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Quantum Statistical Ensemble Resilient to Thermalization Maurizio Coden, Barbara Fresch, and Giorgio J. Moro J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b00274 • Publication Date (Web): 10 May 2016 Downloaded from http://pubs.acs.org on May 13, 2016

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The Journal of Physical Chemistry

Quantum Statistical Ensemble Resilient to Thermalization. Maurizio Coden,† Barbara Fresch,†,‡ and Giorgio J. Moro∗,† †Dipartimento di Scienze Chimiche, Università di Padova 35131 Padova, Italy ‡Départment de Chimie, Université de Liège, B4000 Liège, Belgium E-mail: [email protected] Phone: +39 049 827 5683

1 ACS Paragon Plus Environment


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Abstract The sampling of the wavefunction within a suitable ensemble is an important tool in the statistical analysis of a molecule interacting with its environment. The uniform statistical distribution of quantum pure states in an active space is often the privileged choice. However, such a distribution with constant average populations of eigenstates is not preserved upon the interaction between quantum systems. This appears as a severe methodological shortcoming, as long as a quantum system can be always considered as the result of interactions amongst previously isolated subsystems. In the present work we formulate an alternative statistical ensemble of pure states that is robust with respect to interaction and it is thus preserved when subsystems are merged. It is derived from the condition of invariance of the average populations upon interaction between quantum systems in the same thermal state. These average populations allow a simple identification of the thermodynamic properties of the system. We find that such a statistical distribution is robust with respect to interaction of systems at different temperatures reproducing the thermalization of macroscopic bodies, and for this reason we identify it as the Thermalization Resilient Ensemble (TRE).

1

INTRODUCTION

Nowadays Molecular Dynamics Simulations play an important role in our understanding at the molecular level of condensed matter systems. Such a computational technique relies on different methodologies, basically on efficient numerical solutions of the classical equations of motion, on the one hand, and on the classical statistical mechanics, on the other hand, in order to select system configurations in a well defined thermal state with given temperature. Motivated by the success of Molecular Dynamics Simulations, one wonder whether an analogous procedure can be designed on the ground of a full quantum mechanical description so producing Quantum Dynamics Simulations of material systems. Once the Hamiltonian for the system is chosen, in principle this should be possible by describing the time evolution


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of the system according to the wave-function |Ψ(t)i to be calculated as the solution of the Schrödinger equation for a given initial condition |Ψ(0)i. The observables should then be computed as expectation values of the corresponding operators. It is customary to employ the attribute of pure state to designate the wave-function describing an isolated system, in opposition to mixed state for the statistical density matrix without idempotency. An obstacle to the realization of Quantum Dynamics Simulations derives from the need to solve the time dependent Schrödinger equation, which can be done numerically only for finite representations in Hilbert spaces of not too large dimensions, say of the order of 104 . Because of this limitation, the same detailed description of material systems supplied by Molecular Dynamics Simulations cannot be afforded in a fully quantum mechanical framework. The only practicable way would include a detailed quantum description of some degrees of freedom only for the molecule of interest, with the environment collapsed into a low dimensional model able to capture the main features of the interactions, in the same spirit of spin-boson models. Even in this more realistic formulation, the realization of Quantum Dynamics Simulations is a challenging objective whose attainment would provide a deep understanding of the dynamics of molecules at a fundamental level. In recent years there has been an increasing number of quantum dynamics studies through the numerical solution of the Schrödinger equation for systems of interacting components. They concerned model systems with a sufficiently low dimensional representation to allow the numerical treatment, for instance spin systems 1,2 or boson systems. 3–5 These studies demonstrates that Quantum Dynamics Simulations could be a practicable route, provided that the constraint on the Hilbert space dimensions is satisfied. On the other hand, the development of efficient methodologies to simulate the quantum dynamics of selected molecular states is currently an active and promising research field. 6–8 However, to study phenomena such as dissipation and thermalization, the focus has to be moved from isolated molecules to modular systems made of mutual interacting components, with model Hamiltonians possessing a sufficiently low dimensional representation.



