Highly excited many-particle states in quantum systems such as nuclei, atoms, quantum dots, spin systems, quantum computers etc., can be considered as ``chaotic'' superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is due to a very high level density of many-body states that are easily mixed by a residual interaction between particles (quasi-particles). For such systems, we have derived simple analytical expressions for the time dependence of energy width of wave packets, as well as for the entropy, number of principal basis components and inverse participation ratio, and tested them in numerical experiments. It is shown that the energy width $Δ(t)$ increases linearly and very quickly saturates. The entropy of a system increases quadratically, $S(t) \sim t^2$ at small times, and after, can grow linearly, $S(t) \sim t$, before the saturation. Correspondingly, the number of principal components determined by the entropy, $N_{pc} \sim exp{(S(t))}$, or by the inverse participation ratio, increases exponentially fast before the saturation. These results are explained in terms of a cascade model which describes the flow of excitation in the Fock space of basis components. Finally, a striking phenomenon of damped oscillations in the Fock space at the transition to an equilibrium is discussed.
We investigate the time evolution of wave packets in systems with a mixed phase space where regular islands and chaotic motion coexist. For wave packets started in the chaotic sea on average the weight on a quantized torus of the regular island increases due to dynamical tunneling. This flooding weight initially increases linearly and saturates to a value which varies from torus to torus. We demonstrate for the asymptotic flooding weight universal scaling with an effective tunneling coupling for quantum maps and the mushroom billiard. This universality is reproduced by a suitable random matrix…
In the paper, we employ a wavefunction approach to investigate the evolution of a two-photon wave packet propagating in a one-dimensional waveguide coupled to the Jaynes-Cummings (JC) system. We derive and solve, both analytically and numerically, a set of equations of motion governing the quantum state of the system. That allows us to provide real-time analysis of the evolution of the wave packet two-photon joint spectrum (2PJS) and the excitation dynamics of the JC system in the course of its interaction with the two-photon pulse. We demonstrate that the 2PJS and the spectrum of the wave pac…
The ultimate semiclassical wave packet propagation technique is a complex, time-dependent WBK method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enable…
A new quantum mechanical description of the dynamics of wave packet under continuous measurement is formulated via Bohmian mechanics. The solution to this equation is found through a wave packet approach which establishes a direct correlation between a classical variable with a quantum variable describing the dynamics of the center of mass and the width of the wave packet. The approach presented in this paper gives a comparatively clearer picture than approaches using restrited path integrals and master equation approaches. This work shows how the extremely irregular character of classical cha…
The radiation emitted by a classically pumped three-level $Λ$-type emitter in a resonator cavity featuring both radiative and unwanted losses is studied. In particular, the efficiency of one-photon Fock state excitation of the outgoing wave packet and the spatio-temporal shape of the wave packet are investigated. It is shown that for given shape of the emitter--cavity interaction, adjusting the shape of the pump pulse renders it possible to generate the emission of one-photon Fock state wave packets of desired shape with high efficiency.
Using the quantum trajectories approach we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable…
Within the framework of exact quantum electrodynamics in dispersing and absorbing media, we have studied the quantum state of the radiation emitted from an initially in the upper state prepared two-level atom in a high-$Q$ cavity, including the regime where the emitted photon belongs to a wave packet that simultaneously covers the areas inside and outside the cavity. For both continuing atom--field interaction and short-term atom--field interaction, we have determined the spatio-temporal shape of the excited outgoing wave packet and calculated the efficiency of the wave packet to carry a one-p…
We present a quantitative semiclassical theory for the decay of nondispersive electronic wave packets in driven, ionizing Rydberg systems. Statistically robust quantities are extracted combining resonance assisted tunneling with subsequent transport across chaotic phase space and a final ionization step.
We derive quantum trajectories (also known as stochastic master equations) that describe an arbitrary quantum system probed by a propagating wave packet of light prepared in a continuous-mode Fock state. We consider three detection schemes of the output light: photon counting, homodyne detection, and heterodyne detection. We generalize to input field states that are superpositions and or mixtures of Fock states and illustrate the formalism with several examples.
We describe the quantum mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs. We find that the time scale $τ$ on which this approximation breaks down in a chaotic system is larger than the Ehrenfest times considered previously. In one dimension $τ=\fr{7}{6}λ^{-1}\ln(A/\hbar)$, with $λ$ the Lyapunov exponent and $A$ a typical classical action.
We discover numerically that a moving wave packet in a quantum chaotic billiard will always evolve into a quantum state, whose density probability distribution is exponential. This exponential distribution is found to be universal for quantum chaotic systems with rigorous proof. In contrast, for the corresponding classical system, the distribution is Gaussian. We find that the quantum exponential distribution can smoothly change to the classical Gaussian distribution with coarse graining.
