Minimum Uncertainty Wave Packets Research Articles

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.

Universal quantum uncertainty relations: Minimum-uncertainty wave packet depends on measure of spread

Based on the statistical concept of the median, we propose a quantum uncertainty relation between semi-interquartile ranges of the position and momentum distributions of arbitrary quantum states. The relation is universal, unlike that based on the mean and standard deviation, as the latter may become non-existent or ineffective in certain cases. We show that the median-based one is not saturated for Gaussian distributions in position. Instead, the Cauchy-Lorentz distributions in position turn out to be the one with the minimal uncertainty, among the states inspected, implying that the minimum-uncertainty state is not unique but depends on the measure of spread used. Even the ordering of the states with respect to the distance from the minimum uncertainty state is altered by a change in the measure. We invoke the completeness of Hermite polynomials in the space of all quantum states to probe the median-based relation. The results have potential applications in a variety of studies including those on the quantum-to-classical boundary and on quantum cryptography.

Quantum entanglement of bound particles under free center of mass dispersion

On the basis of the full analytical solution of the overall unitary dynamics, the time evolution of entanglement is studied in a simple bipartite model system evolving unitarily from a pure initial state. The system consists of two particles in one spatial dimension bound by harmonic forces and having its free center of mass initially localized in space in a minimum uncertainty wavepacket. The existence of such initial states in which the bound particles are not entangled is discussed. Galilean invariance of the system ensures that the dynamics of entanglement between the two particles is independent of the wavepacket mean momentum. In fact, as shown, it is driven by the dispersive center of mass free dynamics, and evolves in a time scale that depends on the interparticle interaction in an essential way.

Semiclassical initial value representation with complex trajectories

Semiclassical time-dependent solutions to the Schroedinger equation can be obtained by use of integral formulas known as initial value representations (IVRs). Over the last few decades IVR formulas have established a sucessful reputation, most notably in the field of physical chemistry, for it is often found that the quantum behavior of molecular systems is very well approximated by semiclassical wavefunctions. The IVR input consists in a swarm of independent classical trajectories, each of which obeys the dynamics dictated by the system’s unquantized Hamiltonian. In a recent work [Aguiar et al, Chem. Phys. 370 (2010)] an IVR expression based on complex classical trajectories was formulated. The complex nature of the classical trajectories in this particular formula is a consequence of employing minimum uncertainty wavepackets (coherent states) to represent the time evolution operator upon which the formula is constructed. Here, an improved and simpler version of this complex IVR formula is presented. In order to test its accuracy, we apply it to a simple model system.

An Information Geometric Analysis of Entangled Continuous Variable Quantum Systems

In this work, using information geometric (IG) techniques, we investigate the effects of micro-correlations on the evolution of maximal probability paths on statistical manifolds induced by systems whose microscopic degrees of freedom are Gaussian distributed. Analytical estimates of the information geometric entropy (IGE) as well as the IG analogue of the Lyapunov exponents are presented. It is shown that the entanglement duration is related to the scattering potential and incident particle energies. Finally, the degree of entanglement generated by an s-wave scattering event between minimum uncertainty wave-packets is computed in terms of the purity of the system.

Minimum uncertainty wave packet in relativistic quantum mechanics

In nonrelativistic quantum mechanics in one dimension a wave packet can be constructed for which the minimum uncertainty relation for position and momentum, ΔxΔp=ℏ/2, holds exactly. The wave function of the wave packet is Gaussian and satisfies the Schrödinger equation for a harmonic oscillator potential. We illustrate a similar situation in relativistic quantum mechanics for the one-dimensional Dirac equation.

