Complex saddle trajectories for multidimensional quantum wave packet and coherent state propagation: Application to a many-body system

Annealing approach to root finding

The Newton-Raphson method is a fundamental root-finding technique with numerous applications in physics. In this study, we propose a parameterized variant of the Newton-Raphson method, inspired by principles from physics. Through analytical and empirical validation, we demonstrate that this approach offers increased robustness and faster convergence during root-finding iterations. Furthermore, we establish connections to the Adomian series method and provide a natural interpretation within a series framework. Remarkably, the introduced parameter, akin to a temperature variable, enables an annealing approach. This advancement sets the stage for a fresh exploration of numerical iterative root-finding methodologies.

Reduced dimensional Monte Carlo method: Preliminary integrations

A technique for reducing the number of integrals in a Monte Carlo calculation is introduced. For integrations relying on classical or mean-field trajectories with local weighting functions, it is possible to integrate analytically at least half of the integration variables prior to setting up the particular Monte Carlo calculation of interest, in some cases more. Proper accounting of invariant phase space structures shows that the system's dynamics is reducible into composite stable and unstable degrees of freedom. Stable degrees of freedom behave locally in the reduced dimensional phase space exactly as an analogous integrable system would. Classification of the unstable degrees of freedom is dependent upon the degree of chaos present in the dynamics. The techniques for deriving the requisite canonical coordinate transformations are developed and shown to block diagonalize the stability matrix into irreducible parts. In doing so, it is demonstrated how to reduce the amount of sampling directions necessary in a Monte Carlo simulation. The technique is illustrated by calculating return probabilities and expectation values for different dynamical regimes of a two-degrees-of-freedom coupled quartic oscillator within a classical Wigner method framework.

Controlling quantum chaos: Time-dependent kicked rotor

One major objective of controlling classical chaotic dynamical systems is exploiting the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. In a recent Letter [Phys. Rev. Lett. 130, 020201 (2023)0031-900710.1103/PhysRevLett.130.020201], a generalization of this targeting method to quantum systems was demonstrated using successive unitary transformations that counter the natural spreading of a quantum state. In this paper further details are given and an important quite general extension is established. In particular, an alternate approach to constructing the coherent control dynamics is given, which introduces a time-dependent, locally stable control Hamiltonian that continues to use the chaotic heteroclinic orbits previously introduced, but without the need of countering quantum state spreading. Implementing that extension for the quantum kicked rotor generates a much simpler approximate control technique than discussed in the Letter, which is a little less accurate, but far more easily realizable in experiments. The simpler method's error can still be made to vanish as ℏ→0.

Chaos in the three-site Bose-Hubbard model: Classical versus quantum

We consider a quantum many-body system-the Bose-Hubbard system on three sites-which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multivalued function of energy.

Controlling Quantum Chaos: Optimal Coherent Targeting

One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a preeminent quantum chaotic paradigm, to illustrate the method, and investigate its accuracy. The method's error can be made to vanish as ℏ→0.

Semiclassical propagation of coherent states and wave packets: Hidden saddles

Semiclassical methods are extremely important in the subjects of wave-packet and coherent-state dynamics. Unfortunately, these essentially saddle-point approximations are considered nearly impossible to carry out in detail for systems with multiple degrees of freedom due to the difficulties of solving the resulting two-point boundary-value problems. However, recent developments have extended the applicability to a broader range of systems and circumstances. The most important advances are first to generate a set of real reference trajectories using appropriately reduced dimensional spaces of initial conditions, and second to feed that set into a Newton-Raphson search scheme to locate the exposed complex saddle trajectories. The arguments for this approach were based mostly on intuition and numerical verification. In this paper, the methods are put on a firmer theoretical foundation and then extended to incorporate saddles hidden from Newton-Raphson searches initiated with real trajectories. This hidden class of saddles is relevant to tunneling-type processes, but a hidden saddle can sometimes contribute just as much as or more than an exposed one. The distinctions between hidden and exposed saddles clarifies the interpretation of what constitutes tunneling for wave packets and coherent states in the time domain.