Treating the ideal coherent state as a reference state, the effects due to departure from coherence of an initial wave packet propagating through a nonlinear medium, were examined, specifically in the context of non-classical effects such as revivals, fractional revivals, squeezing and higher-order squeezing during its temporal evolution. Further, these studies were extended to examine the role of quantum entanglement in bipartite systems. The dynamics of quantum expectation values were tracked carefully in various cases and the conditions under which a wave packet spreads chaotically, were in…
We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position-momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure o…
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.
A measure of nonclassicality of quantum states based on the volume of the negative part of the Wigner function is proposed. We analyze this quantity for Fock states, squeezed displaced Fock states and cat-like states defined as coherent superposition of two Gaussian wave packets.
We have performed time-dependent wave packet simulations of realistic Aharonov-Bohm (AB) devices with a quantum dot embedded in one of the arms of the interferometer. The AB ring can function as a measurement device for the intrinsic transmission phase through the quantum dot, however, care has to be taken in analyzing the influence of scattering processes in the junctions of the interferometer arms. We consider a harmonic quantum dot and show how the Darwin-Fock spectrum emerges as a unique pattern in the interference fringes of the AB oscillations.
Stimulated Raman interaction of a classically pumped single three-level $Λ$-type atom in a resonator cavity featuring both radiative and unwanted losses is studied. It is shown that in the regime of stimulated adiabatic Raman passage the excited outgoing wave packet of the cavity-assisted electromagnetic field can be prepared in a one-photon Fock state with high efficiency. In this regime, the spatio-temporal shape of the wave packet does not depend on the interaction shape of the pump field, provided that the interaction of the atom with the pump is time-delayed with respect to the interactio…
An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map. We discuss a possible implementation of such a chaotic quantum computation by using the atomic quantum computer with quantum gates described by the Hartree-Fock equations. In this case, in principle, one can build not only standard linear quantum gates but also nonlinear gat…
The driven non-linear duffing osillator is a very good, and standard, example of a quantum mechanical system from which classical-like orbits can be recovered from unravellings of the master equation. In order to generated such trajectories in the phase space of this oscillator in this paper we use a the quantum jumps unravelling together with a suitable application of the correspondence principle. We analyse the measured readout by considering the power spectra of photon counts produced by the quantum jumps. Here we show that localisation of the wave packet from the measurement of the oscilla…
We experimentally demonstrate coherent control of a quantum system, whose dynamics is chaotic in the classical limit. Interaction of diatomic molecules with a periodic sequence of ultrashort laser pulses leads to the dynamical localization of the molecular angular momentum, a characteristic feature of the chaotic quantum kicked rotor. By changing the phases of the rotational states in the initially prepared coherent wave packet, we control the rotational distribution of the final localized state and its total energy. We demonstrate the anticipated sensitivity of control to the exact parameters…
It is shown that the population Rabi-floppings in a lossless two-level atom, interacting with a monochromatic electromagnetic field, in general are convergent in time. The well-known continuous floppings take place because the restricted choosing of initial conditions, that is when the atom initially is chosen on ground or excited level before the interaction, simultaneously having a definite value of momentum there. The convergence of Rabi-floppings in atomic wave-packet-states is a direct consequence of Doppler effect on optical transition rates (Rabi-frequencies): it gradually leads to ''ir…
Time reversal mirrors have been successfully implemented for various kinds of waves propagating in complex media. In particular, acoustic waves in chaotic cavities exhibit a refocalization that is extremely robust against external perturbations or the partial use of the available information. We develop a semiclassical approach in order to quantitatively describe the refocusing signal resulting from an initially localized wave-packet. The time-dependent reconstructed signal grows linearly with the temporal window of injection, in agreement with the acoustic experiments, and reaches the same sp…
We provide a path integral treatment of the decoherence process induced by a heat bath on a single particle whose dynamics is classically chaotic and show that the decoherence rate is given by the Lyapunov exponent. The loss of coherence is charaterized by the purity, which is calculated semiclassically within diagonal approximation, when the particle initial state is a single gaussian wave packet. The calculation is performed for weak dissipation and in the high temperature limit. This situation allows us to simplify the heat bath description to a single random potential. Although the dissipa…