Emergence of quantum hydrodynamic sound mode of a quantum Brownian particle in a one-dimensional molecular chain

We have theoretically investigated a long-time transport process of a quantum Brownian particle interacting with a thermal phonon field in a one-dimensional molecular chain. A kinetic equation is derived from a quantum Liouville equation in a weak-coupling case by applying a complex spectral representation of Liouvillean. Due to a characteristic Poincar\'e resonance for a quantum one-dimensional system, there are an infinite number of degeneracy for collision invariants. In the hydrodynamic situation, the degeneracy is lifted by the first order of perturbation of the flow term, resulting in a new hydrodynamic mode, i.e., quantum hydrodynamic sound mode. It is found that the time evolution of the quantum hydrodynamic sound mode obeys a macroscopic linear wave equation for the probability distribution of the quantum particle. It is remarkable that the stability of the wave packet of the sound wave increases as temperature increases. As a demonstration, the sound wave of the minimum uncertainty wave packet is theoretically analyzed.

Effect of symmetry in the many-particle Wigner function

An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of $N$ particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.

Spatial Coherence of a Polariton Condensate

We perform Young's double-slit experiment to study the spatial coherence properties of a two-dimensional dynamic condensate of semiconductor microcavity polaritons. The coherence length of the system is measured as a function of the pump rate, which confirms a spontaneous buildup of macroscopic coherence in the condensed phase. An independent measurement reveals that the position and momentum uncertainty product of the condensate is close to the Heisenberg limit. An experimental realization of such a minimum uncertainty wave packet of the polariton condensate opens a door to coherent matter-wave phenomena such as Josephson oscillation, superfluidity, and solitons in solid state condensate systems.

Quantum mechanics and the generalized uncertainty principle

The generalized uncertainty principle has been described as a general consequence of incorporating a minimal length from a theory of quantum gravity. We consider a simple quantum mechanical model where the operator corresponding to position has discrete eigenvalues and show how the generalized uncertainty principle results for minimum uncertainty wave packets.

Decoherence, entanglement, and chaos in the Dicke model

The dynamical properties of quantum entanglement in the Dicke model without rotating-wave approximation are investigated in terms of the reduced-density linear entropy. The characteristic time of decoherence process in the early-time evolution is numerically obtained and it is shown that the characteristic time decreases as the coupling parameter increases. The mean entanglement, which is defined to be averaged over time, is employed to describe the influences of both quantum phase transition and corresponding classical chaos on the behavior of entanglement. For a given energy, initial conditions are taken to be minimum uncertainty wave packets centered at regular and chaotic regions of the classical phase space. It is shown that the entanglement has a distinct change at the quantum phase transition, and that the entanglement for regular initial conditions is smaller than that for chaotic ones in the case of weak coupling, while it fluctuates with small amplitude in strong coupling and for chaotic initial conditions.

Ehrenfest Theorem for the Hamilton-Jacobi Equation

A possible way from quantum mechanics to classical mechanics can be achieved with an exponential substitution used in the Schrodinger equation, and then considering the classical limit. This gives a picture of classicalfluid and an ensemble of classical trajectories. In difference from this approach to the classical limit, while utilising the same substitution, we assume a minimum uncertainty wave packet. It is shown that this approach to the classical limit of quantum mechanics yields a single trajectory traced by the centroid of the minimum uncertainty wave packet. The momentum and the centroid of such packet satisfy the classical Hamilton-Jacobi equation.

Mean field Ehrenfest quantum/classical simulation of vibrational energy relaxation in a simple liquid.

We give a detailed account of the statistical mechanical properties of the mean field Ehrenfest quantum/classical method as applied to liquid phase vibrational energy transfer using a simple harmonic oscillator model Hamiltonian. Depending on the shape of the initial quantum wave packet, a (partial) breakdown of detailed balance is observed, where the frictional response of the classical bath is only correlated to quasiclassical features of the evolving quantum state, i.e., a classical-like fluctuation-dissipation theorem holds. Only in the case of a coherent initial state (minimum uncertainty wave packet) does the mean field method produce physically meaningful results, namely, exponential relaxation (tau=tau(cl)) towards a quasiclassical equilibrium.

On the Relation between the Semiclassical Initial Value Representation and an Exact Quantum Expansion in Time-Dependent Coherent States

The Herman-Kluk (HK) or coherent state initial value representation (IVR) of semiclassical (SC) theory expresses a time-evolving wave function as a sum (integral) over the initial conditions of classically evolved coherent states (aka minimum uncertainty wave packets). An exact expansion of the wave function can also be made in terms of these same coherent states, thinking of them simply as a (time-dependent) basis set. This paper shows the precise set of approximations that are necessary to degenerate the exact expansion into the SC-IVR result. As a byproduct, we also obtain a generalization of the HK prefactor.

