For chaotic classical systems, the distribution of return times to a small region of phase space is universal. We propose a simple tool to investigate multiple returns in quantum systems. Numerical evidence for the baker map and kicked top points, also in the quantum case, at a universal distribution.
Classical predictability and coarse-grained evolution of the quantum baker's map
我们研究如何经典预测的粗粒度演化的量子面包师映射依赖于粗粒度的字符。我们的分析将Brun和Hartle [Phys. Rev. D 60，123503（1999）]的早期工作扩展到混沌映射的情况。为了量化的可预测性，我们比较了一个家庭的粗粒度的退相干历史形式主义的熵增加率。我们发现，熵增加的速率是由表征粗粒化的尺度的数量占主导地位。
We present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros (ref. \QCITE{cite}{}{BV}) and Saraceno (ref. \QCITE{cite}{}{S}). We first construct a natural ``baker covering map'' on the plane $\QTO{mathbb}{\mathbb{R}}^{2}$. We then use as the quantum algebra of observables the subalgebra of operators on $L^{2}(\QTO{mathbb}{\mathbb{R}}) $ generated by $\left\{\exp (2πi\hat{x}) ,\exp (2πi\hat{p}) \right\} $ . We construct a unitary propagator such that as $\hbar \to 0$ the classical dynamics is returned. For Planck's constan…
A Parity-Conserving Canonical Quantization for the Baker's Map
我们在这里提出了一个完整的描述量化的面包师的地图。我们使用的方法与Balazs和Voros [BV]和Saraceno [S]中使用的方法完全不同。我们使用{exp（2 Pi ix），exp（2 Pi ip）}生成的算子作为可观测量的量子代数，并构造一个酉传播子，使得当普朗克常数趋于零时，经典动力学返回。当Planck常数满足1/N且N为偶数的积分条件时，当环面上的波函数满足周期性边界条件时，我们证明了动力学可以归结为N维Hilbert空间上的动力学，并且除了对Planck常数阶数作了小的修正外，酉N × N矩阵传播子与文献[BV]中给出的相同.这种修正被证明是保持对称性x->1-x和p->1-p的经典映射的周期边界条件。
Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some p…
一组量子态，动力学相关的经典周期轨道的混沌映射，被用来作为一个基础，其中的描述其量子版本的本征态大大简化。该集合可以通过包括沿着周期轨道的稳定和不稳定流形的短时间传播来改进，从而导致类似于Vergini的疤痕函数的构造（Jour. A \textbf{33}，（2000）4709）。平均参与率用于量化基础的质量。
The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators till the Ehrenfest time. The growth occurs at the exponentia…
The classical Bernoulli and baker maps are two simple models of deterministic chaos. On the level of ensembles, it has been shown that the time evolution operator for these maps admits generalized spectral representations in terms of decaying eigenfunctions. We introduce the quantum version of the Bernoulli map. We define it as a projection of the quantum baker map. We construct a quantum analogue of the generalized spectral representation, yielding quantum decaying states represented by density matrices. The quantum decaying states develop a quasi-fractal shape limited by the quantum uncertai…
We have studied two complementary decoherence measures purity and fidelity for a generic diffusive noise in two different chaotic systems (the baker and the cat maps). For both quantities, we have found classical structures in quantum mechanics - the scar functions - that are specially stable when subjected to environmental perturbations. We show that these quantum states constructed on classical invariants are the most robust significant quantum distributions in generic dissipative maps.
We study the stability of classical structures in chaotic systems when a dissipative quantum evolution takes place. We consider a paradigmatic model, the quantum baker map in contact with a heat bath at finite temperature. We analyze the behavior of the purity, fidelity and Husimi distributions corresponding to initial states localized on short periodic orbits (scar functions) and map eigenstates. Scar functions, that have a fundamental role in the semiclassical description of chaotic systems, emerge as very robust against environmental perturbations. This is confirmed by the study of other st…
We conjecture that in chaotic quantum systems with escape the intensity statistics for resonance states universally follows an exponential distribution. This requires a scaling by the multifractal mean intensity which depends on the system and the decay rate of the resonance state. We numerically support the conjecture by studying the phase-space Husimi function and the position representation of resonance states of the chaotic standard map, the baker map, and a random matrix model, each with partial escape.
