arXiv:quant-ph/0405011 · Physical Review A

Connection between decoherence and fidelity decay in echo dynamics

T. Gorin, Tomaž Prosen, T. H. Seligman, Walter T. Strunz

Published Oct 1, 2004• 45 citations

Abstract

Entanglement between a quantum system and its environment leads to loss of coherence in the former. In general, the temporal fate of coherences is complicated. Here, we establish the connection between decoherence of a central system and fidelity decay in the environment for a variety of situations, including both energy conserving and dissipative couplings. We show how properties of unitary time evolution of the environment can be inferred from the nonunitary evolution of coherences in the central system. This opens up promising ways for measuring Loschmidt echoes in a variety of situations.

DOI: 10.1103/physreva.70.042105

Related papers

Discrepancies between decoherence and the Loschmidt echo

Bernardo Casabone, García Mata, I., Diego A. Wisniacki

Mar 1, 20106 citations

Discrepancies between decoherence and the Loschmidt echo

B. Casabone, I. García-Mata, D. A. Wisniacki

arXiv:0908.2134Europhysics Letters (EPL)Mar 1, 20109 citations

Fidelity and purity decay in weakly coupled composite systems

Marko Žnidarič, Tomaž Prosen

arXiv:quant-ph/0209145Journal of Physics A Mathematical and GeneralFeb 26, 200372 citations

Decoherence, entanglement decay, and equilibration produced by chaotic environments

Gabriela Barreto Lemos, Fabricio Toscano

arXiv:1108.4024Physical Review EJul 25, 201112 citations

Decoherence from spin environments

Fernando Cucchietti, Juan Pablo Paz, Wojciech Hubert Zurek

arXiv:quant-ph/0312207Physical Review ANov 29, 2005220 citations

Decay of the Loschmidt echo in a time-dependent environment

Fernando Cucchietti, Caio Lewenkopf, Horacio M. Pastawski

arXiv:quant-ph/0601077Physical Review EAug 16, 200611 citations

arXiv:quant-ph/0405011 · Physical Review A

Connection between decoherence and fidelity decay in echo dynamics

T. Gorin, Tomaž Prosen, T. H. Seligman, Walter T. Strunz

Published Oct 1, 2004• 45 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.70.042105

Related papers

Discrepancies between decoherence and the Loschmidt echo

Bernardo Casabone, García Mata, I., Diego A. Wisniacki

The Loschmidt echo and the purity are two quantities that can provide invaluable information about the evolution of a quantum system. While the Loschmidt echo characterizes instability and sensitivity to perturbations, purity measures the loss of coherence produced by an environment coupled to the system. For classically chaotic systems both quantities display a number of —supposedly universal— regimes that can lead to think of them as equivalent quantities. We study the decay of the Loschmidt echo and the purity for systems with finite-dimensional Hilbert space and present numerical evidence of some fundamental differences between them.The Loschmidt echo and the purity are two quantities that can provide invaluable information about the evolution of a quantum system. While the Loschmidt echo characterizes instability and sensitivity to perturbations, purity measures the loss of coherence produced by an environment coupled to the system. For classically chaotic systems both quantities display a number of —supposedly universal— regimes that can lead to think of them as equivalent quantities. We study the decay of the Loschmidt echo and the purity for systems with finite-dimensional Hilbert space and present numerical evidence of some fundamental differences between them.