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The feasibility of numerical solutions of the Schrödinger equation is not, however, the only obstacle. An important issue concerns the rules to be employed for the choice of the initial quantum state |Ψ(0)i of the isolated system. As long as one considers molecular degrees of freedom interacting with a (model) environment, there are no reasons to select a particular quantum state for the overall system and, therefore, a random choice has to be performed amongst a well defined statistical ensemble of pure states. Furthermore, one would like to operate a choice assuring the simulation of the system in a well defined thermal state with given temperature. This necessarily calls for a statistical description like for classical systems. More precisely, the probability distribution on the quantum pure states is required for the system in a given thermal state. Thus, by using Monte Carlo algorithms, one can pick up randomly a pure state to be used as the appropriate initial wave-function for the evolution of the system. Like in Molecular Dynamics Simulations, the statistical tools are essential in order to have a full control of the random choice of system’s states. Quantum Statistical Mechanics in its standard formulations 9,10 deals with mean properties on the basis of the density matrix averaged on the possible realizations of the system and, therefore, does not provide information on the pure state distribution. As a matter of fact the statistics of quantum pure states has been investigated in recent years mainly because of its relevance to the theory and applications of quantum information. 11 Early developments have been addressed to the capability of pure state distributions to describe the macroscopic behavior and to recover average properties in agreement with standard Quantum Statistical Mechanics. 12–17 In this framework an important role has been played by the concept of typicality for the large size limit of the system: different pure states are compatible with a macroscopic (thermodynamical) state as long as they lead to negligible differences on the predicted properties and, therefore, all of them can be considered as typical quantum states of the system. Because of typicality, in the case of macroscopic systems the choice of a specific form for the distribution of pure states does not seem to be an important issue. 13 A completely different situation arises with Quantum Dynamics Simulations when neces-


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sarily small size systems with finite dimensional representations need to be considered. First of all typicality cannot be taken for granted in the absence of the large size limit for the system. Furthermore, in order to avoid subjective choices for the operating conditions, the initial wave-function |Ψ(0)i has to be randomly sampled from a well defined pure state distribution. In this framework a clear and unambiguous definition of the statistical distribution of the quantum pure states is a fundamental methodological requirement. The pure state distribution has to be defined on the basis of a parameterization of the wave-function. The most convenient one is certainly that provided by its expansion coefficients on the eigenstate basis of the Hamiltonian. Furthermore, by adopting their polar representation, one introduces the phases αk and the populations Pk with respect to the eigenstates |Ek i. 18 The populations are the constants of motion of the problem, while the phases, with their linear time dependence, condensate all the dynamical information. In ordinary conditions the time evolution according to the Schrödinger equation produces a uniform distribution with respect to the phases. 19,20 Therefore, the quantum equations of motion leads to a pure state distribution independent of the phases but, at the same time, do not provide information on the dependence on the populations as long as they are the constants of motion. In conclusion, fundamental principles do not impose constraints on the populations and one is obliged to a priori choices for their distribution (the statistical ensemble in the terminology of probability theory 21 ), which corresponds to the statistical weight for different realizations of the quantum pure state identified by a particular set of populations. In order to define a possible population distribution, one necessarily has to invoke guidelines completely different from quantum mechanical principles. The absence of information or constraints on the populations naturally suggests a purely random choice by attributing the same probability to different population sets, very much like with the principle of insufficient reason of classical probability. This generates a uniform distribution of population sets for eigenstates belonging to a given active space. 18 By representing the corresponding wave-



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function as a point on the hyper-dimensional Bloch sphere, such an ensemble corresponds to a purely random choice on the sphere’s surface without attributing any preference to particular directions of the active space. 20 Equivalent forms of this statistical ensemble of pure states has been invoked by several authors. 2,12,16,17,22 It is characterized by a compact formal representation which allows a simple calculation of average properties and a detailed study of typicality in suitable limits. 23–25 Furthermore, a straightforward algorithm is available for sampling the population sets. 18,26,27 Its distinctive feature is the identical mean population for each eigenstate within the active space, which corresponds to the average density matrix of the standard quantum microcanonical statistics. We mention that an alternative form of the random pure state distribution has been proposed by imposing also the constraint of a fixed energy, 28–30 i.e. the expectation value of the Hamiltonian. It might appear rather appealing as the quantum generalization of the classical microcanonical ensemble, however it is unable to generate the canonical distribution for the quantum states of subsystems 23 and, therefore, it is not appropriate for systems in ordinary conditions. It should be emphasized that the uniform statistical distribution of pure quantum states derives from a conventional choice, no privilege to particular directions of the Bloch sphere, without any strict obligation deriving from fundamental principles. This leaves room to alternative proposals of population distribution, which become the central issue of interest if shortcomings arise in the representation of quantum systems according to the uniform distribution. As a matter of fact this happens in the quantum description of the thermalization experiment, that is the attainment of thermal equilibrium between two systems that initially were isolated. 31–33 Let us consider the simpler experiment of this type: two identical systems A and B, initially isolated and described by the uniform statistical distribution, which are brought to interact so leading to a new equilibrium configuration for the overall system A+B. Our analysis of this experiment will show that the populations in the final equilibrium state are no more characterized by an identical average as it was in the statistical ensemble for the initially isolated systems A and B. The same structure of the uniform statistical distribution


