Symplectic Tomography of Schrodinger Cat States of a Trapped Ion

TL;DR

This paper investigates the measurement of nonclassical states of a trapped ion using symplectic tomography, focusing on squeezed and Schrödinger cat states, to better understand their quantum properties.

Contribution

It introduces a method to analyze the marginal distributions of complex nonclassical states of a trapped ion through symplectic tomography.

Findings

Derived explicit forms of marginal distributions for specific nonclassical states.

Demonstrated the applicability of symplectic tomography to trapped ion states.

Provided insights into the quantum features of squeezed and Schrödinger cat states.

Abstract

The marginal distribution of squeezed, rotated and shifted quadrature for two types of nonclassical states of a trapped ion - squeezed correlated states and squeezed even and odd coherent states (squeezed Schrodinger cat states) - is studied.

Equations

$${\omega }^2(t)=1+{\kappa }^2{\sin }^2\Omega t.$$

$${\Psi }_{\alpha }(x,t)={\pi }^{-1/4}[\epsilon (t){]}^{-1/2}\exp \left(\frac{i\dot{\epsilon }(t)x^2}{2\epsilon (t)}\right)\exp \left\{-\frac{|\alpha {|}^2}{2}-\frac{{\alpha }^2{\epsilon }^{\ast }(t)}{2\epsilon (t)}+\frac{\sqrt{2}\alpha x}{\epsilon }\right\},$$

$$\ddot{\epsilon }(t)+{\omega }^2(t)\epsilon (t)=0,$$

$$\epsilon (0)=1;\dot{\epsilon }(0)=i,$$

$${\Psi }_{\alpha }^{(+)}(x,t)=2N^{(+)}{\Psi }_0(x,t)\exp \left\{-\frac{|\alpha {|}^2}{2}-\frac{{\epsilon }^{\ast }(t){\alpha }^2}{2\epsilon (t)}\right\}\cosh \frac{\sqrt{2}\alpha x}{\epsilon (t)};N^{(+)}=\frac{\exp (|\alpha {|}^2/2)}{2\sqrt{\cosh |\alpha {|}^2}},$$

$${\Psi }_{\alpha }^{(-)}(x,t)=2N^{(-)}{\Psi }_0(x,t)\exp \left\{-\frac{|\alpha {|}^2}{2}-\frac{{\epsilon }^{\ast }(t){\alpha }^2}{2\epsilon (t)}\right\}\sinh \frac{\sqrt{2}\alpha x}{\epsilon (t)};N^{(-)}=\frac{\exp (|\alpha {|}^2/2)}{2\sqrt{\sinh |\alpha {|}^2}}.$$

$$\hat{X}=\mu \hat{q}+\nu \hat{p}+\delta ,$$

$$w(X,\mu ,\nu ,\delta )=\int \exp [-ik(X-\mu q-\nu p-\delta )]W(q,p)\frac{dkdqdp}{(2\pi {)}^2}.$$

$$\dot{w}-\mu \frac{\partial }{\partial \nu }w+{\omega }^2(t)\nu \frac{\partial }{\partial \mu }w=0.$$

$$w(X,\mu ,\nu ,\delta ,t)=\frac{1}{\sqrt{2\pi {\sigma }_X(t)}}\exp \left\{-\frac{(X-\bar{X}{)}^2}{2{\sigma }_X(t)}\right\},$$

$${\sigma }_X(t)$$

$$\overline{X}$$

$$W_{\pm }$$

$$w_{\pm }=\mid N^{\pm }{\mid }^2{\pi }^{-1/2}{\left[\mid \dot{\epsilon }{\mid }^2{\nu }^2+\mid \epsilon {\mid }^2{\mu }^2+2\mu \nu \text{mbox}Re(\dot{\epsilon }{\epsilon }^{\ast })\right]}^{-1/2}\left\{w_1+w_2\pm w_3\pm w_4\right\},$$

$$w_1$$

$$w_2$$

$$w_3$$

$$w_4$$

Full text

PACS 03.65 - Quantum mechanics.

Key–words: symplectic tomography, Paul trap, Schrödinger cat states.

