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The average ratio R(t) = ln (Pq(t)/P (t)) as a function of time for different N from N = 2500 (left curve) to N = 129500 (right curve). The horizontal full line corresponds to Pq(t) = P (t). The probability P (t) is given by numerical data obtained with M = 9 · 10 9 orbits for t ≤ 70 and by the fit from Fig.1 (see text) for t > 70. The left insert shows the ratio of the numerically computed classical probability P (t) to the fit function. The deviations from the fit, for t > 70, are due to statistical errors related to finite M. The right insert demonstrates the scaling behaviour of Pq(t) on the variable t/tq where the tq values are determined by condition R(tq) = 0.1.
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We show that the quantum relaxation process in a classically chaotic open dynamical system is characterized by a quantum relaxation time scale t_q. This scale is much shorter than the Heisenberg time and much larger than the Ehrenfest time: t_q ~ g^alpha where g is the conductance of the system and the exponent alpha is close to 1/2. As a result, q...
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We expose two scenarios for the breakdown of quantum multifractality under
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multifractal dimension smoothly goes to the ergodic va...
Citations
... In a certain sense the absorption can be considered as some kind of measurement and it is interesting to understand its influence on EPR characteristics. We note that in the case of quantum evolution of one particle the effects of absorption have been studied in this system in [38][39][40][41][42][43]. The probability that a particle remains inside the system can be considered as a quantum version of Poincaré recurrences [44] which in the classical case of fully chaotic system decays exponentially with time while in the case of divided phase space with stability islands the decay is algebraic (see [45,46] and references therein). ...
... In the following, we compute all quantities (except for P(t) itself ) by first renormalizing the state |ψ(t) → |ψ(t) / |ψ(t) . This model was already extensively studied for the case of one particle in [38,39,42,43]. We note that there are various interesting regimes of Poincare recurrences of entanglement S(t) that requires presentation of various cases given in this section and next two sections. ...
We study numerically the properties of entanglement of two interacting, or noninteracting, particles evolving in a regime of quantum chaos in the quantum Chirikov standard map. Such pairs can be viewed as interacting, on noninteracting, Einstein-Podolsky-Rosen pairs in a regime of quantum chaos. The analysis is done with such tools as the Loschmidt echo of entanglement and the Poincaré recurrences of entanglement in presence of absorption. The obtained results show unusual features of the entropy of entanglement and the spectrum of Schmidt decomposition with their dependence on interactions at different quantum chaos regimes.
... In a certain sense the absorption can be considered as some kind of measurement and it is interesting to understand its influence on EPR characteristics. We note that in the case of quantum evolution of one particle the effects of absorption have been studied in this system in [38,39,40,41,42,43]. The probability that a particle remains inside the system can be considered as a quantum version of Poincaré recurrences [44] which in the classical case of fully chaotic system decays exponentially with time while in the case of divided phase space with stability islands the decay is algebraic (see [45,46] and Refs. ...
... In the following, we compute all quantities (except for P (t) itself) by first renormalizing the state |ψ(t) → |ψ(t) / |ψ(t) . This model was already extensively studied for the case of one particle in [38,39,42,43]. We start with the initial entangled state: ...
We study numerically the properties of entanglement of two interacting, or noninteracting, particles evolving in a regime of quantum chaos in the quantum Chirikov standard map. Such pairs can be viewed as interacting, on noninteracting, Einstein-Podolsky-Rosen pairs in a regime of quantum chaos. The analysis is done with such tools as the Loschmidt echo of entanglement and the Poincar\'e recurrences of entanglement in presence of absorption. The obtained results show unusual features of the entropy of entanglement and the spectrum of Schmidt decomposition with their dependence on interactions at different quantum chaos regimes.
... Quantum chaos is also invoked as an explanation of thermalizing behavior in closed quantum many-body systems [23,[73][74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92] and the emergence of irreversibility [93] resulting from the establishment of a universal pattern of entanglement corresponding to that of a random state in the Hilbert space. The sensitivity of the dynamics can also be captured by the Loschmidt Echo (LE), and it has been shown that for bipartite systems LE and OTOCs coincide [94]. ...
