Propagator for the general time-dependent harmonic oscillator with application to an ion trap

Abstract

We present the simplest possible formula for the propagator of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second-order ordinary differential equation corresponding to the classical quadratic Hamiltonian. We give an example for the case of the Paul trap.

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

“…Also in this Table, In particular ĥ 6 ,ĥ 9 ,ĥ 12 or ĥ 7 ,ĥ 10 ,ĥ 13 form the SU (1, 1) Lie algebra that has been used to study KanaiCaldirola Hamiltonians through the Lie algebraic approach [27,58]. The sub-algebras ĥ 1 ,ĥ 2 ,ĥ 4 ,ĥ 6 ,ĥ 9 ,ĥ 12 or ĥ 1 ,ĥ 3 ,ĥ 5 ,ĥ 7 ,ĥ 10 ,ĥ 13 correspond to the generalised one-dimensional harmonic oscillator [1,35] along the x and y axis respectively. One can easily express Hamiltonian (32) as a linear combination of the L 15 elementŝ…”

“…Even though the most general form of the one-dimensional quadratic Hamiltonian (1) has been treated through the Lewis Riesenfeld theory [21,35,49] and linear invariants [1], the two-dimensional quadratic Hamiltonian (twoDQH) has only been studied for a limited number of special cases. One of these corresponds to a charged particle subject to a constant uniform magnetic field and a quadratic potential [50] whose propagator was calculated by means of the path integral method.…”

“…In many applications as radio-frequency ion traps [1][2][3][4][5][6][7][8], quantum optics [9][10][11][12], cosmology [13,14], quantum field theory [15], quantum dissipation [16][17][18][19][20][21][22], magneto transport in lateral heterostructures [23][24][25][26] and even gravitational waves [27] the time evolution of particles in quadratic potentials is frequently examined. The one-dimensional, generalized time-dependent quadratic Hamiltonian is given bŷ H = a 1 (t) + a 2 (t)x + a 3 (t)p + a 4 (t)x 2 + a 5 (t)p 2 + a 6 (t) (xp +px) ,…”

“…an oscillator with time-varying frequency [21,32,33]. The time evolution generated by the most general version of (1) has been studied by means of Lewis and Riesenfeld [34] invariants [35] and through linear invariants [1]. Combining two one-dimensional generalized quadratic Hamiltonians along the x and y coordinates and adding cross terms for the position and momentum operators one arrives to the most general form of the two-dimensional quadratic Hamiltonian H = a 1 (t) + a 2 (t)x + a 3 (t)ŷ + a 4 (t)p x + a 5 (t)p y + a 6 (t)x 2 + a 7 (t)ŷ 2 + a 8 (t)xŷ + a 9 (t)p 2 x + a 10 (t)p 2 y + a 11 (t)p xpy + a 12 (t) (xp x +p xx ) + a 13 (t) (ŷp y +p yŷ ) + a 14 (t)xp y + a 15 (t)ŷp x , (2) where newly the position and momentum operators along the x and y axes follow the standard commutation relations [x,ŷ] = [p x ,p y ] = 0 and [x,p x ] = [ŷ,p y ] = i .…”

“…The aim of this paper is therefore to develop a systematic method based on the Lie algebraic approach mainly with the purpose of obtaining the evolution operator of the two-dimensional, generalized time-dependent quadratic Hamiltonian presented in (1) . Although most of this paper is devoted to the Lie algebra of (2), that consists of 15 generators, the presentation on the Lie algebraic approach is general enough to be applied to systems whose Hamiltonians can be expanded in terms of an arbitrarily large number of generators.…”