We investigate how decoherence affects the short-time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model. For a wide range of parameters, the distance between the corresponding phase-space distributions depends on a single parameter $χ$ that relates an effective Planck constant $\hbar_{\rm eff}$, the Lyapunov coeffficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on $\hbar_{\rm eff}$, in agreement with previous estimations of the separation time for Hamiltonian systems. However, for $χ\lesssim 1$, the separation remains small, going down with $\hbar_{\rm eff}^2$, so the concept of separation time loses its meaning.
We study the dynamics of a quantum spin ensemble controlled by trains of ultrashort pulses. To model disturbances of the kicks, we consider that the spins are submitted to different kick trains which follow regular, random, stochastic or chaotic dynamical processes. These are described by dynamical systems on a torus. We study the quantum decoherence and the population relaxation of the spin ensemble induced by these classical dynamical processes disturbing the kick trains. For chaotic kick trains we show that the decoherence and the relaxation processes exhibit a signature of chaos directly r…
Entropy production in quantum (field) systems requiring environment-induced decoherence is described in a Gaussian variational approximation. The new phenomenon of Semiquantum Chaos is reported. (Presented at the International Conference on Nonlinear Dynamics, Chaotic and Complex Systems, Zakopane (Poland), 7-12.11.95.)
We investigate the sensitivity of quantum systems that are chaotic in a classical limit, to small perturbations of their equations of motion. This sensitivity, originally studied in the context of defining quantum chaos, is relevant to decoherence in situations when the environment has a chaotic classical counterpart.
We study analytically and numerically the behavior of quantum entanglement in a quantum computer operating an efficient algorithm for quantum chaos. Our results show that in an ideal algorithm the entanglement decays exponentially with the diffusive relaxation rate induced by classical chaos. This decay goes down to a residual level which drops exponentially with the number of qubits n_q. Decoherence destroys the residual entanglement with a rate exponential in n_q.
We analyze the interplay of chaos, entanglement and decoherence in a system of qubits whose collective behaviour is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement…
In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem we aim to distinguish effects of the two types of dynamics from those depending on the choice of the wave packet. To isolate the former we introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find at intermediate times a weak effect of spectral correlations that is reminiscent of the correlation hole.
We numerically investigate decoherence of a two-spin system (central system) by a bath of many spins 1/2. By carefully adjusting parameters, the dynamical regime of the bath has been varied from quantum chaos to regular, while all other dynamical characteristics have been kept practically intact. We explicitly demonstrate that for a many-body quantum bath, the onset of quantum chaos leads to significantly faster and stronger decoherence compared to an equivalent non-chaotic bath. Moreover, the non-diagonal elements of the system's density matrix decay differently for chaotic and non-chaotic ba…
The decay of the overlap between a wave packet evolved with a Hamiltonian H and the same state evolved with H}+$Σ$ serves as a measure of the decoherence time $τ_ϕ$. Recent experimental and analytical evidence on classically chaotic systems suggest that, under certain conditions, $τ_ϕ$ depends on H but not on $Σ$. By solving numerically a Hamiltonian model we find evidence of that property provided that the system shows a Wigner-Dyson spectrum (which defines quantum chaos) and the perturbation exceeds a crytical value defined by the parametric correlations of the spectra.
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of th…
The famous Nils Bohr's quantum-classical correspondence principle states that the classical mechanics is a limiting case of the more general quantum mechanics. This implies that ``under certain conditions" quantum laws of motion become equivalent to classical laws. One of the conditions is fairly obvious: the corresponding classical action should be very large as compared with the Planck's constant. But this condition is not sufficient. Suppression of effects of quantum interference ("decoherence") is the phenomenon of primary importance for understanding of the Bohr's principle. Being, in ess…
Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is an extremely well-developed and sophisticated field. By contrast, very little work has been done in looking at quantum versions of systems which classically exhibit {\it dissipative} chaos. Using the decoherence formalism of Gell-Mann and Hartle, I find a quantum mechanical analog of one such system, the forced damped Duffing oscillator. I demonstrate the classical limit of the system, and discuss its decoherent histories. I show that using decoherent histories, one can define not only the quantum map of an entire de…
The effect of introducing measuring devices in a ``quantum pinball'' system is shown to lead to a chaotic evolution for the particle position as defined in Bohm's approach to Quantum Mechanics.
A multi-particle extension of the Arnol'd Cat Hamiltonian system is defined and examined. We propose to compute its Alicki-Fannes quantum dynamical entropy, to validate (or disprove) the validity of the decoherence approach to quantum chaos. A first set of numerical experiments is presented and discussed.
This study was started to know mysterious classicality of nuclei. This time, I found a new rule for decoherence. I used a model without chaos. As a result, it was shown that not only the intersection of classical trajectories but also branching of classical trajectories are needed for decoherence. In other words, it was shown that interactions between a main system and environments have to make enough branchings of classical trajectories of the main system for decoherence.
