Proposal for Measurement of Harmonic Oscillator Berry Phase in Ion Traps

Abstract

We propose a scheme for measuring the Berry phase in the vibrational degree of freedom of a trapped ion. Starting from the ion in a vibrational coherent state we show how to reverse the sign of the coherent state amplitude by using a purely geometric phase. This can then be detected through the internal degrees of freedom of the ion. Our method can be applied to preparation of Schrödinger cat states.Pacs When the Hamiltonian of a quantum system is varied adiabatically in a cyclic fashion, the state of the system acquires a geometrical phase in addition to the usual dynamical phase. This effect, discovered by Berry [1], (and generalized in various ways [2]) has been widely tested for 2-state systems [3] and attracted interest from a variety of fields [4,5]. However, the Berry phase has not been experimentally measured for quantum harmonic oscillators, though some theoretical calculations exist [6][7][8][9][10]. The reason for this might be the fact that for the simplest case, namely for adiabatic displacement of an oscillator state in phase space, the Berry phase is independent of the state [6], and thereby undetectable. However, when a squeezing Hamiltonian is switched on, and the squeezing parameter is varied, there would be a detectable Berry phase [6][7][8]. For an initial Fock state |n which undergoes squeezing, the Berry phase after a cycle is −(n + 1/2) times the classical Hannay angle [6,7]. The squeezed states of the electromagnetic field would have been a natural candidate to test this kind of phase, but they are not stable enough for an adiabatic evolution. However, the vibrational mode of a trapped ion has been a fertile ground for the preparation of long lived nonclassical states of a harmonic oscillator [11][12][13][14][15]. In this letter, we derive a Berry phase formula for a certain adiabatic evolution of a joint state of the internal levels of a trapped ion and its vibrational motion. Despite being the phase gained by a joint state, its value is fundamentally dependent on the harmonic oscillator nature of the vibrational mode. We propose a scheme to detect this phase which is feasible with current technology. It is worthwhile to mention that based on calculations of Ref. [8] (which where tested in systems other than quantum harmonic oscillators [16]), there has been an earlier attempt to detect the nonadiabatic geometric phase in ion traps by applying a set of four squeezes to the vibrational state [17]. Here we propose a way of detecting the harmonic oscillator version of Berry's original adiabatic geometric phase.Consider the HamiltonianwhereIn the above, |e and |g are two states of a qubit, a † and a are the creation and annihilation operators of a harmonic oscillator, g a and g b are unequal positive interaction strengths (say g a > g b ) and φ is an arbitrary phase factor. The motivation for choosing this Hamiltonian is the possibility of its physical implementation and this will be described later. If the phase φ is slowly varied over a complete loop (so that the adiabatic approximation h...

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