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Bulletin of the American Physical Society

Time-reversal symmetric expansion of the time evolution operator of open quantum systems

Naomichi Hatano, Gonzalo E. Ordonez

Published Mar 3, 2014• 0 citations
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The arrow of time in open quantum systems and dynamical breaking of the resonance–anti-resonance symmetry

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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.

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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 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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Time Evolution in Closed Quantum Systems

́Angel Rivas, Susana F. Huelga

SpringerBriefs in physics•Sep 29, 2011•4 citations
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.
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.
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.