Tunneling out of a time-dependent well

Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: One class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like i.e the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this article we discuss and develop analytic methods for solving time-dependent tunneling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunneling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunneling phenomena in Bose-Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.Comment: 11 pages, 3 figure

Exact results for `bouncing' Gaussian wave packets

We consider time-dependent Gaussian wave packet solutions of the Schrodinger equation (with arbitrary initial central position, x_0, and momentum, p_0, for an otherwise free-particle, but with an infinite wall at x=0, so-called bouncing wave packets. We show how difference or mirror solutions of the form psi(x,t)-psi(-x,t) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate _t explicitly which illustrates how the free-particle kinetic (and hence total) energy is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, _t, and momentum, _t, and their relation to the impulsive force during the `collision' with the wall. Finally, the x_0,p_0 --> 0 limit is shown to reduce to a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands on the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.Comment: 14 pages, 1 embedded .eps figur

Numerical approach to the dynamical Casimir effect

The dynamical Casimir effect for a massless scalar field in 1+1-dimensions is studied numerically by solving a system of coupled first-order differential equations. The number of scalar particles created from vacuum is given by the solutions to this system which can be found by means of standard numerics. The formalism already used in a former work is derived in detail and is applied to resonant as well as off-resonant cavity oscillations.Comment: 15 pages, 4 figures, accepted for publication in J. Phys. A (special issue: Proceedings of QFEXT05, Barcelona, Sept. 5-9, 2005

Exact propagators for atom-laser interactions

A class of exact propagators describing the interaction of an $N$-level atom with a set of on-resonance $\delta$-lasers is obtained by means of the Laplace transform method. State-selective mirrors are described in the limit of strong lasers. The ladder, V and $\Lambda$ configurations for a three-level atom are discussed. For the two level case, the transient effects arising as result of the interaction between both a semi-infinite beam and a wavepacket with the on-resonance laser are examined.Comment: 13 pages, 6 figure

Self-interference of a single Bose-Einstein condensate due to boundary effects

A simple model wavefunction, consisting of a linear combination of two free-particle Gaussians, describes many of the observed features seen in the interactions of two isolated Bose-Einstein condensates as they expand, overlap, and interfere. We show that a simple extension of this idea can be used to predict the qualitative time-development of a single expanding BEC condensate produced near an infinite wall boundary, giving similar interference phenomena. We also briefly discuss other possible time-dependent behaviors of single BEC condensates in restricted geometries,such as wave packet revivals.Comment: 8 pages, no figures, to appear in Physica Script

Exact solution for the energy density inside a one-dimensional non-static cavity with an arbitrary initial field state

We study the exact solution for the energy density of a real massless scalar field in a two-dimensional spacetime, inside a non-static cavity with an arbitrary initial field state, taking into account the Neumann and Dirichlet boundary conditions. This work generalizes the exact solution proposed by Cole and Schieve in the context of the Dirichlet boundary condition and vacuum as the initial state. We investigate diagonal states, examining the vacuum and thermal field as particular cases. We also study non-diagonal initial field states, taking as examples the coherent and Schrodinger cat states.Comment: 10 pages, 8 figure

Exact closed form analytical solutions for vibrating cavities

For one-dimensional vibrating cavity systems appearing in the standard illustration of the dynamical Casimir effect, we propose an approach to the construction of exact closed-form solutions. As new results, we obtain solutions that are given for arbitrary frequencies, amplitudes and time regions. In a broad range of parameters, a vibrating cavity model exhibits the general property of exponential instability. Marginal behavior of the system manifests in a power-like growth of radiated energy.Comment: 17 pages, 7 figure

Squeezing and photon distribution in a vibrating cavity

We obtain explicit analytical expressions for the quadrature variances and the photon distribution functions of the electromagnetic field modes excited from vacuum or thermal states due to the non-stationary Casimir effect in an ideal one-dimensional Fabry--Perot cavity with vibrating walls, provided the frequency of vibrations is close to a multiple frequency of the fundamental unperturbed electromagnetic mode.Comment: 20 pages, LaTex2e, iopart document class, 2 ps figures, accepted for publication in J. Phys.