Quantum thermalization and equilibrium state with multiple temperatures
A large class of isolated quantum system in a pure state can equilibrate and serve as a heat bath. We show that once the equilibrium is reached, any of its subsystems that is much smaller than the isolated system is thermalized such that the subsystem is governed by the Gibbs distribution. Within this theoretical framework, the celebrated superposition principle of quantum mechanics leads to a prediction of a thermalized subsystem with multiple temperatures when the isolated system is in a superposition state of energy eigenstates of multiple distinct energy scales. This multiple-temperature state is at equilibrium, completely different from a non-equilibrium state that has multiple temperatures at different parts. Feasible experimental schemes to verify this prediction are discussed.
Standard textbook always starts its discussion of quantum statistical mechanics with the microcanonical ensemble where every energy eigenstate in a narrow energy range is equally possible. This ensemble is usually established with two postulates, equal a priori probability and random phases, which are either explicitly or inexplicitly stated in standard textbooks (1); (2). Interestingly, von Neumann laid down a very different foundation for quantum statistical physics in a 1929 paper (3), where he proved both the quantum ergordic theorem and the quantum H-theorem “in full rigor and without disorder assumptions.” For some reasons, von Neumann’s work had been forgotten for a long time (4); all textbooks start with “disorder assumptions” (or postulates) .
There have been renewed interests in the foundation of quantum statistical mechanics (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30); (31); (32); (33); (34); (35); (36). Many physicists now agree with von Neumann that thermodynamics can be derived from the dynamics of a truly isolated quantum system without the postulates. These increased efforts to address the issue of equilibration of an isolated quantum system are partly due to that an isolated quantum system can now be achieved and sustained experimentally for a reasonable long time in a highly excited state with ultra-cold atoms (37). Great progress has been made. However, there is much more to be desired. For example, will the improved understanding on the foundation of quantum statistics result in new physics?
The quantum ergodic theorem proved by von Neumann is mathematically an inequality (3). A different version of this inequality, which is more practical and well defined, was proved by Reimann (9). According to these two inequalities (or quantum ergodic theorem), these quantum systems will equilibrate in the sense that fluctuations are very small almost all the time. Both inequalities can only be applied to systems where there are no degenerate energy-gaps. In this work we first show that this quantum ergodic theorem may be applied to a broader class of systems, which include, for example, quantum chaotic systems (38); (39); (40). This is done by example. We numerically study the dynamics of a quantum chaotic system, the Henon-Heiles system (41), which has no bound states and does not satisfy the non-degenerate energy-gap condition established by von Neumann and Reimann. Nevertheless, we find that the Henon-Heiles system still equilibrates in the sense of small fluctuations. Furthermore, our numerical results show that the equilibration is also accompanied by an entropy approaching maximum. This is in agreement in spirit with von Neumann’s quantum H-theorem (3).
We then prove analytically that a subsystem of an isolated quantum system at equilibrium is thermalized such that it is described by the Gibbs distribution. Here we distinguish between equilibration and thermalization: a system equilibrates if its overall features no longer change with time while it still evolves microscopically. Thermalization is only for a subsystem that is described by the Gibbs distribution. A thermalized system must be at equilibrium but not vice versa.
A natural and surprising outcome of this theoretical framework is that a subsystem can thermalize with multiple distinct temperatures. This can happen when the isolated system is in a superposition of energy eigenstates that concentrate around different energy scales. This thermalized system with multiple temperatures appears unavoidable for two reasons. (1) According to the quantum ergodic theorem (3); (9), the equilibrated state has the same energy distribution as the initial state. One can manipulate the energy distribution by choosing a suitable initial condition. (2) There is no a priori reason that an initial state must be in a state which is composed only of energy eigenstates from a narrow energy range. We emphasize that this multi-temperature state is at equilibrium where both hot and cold exist in one system: (i) it is completely different from a state that is out of equilibrium and has different temperatures at different parts of the system or for its different degrees of freedom. (ii) It is also different from an ensemble of systems where some systems have higher temperatures while others have lower temperatures. We discuss feasible experimental schemes with ultra-cold atoms and nuclear spins to confirm our predication.