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characterized by identical average populations within the active space, is not preserved in the thermalization experiment. This is a severe shortcoming from a methodological point of view, since isolated system can be always considered as the result of interactions amongst previously independent systems. In this work we intend to find and characterize a statistical ensemble for populations that overcomes the drawbacks of the uniform distribution of pure states. From the methodological point of view, the main issue concerns what a guideline should be assumed in order to recognize such an ensemble, taking into account that quantum dynamical laws does not provide any specific information about the values to attribute to the populations. Our choice is that of starting precisely from the failures of the uniform statistical distribution. We intend to identify a statistical ensemble resilient to thermalization, or at least to its simpler realizations, in the meaning that the population distribution should maintain its structure after thermalization. This requirement will be employed as the guideline for the derivation of such a statistical ensemble, in the following called as Thermalization Resiliant Ensemble (TRE). It should provide a convenient framework for treating the interactions between quantum systems, as long as the structure of the statistical distribution is preserved and the identification of thermodynamical properties is assured. In perspective it could be the privileged statistical ensemble to implement in Quantum Dynamics Simulations. In order to follow a simple procedure leading to the definition and the full characterization of TRE, we identify two separate objectives: i) the thermal state dependence of the average populations, and ii) the full statistical distribution on the population set. The present work is dedicated to the attainment of the first objective. We will show that the guideline of invariance applied to the simplest experiment of thermalization is sufficient to recognize the energy dependence of the mean populations parametrically dependent on the thermal state. In a future contribution we will present the full population distribution to be identified on the basis of TRE mean populations of this work. 34 The paper is organized as follow. In Section 2 the ensemble of theoretical methods



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employed by us are described. First, in subsection 2.1, the main definitions for statistical ensembles of pure state are specified according to populations Pk and phases αk of the eigenstates |Ek i, 18 in relation to equilibrium properties given as time average of expectation values. On this basis we define also the parametric dependence on the thermal state of the mean populations. In subsection 2.2 we introduce the population composition rule that is employed to describe the effects of interactions between quantum systems. The derivation of the invariant form for population averages is tackled in the following subsection 2.3. By considering the simplest case of two identical systems at the same thermal state, the explicit functional form is found for TRE mean populations with an exponential dependence on the energy. In Section 3 we discuss the general properties of TRE statistics. In particular the thermodynamic properties of systems described according to TRE mean populations are evaluated in subsection 3.1, in order to identify the quantum counterpart of the temperature. In the following subsection 3.2 the resilience of TRE average populations in a generic thermalization experiment is verified for model systems specified through the density of states. Section 4 summarizes the key findings of this work.

2 2.1

THEORETICAL METHODS Statistical Ensembles for Isolated Quantum Systems.

We consider an isolated quantum system described by a pure state, that is by the time dependent wave-function |Ψ(t)i belonging to the Hilbert space H. A general parameterization of the quantum state is provided by the set of coefficients ck (t) = hEk |Ψ(t)i of the wave-function expansion |Ψ(t)i =

X

ck (t) |Ek i

k


(1)

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ˆ of the system along the eigenstates of the Hamiltonian H

ˆ |Ek i = Ek |Ek i , H

hEk |Ek0 i = δk,k0

(2)

for k = 1, 2, 3, · · · and the eigenvalues ordered with increasing magnitude, Ek 6 Ek+1 . The full set of coefficients ck (t) identifies the quantum pure state at a given time. The complex coefficients ck (t) are conveniently replaced by real parameters obtained from their polar representations ck (t) =

p Pk e−iαk (t)