Mar 1, 20106 citations

Discrepancies between decoherence and the Loschmidt echo

B. Casabone, I. García-Mata, D. A. Wisniacki

arXiv:0908.2134Europhysics Letters (EPL)Mar 1, 20109 citations

Fidelity and purity decay in weakly coupled composite systems

Marko Žnidarič, Tomaž Prosen

We study the stability of unitary quantum dynamics of composite systems (for example: central system + environment) with respect to weak interaction between the two parts. Unified theoretical formalism is applied to study different physical situations: (i) coherence of a forward evolution as measured by the purity of the reduced density matrix, (ii) stability of time evolution with respect to small coupling between subsystems and (iii) Loschmidt echo measuring dynamical irreversibility. Stability has been measured either by fidelity of pure states of a composite system, or by the so-called reduced fidelity of reduced density matrices within a subsystem. Rigorous inequality among fidelity, reduced fidelity and purity is proved and a linear response theory is developed expressing these three quantities in terms of time correlation functions of the generator of interaction. The qualitatively different cases of regular (integrable) or mixing (chaotic in the classical limit) dynamics in each of the subsystems are discussed in detail. Theoretical results are demonstrated and confirmed in a numerical example of two coupled kicked tops.We study the stability of unitary quantum dynamics of composite systems (for example: central system + environment) with respect to weak interaction between the two parts. Unified theoretical formalism is applied to study different physical situations: (i) coherence of a forward evolution as measured by the purity of the reduced density matrix, (ii) stability of time evolution with respect to small coupling between subsystems and (iii) Loschmidt echo measuring dynamical irreversibility. Stability has been measured either by fidelity of pure states of a composite system, or by the so-called reduced fidelity of reduced density matrices within a subsystem. Rigorous inequality among fidelity, reduced fidelity and purity is proved and a linear response theory is developed expressing these three quantities in terms of time correlation functions of the generator of interaction. The qualitatively different cases of regular (integrable) or mixing (chaotic in the classical limit) dynamics in each of the subsystems are discussed in detail. Theoretical results are demonstrated and confirmed in a numerical example of two coupled kicked tops.

arXiv:quant-ph/0209145Journal of Physics A Mathematical and GeneralFeb 26, 200372 citations

Decoherence, entanglement decay, and equilibration produced by chaotic environments

Gabriela Barreto Lemos, Fabricio Toscano

We investigate decoherence in quantum systems coupled via dephasing-type interactions to an arbitrary environment with chaotic underlying classical dynamics. The coherences of the reduced state of the central system written in the preferential energy eigenbasis are quantum Loschmidt echoes, which in the strong coupling regime are characterized at long time scales by fluctuations around a constant mean value. We show that due to the chaotic dynamics of the environment, the mean value and the width of the Loschmidt-echo fluctuations are inversely proportional to the quantity we define as the effective Hilbert-space dimension of the environment, which in general is smaller than the dimension of the entire available Hilbert space. Nevertheless, in the semiclassical regime this effective Hilbert-space dimension is in general large, in which case even a chaotic environment with few degrees of freedom produces decoherence without revivals. Moreover we show that in this regime the environment leads the central system to equilibrate to the time average of its reduced density matrix, which corresponds to a diagonal state in the preferential energy eigenbasis. For the case of two uncoupled, initially entangled central systems that interact with identical local quantum environments with chaotic underlying classical dynamics, we show that in the semiclassical limit the equilibration state is arbitrarily close to a separable state. We confirm our results with numerical simulations in which the environment is modeled by the quantum kicked rotor in the chaotic regime.We investigate decoherence in quantum systems coupled via dephasing-type interactions to an arbitrary environment with chaotic underlying classical dynamics. The coherences of the reduced state of the central system written in the preferential energy eigenbasis are quantum Loschmidt echoes, which in the strong coupling regime are characterized at long time scales by fluctuations around a constant mean value. We show that due to the chaotic dynamics of the environment, the mean value and the width of the Loschmidt-echo fluctuations are inversely proportional to the quantity we define as the effective Hilbert-space dimension of the environment, which in general is smaller than the dimension of the entire available Hilbert space. Nevertheless, in the semiclassical regime this effective Hilbert-space dimension is in general large, in which case even a chaotic environment with few degrees of freedom produces decoherence without revivals. Moreover we show that in this regime the environment leads the central system to equilibrate to the time average of its reduced density matrix, which corresponds to a diagonal state in the preferential energy eigenbasis. For the case of two uncoupled, initially entangled central systems that interact with identical local quantum environments with chaotic underlying classical dynamics, we show that in the semiclassical limit the equilibration state is arbitrarily close to a separable state. We confirm our results with numerical simulations in which the environment is modeled by the quantum kicked rotor in the chaotic regime.

