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Europhysics Letters (EPL)

Symplectic invariant method for time-dependent quadratic Hamiltonians

Charles Antoine

Published Nov 1, 2012• 0 citations
Publisher

Abstract

\n\nUsing the ABCD formalism of atom optics, the generating function method and the invariant operator method (Liouville-von Neumann picture), I show in this letter how the quantum time evolution of any wave packet can be entirely determined by a natural symplectic invariant, which provides both the (non-Hermitian) invariant operator and the adequate eigenvalue. Within this framework, the quantum propagator does not appear any more as the fundamental kernel of integration but merely as a particular property of this invariant operator. The symplectic invariant method is interpreted in terms of creation/annihilation operators of N-dimensional Hermite-Gaussian modes.\n

DOI
10.1209/0295-5075/100/30006

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\n\nUsing the ABCD formalism of atom optics, the generating function method and the invariant operator method (Liouville-von Neumann picture), I show in this letter how the quantum time evolution of any wave packet can be entirely determined by a natural symplectic invariant, which provides both the (non-Hermitian) invariant operator and the adequate eigenvalue. Within this framework, the quantum propagator does not appear any more as the fundamental kernel of integration but merely as a particular property of this invariant operator. The symplectic invariant method is interpreted in terms of creation/annihilation operators of N-dimensional Hermite-Gaussian modes.\n
The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in L2(Rn)L^2(\mathbb R^n)L2(Rn) corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.