(3)

where Pk and αk (t) are the population and the phase, respectively, of the k-th eigenstate |Ek i. The Schrödinger evolution of the pure states implies that the populations Pk are time independent, i.e., they represent the constants of motion of the dynamical problem, while the phases are linearly dependent on the time, αk (t) = αk (0) + Ek t/~, and bring all the P information about the system dynamics. Populations are normalized as k Pk = 1. By starting from these essential definitions and focusing on a single realization of a quantum systems, we wonder how to choose a specific wave function amongst all the normalized elements of the Hilbert space. A statistical description of the system should provide the answer. However, conventional quantum statistical mechanics through the so-called microcanonical ensemble 9,10 does not address this issue. Let us summarize the basic definitions of the quantum microcanonical ensemble in the following way. Pure states belonging to a finite dimensional active subspace HN ⊂ H, N being its dimension, are defined as linear combinations of eigenstates |Ek i with energies Ek within fixed boundaries Emin and Emax :

|Ek i ∈ HN

if

Emin < Ek < Emax

(4)

The corresponding pure state density matrix reads

ρˆ(t) := |Ψ(t)i hΨ(t)| 9 ACS Paragon Plus Environment

(5)


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The microcanonical density matrix is introduced as the average of the pure state density matrix on a suitable ensemble of realizations of quantum states. Moreover, it is assumed that its relevant part, that corresponding to the active space, is proportional to the unit matrix. The microcanonical density matrix is then employed to evaluate expectation values representing average properties, for instance in relation to macroscopic observables. Therefore, the microcanonical quantum ensemble allows the calculation of average properties, but it does not specify the probability for the pure states corresponding to the single realizations of the quantum system. The sampling of single realizations has a central role in quantum computation and quantum information research 11 when single quantum states have to be manipulated, and only in recent years it has been specifically addressed. 13,16,17 Moreover, it is a necessary step for the study of the dynamical properties of isolated quantum systems. The absence of a priori informations about the quantum state suggests quite naturally a random sampling of the wave function. Therefore, one should introduce a statistical ensemble 21 for quantum pure states characterized by a suitable probability distribution on the independent parameters which identify the wave-function. The set of populations and phases represents a convenient choice for them, because a simple homogeneous distribution on the phases 19,20 is recovered from the unitary evolution if the energy eigenvalues are rationally independent. 35 Notice that this result does not require the ergodic hypothesis and it derives directly from the linear time dependence of the phases. We emphasize that the condition of rational independence of the energy eigenvalues is rather mild and realistic as long as for material systems one should consider the effects of different types of interactions, depending on inter-particle distances, leading to an energy spectrum with at least a partial random character. 36–41 The stochasticity entailed by a complex interaction pattern is well supported by the successful characterization of the energy spectra of different many-body systems by means of a suitable ensemble of random matrices. Initially introduced to model the spectra of heavy atoms, 42 random matrices also describe vibrational energies of large molecules 43


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and the interaction of a system with a generic environment. 23,44,45 The presence of such a stochastic character into the system Hamiltonian excludes the rational dependence amongst energy eigenvalues, which is the only condition required by our analysis. In particular, the presence of localized eigenstates that is often invoked as a deviation from ergodicity in quantum system 46 does not affect the homogeneous distribution on the phase parameters. The quantum dynamics determines the distribution on the phases only, and not on the populations which are the constants of motion. Therefore the population distribution must be assumed a priori on the ground of general considerations different from the dynamical law. On the other hand the populations determine the system’s equilibrium properties. Given ˆ its expectation value the operator A,

ˆρ(t)} A(t) = hΨ(t)| Aˆ |Ψ(t)i = Tr{Aˆ

(6)

describes the corresponding observable, and from Eq. (3) one derives that the phases are the only responsible of the time dependence of the observable. In complex systems the superposition of oscillations produced by different phases generates a fluctuating type of evolution for the observable (see for instance Fig. 1 of ref. 24 ). Thus the equilibrium value A¯ of the observable is conventionally identified with the asymptotic time-average of the expectation value, which is determined by the populations, 20 1 A¯ := lim T →∞ T