Equilibration of an isolated quantum chaotic system. The inequality proved for the quantum ergodic theorem by Reimann (9) was slightly modified by Short et al. (14); (15). This inequality for an observable now reads
where with being the wave function of the isolated system and . is the energy eigenstate of the system. The subscript in represents an averaging over a time period much longer than the characteristic time scale of the system. is the maximal value of regarding all the states in the Hilbert space. The effective dimension indicates how widely the state is spread over the energy eigenstates. This inequality holds for a large class of quantum systems that satisfies the non-degenerate energy-gap conditionin (3); (9).
If a quantum system is in a typical state of high energy, then should be large since the density of states usually increases tremendously with energy. This means that the right hand side of Eq.(1) is small. Therefore, this inequality tells us two things: (i) An isolated quantum system in a high-energy state will eventually relax to a state where an observable will fluctuate in small amplitude around its averaged value. (ii) Although the isolated quantum system is described by a wave function, the expectation value of any observable at almost any moment can be computed with , that is, .
Remarks are warranted here. (1) is different from the standard micro-canonical density matrix in textbooks (1); (2): the coefficients ’s are determined by the initial condition and they are not necessarily equal to each other and distributed in a narrow energy range. (2) The coefficients ’s do not change with time; therefore, the energy distribution of the final equilibrated state is the same as the initial state.
For many physicists, small fluctuations already imply equilibrium; however, for others, a state is equilibrated only when its entropy is maximized. For the latter group, even though ground states and other eigenstates have no fluctuations, they can not be regarded as equilibrium. Von Neumann belongs to the latter group. By introducing an entropy for a pure state (3), he proved the quantum H-theorem , which demands that a low-entropy state evolve into a high-entropy state. We address this entropy issue with an example by defining a special entropy for pure states in the single particle Henon-Heiles system (41). This is in spirit similar to the entropy for a pure state introduced by von Neumann, for which there is no known practical procedure to compute so far. Note that this entropy for a pure state introduced by von Neumann in 1929 is different from the usual von Neumann entropy, which is zero for all pure states.
We emphasize that it is reasonable to use a single-particle quantum chaotic system for illustration. When expressed in the form of matrix, there is no essential difference between one-body Hamiltonian and many-body Hamiltonian as long as they belong to the same class of random matrix (38). This is particularly true for the system’s energy spectrum, which appears to be the only factor in determining whether the system equilibrates or not (3); (9). We expect that the one-body and many-body systems share many dynamical features when their corresponding matrices belong to the same class. More discussion on this point can be found in Ref.(42).
The Hamiltonian of the system for the Henon-Heiles system is , which has three saddle points located at a distance from the origin. These three points are the corners of the energy triangular contour with potential as shown in Fig.1(a1-a5). The momentum corresponding to the saddle point energy is as indicated by the circle in Fig.1(b1-b5). In our numerical simulation we set , , and .
The initial condition is a highly localized Gaussian wave packet as shown in Fig.1 (a1,b1) so that the system energy is high. This wave packet is centered at and in the real and momentum spaces, respectively. A classical particle with and has energy and its motion is fully chaotic.
We numerically solve the Schrödinger equation and the dynamical evolution of the wave packet is illustrated in Fig.1(a1-a4,b1-b4). As the wave packet evolves, it begins to spread out and distort in shape. Eventually it reaches an equilibrium state, where the wave packet spreads out all over the classically allowed region in the real space and the momentum space. This overall feature will no longer change even though the details of the wave packet still change in the following dynamical evolution. For comparison, we have calculated and by long-time averaging, i.e., the equilibrium state obtained by Reimann (9). The results are shown in Fig.1(a5,b5). It is clear that and are very similar to the wave packet at with the same overall feature that the wave function spread all over both the triangular spatial region and the circled momentum region except for some fluctuations.
