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Journal of Mathematical Physics

Dynamical symmetry breaking in the Lewis–Riesenfeld oscillator

Neil J. Günther

Published Feb 1, 1980• 1 citation
Publisher

Abstract

Dynamical realization is given to the generator responsible for breaking the usual SU(3) invariance symmetry in the case of the Lewis–Riesenfeld time-dependent oscillator: H (t) = 1/2 𝒥3i[pipi+w2(t) qiqi]. This is achieved by finding a simple expression for the time-dependent Hamiltonian in the interaction picture: H=Ho+gH1. The breaking interaction is shown to transform as (6⊕6̄) of SU(3). Time-dependent dilations relate the broken Hamiltonian to the Lewis–Riesenfeld form. This scaling generates the algebra of the Weyl group and its role in oscillator noninvariance symmetries is considered.

DOI
10.1063/1.524412

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Dynamical realization is given to the generator responsible for breaking the usual SU(3) invariance symmetry in the case of the Lewis–Riesenfeld time-dependent oscillator: H (t) = 1/2 𝒥3i[pipi+w2(t) qiqi]. This is achieved by finding a simple expression for the time-dependent Hamiltonian in the interaction picture: H=Ho+gH1. The breaking interaction is shown to transform as (6⊕6̄) of SU(3). Time-dependent dilations relate the broken Hamiltonian to the Lewis–Riesenfeld form. This scaling generates the algebra of the Weyl group and its role in oscillator noninvariance symmetries is considered.
Elaboration is given to a previous discussion concerning dynamical symmetries of the Lewis–Riesenfeld time-dependent harmonic oscillator. A boson operator formalism is used to define a generator of impicitly time-dependent unitary transformations. Bilinear combinations of the transformed boson operators are shown to span the symplectic algebra Sp(2n,IR) regarded as a larger noninvariance group for the Lewis–Riesenfeld Hamiltonian. The multiplet structure of the embeddings, Sp(2n) ↓SU(n), and SU (n+1) ↓SU(n), is determined for the case n=3, using established branching rules. Boson operator realizations are presented for the multiplet structures in each SU(n) declension of these dynamical groups.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
Here we first present an alternative formulation of the Lewis & Riesenfeld theorem for solving the Schrödinger equation with nonautonomous Hermitian and pseudo-Hermitian Hamiltonians. We then employ this framework to characterize the spontaneous breaking of time-dependent antilinear symmetries of these Hamiltonians. We demonstrate that, under unbroken antilinear symmetries, the Lewis & Riesenfeld phases are real and odd functions of time, which allows us to recover the well-known real spectra of time-independent pseudo-Hermitian Hamiltonians. However, in the spontaneously broken regime, imaginary components of the Lewis & Riesenfeld phases arise, leading to coalescence effects analogous to those in the time-independent scenario. Finally, we present an illustrative example of unbroken and broken PT-symmetry for a time-dependent Hamiltonian modeling the non-Hermitian dynamical Casimir effect.
C -operators were introduced as involution operators in non-Hermitian theories that commute with the time-independent Hamiltonians and the parity/time-reversal operator. Here we propose a definition for time-dependent C ( t ) -operators and demonstrate that for a particular signature they may be expanded in terms of time-dependent biorthonormal left and right eigenvectors of Lewis-Riesenfeld invariants. The vanishing commutation relation between the C -operator and the Hamiltonian in the time-independent case is replaced by the Lewis-Riesenfeld equation in the time-dependent scenario. Thus, C ( t ) -operators are always Lewis-Riesenfeld invariants, whereas the inverse is only true in certain circumstances. We demonstrate the working of the generalities for a non-Hermitian two-level matrix Hamiltonian. We show that solutions for C ( t ) and the time-dependent metric operator may be found that hold in all three PT -regimes, i.e., the PT -regime, the spontaneously broken PT -regime and at the exceptional point.