We consider a model Hamiltonian describing the interaction of a single-mode radiation field with the atoms of a nonlinear medium, and study the dynamics of entanglement for specific non-entangled initial states of interest: namely, those in which the field mode is initially in a Fock state, a coherent state, or a photon-added coherent state. The counterparts of near-revivals and fractional revivals are shown to be clearly identifiable in the entropy of entanglement. The ``overlap fidelity'' of the system is another such indicator, and its behaviour corroborates that of the entropy of entanglem…
Integrable quantum systems of finite size are generically robust against weak enough integrability-breaking perturbations, but become quantum chaotic and thermalizing if the integrability-breaking is strong enough. We argue that the onset of quantum chaos can be described as a Fock-space delocalization process, with the eigenstates of the integrable system being taken as the "Fock states". The integrability-breaking perturbation introduces hopping in this Fock space, and chaos sets in when this hopping delocalizes the many-body eigenstates in this space. Depending on the range of the dominant …
We investigate the recurrence properties of the time series of quantum mechanical expectation values, in terms of two representative models for a single-mode radiation field interacting with a nonlinear medium. From recurrence-time distributions, return maps and recurrence plots, we conclude that the dynamics of appropriate observables pertaining to the field can vary from quasiperiodicity to hyperbolicity, depending on the extent of the nonlinearity and of the departure from coherence of the initial state of the field. We establish that, in a simple bipartite model in which the field is effec…
We present a time dependent quantum calculation of the scattering of a few-photon pulse on a single atom. The photon wave packet is assumed to propagate in a transversely strongly confined geometry, which ensures strong atom-light coupling and allows a quasi 1D treatment. The amplitude and phase of the transmitted, reflected and transversely scattered part of the wave packet strongly depend on the pulse length (bandwidth) and energy. For a transverse mode size of the order of $λ^2$, we find nonlinear behavior for a few photons already, or even for a single photon. In a second step we study the…
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 pre…
The dynamics of a (quasi)one-dimensional interacting atomic Bose-Einstein condensate in a tilted optical lattice is studied in a discrete mean-field approximation, i.e., in terms of the discrete nonlinear Schrödinger equation. If the static field is varied the system shows a plethora of dynamical phenomena. In the strong field limit we demonstrate the existence of (almost) non-spreading states which remain localized on the lattice region populated initially and show coherent Bloch oscillations with fractional revivals in the momentum space (so called quantum carpets). With decreasing field, th…
Brooding over bosons, wave packets and Bose - Einstein correlations, we present a generic quantum mechanical system that contains arbitrary number of bosons characterized by wave-packets and that can undergo a Bose-Einstein condensation either by cooling, or increasing the number density of bosons, or by increasing the overlap of the multi-boson wave-packet states, achieved by changing the size of the single-particle wave-packets. We show that the n-particle correlations may mimic coherent or chaotic behaviour for certain limiting wave-packet sizes. Effects of complete n-particle symmetrizatio…
Superradiance in an ensemble of atoms leads to the collective enhancement of radiation in a particular mode shared by the atoms in their spontaneous decay from an excited state. The quantum aspects of this phenomenon are highlighted when such collective enhancement is observed in the emission of a single quantum of light. Here we report a further step in exploring experimentally the nonclassical features of superradiance by implementing the process not only with single excitations, but also in a two-excitations state. Particularly we measure and theoretically model the wave-packets correspondi…
Wave packet motion in a quantized electromagnetic field: Analytic results
ELI ALPS, The Extreme Light Infrastructure ERIC（ELI ALPS，极端光基础设施欧洲研究基础设施公司）；Department of Theoretical Physics, University of Szeged（塞格德大学理论物理系）；Bolyai Institute, University of Szeged（塞格德大学博莱研究所）；Department of Basic Sciences, Faculty of Mechanical Engineering and Automation, John von Neumann University（约翰·冯·诺依曼大学机械与自动化学院基础科学系）；Faculty of Physics, Babeş-Bolyai University（巴贝什-博约伊大学物理学院）
Coherent motion of cold atoms in a standing-wave field is interpreted as a propagation in two optical potentials. It is shown that the wave-packet dynamics can be either regular or chaotic with transitions between these potentials after passing field nodes. Manifestations of de Broglie-wave chaos are found in the behavior of the momentum and position probabilities and the Wigner function. The probability of those transitions depends on the ratio of the squared detuning to the Doppler shift and is large in that range of the parameters where the classical motion is shown to be chaotic.
