Time-dependent Quadratic Hamiltonian System Research Articles

TIME-DEPENDENT WIGNER DISTRIBUTION FUNCTION EMPLOYED IN SQUEEZED SCHRÖDINGER CAT STATES: |Ψ(t)〉 = N-1/2(|β〉+eiϕ|-β〉)

The Wigner distribution function (WDF) for the time-dependent quadratic Hamiltonian system is investigated in the squeezed Schrödinger cat states with the use of Lewis–Riesenfeld theory of invariants. The nonclassical aspects of the system produced by superposition of two distinct squeezed states are analyzed with emphasis on their application into special systems beyond simple harmonic oscillator. An application of our development to the measurement of quantum state by reconstructing the WDF via Autler–Townes spectroscopy is addressed. In addition, we considered particular models such as Cadirola–Kanai oscillator, frequency stable damped harmonic oscillator, and harmonic oscillator with time-variable frequency as practical applications with the object of promoting the understanding of nonclassical effects associated with the WDF.

Dynamics of SU(1, 1) coherent states for the time-dependent quadratic Hamiltonian system

The dynamics of SU(1, 1) coherent states introduced by Perelomov are investigated for the time-dependent quadratic Hamiltonian system. SU(1, 1) generators we employed are closely related to the invariant operator theory while those of the previous work of Gerry et al. [C.C. Gerry, P.K. Ma, E.R. Vrscay, Phys. Rev. A 39 (1989) 668] are associated to the simple harmonic oscillator. This is the main difference between the two approaches. The merit of the method used in this paper is that it admits wide sphere of analytical description for quantum features of time-dependent quadratic Hamiltonian system. Our development is applied to the Caldirola–Kanai oscillator and compared the corresponding results with those of the Gerry et al. after correcting some miscalculations of theirs. We showed that the results of our theory are in good agreement with the results of the corrected work of Gerry et al. even if the form of the SU(1, 1) generators we employed are somewhat different from those of their work. The nontrivial zero-point energy plays a dominant role in the very low energy limit ( ξ ˜ 0 → 0 ) for the Caldirola–Kanai oscillator, leading the system to exhibit pure quantum effects as expected. On the other hand, it turn out for sufficiently high energy limit ( ξ ˜ 0 → 1 ) that the characteristic feature of dissipating quantum energy become very much the same as that of the classical energy.

Time-dependent Wigner distribution function employed in coherent Schrödinger cat states: |Ψ(t)⟩=N−1/2(|α⟩+eiϕ|-α⟩)

The Wigner distribution function for the time-dependent quadratic Hamiltonian system in the coherent Schrödinger cat state is investigated. The type of state we consider is a superposition of two coherent states, which are by an angle of π out of phase with each other. The exact Wigner distribution function of the system is evaluated under a particular choice of phase, δc, q. Our development is employed for two special cases, namely, the Caldirola–Kanai oscillator and the frequency stable damped harmonic oscillator. On the basis of the diverse values of the Wigner distribution function that were plotted, we analyze the nonclassical behavior of the systems.

SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schrodinger cat states defined as an eigenstate of \(\hat{K}_{-}^{2}\) are also studied. We applied our development to two-mode Caldirola-Kanai oscillator which is a typical example of the time-dependent quadratic Hamiltonian system. The time evolution of the quadrature distribution for the probability density in the coherent states are analyzed for the two-mode Caldirola-Kanai oscillator by plotting relevant figures.

Algebraic structure of the Feynman propagator and a new correspondence for canonical transformations

We investigate the algebraic structure of the Feynman propagator with a general time-dependent quadratic Hamiltonian system. Using the Lie-algebraic technique, we obtain a normal-ordered form of the time-evolution operator, and then the propagator is easily derived by a simple “integration within ordered product” technique. It is found that this propagator contains a classical generating function, which demonstrates a new correspondence between classical and quantum mechanics.

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed.

SU(1,1) LIE ALGEBRA APPLIED TO THE TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM PERTURBED BY A SINGULARITY

We realized SU (1,1) Lie algebra in terms of the appropriate SU (1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU (1,1) Lie algebra. Various expectation values in two kinds of the generalized SU (1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola–Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU (1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU (1,1) coherent states for the CKOPS are very similar to the classical trajectory.

WKB Wave Function of the General Time-Dependent Quadratic Hamiltonian System

We derived the WKB wave function for the general time-dependent quadratic Hamiltonian system using a unitary transformation method. We applied our research to sinusodially drived Caldirola–Kanai oscillator and confirmed that the time evolution of our approximated WKB wave function is similar to that of the exact one. This wave function can be used to analyze the interference between the probability amplitudes contributed by the area of overlap in phase space of quantum states.

Relations of canonical and unitary transformations for a general time-dependent quadratic Hamiltonian system

We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum ${\mathrm{p}}_{\mathrm{k}}$ are equal to those of the other Q and ${\mathrm{P}}_{\mathrm{k}}$ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.