ZT

dt A(t) = Tr{Aˆρ¯ˆ} =

X

Pk Akk

(7)

k

0

where ρ¯ˆ is the time averaged density matrix

ρ¯ˆ =

X

|Ek i Pk hEk |

(8)

k

and Akk := hEk | Aˆ |Ek i are the operator’s diagonal elements in the energy representation. In conclusion, different realizations of a quantum pure state correspond to different sets of 11 ACS Paragon Plus Environment


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phases and populations. However, a difference in the phases is equivalent to a shift of the time axis. 20 Therefore, different populations characterizes distinct pure state realizations with different equilibrium properties. In order to describe the statistical ensemble, 21 we need to specify the independent stochastic variables as the set of positive and normalized populations P := (P1 , P2 , P3 , · · · ), with P k Pk = 1 and Pk ≥ 0 ∀k, and the corresponding probability density p(P ) normalized on the existence domain of populations P . Furthermore, for a given quantum system specified through its Hilbert space and its Hamiltonian, one should consider different possible distributions in correspondence to different thermodynamic states if the the system is macroscopic, or more generally in correspondence to different thermal states if small size quantum systems are examined. Let us denote with ζ the parameter (or the set of parameters) used to identify the thermal state. Then the population distribution will be denoted as pζ (P ) in order to highlight its parametric dependence on the thermal state ζ. Once this probability density has been defined one can calculate the ensemble average of a function h(P ) of populations Z hhiζ =

with the constraints Pk ≥ 0 and

P

k

dP h(P )pζ (P )

(9)

Pk = 1 determining the integration domain. We denote

such an average as hhiζ to emphasize its dependence on the thermal state ζ brought by the choice of the probability density. As an example one could consider the ensemble average of the equilibrium value A¯ Eq. (7) of an observable

¯ζ= hAi

X hPk iζ Akk

(10)

k

with hEiζ =

P

k hPk iζ Ek

in the case of the expectation value of the Hamiltonian, i.e., the


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energy. Then the ensemble average of a function f (E) of the energy becomes

hf (E)iζ =

X

hPk iζ f (Ek )

(11)

k

The analysis of thermalization experiments is conveniently performed by adopting the representation in the continuum of the system’s properties. This corresponds to specify Eq. (11) as an energy integral by introducing the density of states to take into account the multiplicity of energy eigenvalues. Furthermore, one needs to invoke the existence of an energy function P˜ζ (E) for a given thermal state ζ, which represents the extension to the continuum of the average populations such that

hPk iζ = P˜ζ (Ek )

(12)

Then the average of f (E) can be specified as Z+∞ hf (E)iζ = dE g(E)P˜ζ (E) f (E)

(13)

−∞

where g(E) is the density of (energy eigen-)states

g(E) =

X

δ(E − Ek )

(14)

k

In the presence of a dense structure of energy eigenstates, the density of states can be replaced by its continuous extension g˜(E) Z+∞ Z+∞ hf (E)iζ = dE g˜(E)P˜ζ (E) f (E) = dE ρ˜ζ (E) f (E) −∞

−∞


(15)


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which allows the identification of probability density on the energy

ρ˜ζ (E) := g˜(E)P˜ζ (E)

(16)

for the thermal state ζ. Notice that we employ symbols with a tilde to denote the functions representing the extensions to the continuum of system’s properties. The function ρ˜ζ (E) condensates the informations on the distribution of energy states and on their average populations. Its normalization Z+∞ Z+∞ dE g˜(E)P˜ζ (E) = 1 dE ρ˜ζ (E) = −∞

(17)

−∞

implies that function P˜ζ (E) should be considered implicitly as a functional of the density of states. More specifically, different systems at the same thermal state ζ cannot be described by the same energy function P˜ζ (E) for the average populations, because in general they are characterized by different density of states g˜(E). In order to preserve the normalization Eq. (17) they have to be described by different functions P˜ζ (E). As an example, we consider here the uniform statistical distribution of pure states often employed for the sampling of quantum states. 2,12,16–18,22–25 The uniform distribution is defined on an active space, i.e. the subspace spanned by the eigenstates with non vanishing populations. An obvious choice would be the same active space HN in Eq. (4) of the standard quantum microcanonical ensemble for given boundaries Emin and Emax . By assuming that pure states of the Bloch sphere relative to the active space are equiprobable, one derives a constant probability density Pζ = (N − 1)!, where N is the dimension of the active space. Then the same average hPk iζ = 1/N is recovered for the populations within the active space, since the probability density is invariant with respect to exchange of the populations. In other words the different energy eigenvectors within the active space play an identical role. Vanishing averages are attributed to populations for eigenstates outside the