We study the time evolution of two wave packets prepared at the same initial state, but evolving under slightly different Hamiltonians. For chaotic systems, we determine the circumstances that lead to an exponential decay with time of the wave packet overlap function. We show that for sufficiently weak perturbations, the exponential decay follows a Fermi golden rule, while by making the difference between the two Hamiltonians larger, the characteristic exponential decay time becomes the Lyapunov exponent of the classical system. We illustrate our theoretical findings by investigating numerical…
We report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two correspondi…
We prove a quantum version of the second law of thermodynamics: the (quantum) Boltzmann entropy increases if the initial (zero time) density matrix decoheres, a condition generally satisfied in Nature. It is illustrated by a model of wave-packet reduction, the Coleman-Hepp model, along the framework introduced by Sewell in his approach to the quantum measurement problem. Further models illustrate the monotonic-versus-non-monotonic behavior of the quantum Boltzmann entropy in time. As a last closely related topic, decoherence, which was shown by Narnhofer and Thirring to enforce macroscopic pur…
Resonant systems emerge as weakly nonlinear approximations to problems with highly resonant linearized perturbations. Examples include nonlinear Schroedinger equations in harmonic potentials and nonlinear dynamics in Anti-de Sitter spacetime. The classical dynamics within this class of systems can be very rich, ranging from fully integrable to chaotic as one changes the values of the mode coupling coefficients. Here, we initiate a study of quantum infinite-dimensional resonant systems, which are mathematically a highly special case of two-body interaction Hamiltonians (extensively researched i…
We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of quantum kicked rotator. This main model of quantum chaos, which allows for a transition from regular to to chaotic behavior in the classical limit, may be realized in experiments with cold atoms. We study the generation of squeezed states in the quasiclassical limit and in a time interval when quantum-classical correspondence is yet well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov parameter of stochast…
Virginia Tech（弗吉尼亚理工大学）；Ludwig-Maximilians-Universität（慕尼黑大学）；Excellence Cluster ORIGINS（ORIGINS卓越集群）；University of Copenhagen（哥本哈根大学）；Indian Institute of Technology Delhi（德里印度理工学院）
Particles in space periodic potentials constitute standard models for investigation of crystalline phenomena in solid state physics. Time periodicity of periodically driven systems is a close analogue of space periodicity of solid state crystals. There is an intriguing question if solid state phenomena can be observed in the time domain. Here we show that wave-packets localized on resonant classical trajectories of periodically driven systems are ideal elements to realize Anderson localization or Mott insulator phase in the time domain. Uniform superpositions of the wave-packets form stationar…
Quantum mechanical spreading of a particle hopping on tight binding lattices can be suppressed by the application of an external ac force, leading to periodic wave packet reconstruction. Such a phenomenon, referred to as dynamic localization (DL), occurs for certain magic values of the ratio $Γ=F_0/ ω$ between the amplitude $F_0$ and frequency $ω$ of the ac force. It is generally believed that in the low-frequency limit ($ω\rightarrow 0$) DL can be achieved for an infinitesimally small value of the force $F_0$, i.e. at finite values of $Γ$. Such a normal behavior is found in homogeneous lattic…
We determine the ground state energy of atoms and quantum dots whose number N of electrons is large. We show that the dominant terms of the energy are those given by a semiclassical Hartree-Fock theory. Correlation effects appear at the order N ln N for atoms and the order N for quantum dots. We compute them. The semiclassical Hartree-Fock theory creates oscillations in the ground state energy as a function of N . These oscillations reflect the dynamics of a classical particle moving in the presence of the Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but we present …
Although highly successful, the Truncated Wigner Approximation (TWA) does not account for genuine many-body quantum interference between different solutions of the mean-field equations of a bosonic many-body (MB) system. This renders the TWA essentially classical, where a large number of particles formally takes the role of the inverse of Planck's constant $\hbar$. The failure to describe genuine interference phenomena, such as localization and scarring in Fock space, can be seen as a virtue of this quasiclassical method, which thereby allows one to identify genuine quantum effects when being …
Quantum mechanics allows the emergence of nonstatic quantum light waves in the Fock state even in a transparent medium of which electromagnetic parameters do not vary over time. Such wave packets become broad and narrow in turn periodically in time in the quadrature space. We investigate the effects of wave nonstaticity arisen in a static environment on the behavior of accompanying geometric phases in the Fock states. In this case, the geometric phases appear only when the measure of nonstaticity is not zero and their time behavior is deeply related to the measure of nonstaticity. While the dy…
We study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled-tops, where chaos and entanglement arise from the same physical mechanism. The long-time averaged entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section -- it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take local…
The quantum photodynamics of a simple diatomic molecule with a permanent dipole immersed within an optical cavity containing a quantized radiation field is studied in detail. The chosen molecule under study, lithium fluoride (LiF), is characterized by the presence of an avoided crossing between the two lowest $^1Σ$ potential energy curves (covalent-ionic diabatic crossing). Without field, after prompt excitation from the ground state $1\; ^1Σ$, the excited nuclear wave packet moves back and forth in the upper $2\; ^1Σ$ state, but in the proximity of the avoided crossing, the non-adiabatic coup…
Quantum chaos is the study of quantum systems whose classical description is chaotic. How does chaos manifest itself in the quantum world? In this spirit, we study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled-tops, where chaos and entanglement arise from the same physical mechanism. The long-time entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section - it is nearly uniform in the chaotic sea, and reproduces the detailed structure of …
We approach quantum dynamics in one spatial dimension from a systematic study of moments starting from the dynamics of the mean position. This is complementary to the approach of Brizuela whose starting point was generalized recursion relations between moments. The infinite set of coupled equations is truncated which allows us to use the techniques used in the study of dynamical systems. In particular we predict for what initial variance the purely quartic oscillator will time develop with minimal change in the shape of the initial packet and what the frequency of oscillation of the mean posit…