... As we discussed earlier, many quantities have been proposed towards a diagnostic of quantum chaos [23,88,98,126,127], forming an intricate web [98] that necessitates a unifying framework. ...
We present a systematic construction of probes into the dynamics of
isospectral ensembles of Hamiltonians by the notion of Isospectral twirling,
expanding the scopes and methods of ref.[1]. The relevant ensembles of
Hamiltonians are those defined by salient spectral probability distributions.
The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic
Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal
Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the
Isospectral twirling of several classes of important quantities in the analysis
of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCs,
Entanglement, Tripartite mutual information, coherence, distance to equilibrium
states, work in quantum batteries and extension to CP-maps. Moreover, we
perform averages in these ensembles by random matrix theory and show how these
quantities clearly separate chaotic quantum dynamics from non chaotic ones.
... Quantum chaos is also invoked as an explanation of thermalizing behavior in closed quantum many-body systems [17,[65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84] and the emergence of irreversibility [85] resulting from the establishment of a universal pattern of entanglement corresponding to that of a random state in the Hilbert space. The sensitivity of the dynamics can also be captured by the Loschmidt Echo (LE), and it has been shown that for bipartite systems LE and OTOCs coincide [86]. ...
... As we discussed earlier, many quantities have been proposed towards a diagnostic of quantum chaos [17,80,90,116,117], forming an intricate web [90] that necessitates a unifying framework. ...
In this paper, we present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref.[1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCS, Entanglement, Tripartite mutual information, coherence, distance to equilibrium states, work in quantum batteries and extension to CP-maps. Moreover, we perform averages in these ensembles by random matrix theory and show how these quantities clearly separate chaotic quantum dynamics from non chaotic ones.
... The dynamical signatures of quantum chaos in the presence of dissipation and decoherence have been taken up in, e.g. [21][22][23][24]. In the context of open chaotic scattering, the time evolution of decay functions or survival probabilities have been studied using random matrix theory [25][26][27]. ...
... Here we give some heuristic justification for this statement. In [23], it was argued that for open chaotic systems, the survival probability, which is known classically to exhibit exponential decay, experiences significant quantum fluctuations on an intermediate time scale that can be parametrically smaller than the Heisenberg time, but still diverging with N. The classical dynamics then holds for times up to this quantum time scale, which is set by the mean spacing between scattering resonances on the complex plane (as opposed to the energy level spacing which sets the Heisenberg time). In our asymptotic analysis, we work with the limit in which the mean spacing of the Lindblad eigenvalues tends to zero. ...
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... The dynamical signatures of quantum chaos in the presence of dissipation and decoherence have been taken up in, e.g. [19][20][21][22]). In the context of open chaotic scattering, the time evolution of decay functions or survival probabilities have been studied using random matrix theory [23][24][25]. ...
... We only have a heuristic justification for this. In Ref. [21], it was argued that for open chaotic systems, the survival probability, which is known classically to exhibit exponential decay, experiences significant quantum fluctuations on an intermediate time scale that can be parametrically smaller than the Heisenberg time, but still diverging with N . The classical dynamics then holds for times up to this quantum time scale, which is set by the mean spacing between scattering resonances on the complex plane (as opposed to the energy level spacing which sets the Heisenberg time). ...
... LG 11 L † −LG 21 −LG 22 LG ...
We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the long-time behavior of a dissipative form factor, which is the trace of the propagator, and use this as a diagnostic for the existence or absence of a spectral gap in the distribution of eigenvalues of the Lindblad generator. We find that simple generators with a single jump operator are slowly mixing, and relax algebraically in time, due to the closing of the spectral gap in the thermodynamic limit. Introducing additional jump operators or a Hamiltonian opens up a spectral gap which remains finite in the thermodynamic limit, leading to exponential relaxation and thus rapid mixing. We use the method of moments and introduce a novel diagrammatic expansion to determine exactly the form factor to leading order in Hilbert space dimension N. We also present numerical support for our main results.
... [15,16] for reviews, and Refs. [14][15][16][17][18][19][21][22][23][24][25][26][27][28][29][30][31][32][33][34] for some examples. ...