Corrected Maslov index for complex saddle trajectories

Saddle-point approximations, extremely important in a wide variety of physical contexts, require the analytical continuation of canonically conjugate quantities to complex variables in quantum mechanics. An important component of this approximation's implementation is arriving at the phase correction attributable to caustics, which involves determinantal prefactors. The common prescription of using the inverse of half a certain determinant's total accumulated phase sometimes leads to sign errors. The root of this problem is traced to the zeros of the determinants at complex times crossing the real time axis. Deformed complex time contours around the zeros can repair the sign errors that sometimes occur, but a much more practical way is given that links saddles back to associated real trajectories and avoids the necessity of locating the complex time zeros of the determinants.

Eigenstate thermalization scaling in approaching the classical limit

According to the eigenstate thermalization hypothesis (ETH), the eigenstate-to-eigenstate fluctuations of expectation values of local observables should decrease with increasing system size. In approaching the thermodynamic limit-the number of sites and the particle number increasing at the same rate-the fluctuations should scale as ∼D^{-1/2} with the Hilbert space dimension D. Here, we study a different limit-the classical or semiclassical limit-by increasing the particle number in fixed lattice topologies. We focus on the paradigmatic Bose-Hubbard system, which is quantum-chaotic for large lattices and shows mixed behavior for small lattices. We derive expressions for the expected scaling, assuming ideal eigenstates having Gaussian-distributed random components. We show numerically that, for larger lattices, ETH scaling of physical midspectrum eigenstates follows the ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via a different exponent. We examine several plausible mechanisms for this anomalous scaling.

Semiclassical evaluation of expectation values

Semiclassical mechanics allows for a description of quantum systems which preserves their phase information, and thus interference effects, while using only the system's classical dynamics as an input. In particular one of the strengths of a semiclassical description is to present a coherent picture which (to negligible higher-order ℏ corrections) is independent of the particular canonical coordinates used. However, this coherence relies heavily on the use of the stationary phase approximation. It turns out, however, that in some important cases, a brutal application of stationary phase approximation washes out all interference, and thus quantum, effects. In this paper, we address this issue in detail in one of its simplest instantiations, namely the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. The efficiency of our approach is based on the fact that we treat analytically all the integrals that can be performed within the stationary phase approximation, implying that the remaining integrals are simple integrals, in the sense that the integrand has no significant variations on the quantum scale (and thus they are very easy to perform numerically). This to be contrasted with other approaches such as the ones based on initial value representation, popular in chemical and molecular physics, which avoid a root search for the classical dynamics, but at the cost of performing numerically integrals whose evaluation requires a sampling on the quantum scale, and which are therefore not well designed to address the deep semiclassical regime. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.

Enhancement of Many-Body Quantum Interference in Chaotic Bosonic Systems: The Role of Symmetry and Dynamics

Although highly successful, the truncated Wigner approximation (TWA) leaves out many-body quantum interference between mean-field Gross-Pitaevskii solutions as well as other quantum effects, and is therefore essentially classical. Turned around, if a system's quantum properties deviate from TWA, they must be exhibiting some quantum phenomenon, such as localization, diffraction, or tunneling. Here, we examine a particular interference effect arising from discrete symmetries, which can significantly enhance quantum observables with respect to the TWA prediction, and derive an augmented TWA in order to incorporate them. Using the Bose-Hubbard model for illustration, we further show strong evidence for the presence of dynamical localization due to remaining differences between the TWA predictions and quantum results.

Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-Heisenberg time scales

Despite considerable progress during the past decades in devising a semiclassical theory for classically chaotic quantum systems a quantitative semiclassical understanding of their dynamics at late times (beyond the so-called Heisenberg time T_{H}) is still missing. This challenge, corresponding to resolving spectral structures on energy scales below the mean level spacing, is intimately related to the quest for semiclassically restoring unitary quantum evolution. Guided through insights for quantum graphs we devise a periodic-orbit resummation procedure for spectra of quantum chaotic systems invoking periodic-orbit self-encounters as the structuring element of a hierarchical phase space dynamics. Quantum unitarity is reflected in real-valued spectral determinants with zeros giving discrete energy levels. We propose a way to purely semiclassically construct such real spectral determinants based on two major underlying mechanisms. (i) Complementary contributions to the spectral determinant from regrouped pseudo-orbits of duration T<T_{H} and T_{H}-T are complex conjugate to each other. (ii) Contributions from long periodic orbits involving multiple traversals along shorter orbits cancel out. We furthermore discuss implications for interacting N-particle quantum systems with a chaotic classical large-N limit that have recently attracted particular interest in the context of many-body quantum chaos.

Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators

Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τ_{E} in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τ_{E}=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τ_{E}, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems.