbative equation for different values of the coupling parameter.共a兲 and 共b兲 correspond to ⌬⫽0.1 and 共c兲 to ⌬⫽0.5 and

␭⫽0.1. 共d兲 Comparison between

equations. This is the consequence of the fact that the ␭2t scaling that we use amplifies the non-Markovian short time behavior 共that occurs on time of order of the characteristic time scale of the environment 2␲兲 when ␭ increases. Figures 10共c兲 and 10共d兲 show us the long time behavior of the Mar-kovian and non-MarMar-kovian versions of our equation. We see that the equilibrium value of

具

␴ˆz

典

depends on␭ for the non-Markovian equation. This is not the case for the non-Markovian equation. The differences between the equilibrium values are small. But, comparing Figs. 10共c兲 with plot 10共d兲, one no-tices that when⌬ is larger, the differences are larger. This is a consequence of the error made on short-time dynamics using the Markovian approximation. Because this error is more important when␭ is large, the consequence on the long time dynamics is more important, even if globally small.

We now compare the non-Markovian version 共33兲 and 共34兲 of our perturbative equation with the exact von Neu-mann equation共26兲, staying in the parameter domain A. Fig-ures 11共a兲 and 11共b兲 show the time evolution described by both equations at different␭ values using the ␭2t time scal-ing. In Fig. 11共a兲 the Markovian and the non-Markovian ver-sions of our equation are so close that we only plotted the second one. We see that the non-Markovian equation is valid not only for values of ␭ below an upper bound, but also above a lower bound. When ␭ is too small, the non-Markovian version of our perturbative equation does not fit with the exact result. The exact initial dynamics is well re-produced by our equation, but the relaxation process to the equilibrium value is not reproduced. This is due to the dis-crete nature of the spectrum and therefore depends on the number N of states and disappears in the limit N→⬁. It corresponds to the border between the localized and Lorent-zian regimes of the SOE. This is one of the main results of this paper. This phenomena is of course again related to the effect of the perturbation between the levels in the total spec-trum that we already observed in the spacing distribution, in the SOE, and in the ATPK. ␭ has to be large enough (␭2 ⬎1/N) to ‘‘mix’’ the nonperturbed levels in order for all the states inside the total unperturbed energy shell to be mixed together. Remember that the equilibrium value of our Mar-kovian perturbative equation is given by a microcanonical distribution in the total unperturbed energy shell 关see Eq. 共37兲兴. One also sees in Fig. 11共b兲 that our perturbative equa-tion loses again its validity above a certain value of ␭. It happens at value of␭ that cannot be considered as perturba-tive any more. It also corresponds to a value of the coupling parameter corresponding in the SOE to the transition in the Lorentzian regime when the width of the Lorentzian starts to be larger than the typical variation energy scale of the envi-ronment density of states.

In Fig. 11共c兲 we deal with the parameter domain B 共see Fig. 1兲. We again compare the non-Markovian version of our perturbative equation with the exact dynamics given by the von Neumann equation. In this case, the system dynamics共of period 0.4␲兲 is faster than the environment one 共of period 2␲兲. We are in a highly non-Markovian situation. The Mar-kovian version of our equation 共that describes no evolution in this case兲 completely misses the observed behavior of damped oscillations. But the non-Markovian equation

repro-duces this behavior with a very high accuracy. The frequency of the oscillations corresponds to the system dynamics ones and the damping of these oscillations occur on a time scale corresponding to the environment characteristic time scale.

Finally, in Fig. 11共d兲 we are in the high coupling param-eter domain C 共see Fig. 1兲. We see that when ␭ becomes large enough to make the coupling term dominant in the total Hamiltonian, the dynamics obey the Bessel behavior derived in Eq.共53兲. It is important to notice that this behavior scales in time according to␭t.

A summary of the validity of the different approximated equations 关Markovian 共36兲–共38兲 and non-Markovian 共33兲– 共34兲 versions of our perturbative equation and the strong coupling Bessel equation共53兲兴 and of the different scalings, depending on the regime that one considers, is represented in Fig. 12.

D. Average versus individual realizations

An interesting point is the comparison, for the system dynamics, between the averaged 共random matrix ensemble averaged or microcanonically averaged兲 dynamics and the dynamics of an individual realization within the statistical ensembles. This latter corresponds to a dynamics generated by an initial condition that corresponds to a pure state and without any random-matrix average共␹⫽1兲.