We study the decoherence of a one-particle system, whose classical correpondent is chaotic, when it evolves coupled to a weak quenched environment. This is done by analytical evaluation of the Loschmidt Echo, (i.e. the revival of a localized density excitation upon reversal of its time evolution), in presence of the perturbation. We predict an exponential decay for the Loschmidt Echo with a (decoherence) rate which is asymptotically given by the mean Lyapunov exponent of the classical system, and therefore independent of the perturbation strength, within a given range of strengths. Our results…
We study the effects of dissipation and decoherence induced on a harmonic oscillator by the coupling to a chaotic system with two degrees of freedom. Using the Feynman-Vernon approach and treating the chaotic system semiclassically we show that the effects of the low dimensional chaotic environment are in many ways similar to those produced by thermal baths. The classical correlation and response functions play important roles in both classical and quantum formulations. Our results are qualitatively similar to the high temperature regime of the Caldeira-Leggett model.
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 times 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 ef…
The environment -- external or internal degrees of freedom coupled to the system -- can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i. e., quantum dynamics of systems which are classically chaotic).
We study the ultimate limits to the decoherence rate associated with dephasing processes. Fluctuating chaotic quantum systems are shown to exhibit extreme decoherence, with a rate that scales exponentially with the particle number, thus exceeding the polynomial dependence of systems with fluctuating $k$-body interactions. Our findings suggest the use of quantum chaotic systems as a natural test-bed for spontaneous wave function collapse models. We further discuss the implications on the decoherence of AdS/CFT black holes resulting from the unitarity loss associated with energy dephasing.
Violation of correspondence principle may occur for very macroscopic byt isolated quantum systems on rather short timescales as illustrated by the case of Hyperion, the chaotically tumbling moon of Saturn, for which quantum and classical predictions are expected to diverge on a timescale of approximately 20 years. Motivated by Hyperion, we review salient features of ``quantum chaos'' and show that decoherence is the essential ingredient of the classical limit, as it enables one to solve the apparent paradox caused by the breakdown of the correspondence principle for classically chaotic systems…
We propose a strategy to suppress decoherence of a solid-state qubit coupled to non-Markovian noises by attaching the qubit to a chaotic setup with the broad power distribution in particular in the high-frequency domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-frequency components of our control are generated by the chaotic setup driven by a low-frequency field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide…
Quantum chaos imposes universal spectral signatures that govern the thermofield dynamics of a many-body system in isolation. The fidelity between the initial and time-evolving thermofield double states exhibits as a function of time a decay, dip, ramp and plateau. Sources of decoherence give rise to a nonunitary evolution and result in information loss. Energy dephasing gradually suppresses quantum noise fluctuations and the dip associated with spectral correlations. Decoherence further delays the appearance of the dip and shortens the span of the linear ramp associated with chaotic behavior. …
We present evidence that decoherence can produce a smooth quantum-to-classical transition in nonlinear dynamical systems. High-resolution tracking of quantum and classical evolutions reveals differences in expectation values of corresponding observables. Solutions of master equations demonstrate that decoherence destroys quantum interference in Wigner distributions and washes out fine structure in classical distributions bringing the two closer together. Correspondence between quantum and classical expectation values is also re-established.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well-known that quantum coherence can obliterate many chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact many classical chaotic behavior. How effective decoherence works to sust…
We investigate the decoherence properties of a central system composed of two spins 1/2 in contact with a spin bath. The dynamical regime of the bath ranges from a fully integrable integrable limit to complete chaoticity. We show that the dynamical regime of the bath determines the efficiency of the decoherence process. For perturbative regimes, the integrable limit provides stronger decoherence, while in the strong coupling regime the chaotic limit becomes more efficient. We also show that the decoherence time behaves in a similar way. On the contrary, the rate of decay of magnitudes like lin…
Chaotic evolutions exhibit exponential sensitivity to initial conditions. This suggests that even very small perturbations resulting from weak coupling of a quantum chaotic environment to the position of a system whose state is a non-local superposition will lead to rapid decoherence. However, it is also known that quantum counterparts of classically chaotic systems lose exponential sensitivity to initial conditions, so this expectation of enhanced decoherence is by no means obvious. We analyze decoherence due to a "toy" quantum environment that is analytically solvable, yet displays the cruci…
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, quantum ergodic regime and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates and per gate dissipation rate induced b…
We study the differences between the process of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models wich contain both regular and chaotic environments. In all cases the system of interest is a "quantum walker", i.e. a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment wich has a D dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) timescale t*, wich scales with the dimensionality of t…
We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system--environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.
