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Self-generated breaking of time-reversal symmetry in open quantum systems

Gonzalo E. Ordonez, Naomichi Hatano

Published Oct 5, 2016• 0 citations
OpenAlex

Abstract

We show how time-reversal symmetry is automatically broken in simple open quantum systems, modeled by a tight-binding Hamiltonian as well as by the Friedrichs model. We start with an explicitly time-reversal symmetric expansion of unity involving discrete eigenstates of the Hamiltonian. This expansion includes complex-conjugate pairs of resonant and anti-resonant states. An initially time-reversal-symmetric state contains equal contributions from the resonant and anti-resonant states. However, as it evolves in time, the anti-resonant contribution is automatically suppressed, breaking the time-reversal symmetry of the initial state. We show that there is a time-scale for this breaking of time-reversal symmetry, which we associate with the Zeno time. We also compare the time-reversal symmetric expansion with an asymmetric expansion used previously by several researchers. We show how the present time-reversal symmetric expansion bypasses the non-Hilbert nature of the resonant and anti-resonant states, which previously introduced divergences into the asymmetric expansion.

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We consider open quantum systems modeled as discrete lattices. Using a simple model of a single-site coupled to two leads as an example, we show that the time evolution of these systems can be analyzed in terms of an explicitly time-reversal symmetric resolution of unity. This resolution of unity includes both resonant states, which decay in the future, and anti-resonant states, which decay in the past. We show that a time-reversal invariant state contains both resonant and anti-resonant components with equal weights. However, this symmetry is automatically broken as the system evolves in time, with the resonant component becoming much larger than the anti-resonant component for t > 0 (and vice versa for t < 0). We argue that irreversibility is a manifestation of this symmetry breaking. We also compare our present approach with the subdynamics approach developed by Prof. Prigogine and collaborators. Finally, we suggest an extension of our present approach from the level of wave functions to the level of density matrices.

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Naomichi Hatano, Gonzalo Ordonez

We present a new complete set of states for a class of open quantum systems, to be used in expansion of the Green’s function and the time-evolution operator. A remarkable feature of the complete set is that it observes time-reversal symmetry in the sense that it contains decaying states (resonant states) and growing states (anti-resonant states) parallelly. We can thereby pinpoint the occurrence of the breaking of time-reversal symmetry at the choice of whether we solve Schrödinger equation as an initial-condition problem or a terminal-condition problem. Another feature of the complete set is that in the subspace of the central scattering area of the system, it consists of contributions of all states with point spectra but does not contain any background integrals. In computing the time evolution, we can clearly see contribution of which point spectrum produces which time dependence. In the whole infinite state space, the complete set does contain an integral but it is over unperturbed eigenstates of the environmental area of the system and hence can be calculated analytically. We demonstrate the usefulness of the complete set by computing explicitly the survival probability and the escaping probability as well as the dynamics of wave packets. The origin of each term of matrix elements is clear in our formulation, particularly, the exponential decays due to the resonance poles.

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Explicit, spontaneous and dynamical breakdown of the time reversal symmetr

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The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial manner. When open systems are considered then the initial conditions of the environment must be included in the effective dynamics. This is achieved by means of a generalized $\epsilon$-prescription where the non-uniform convergence of the limit $\epsilon\to0$ leaves behind a spontaneous breakdown of the time reversal symmetry.

•Mar 29, 2015•0 citations
We show how time-reversal symmetry is automatically broken in simple open quantum systems, modeled by a tight-binding Hamiltonian as well as by the Friedrichs model. We start with an explicitly time-reversal symmetric expansion of unity involving discrete eigenstates of the Hamiltonian. This expansion includes complex-conjugate pairs of resonant and anti-resonant states. An initially time-reversal-symmetric state contains equal contributions from the resonant and anti-resonant states. However, as it evolves in time, the anti-resonant contribution is automatically suppressed, breaking the time-reversal symmetry of the initial state. We show that there is a time-scale for this breaking of time-reversal symmetry, which we associate with the Zeno time. We also compare the time-reversal symmetric expansion with an asymmetric expansion used previously by several researchers. We show how the present time-reversal symmetric expansion bypasses the non-Hilbert nature of the resonant and anti-resonant states, which previously introduced divergences into the asymmetric expansion.
Abstract Open quantum systems are often represented by non-Hermitian effective Hamiltonians that have complex eigenvalues associated with resonances. In previous work we showed that the evolution of tight-binding open systems can be represented by an explicitly time-reversal symmetric expansion involving all the discrete eigenstates of the effective Hamiltonian. These eigenstates include complex-conjugate pairs of resonant and anti-resonant states. An initially time-reversal-symmetric state contains equal contributions from the resonant and anti-resonant states. Here we show that as the state evolves in time, the symmetry between the resonant and anti-resonant states is automatically broken, with resonant states becoming dominant for t > 0 and anti-resonant states becoming dominant for t < 0 . Further, we show that there is a time-scale for this symmetry-breaking, which we associate with the ‘Zeno time’. We also compare the time-reversal symmetric expansion with an asymmetric expansion used previously by several researchers. We show how the present time-reversal symmetric expansion bypasses the non-Hilbert nature of the resonant and anti-resonant states, which previously introduced exponential divergences into the asymmetric expansion.
It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .
We consider open quantum systems modeled as discrete lattices. Using a simple model of a single-site coupled to two leads as an example, we show that the time evolution of these systems can be analyzed in terms of an explicitly time-reversal symmetric resolution of unity. This resolution of unity includes both resonant states, which decay in the future, and anti-resonant states, which decay in the past. We show that a time-reversal invariant state contains both resonant and anti-resonant components with equal weights. However, this symmetry is automatically broken as the system evolves in time, with the resonant component becoming much larger than the anti-resonant component for t > 0 (and vice versa for t < 0). We argue that irreversibility is a manifestation of this symmetry breaking. We also compare our present approach with the subdynamics approach developed by Prof. Prigogine and collaborators. Finally, we suggest an extension of our present approach from the level of wave functions to the level of density matrices.
We present a new complete set of states for a class of open quantum systems, to be used in expansion of the Green’s function and the time-evolution operator. A remarkable feature of the complete set is that it observes time-reversal symmetry in the sense that it contains decaying states (resonant states) and growing states (anti-resonant states) parallelly. We can thereby pinpoint the occurrence of the breaking of time-reversal symmetry at the choice of whether we solve Schrödinger equation as an initial-condition problem or a terminal-condition problem. Another feature of the complete set is that in the subspace of the central scattering area of the system, it consists of contributions of all states with point spectra but does not contain any background integrals. In computing the time evolution, we can clearly see contribution of which point spectrum produces which time dependence. In the whole infinite state space, the complete set does contain an integral but it is over unperturbed eigenstates of the environmental area of the system and hence can be calculated analytically. We demonstrate the usefulness of the complete set by computing explicitly the survival probability and the escaping probability as well as the dynamics of wave packets. The origin of each term of matrix elements is clear in our formulation, particularly, the exponential decays due to the resonance poles.
The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial manner. When open systems are considered then the initial conditions of the environment must be included in the effective dynamics. This is achieved by means of a generalized ϵ\epsilonϵ-prescription where the non-uniform convergence of the limit ϵ→0\epsilon\to0ϵ→0 leaves behind a spontaneous breakdown of the time reversal symmetry.