Time Dependent Quantum Mechanics with Correlated Coherent States

Zoltán Papp

Giovanni Pecci, Piero Naldesi, Anna Minguzzi +1 more

In quantum mechanics, each particle is described by a complex valued wave function characterized by amplitude and phase. When many particles interact each other, cooperative phenomena give rise to a quantum many-body state with a specific quantum coherence. What is the interplay between particle's phase coherence and many-body quantum coherence? Over the years, such question has been object of profound analysis in quantum physics. Here, we demonstrate how state of the art quantum technology allows to explore the question operatively. To this end, we study the time-dependent interference formed by releasing an interacting degenerate Fermi-gas from a specific matter-wave circuit in an effective magnetic field. Particle phase coherence, indicated by the first-order correlator, and many-body quantum coherence, indicated by the noise correlator, are displayed as distinct features of the interferogram. Particle phase coherence produces spiral interference of the Fermi orbitals at intermediate times. Many-body quantum coherence emerges at long times. The coupling between particles coherence and many-body coherence is reflected in a specific dependence of the interference pattern on the effective magnetic field.

Sanjib Dey, Andreas Fring

We investigate a quantum mechanical system on a noncommutative space for which the structure constant is explicitly time dependent. Any autonomous Hamiltonian on such a space acquires a time-dependent form in terms of the conventional canonical variables. We employ the Lewis-Riesenfeld method of invariants to construct explicit analytical solutions for the corresponding time-dependent Schr\"odinger equation. The eigenfunctions are expressed in terms of the solutions of variants of the nonlinear Ermakov-Pinney equation and discussed in detail for various types of background fields. We utilize the solutions to verify a generalized version of Heisenberg's uncertainty relations for which the lower bound becomes a time-dependent function of the background fields. We study the variance for various states, including standard Glauber coherent states with their squeezed versions and Gaussian Klauder coherent states resembling a quasiclassical behavior. No type of coherent state appears to be optimal in general with regard to achieving minimal uncertainties, as this feature turns out to be background field dependent.

Jeong Ryeol Choi

We obtained the uncertainty relation in squeezed states for a time-dependent oscillator. The uncertainty relation in coherent states is same as that of the number states with n=0. However, the uncertainty relation in squeezed states does not satisfy this property and depends on squeezing parameter c. For instance, the uncertainty relation is ℏ/2 which is the minimum value as far as quantum mechanics permits for c=1, same as that in coherent state for c=±∞, and infinity for c=-1. If the time-dependency of the Hamiltonian for the system vanishes, the uncertainty relation in squeezed states will no longer depend on c and becomes the same as that in number state with n=0, like the uncertainty relation in coherent states.