To demonstrate the system has relaxed to an equilibrium state, we define an entropy as . This entropy indicates how wide spread the wave function is in the classically-allowed region. The time evolution of is shown in Fig.2(a1,a2), where we see clearly the entropies quickly saturate and reach the maximum values, indicating that an equilibrium state is reached. Note that the relaxation times in both the real and momentum spaces are about the same. However, it must be pointed out that this definition of entropy applies only for some special systems and do not apply for a general quantum system. It is in spirit in accordance with the entropy for a pure state introduced for a general system by von Neumann (3).
The equilibrium state reached is consistent with the inequality Eq.(1). To check the inequality numerically, one needs to compute energy eigenstates of the system. As it is difficult to compute the eigenstates for the Henon-Heiles system, we have turned to the ripple billiard system studied in Ref.(25); (43) to verify the inequality. The verification is successful and the detailed computation can found in Ref.(42). We only mention here that for a similar Gaussian wave packet in the ripple billiard system.
Note that the quantum ergodic theorem was originally proved by von Neumann and Reimann for systems that have no degenerate energy gaps (3); (9). It was later generalized to systems that have limited amount of degeneracy (15). However, it is still not clear that how these degeneracy conditions are related to the integrability of the systems. Our numerical simulations here and elsewhere (25); (42) suggest that this theorem may be applied in a much broader class of quantum systems, which include quantum chaotic systems.
Thermalization of subsystems. We have shown that a large class of truly isolated quantum systems, including chaotic systems, can relax to an equilibrium state. Now we decompose an equilibrated isolated quantum system into two parts, subsystem and thermal bath . We consider an operator , which traces out the thermal bath and gives the density operator for the small subsystem . Note the subsystem is small compared with system but still large on the microscopic level. Based on our equilibration picture, the expectation value of should also equilibrate. We shall show that due to the coupling to the rest of the system, these equilibrated subsystems are also thermalized so that they are described by the Gibbs distribution. The derivation of Gibbs distribution for a subsystem has been considered before with the assumption that the isolated system is in a pure state composed of energy eigenstates from a small energy interval (28); (32). We show that this assumption is not necessary and when the pure state is composed of energy eigenstates of different energy scales, the subsystem is thermalized with multiple temperatures.
We write the Hamiltonian of the isolated system as , where is the weak interaction between system and thermal bath . Suppose that the composite system is described by a wave function , where ’s are the energy eigenstates of the composite system. By tracing out thermal bath , we obtain the density operator for system , . The system will eventually equilibrate; as an observable, will be close to its long time average, i.e. .
We expand the eigenstate as follows
where and are energy eigenstates of system and thermal bath , respectively. The prime above indicates that the summation is only over eigen-energies satisfying
where is the interaction energy that is usually very small compared to and when long-range interaction is negligible, e.g., gravity, in the system. Two remarks are warranted. (i) The approximation made in Eq.(2) is justified. The equality holds when there is no coupling . We expect it hold when the weak interaction is turned on. (ii) The weak interaction can drive the system to a state with ’s randomly uniformly distributed on the sphere . This random distribution is similar to the idea of ”typicality” (26); (27). The connection between interaction and randomness is widely acknowledged since the details of the interaction is irrelevant to the statistical properties (44); (45). As a result, the average value of is , where is the degeneracy brought by the combination of states. We emphasize that this degeneracy is different from the intrinsic degeneracy of the system and it is due to the existence of in Eq.(3) .
With the approximation made in Eq.(2), we now proceed with our derivation,
The central limit theorem gives the results of the first summation as and the second summation as where is the degeneracy of the thermal bath and we have used that is much smaller than so that .
With the standard argument for the Gibbs distribution (2), we arrive finally at
where defines the temperature of the total system for eigenstate . In this way, we have proved that a subsystem of an isolated equilibrated system is thermalized.