We study the open system dynamics of a circuit QED model operating in the ultrastrong coupling regime. If the resonator is pumped periodically in time the underlying classical system is chaotic. Indeed, the periodically driven Jaynes-Cummings model in the Born-Oppenheimer approximation resembles a Duffing oscillator which in the classical limit is a well-known example of a chaotic system. Detection of the field quadrature of the output field acts as an effective position measurement of the oscillator. We address how such detection affects the quantum chaotic evolution in this bipartite system.…
We study the decoherence process for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system--environment coupling strength). For longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is inv…
In this work we derive Born's rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to a environement made of "qubits" (i.e., Bohmian pointers) we show that entanglement together with deterministic chaos lead to a fast relaxation from any statistitical distribution $ρ(x)$ (of finding a particle at point $x$) to the Born probability law $|Ψ(x)|^2$. Our model is discussed in the context of Boltzmann's kinetic theory and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.
Environment-induced decoherence has long been recognised as being of crucial importance in the study of chaos in quantum systems. In particular, the exact form and strength of the system-environment interaction play a major role in the quantum-to-classical transition of chaotic systems. In this work we focus on the effect of varying monitoring strategies, i.e. for a given decoherence model and a fixed environmental coupling, there is still freedom on how to monitor a quantum system. We show here that there is a region between the deep quantum regime and the classical limit where the choice of …
We study the behavior of an open quantum system, with an $N$--dimensional space of states, whose density matrix evolves according to a non--unitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's and Harper's map are the two examples we consider). A non--unitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large $N$) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail b…
Using recent results in the field of quantum chaos we derive explicit expressions for the time scale of decoherence induced by the system-environment entanglement. For a generic system-environment interaction and for a generic quantum chaotic system as environment, conditions are derived for energy eigenstates to be preferred states in the weak coupling regime. A simple model is introduced to numerically confirm our predictions. The results presented here may also help understanding the dynamics of quantum entanglement generation in chaotic quantum systems.
We show that some classically chaotic quantum systems uncoupled from noisy environments may generate intrinsic decoherence with all its associated effects. In particular, we have observed time irreversibility and high sensitivity to small perturbations in the initial conditions in a quasiperiodic version of the kicked rotor. The existence of simple quantum systems with intrinsic decoherence clarifies the quantum--classical correspondence in chaotic systems.
Dynamical irreversibility and local decoherence in quantum many-body chaos
Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）；Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS)（基础科学研究院复杂系统理论物理中心）
Driven chaotic systems are of interest in mesoscopic physics, as well as in nuclear, atomic and molecular physics. Such systems [coordinates $(Q,P)$]$ tend to absorb energy. This irreversible effect is known as dissipation. "Driving" means that a parameter $x$ is changed in time. More generally, $x$ may be a dynamical variable. In such case the interaction of $(x,p)$ with the environmental degrees of freedom $(Q,P)$ leads to dephasing as well as to dissipation. We introduce a general framework for the analysis of dissipation and dephasing, and we clarify the tight connection to recent studies …
The following statements belonging to the folklore of the theory of environmental decoherence are shown to be incorrect: 1) linear coupling to harmonic oscillator bath is a universal model of decoherence, 2) chaotic environments are more efficient decoherers.
The population decay due to a small opening in an otherwise closed cavity supporting chaotic classical dynamics displays a quantum correction on top of the classical exponential form, a pure manifestation of quantum coherence that acquires a universal form and can be explained by path interference. Being coherent, such enhancement is prone to decoherence effects due to the coupling of the system to an external environment. We study this interplay between incoherent and coherent quantum corrections to decay by evaluating, within a Caldeira-Legget scenario, off-diagonal contributions to the deco…
We address the issue of the quantum-classical correspondence in chaotic systems using, as recently done by Zurek [e-print quant-ph/9802054], the solar system as a whole as a case study: this author shows that the classicality of the planetary motion is ensured by the environment-induced decoherence. We show that equivalent results are provided by the theories of spontaneous fluctuations and that these latter theories, in some cases, result in a still faster process of decoherence. We show that, as an additional benefit, the assumption of spontaneous fluctuation makes it possible to genuinely d…
The manner in which unpredictable chaotic dynamics manifests itself in quantum mechanics is a key question in the field of quantum chaos. Indeed, very distinct quantum features can appear due to underlying classical nonlinear dynamics. Here we observe signatures of quantum nonlinear dynamics through the direct measurement of the time-evolved Wigner function of the quantum-kicked harmonic oscillator, implemented in the spatial degrees of freedom of light. Our setup is decoherence-free and we can continuously tune the semiclassical and chaos parameters, so as to explore the transition from regul…
This paper is motivated by the suggestion [W. Zurek, Physica Scripta, T76, 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become non-classical within 20 years, but for the effects of environmental decoherence. The dynamics of quantum and classical probability distributions are compared for a satellite rotating perpendicular to its orbital plane, driven by the gravitational gradient. The model is studied with and without environmental decoherence. Without decoherence, the maximum quantum-classical (QC) differences in its average angular momentum scale as hbar^{2/3} for ch…
A popular model of decoherence based on the linear coupling to harmonic oscillator heat baths is analized and shown to be inappropriate in the regime where decoherence dominates over energy dissipation, called pure decoherence regime. The similar mechanism essentially related to the energy conservation implies that, on the contrary to the recent conjectures, chaotic environments can be less efficient decoherers than regular ones. Finally, the elastic scattering mechanism is advocated as the simplest source of pure decoherence.