We see in Figs. 13共c兲–13共f兲 the dynamics of a system with small energy spacing ⌬⫽0.1 for different values of ␭2_{N. The solid line represents the random-matrix and} micro-canonically averaged dynamics. The dashed lines depict some of the individual members of the random-matrix en-semble corresponding to an initial pure state inside the total unperturbed energy shell. We see that the larger ␭2N is, the closer the individual trajectories are from the averaged tra-jectory. We therefore have, when ␭2N is large enough, that

the individual realizations are self-averaging in the random-matrix and microcanonical ensembles. In order to quantify this behavior, we plotted in Fig. 13共a兲 the variance between the individual trajectories and the averaged trajectory as a function of time for different values of␭2N. We observe that this variance decreases as␭2N→⬁. In Fig. 13共b兲 we show that the asymptotic value of the variance decreases with a power-law dependence with respect to␭2N.

We again see the relation with the SOE regimes. In the localized regime 关Fig. 13共c兲兴 the dynamics is governed by very different individual trajectories oscillating with a very few frequencies that differ from one individual trajectory to another. This is a consequence of the fact that the perturbed levels are still close to the nonperturbed ones and are only slightly affected by neighboring nonperturbed levels. In the Lorentzian regime 关Figs. 13共d兲 and 13共e兲兴, each individual trajectory follows roughly the averaged trajectory and con-tains a very large number of different frequencies. This shows that the interaction ‘‘mixes’’ many of the nonper-turbed levels, deleting the discrete structure of the spectrum. Therefore, we can say that our master equation 共31兲 or 共32兲 holds with a given accuracy for a majority of individual trajectories if ␭ is small enough satisfying ␭⭓CN⫺␯ with

␯⬍1

2 and a constant C⬎0, in the limit N→⬁.

V. THE VERY LONG TIME BEHAVIOR

We here focus on the very long time behavior of our model, in other words on its equilibrium properties.

A. Equilibrium values of the system observables Let us consider the spin observable ␴ˆz of the two-level system. This observable evolves in time according to

具␴

ˆz

典

共t兲⫽Tr␳ˆ共0兲eiH ˆ tott␴_ˆ ze⫺iHˆtott ⫽

兺

␣,␣_⬘

具␣

兩␳ ˆ共0兲兩␣⬘典具␣⬘兩␴ˆz兩␣典ei(E␣⫺E␣⬘)t. 共54兲 The time-averaged value of

具␴

ˆz

典

(t) is obtained perform-ing the followperform-ing time average:

具

␴ˆz

典

⬁⫽ lim T→⬁ 1 T

冕

0 T dt

具␴

ˆz

典

共t兲⫽

兺

␣

具␣

兩␳ˆ共0兲兩␣典具␣兩␴ˆz兩␣

典

. 共55兲 We see that this time averaged value clearly depends on the initial condition.

FIG. 13. In all the figures

⌬⫽0.1, N⫽500, ⑀⫽0, and ␦⑀ ⫽0.05. 共a兲 Variance between

indi-vidual trajectories and the aver-aged one共␹⫽100兲 as a function of time for different values of␭2N.

共b兲 Power-law dependence

be-tween the equilibrium value of this variance and␭2_{N. 共c兲–共f兲}

In-dividual trajectories of the en-semble共dashed lines兲 and the en-semble averaged trajectory 共solid line兲. In 共c兲 ␭2_{N⫽0.1, in 共d兲}

␭2

N⫽1, in 共e兲 ␭2N⫽10, and in

共f兲 ␭2

An important result, that holds everywhere on the param-eter space and for all kinds of initial conditions, is that the observed equilibrium value given by the exact von Neumann equation corresponds very well to the time-averaged value 共55兲. Therefore, the study of the equilibrium properties of systems as ours is equivalent to studying the time averaged quantities 共55兲.