... See Refs. [14][15][16][17][18][19][21][22][23][24][25][26][27][28][29][30][31][32][33][34] for examples. While these heuristic models have stimulated many findings in the recent past, they do not represent a step by step reduction of a scattering system in the sense of Ref. [20] and it remains unclear to which degree they faithfully represent resonances. ...
... [15,16] for a review, and Refs. [14][15][16][17][18][19][21][22][23][24][25][26][27][28][29][30][31][32][33][34] for examples. ...
We consider quantum maps induced by periodically-kicked scattering systems and discuss the computation of their resonance spectra in terms of complex scaling and sufficiently weak absorbing potentials. We also show that strong absorptive and projective openings, as commonly used for open quantum maps, fail to produce the resonance spectra of kicked scattering systems, even if the opening does not affect the classical trapped set. The results are illustrated for a concrete model system whose dynamics resembles key features of ionization and exhibits a trapped set which is organized by a topological horseshoe at large kick strength. Our findings should be useful for future tests of fractal Weyl conjectures and investigations of dynamical tunneling.
... Another characteristic length scale can be pointed out by comparing the orders of magnitude of C 2 (t) and This time scale appeared previously in the context of weak localization in classically chaotic [84,85] and disordered [45] systems, but its role in the analysis of fluctuations of transport properties has never been identified. Because for short pulses in the long-time limit we can write C 2 (t) ∼ (t p /t q )(t/t q ) and C 3 (t) ∼ (t p /t q )(t/t q ) 3 , the formally next-order contribution (in 1/g expansion) to the intensity correlation function, C 3 (t) becomes larger than C 2 (t) when t > t q . ...
This thesis is devoted to the study of coherent effects that arise when a wave propagates in a strongly disordered medium. Several aspects of their dynamics are addressed, with special interest in mesoscopic correlations and Anderson localization.
First, we demonstrate a generalized form of the self-consistent theory of Anderson localization, adapted to the description of finite open media. This theory introduces a diffusion coefficient that depends on position. The importance of this property is highlighted through the investigation of the phenomenon of transverse confinement of waves in disordered media. The theory is also confronted to predictions of the scaling theory of localization.
The second part of the thesis focuses on the dynamics of intensity speckle patterns that arises from the propagation of short pulses in a disordered waveguide. We study the two-point intensity correlation function, which contains the information about the universal conductance fluctuations, well known in mesoscopic conductors. In the dynamic situation, conductance fluctuations acquire a time dependence, are enhanced at long times and lose their universal nature. These results are confirmed by a microwave experiment.
Finally, we investigate the physics of matter-wave speckle patterns, resulting from the expansion of a Bose-Einstein condensate in a random potential. They are found to exhibit long-range correlations that grow with time, and can take negative values. We interpret these results in terms of a random displacement of the center of mass of the condensate. The role of atomic interactions during the expansion of the condensate is discussed.
... Kicked rotator with absorbing boundaries. In [3,7] a kicked rotator with absorbtion was used to model the process of ionization. The classical kicked rotator is Chirikov's standard map on the cylinder, which is a paradigmatic model for transitions from regular to chaotic motion [9]. ...
We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.
... Among the physical models, two-dimensional billiards with attached leads[203][204][205], simplified models of atomic and molecular systems[206][207][208], the kicked rotor with absorbing boundary condition[209][210][211], and scattering on graphs[212,213] could be mentioned. ...
In this work, we discuss the resonance states of a quantum particle in a periodic potential plus a static force. Originally this problem was formulated for a crystal electron subject to a static electric field and it is nowadays known as the Wannier-Stark problem. We describe a novel approach to the Wannier-Stark problem developed in recent years. This approach allows to compute the complex energy spectrum of a Wannier-Stark system as the poles of a rigorously constructed scattering matrix and solves the Wannier-Stark problem without any approximation. The suggested method is very efficient from the numerical point of view and has proven to be a powerful analytic tool for Wannier-Stark resonances appearing in different physical systems such as optical lattices or semiconductor superlattices. Comment: 94 pages, 41 figures, typos corrected, references added