We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength $ε$. The critical border $ε_c$ between two phases drops polynomially with the number of qubits $n_q$ as $ε_c \sim n_q^{-3/2}$. In the regular phase $(ε< ε_c)$ the algorithm remains robust against imperfections showing the efficiency gain $…
In the long quest to identify and compensate the sources of decoherence in many-body systems far from the ground state, the varied family of Loschmidt echoes (LEs) became an invaluable tool in several experimental techniques. A LE involves a time-reversal procedure to assess the effect of perturbations in a quantum excitation dynamics. However, when addressing macroscopic systems one is repeatedly confronted with limitations that seem insurmountable. This led to formulate the \textit{central hypothesis of irreversibility} stating that the time-scale of decoherence, $T_3$, is proportional to th…
Using the methods of quantum trajectories we study effects of dissipative decoherence on the accuracy of the Grover quantum search algorithm. The dependence on the number of qubits and dissipation rate are determined and tested numerically with up to 16 qubits. As a result, our numerical and analytical studies give the universal law for decay of fidelity and probability of searched state which are induced by dissipative decoherence effects. This law is in agreement with the results obtained previously for quantum chaos algorithms.
In a recent paper [Nature 412, 712 (2001)], Zurek has argued that (1) time evolution typically causes chaotic quantum systems to generate structure that varies on the scale of phase-space volume elements of size $(\hbar^2/A)^d$, where A is a classical action characteristic of the state and d is the number of degrees of freedom, and that (2) this structure implies that a small change in a phase-space coordinate X by an amount $δX \sim \hbar X/A$ generically results in an orthogonal state. While we agree with (1), we argue that (2) is not correct if the number of degrees of freedom is small. Our…
Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $ς={\rm Tr}ρ^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $\bar M(t)$. It is given by the average squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $ς(t)$ and $\bar M(t)$. In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classical…
We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equa…
We show that classical-quantum correspondence of center of mass motion in two coupled delta-kicked rotors can be obtained from intrinsic decoherence of the system itself which occurs due to the entanglement of the center of mass motion to the internal degree of freedom without coupling to external environment.
Based on the connection between the spectral form factor and the probability to return, the origin of the $1/f^α$-noise in fully chaotic and fully integrable systems is traced to the quantum interference between invariant manifolds of the classical dynamics and the dimensionality of those invariant manifolds. This connection and the order-to-chaos transition are analyzed in terms of the statistics of Floquet's quasienergies of a classically chaotic driving non-linear system. An immediate prediction of the connection established here is that in the presence of decoherence, the spectral exponent…
A direct classical analog of the quantum dynamics of intrinsic decoherence in Hamiltonian systems, characterized by the time dependence of the linear entropy of the reduced density operator, is introduced. The similarities and differences between the classical and quantum decoherence dynamics of an initial quantum state are exposed using both analytical and computational results. In particular, the classicality of early-time intrinsic decoherence dynamics is explored analytically using a second-order perturbative treatment, and an interesting connection between decoherence rates and the stabil…
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 on 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 ev…
The quantum-classical correspondence for dynamics of the nonlinear classically chaotic systems is analysed. The problem of quantum chaos consists of two parts: the quasiclassical quantisation of the chaotic systems and attempts to understand the classical chaos in terms of quantum mechanics. The first question has been partially solved by the Gutzwiller semiclassical trace formula for the eigenvalues of chaotic systems, while the classical chaos may be derived from quantum equations only introducing the decoherence process due to interaction with system's environment or intermediate frequent m…
A direct classical analog of quantum decoherence is introduced. Similarities and differences between decoherence dynamics examined quantum mechanically and classically are exposed via a second-order perturbative treatment and via a strong decoherence theory, showing a strong dependence on the nature of the system-environment coupling. For example, for the traditionally assumed linear coupling, the classical and quantum results are shown to be in exact agreement.
I give a pedagogical overview of decoherence and its role in providing a dynamical account of the quantum-to-classical transition. The formalism and concepts of decoherence theory are reviewed, followed by a survey of master equations and decoherence models. I also discuss methods for mitigating decoherence in quantum information processing and describe selected experimental investigations of decoherence processes.