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active space. Because of the uniform distribution within the active space, the thermal state can be modified only by changing HN . Thus the thermal state is parameterized by the energy boundaries, ζ = (Emin , Emax ), with distinct active spaces having different distributions on the populations. Correspondingly, the function P˜ζ (E) describing population averages is constant within the energy range of the active space and vanishes outside

P˜ζ (E) =

   constant

for Emin < E < Emax

 0

otherwise

(18)

with the constant determined by the normalization Eq. (17). When multiplied with g˜(E) to generate the energy probability density ρ˜ζ (E), the function P˜ζ (E) simply selects the portion of the density of states belonging to the active space. Fig. 1 illustrates this behavior for a particular situation with the following density of states  M  A E   Es Es g˜(E) =   0

for E > 0 (19) otherwise

which is often employed in the statistical mechanics of ideal systems. 10 Notice that it is implicitly assumed that E = 0+ is the ground state. The parameter Es represents the energy unit. For calculations we use an unitary value for the constant A. In the past we have analyzed in detail the uniform statistical distribution of populations in an active space without the lower boundary and the energy eigenvalues in the range Ek < Emax , the so-called Random Pure State Ensemble (RPSE). 18,23–25,47 Such a procedure leads to a thermal state specified by a single parameter, ζ = Emax , which can be identified with the internal energy in the large size limit of the system. 24 For the general case of an active space with both boundaries, ζ = (Emin , Emax ), one can verify in the macroscopic limit the equivalence of Emax with the internal energy, and that the thermodynamic parameters are independent of the lower energy boundary Emin 24 . Therefore, the use of RPSE statistics



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∼

Pζ(E)

∼

g (E) ∼ ρ (E) ζ

0

2

4

6

8

10

12

14

16

E/Es Figure 1: Average population function P˜ζ (E) (green continuous line) and energy probability density ρ˜ζ (E) (red dotted line) for the uniform statistical distribution with density of states g˜(E) Eq. (19) (blue dashed line) for M = 4, Emin = 10Es and Emax = 11Es . These functions have been scaled by suitable multiplicative factors in order to allow a visual comparison. appears to be more straightforward. On the other hand we emphasize that the formal properties of population distribution analyzed in detail in ref. 18,23–25 depends only on the dimension of the active space and, therefore, they can be employed also when dealing with the active space of Eq. (4).

2.2

Population’s Composition Rule for Interacting Systems

In this section we analyze the average populations for the composite system (A + B) deriving from the interaction between system A and system B , which are initially isolated. The formalism introduced above is adopted to describe statistically the pure state (wave-function) ˆ A and of the isolated system A with reference to its Hilbert space HA , its Hamiltonian H


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its density of states g A (E) =

P

kA

δ(E − EkAA ) where EkAA denotes the corresponding eigen-

ˆ A |E AA i = E AA |E AA i with |E AA i ∈ HA . Furthermore, we assume that the isolated energies: H k k k k system A is characterized by well-defined ensemble average populations for the thermal state ζ A described by the function P˜ζAA (E). Correspondingly, one can introduce the probability Ã density ρÃ g A (E) on the energy variable E by invoking the extension to the ζ A (E) = Pζ A (E)˜ continuum of the density of states, gÃ (E). An analogous formal description, with B replacing the superscript A, is adopted for the pure states of isolated system B in the thermal state ζ B and with average populations described by function P˜ζBB (E). Let us now consider the composite system deriving from system A and system B in the presence of a weak interaction. As in the classical setting, a coupling should be considered to be weak as long as it does not change the total energy in a noticeable way. The Hamiltonian ˆ = H Â + H ˆ B + λH ˆ int and the physically relevant weak of the composite system reads H int int A A B B coupling limit is defined by the condition λ(Emax − Emin )

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