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Physical Review A

Unitary equivalence and phase properties of the quantum parametric and generalized harmonic oscillators

Mustapha Maamache, Jean-Pierre Provost, Gérard Vallée

Published Mar 1, 1999• 24 citations
Publisher

Abstract

A quantum parametric harmonic oscillator with a unitary evolution but not invariant under time reversal, is shown to be unitarily equivalent to a quantum generalized harmonic oscillator. As a consequence the phase properties of the two systems, in particular their geometric phases, are identical. Equivalent results for the corresponding classical systems are deduced from the study of the time evolution of the appropriate squeezed coherent states.

DOI
10.1103/physreva.59.1777

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A quantum parametric harmonic oscillator with a unitary evolution but not invariant under time reversal, is shown to be unitarily equivalent to a quantum generalized harmonic oscillator. As a consequence the phase properties of the two systems, in particular their geometric phases, are identical. Equivalent results for the corresponding classical systems are deduced from the study of the time evolution of the appropriate squeezed coherent states.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.
Time-evolution of squeezed coherent states of a generalized Caldirola-Kanai type quantum parametric oscillator is found explicitly using the exact evolution operator obtained by the Wei-Norman algebraic approach. Properties of these states are investigated according to the parameters of the unitary squeeze operator and the time-variable parameters of the generalized quadratic Hamiltonian. As an application, we consider exactly solvable quantum models with specific frequency modification for which the corresponding classical oscillator is in underdamping case and driving forces are of sinusoidal type. For each model we explicitly provide the evolution of the squeezed coherent states and discuss their behavior.
We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T-periodic, the corresponding class of GEQS includes a system with a T-periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian H(t) = e −iKt H0e iKt. We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator. PACS numbers: 03.65.Bz 1
We introduce the notion of the geometrically equivalent quantum systems (GEQSs) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQSs. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T -periodic, the corresponding class of GEQSs includes a system with a T -periodic Hamiltonian. We apply our general results to study the classes of GEQSs that include a system with a cranked Hamiltonian H ( t ) = e -i Kt H 0 e i Kt . We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator.
The unitary operator that transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different timescale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.
The evolution operator of a Caldirola-Kanai type quantum parametric oscillator with a generalized quadratic Hamiltonian is obtained using the Wei-Norman Lie algebraic approach, and time evolution of the eigenstates of a harmonic oscillator and Glauber coherent states is found explicitly. Behavior of this oscillator is investigated under the influence of the external mixed term B(t)(q^p^+p^q^)/2, which affects the squeezing properties of the wave packets, and linear terms D0(t)q^, E0(t)p^ responsible for their displacement in time. According to this, we construct all exact quantum models with different parameters B(t), for which the structure of the Caldirola-Kanai oscillator in position space is preserved. Then, for each model, we obtain explicit solutions and analyze the squeezing and displacement properties of the wave packets according to the frequency modification by B(t) and periodic forces in the corresponding classical equation of motion.