Recently, in [1] it was shown that the steady state of a trapped ion irradiated by a bichromatic laser field is a superposition of two coherent states which is the even/odd coherent state introduced in [2] (Schrödinger cat states). The theory of an ion in a Paul trap was developed in [3, 4] where the trapped ion was described by the model of quantum oscillator with a periodically varying frequency. In [6], a procedure was formulated to obtain the Wigner function of the quantum system in terms of the marginal distribution of rotated quadrature, which may be measured by a balanced homodyne detector and the scheme called optical tomography has been used experimentally [7]. In [8], a symplectic-tomography procedure was suggested in which the quantum states were measured by measuring the marginal distribution for squeezed, rotated, and shifted quadrature. In [9], a new equation in quantum mechanics was introduced describing time evolution of this marginal distribution which has completely classical form but contains all the information about the quantum system. The aim of this work is to consider two important types of nonclassical states of a trapped ion (squeezed and correlated states of the ion in a Paul trap [3–5] and even and odd coherent states of the ion irradiated by bichromatic laser field [1]) within the framework of the symplectic-tomography procedure and using new quantum evolution equation.

An ion in a Paul trap is described by the model of parametric oscillator. For the trapped ion, the time–dependence of the frequency is taken to be periodic [3]:

[TABLE]

It is easy to show that packet solutions to the Schrödinger equation may be introduced and interpreted as coherent states  $(\hbar=1)\,:$

[TABLE]

where the classical complex trajectory satisfies the equation

[TABLE]

with initial conditions

[TABLE]

where $~{}\alpha\,$ is a complex number.

Other normalized solutions to the Schrödinger equation are a squeezed even coherent state and a squeezed odd coherent state [2] (squeezed Schrödinger cat states)

[TABLE]

[TABLE]

In [8], it was shown that for the generic linear combination of quadratures which is a measurable observable

[TABLE]

where $\,\hat{q}\,$ and $\,\hat{p}\,$ are the position and momentum, respectively, the marginal distribution $w\,(X,\,\mu,\,\nu,\,\delta)\,$ (normalized with respect to the $X$ variable) depending upon three extra real parameters $\mu\,,\nu\,,\delta\,$ is related to the state of the quantum system expressed in terms of its Wigner function $W\,(q,p)\,$ as follows

[TABLE]

The physical meaning of the parameters $\mu,\,\nu,\,\delta\,$ is that they describe ensemble of shifted, rotated, and scaled reference frames in which the position $X$ is measured. This formula can be inverted and the Wigner function of state can be expressed in terms of the marginal distribution [8]. In [9], it was shown that for Hamiltonian systems the marginal distributions satisfy the quantum time-evolution equation, which for a trapped ion takes the form

[TABLE]

Calculating the integral (4) one can show that for generic Gaussian packets of a trapped ion (also for particular case (1) ) the marginal distribution is

[TABLE]

where the dispersion of the symplectic observable $X$ and the mean value of the observable depend on the time and the parameters as follows:

[TABLE]

One can check that the normalized marginal distribution (6) with parameters (7) and (8) satisfy the evolution equation (5).

Now we will discuss the marginal distribution for nonclassical states of the parametric oscillator, namely, even and odd coherent states [2]. The Wigner function for even and odd coherent states is

[TABLE]

The marginal distribution of a trapped ion in even/odd coherent states is

[TABLE]

where

[TABLE]

In the present work, we calculated the marginal distribution of a symplectic observable (which is a generic linear quadrature) for nonclassical states of a trapped ion modeled by a parametric quantum oscillator. Measurements of the marginal distribution give the possibility of measuring the quantum states. In the case of the particular choice of the parameters $\mu=\cos\varphi;\,\nu=\sin\varphi,$ the measurement is reduced to finding the marginal distribution for homodyne output and reconstructing the Wigner function by means of the Radon transform of the optical-tomography scheme [6, 7]. The distribution found for generic linear quadrature satisfies a new classical-like equation of quantum dynamics introduced in the symplectic-tomography scheme.

Acknowledgments

This work was partially supported by the Russian Foundation for Basic Research (Project No. 96–02–18623) and by the RF State Program “Optics. Laser Physics.” O.V.M. would like to express her sincere appreciation to the Organizing Committee of the Second International Symposium on Fundamental Problems in Quantum Physics for invitation and Professor M. Ferrero and Dr. S.F. Huelga for hospitality.