Let us note that the standard initial condition we used till now is a microcanonical distribution around energy⑀for the environment and an upper state for the system formally given by ␳ˆ共0兲⫽兩1

典具

1兩丢␦共HˆB⫺⑀兲 n共⑀兲 ⫽

兺

n ␦共En⫺⑀兲 n共⑀兲 兩1n

典具

1n兩. 共56兲 Therefore, we have that

具␣

兩␳ˆ共0兲兩␣

典

⫽

兺

n

␦共En⫺⑀兲

n共⑀兲 兩

具␣

兩1n

典

兩

2_{. 共57兲}

An interesting point is to understand when the Markovian perturbative equation gives the correct equilibrium value. We want, therefore, to compare 共55兲 with 共37兲. This is done in Fig. 14 where we plotted the time averaged value

具␴

ˆ_z

典

_⬁as a function of the initial energy of the environment⑀关the initial value of the total system being共56兲兴. The different curves in Figs. 14共a兲 and 14共b兲 correspond to different values of ␭. We compare these curves to the␭ independent curves given by the Markovian perturbative equation共37兲. As expected from our precedent study of the dynamics, we find again that, when␭ is too small, the ‘‘mixing’’ between the nonperturbed levels is not sufficient inside the microcanonical energy shell and the Markovian perturbative results overestimate the equilibrium values. If ␭ is too large, the Markovian

pertur-bative equation gives again bad results. The Markovian per-turbative equilibrium values are correct in a characteristic region of␭. The beginning of this region corresponds, in the SOE, to the critical value of␭ at which the transition occurs from the localized to the Lorentzian regimes. The end of this region corresponds to␭ values that cannot be considered any more as perturbative. Figures 14共c兲 and 14共d兲 show that

具

␴ˆ_z

典

_⬁ scales like ␭2_{N. This again confirms our precedent} analysis.

B. Thermalization of the system

One of the important questions is to understand the con-ditions under which the system thermalizes under the effect of a weak contact with the environment, or in other words, under which conditions the system relaxes to a canonical distribution corresponding to the microcanonical temperature of the environment.

We begin by recalling these conditions in the general case of a small system weakly interacting with its environment. The isolated total system is composed of the system and the environment and has the total energy

Etot⫽⑀⫹e. 共58兲

⑀ is the energy of the environment and e the energy of the system. We suppose that the contribution of the interaction energy between the environment and the system is negligible compared to the total energy. The microcanonical environ-ment entropy is defined as

SB共⑀兲⫽k ln ⍀B共⑀兲, 共59兲 where ⍀_B(⑀) is the number of states of the environment available at energy⑀. This number can be related to the den-sity of states of the environment nB(⑀) using the fact that

FIG. 14. Comparison between the equilibrium value of 具␴ˆz典⬁

given by the Pauli equation and the exact values given by the time averaged value for different val-ues of␭. These equilibrium values are depicted as a function of the initial microcanonical energy⑀ of the environment. ␦⑀⫽0.05 in all figures. The parameter values are

共a兲 ⌬⫽0.01 and N⫽500; 共b兲 ⌬⫽0.5 and N⫽500; 共c兲 ⌬⫽0.01

⍀B(⑀)⫽nB(⑀)␦⑀, where ␦⑀ is a small energy interval but contains many states of the environment. The microcanonical temperature of the environment is given by

1 TB共⑀兲 ⫽

dSB共⑀兲

d⑀ . 共60兲

It can be expanded in the system energy as

TB共⑀⫽Etot⫺e兲⫽TB共⑀⫽Etot兲⫺e

dTB共⑀⫽Etot兲 d⑀ ⫹•••,

共61兲 because we suppose that the system energy is much smaller than the environment energy. The specific heat capacity of the environment is 1 C_vB共⑀兲 ⫽ dTB共⑀兲 d⑀ . 共62兲 If the condition 兩CvB共⑀兲兩Ⰷ

冏

e TB共⑀兲

冏

共63兲 is satisfied, the temperature expansion can be truncated as follows:

TB共⑀⫽Etot⫺e兲⫽TB共⑀⫽Etot兲. 共64兲 Therefore, we understand that if Eq. 共63兲 is satisfied, the environment plays the role of a heat bath because its tem-perature is almost not affected by the system energy.