Thermalized state with multiple temperatures. Here we examine Eq. (6) for two typical cases: (i) The coefficients of the composite system have a single sharp peak distribution around energy . This case is considered by others (28) in different formalisms. For this case, the density matrix in Eq.(6) is reduced to . This is exactly the typical Gibbs distribution discussed in all textbook on statistical mechanics. (ii) The coefficients have two well-separated sharp peaks around two energies and . In other words, the composite system (or the heat bath) is in a superposition of numerous eigenstates centered around two very different energy scales. In this case, the thermalized system has two temperatures, for and for , with the following density matrix
where and are the weight of the two distribution peaks. The following is a list of key points for a good understanding of this quantum equilibrium state with two different temperatures.
When the quantum heat bath is in a superposition of states with two well-separated energy scales, each particle in the subsystem always feel different energy scales simultaneously when it exchanges energy with the heat bath. This leads to a thermalized state with two different temperatures.
When a system is in such a state, it consists of two parts, one hot and one cold. However, one can not tell which particle belongs to the hot part and which particle is in the cold part. This is similar to liquid helium. It consists of a superfluid part and a normal fluid part; but no single helium atom can be assigned to either the superfluid part or the normal fluid part.
When an ideal gas is thermalized to such an equilibrium state with two temperatures, each particle in the gas can be roughly viewed as in a superposition state of two different momenta. This is impossible in a classical ideal gas, where each particle has a definite momentum.
Since the total system is isolated, the coefficients s are constants of motion and only depend on the initial condition. As a result, the two peaks in the initial distribution of ’s will remain intact during the whole dynamical process. In other words, the system is stable with the double-peak energy distribution.
If the total system is a superposition of just two different energy eigenstates, the total system is not in an equilibrium state. This is because in this case and the left hand side of the inequality is large. To ensure equilibration, we need , that is to have large number of eigenstates concentrating around two different energy scales.
This state does not describe a statistical ensemble of systems, where some systems are cold and some systems are hot.
Our state is an equilibrium state with multiple temperatures; it is completely different from the usual non-equilibrium state which has different temperatures for different parts.
Our state is not a Schrödinger cat state (47); it does not collapse upon measurement.
For most of the systems that we have encountered in nature or studied in experiment, they are in contact with classical heat bath. However, with the advance of technology, we can now create large quantum systems which can serve as quantum heat bath. Two such examples are Bose-Einstein condensates (BECs) and nuclear spins in a quantum dot, where feasible experiments can be set up to test our prediction. (i) Consider a two-species BEC. One species with larger population is trapped optically in an uneven double-well potential (46) while the smaller species is trapped in a single well potential. The larger species serves as a heat bath with two energy peaks due to uneven double-well potential. By exchanging energy with the larger species, the smaller species should develop a double-peak distribution in momentum space, signaling the existence of two temperatures. Since the uneven double-well has to be kept for the double-peaked energy distribution, the state realized here is not strictly equilibrium and might be more accurately called stationary. Now a two-species BEC has been realized just recently in experiment (48) (ii) Due to the weak coupling to the enviornment, nuclear spins in a quantum dot can be regarded as quantum bath for a long time (49); (50); (51). With the feedback technique that has been demonstrated both theoretically and experimentally (52), one should be readily design a scheme that can put these nuclear spins in a superposition state of two different energy scales and use the electron spin to probe such a state (53).
Note that Fine et al. have also abandoned the transitional microcanonical ensemble and replaced it with “quantum microcanonical” ensemble (56); (55); (54). This is fundamentally different from our approach, where no assumption for an ensemble is needed.
Acknowledgements.We acknowledge helpful discussion with Hongwei Xiong , Qian Niu, and Wang Yao. This work is supported by the NBRP of China (2012CB921300,2013CB921900) and the NSF of China (10825417,11274024,11334001) and the RFDP of China (20110001110091).
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