Charge in a magnetic field: quasi-classical states

We propose an elementary approach for introducing the quasi-classical quantum states of a spinless charged particle in a uniform magnetic field. By exploiting the similarity with the case of the two-dimensional harmonic oscillator as well as a property of the solutions of the pertinent Schrödinger equation, we derive two basic solutions. One of them is a stationary, minimum-uncertainty wavepacket centred at an arbitrary point. This corresponds to a classical particle at rest. The other solution is a minimum-uncertainty wavepacket that rotates at the classical angular speed. The expected value of the energy agrees with the classical prediction within the zero-point energy. These results are obtained without any knowledge of the energy eigenstates. An appendix suggests a brief and self-contained procedure for writing the Hamiltonian of a charged particle under the Lorentz force.

Optimal pump-dump control: phase-locked versus phase-unlocked schemes

We report a unified theoretical framework for the study of the pump-dump control with either a single coherent field or a pair of phase-unlocked coherent fields in both the strong and weak response regimes, and in terms of both the Liouville-space density matrix dynamics and the Hilbert-space wave function evolution. Shown are also the close relations between the pump-dump control kernels in the phase-locked (i.e. the single coherent field) and the phase-unlocked control schemes in the strong response regime. These strong field control kernels can further be linearized in the case of pure state control in the weak pump-dump response regime. In this case, the optimal control theory reduces to the eigen problem of a certain specially constructed Hermitian matrix, from which even the globally optimal pump-dump control fields in either the phase-locked or the phase-unlocked control scheme can be identified. The common key quantity in both of the control schemes is a Hilbert-space pump-dump control response function, B( τ, τ′), which shares a great amount of information mutually with the optical resonant Raman spectroscopies. Numerical examples of pump-dump controlling I 2 vibration onto an eigenstate and onto a minimum uncertainty wave packet in the ground electronic X state are presented to further elucidate the control mechanisms in the phase-locked and phase-unlocked schemes in the weak response regime.

Quantum Dynamical Manifestation of Chaotic Behavior in the Process of Entanglement

Manifestation of chaotic behavior is found in an intrinsically quantum property. The entanglement process, quantitatively expressed in terms of the reduced density linear entropy, is studied for the $N$-atom Jaynes-Cummings model. For a given energy, initial conditions are prepared as minimum uncertainty wave packets centered at regular and chaotic regions of the classical phase space. We find for short times a faster increase in decoherence for the chaotic initial conditions as compared to regular ones, which have oscillatory increase.

Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time

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). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time tℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale tℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.

Coherent states for the hydrogen atom

We construct wave packets for the hydrogen atom labelled by the classical action-angle variables with the following properties. i) The time evolution is exactly given by classical evolution of the angle variables. (The angle variable corresponding to the position on the orbit is now non-compact and we do not get exactly the same state after one period. However the gross features do not change. In particular the wave packet remains peaked around the labels.) ii) Resolution of identity using this overcomplete set involves exactly the classical phase space measure. iii) Semi-classical limit is related to Bohr-Sommerfield quantization. iv) They are almost minimum uncertainty wave packets in position and momentum.

Many particle simulations of the quantum electron gas using momentum-dependent potentials

A simple quasiclassical model of the quantum electron gas based on a quasiclassical dynamics with an effective momentum-dependent Hamiltonian is developed. The quantum mechanical effects corresponding to the Pauli and the Heisenberg principles are modeled by constraints in the Hamiltonian. By using the concept of minimum uncertainty wave packets, momentum-dependent effective potentials are derived. Monte Carlo and molecular dynamics calculations are carried out for ensembles of 64--512 electrons and calculations of the internal energy at several temperatures are given. A comparison with quantum Monte Carlo calculations of the mean energy for low temperatures and with Pad\'e approximations for several finite temperatures show that at least in thermodynamic equilibrium our simple model yields reasonable results.