We suppose further that the interaction between the sys-tem and the environment, even if small, is able to make the total probability distribution microcanonical on the total en-ergy shell at enen-ergy Etot. We suppose also that the energy levels are discrete and, therefore, that E_tot⫽E_s⫹E_b where s and b are discrete index’s, respectively, for the system and the environment. Therefore, the probability PS(Es) for the system being at energy Es is given by

PS共Es兲⫽

⍀B共Eb⫽Etot⫺Es兲 ⍀tot共Etot兲

, 共65兲

where ⍀B(Eb) is the number of states of the environment available at energy Eb, and⍀tot(Etot) the number of states of the total system available at energy Etot. Using Eqs.共60兲 and 共63兲, one gets that

PS共Es兲⫽

e(1/k)SB(Eb⫽Etot⫺Es) ⍀tot共Etot兲 ⯝e

(1/k)S_B(E_b⫽E_tot)⫺关E_s/kT_B(E_B⫽E_tot)兴 ⍀tot共Etot兲

. 共66兲

Using the normalization PS(Es)⫽1, one finally gets the well-known canonical probability distribution for the system

PS共Es兲⫽

e⫺[Es/kTB(EB⫽Etot)]

Z , 共67兲

where Z⫽兺se⫺[Es/kTB(EB⫽Etot)].

We conclude that two conditions are necessary in order to thermalize the system to a canonical probability distribution due to the contact with the environment: a large heat capacity of the environment兩C_vB(⑀)兩Ⰷ兩e/TB(⑀)兩 and a microcanoni-cal distribution on the total energy shell.

Let us apply this result to the spin-GORM model. The density of states of the environment is

n_B共⑀兲⫽4N

␲

冑

1 4⫺⑀

2_{. 共68兲}

Therefore the microcanonical temperature of the environ-ment is

TB共⑀兲⫽

⑀2_⫺1 4

k⑀ 共69兲

and the heat capacity is

C_vB共⑀兲⫽ k⑀

⑀2_⫹1 4

. 共70兲

The canonical distribution of the populations of the system, at the microcanonical temperature of the environment, is given by

具␴

ˆz

典

T_B(⑀) can _⫽⫺tanh ⌬ 2kTB共⑀兲 ⫽⫺ tanh ⌬⑀ 2共⑀2⫺1 4兲 . 共71兲

The first condition共63兲 in order to thermalize the system to a canonical probability distribution becomes

兩CvB共⑀兲TB共⑀兲兩⫽

冏

⑀2_⫺1 4 ⑀2_⫹1 4

冏

Ⰷ⌬ 共72兲

and is depicted in Fig. 15. The second condition to have a microcanonical distribution on the total energy shell is satis-fied 共as we discussed in Sec. V A兲 when ␭2N⬎1. In this FIG. 15. Representation of the region where the thermalization condition兩(⑀2⫺14)/(⑀

2_⫹1

case, the populations of the system obey the microcanonical equilibrium value of the Markovian version of our perturba-tive equation共37兲, i.e.,

具␴

ˆz

典

_⑀ micro_⫽

冑

1 4⫺

冉

⑀⫺ ⌬ 2

冊

2 ⫺

冑

1 4⫺

冉

⑀⫹ ⌬ 2

冊

2

冑

1 4⫺

冉

⑀⫺ ⌬ 2

冊

2 ⫹

冑

1 4⫺

冉

⑀⫹ ⌬ 2

冊

2. 共73兲

If the two conditions are satisfied, then Eqs. 共71兲 and 共73兲 should be equal. The comparison between Eqs.共71兲 and 共73兲 can be seen in Fig. 16 for different system energies. We see

that the smaller the system energy is the better the compari-son is. Therefore, we can conclude that under these two con-ditions 关兩(⑀2⫺14)/(⑀

2_⫹1

4)兩Ⰷ⌬ and ␭

2_{N⬎1], the random} matrices of the spin-GORM model can model an environ-ment that behaves as a heat bath.

C. Thermalization of the total system

Until now, we have chosen initial conditions where the system is in the upper state with a microcanonical environ-ment at a given energy, like in Eqs. 共56兲 or 共57兲. We now want to consider initial conditions where the system is again in the upper state but where the environment is at a given canonical temperature: ␳ˆ共0兲⫽兩1

典具

1兩丢e ⫺␤BHˆB ZB ⫽

兺

n e⫺␤BEn ZB 兩1n

典具

1n兩. 共74兲 Therefore

具␣

兩␳ˆ共0兲兩␣

典

⫽

兺

n e⫺␤BEn ZB 円

具␣

兩1n

典

円2_{. 共75兲} It is important to notice that there is no statistical equiva-lence between the canonical and the microcanonical en-sembles in the spin-GORM model共see Appendix B兲. There-fore, it is interesting to ask how the probability distribution looks like at equilibrium after the interaction.