arXiv:1108.4024Physical Review EJul 25, 201112 citations

Decoherence from spin environments

Fernando Cucchietti, Juan Pablo Paz, Wojciech Hubert Zurek

We examine two exactly solvable models of decoherence---a central spin-system, (i) with and (ii) without a self-Hamiltonian, interacting with a collection of environment spins. In the absence of a self-Hamiltonian we show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths can lead to a universal (Gaussian) suppression of coherence between pointer states. On the other hand, we show that when the dynamics of the central spin is dominant a different regime emerges, which is characterized by a non-Gaussian decay and a dramatically different set of pointer states. We explore the regimes of validity of the Gaussian decay and discuss its relation to the spectral features of the environment and to the Loschmidt echo (or fidelity).We examine two exactly solvable models of decoherence---a central spin-system, (i) with and (ii) without a self-Hamiltonian, interacting with a collection of environment spins. In the absence of a self-Hamiltonian we show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths can lead to a universal (Gaussian) suppression of coherence between pointer states. On the other hand, we show that when the dynamics of the central spin is dominant a different regime emerges, which is characterized by a non-Gaussian decay and a dramatically different set of pointer states. We explore the regimes of validity of the Gaussian decay and discuss its relation to the spectral features of the environment and to the Loschmidt echo (or fidelity).

arXiv:quant-ph/0312207Physical Review ANov 29, 2005220 citations

Decay of the Loschmidt echo in a time-dependent environment

Fernando Cucchietti, Caio Lewenkopf, Horacio M. Pastawski

We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation $V(\mathbf{q},t)$ with typical strength $\ensuremath{\hbar}∕{\ensuremath{\tau}}_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length ${\ensuremath{\xi}}_{0}$ and a correlation time ${\ensuremath{\tau}}_{0}$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi ``golden rule,'' $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}∕{\ensuremath{\tau}}_{V}^{2}$, where ${\ensuremath{\tau}}_{c}\ensuremath{\sim}\mathrm{min}[{\ensuremath{\tau}}_{0},{\ensuremath{\xi}}_{0}∕v]$ and $v$ is the typical particle velocity. Whenever the rate $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence one can protect the system against decoherence.We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation $V(\mathbf{q},t)$ with typical strength $\ensuremath{\hbar}∕{\ensuremath{\tau}}_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length ${\ensuremath{\xi}}_{0}$ and a correlation time ${\ensuremath{\tau}}_{0}$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi ``golden rule,'' $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}∕{\ensuremath{\tau}}_{V}^{2}$, where ${\ensuremath{\tau}}_{c}\ensuremath{\sim}\mathrm{min}[{\ensuremath{\tau}}_{0},{\ensuremath{\xi}}_{0}∕v]$ and $v$ is the typical particle velocity. Whenever the rate $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence one can protect the system against decoherence.

arXiv:quant-ph/0601077Physical Review EAug 16, 200611 citations

We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation

V(\mathbf{q},t)

with typical strength $\ensuremath{\hbar}∕{\ensuremath{\tau}}_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length ${\ensuremath{\xi}}_{0}$ and a correlation time ${\ensuremath{\tau}}_{0}$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi ``golden rule,'' $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}∕{\ensuremath{\tau}}_{V}^{2}$, where ${\ensuremath{\tau}}_{c}\ensuremath{\sim}\mathrm{min}[{\ensuremath{\tau}}_{0},{\ensuremath{\xi}}_{0}∕v]$ and

v

is the typical particle velocity. Whenever the rate $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence one can protect the system against decoherence.