Fluctuation Theorem for Many-Body Pure Quantum States

Abstract

We prove the second law of thermodynamics and the nonequilibirum fluctuation theorem for pure quantum states. The entire system obeys reversible unitary dynamics, where the initial state of the heat bath is not the canonical distribution but is a single energy-eigenstate that satisfies the eigenstatethermalization hypothesis (ETH). Our result is mathematically rigorous and based on the Lieb-Robinson bound, which gives the upper bound of the velocity of information propagation in many-body quantum systems. The entanglement entropy of a subsystem is shown connected to thermodynamic heat, highlighting the foundation of the information-thermodynamics link. We confirmed our theory by numerical simulation of hard-core bosons, and observed dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our result reveals a universal scenario that the second law emerges from quantum mechanics, and can experimentally be tested by artificial isolated quantum systems such as ultracold atoms.Introduction. Although the microscopic laws of physics do not prefer a particular direction of time, the macroscopic world exhibits inevitable irreversibility represented by the second law of thermodynamics. Modern researches has revealed that even a pure quantum state, described by a single wave function without any genuine thermal fluctuation, can relax to macroscopic thermal equilibrium by a reversible unitary evolution [1][2][3][4][5][6][7][8][9][10][11]. Thermalization of isolated quantum systems, which is relevant to the zeroth law of thermodynamics, is now a very active area of researches in theory [1-6], numerics [10][11][12][13][14][15][16], and experiments [17][18][19][20][21][22][23]. Especially, the concepts of typicality [9,[24][25][26] and the eigenstate thermalization hypothesis (ETH) [10][11][12][27][28][29][30][31][32][33][34][35][36] have played significant roles.However, the second law of thermodynamics, which states that the thermodynamic entropy increases in isolated systems, has not been fully addressed in this context. We would emphasize that the informational entropy (i.e., the von Neumann entropy) of such a genuine quantum system never increases, but is always zero [37]. In this sense, a fundamental gap between the microscopic and macroscopic worlds has not yet been bridged: How does the second law emerge from pure quantum states?In a rather different context, a general theory of the second law and its connection to information has recently been developed even out of equilibrium [38][39][40][41], which has also been experimentally verified in laboratories [42][43][44][45]. This has revealed that information contents and thermodynamic quantities can be treated on an equal footing, as originally illustrated by Szilard and Landauer in the context of Maxwell's demon [46,47]. This research direction invokes a crucial assumption that the heat bath is, at least in the initial time, in the canonical distribution [48]; this special initial condition effectively breaks the

Entropy dynamics of a dephasing model in a squeezed thermal bath

“…Recently, it was noticed that the entropy production in conventional thermodynamics can be understood as the the correlation generation between an open quantum system and its thermal reservoir [13][14][15][16][17][18][19][20][21][22][23]. For example, for a thermal state [21,24].…”

Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization

“…As is the case for the conventional ETH, we can also consider the weaker version of the 2-ETH, which states that the fraction of the eigenstates that do not satisfy the 2-ETH vanishes in the thermodynamic limit. We expect that weak 2-ETH is true even in the integrable case, as is the case of the weak 1-ETH [89][90][91]. However, this is not clear from our numerical data because of the finite-size effect.…”

Anomalous Behavior of Magnetic Susceptibility Obtained by Quench Experiments in Isolated Quantum Systems

“…For the parameters presented here, Eqs. (5) and (6) hold in all three cases, while Eq. (4) only in the XYZ one.…”

“…Introduction-. Ultracold atoms [1,2] and molecules [3][4][5] in optical lattices offer nearly ideal playgrounds for studying quantum many-body systems experimentally. Various model systems [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] are realized on the optical lattices with various geometry [22][23][24][25][26][27] and with tunable physical parameters [2,[28][29][30][31].…”

“…Since a smooth function of E ν /N can be regarded as linear within the narrow region |δE ν | < ∼ T √ c h N , any system satisfying the strong ETH also satisfies condition (8), while notice that the converse is not necessarily true. By contrast, the ordinary weak ETH [53][54][55] requires only that ν|δσ z 0 |ν = o(1) for almost all ν in the same energy region. For this reason, some models that satisfy the ordinary weak ETH do not satisfy Eq.…”