One can clarify this point by plotting

具

␣兩␳ˆ (0)兩␣典 versus energy. One uses the following energy representation:

P共␧兲⫽

兺

␣ ␦共E␣⫺␧兲

具

␣兩␳ˆ共0兲兩␣典. 共76兲

If the total system thermalizes and reaches a canonical distribution for the total system at an effective temperature

␤eff⫺1, one would have that P共␧兲⫽e ⫺␤eff␧ Ztot , 共77兲 because

具

␣兩␳ˆ共0兲兩␣典⫽e ⫺␤effE␣ Ztot . 共78兲

As we shall see, it is the case if, again,␭ is large enough to induce ‘‘mixing’’ between the states␭2N⬎1.

Indeed, one sees in Fig. 17共a兲 that, for ␭⫽0, the states of the total system corresponding to the upper level of the sys-tem are exponentially populated and the ones corresponding to the lower level are not. When the interaction is turned on and increased, one can notice that the probability distribution starts to accumulate around a mean effective canonical dis-tribution. As expected, this accumulation becomes significant when␭2N⬎1 and, in this case, the total system can be con-sidered as having thermalized. One can calculate the final effective temperature that the total system has reached after interaction. This effective temperature is depicted in Fig. 17共b兲 as a function of ␭. The correlation coefficient indicates

whether the exponential fit of the final effective temperature is good or not. The effective temperature obeys the following law:␤e f f⫽␤i/(1⫹␭2). We also show in Fig. 17共c兲 the com-parison between the time-averaged value of

具␴

ˆz

典

and his canonical average computed with the effective temperature. One sees that, when ␭2N⬎1, both coincide.

This thermalization is not statistical in the sense of the equivalence between the ensembles. It is an intrinsic ther-malization due to the complexity of the interaction between the states. This thermalization appears at a critical value of the coupling parameter when the interaction term becomes of the order of or larger than the mean level spacing of the total system.

VI. CONCLUSIONS

In this paper, we have studied a system made of two parts: a two-level system interacting in a nondiagonal way with a complex environment modeled by Gaussian orthogonal ran-dom matrices. We began our study by analyzing the spectral properties of this model. We investigated the spectrum on a large energy scale with the averaged smooth density of states and on a finer energy scale with the eigenvalue diagrams, the shape of the eigenstates共SOE兲, the spacing distribution, and the asymptotic transition probability kernel 共ATPK兲. We found a global repulsion as well as avoided crossings be-tween the eigenvalues when the coupling parameter ␭ was increased. We also showed the existence of three regimes 共easy to distinguish in the SOE兲 that are important to de-scribe the different qualitative behaviors of the model: the localized regime when the interaction between the levels is weaker than the mean level spacing ␭2Nⱗ1 共giving rise to very narrow SOE兲, the Lorentzian regime when the interac-tion between the levels becomes larger than the mean level spacing ␭2N⬎1 共giving rise to a Lorentzian SOE兲, and the delocalized regime for very large ␭ 共giving rise to SOE spread over the whole spectrum兲.

After the spectral study, we started the study of the dy-namics of the system populations. We defined different do-mains in the parameter space共see Fig. 1兲 and related them to the different relaxation behaviors of the system population induced by the interaction with the environment. For each of these domains we tested the validity of approximated popu-lation evolution equations. In the strong coupling limit, we identified a population relaxation regime described by a Bessel function:⬃2关J1(␭t)/␭t兴 共53兲 that scales in time ac-cording to ␭t and reaches an equilibrium distribution corre-sponding to the same probability of being in the upper and lower states of the system. In the small coupling limit and for small system energy, we obtained a Pauli-type equation 共36兲–共38兲 describing an exponential relaxation of the system population that scales in time according to ␭2t and that reaches an equilibrium distribution depending on the system energy. The equilibrium value corresponds to a microcanoni-cal probability distribution of being in a nonperturbed state of the total system inside the total energy shell. Finally, we showed the necessity of taking into account the non-Markovian effects in the dynamics 共which are important when the system energy becomes non-negligible in front of FIG. 17. In all the figures⌬⫽0.01, N⫽500, and␤⫽2. 共a兲

Prob-ability P(␧)⫽Dist(␧) of being in an eigenstate of the total system at equilibrium starting from the initial condition ␳ˆ(0)⫽兩1典具1兩

丢e⫺␤HˆB_/Z

Bwith 1/␤⫽0.5. 共b兲 Effective temperature of the