Vibrational pair cat states

Abstract

We propose a scheme for generating mesoscopic superpositions of two distinguishable pair coherent states of motion in a two-dimensional ion trap. In our scheme, the trapped ion is excited bichromatically by five laser beams along different directions in the X-Y plane of the ion trap. Four of these have the same frequency and can be derived from the same source, reducing the demands on the experimentalist. We show that if the initial vibrational state is given by a two-mode Fock state, as demonstrated in recent experiments, these highly correlated two-mode ''Schrödinger cat'' states are realized when the system reaches its steady state, which is indicated by the extinction of the fluorescence emitted by the ion. ͓S1050-2947͑96͒02611-X͔ PACS number͑s͒: 42.50. Vk, 42.50.Dv, 32.80.Pj Ever since Schrödinger suggested his famous cat experiment in 1935 ͓1͔, superpositions of macroscopically distinguishable quantum states ͑which are also known as Schrö-dinger cat states͒ have become a longstanding exemplar of the peculiarities which occur in the interpretation of quantum reality. To explore this subtlety and to gain insight into the fundamental issues of quantum theory, a number of schemes have been proposed for the realization of such states. In quantum optics, the Schrödinger cat states are usually described as superpositions of different coherent states ͓2͔, as coherent states are the closest quantum states to a classical description of the field of definite complex amplitude. Specifically, the archetype of a Schrödinger cat state is given by the superposition ͉⌿͘ cat ϭN͓͉␣͘ϩexp(i)͉Ϫ␣͘], where ͉␣͘ is a coherent state of the single-mode quantized field and N is a normalization coefficient. In particular, these states are referred to the even, odd, and Yurke-Stoler ͓3͔ coherent states when ϭ0,, and /2, respectively. They have been extensively studied and shown to exhibit nonclassical properties such as squeezing and sub-Poissonian statistics ͓2͔.In a recent paper Gerry and Grobe ͓4͔ have proposed a two-mode generalization of Schrödinger cat states which are defined as superpositions of two different pair coherent states ͑PCS͒. For two independent boson annihilation operators â and b , a pair coherent state ͉,q͘ is defined as an eigenstate of both the pair annihilation operator â b and the number difference operator Q ϭâwhere is a complex number and q is the ''charge'' which is a fixed integer. Without loss of generality, we may set qу0 and the PCS can be explicitly expanded as a superposition of the two-mode Fock states, i.e.,where N q ϭ͓͉͉ Ϫq I q (2͉͉)͔ Ϫ1/2 is the normalization constant (I q is the modified Bessel function of the first kind of order q). Pair coherent states are regarded as an important type of correlated two-mode state with prominent nonclassical properties such as sub-Poissonian statistics, strong intermode correlation in the number fluctuations, squeezing of quadrature variances, and violations of Cauchy-Schwarz inequalities ͓5͔.The correlated two-mode Schrödinger cat states ͉,q,͘ are defined as...

Nonclassical Properties of a Class of Nonlinear Pair Coherent State

“…Moreover, their some nonclassical properties have been investigated. The differences in phenomenon between two classes of finite-dimensional PCSs [23,24], the standard PCSs [22] and NLPCSs [25] have also been found. Nonlinear extensions of the single-mode squeezed vacuum and squeezed coherent states are constructed.…”

“…Another scheme has been suggested for generating vibrational pair coherent states via the quantized motion of a trapped ion in a two-dimensional trap [22]. As an important extension of PCS in the finite dimensional Hilbert space, the finite dimensional PCS and the finite dimensional nonlinear PCS (NLPCS) have been introduced [23,24].…”

Decoherence and robustness of parity-dependent entanglement in the dynamics of a trapped ion

“…If, in fact, an ion confined in a miniaturised Paul trap is exposed to suitably configured laser beams, the 3D harmonic motion of the ionic center of mass gets entangled with the internal degrees of freedom. Some peculiar aspects of the vibronic response stemming from such a dissipation-free situation have been successfully exploited for realizing experimentally Fock states, coherent states, squeeezed states and Schrödinger cat-like states [9,10,11], presenting theoretical schemes for engineering several nonclassical state [13,14,16,15,17,18,19,20,21,22,23], implementing quantum logic gates [8,24,25], realising tomographic reconstructions of the density matrix of the system [26,27,28,29], and, more in general, discovering quantum effects characterising the ionic quantized oscillatory motion [30,31,32,33,34,35]. Many hamiltonian models have been reported so far in the literature to describe physical properties of trapped ions.…”

Proposal for Measurement of Harmonic Oscillator Berry Phase in Ion Traps

“…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.…”

“…(1) can be physically realized. Recently, ion traps have been a very active field of both experimental [11][12][13][14] and theoretical [15,19,20,22] research. Consider a single Λ threelevel ion with two hyperfine ground states and an excited state (such as 9 Be + [11]) in a harmonic trap of frequency ν .…”