1 Rapid Generation of a Macroscopic Schrödinger Cat State of Atoms with Parity -Independent Orientation

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Rapid Generation of a Macroscopic Schrödinger Cat

State of Atoms with Parity-Independent Orientation

Jinyang Li1, Gregório R. M. da Silva1, Schuyler Kain1, Selim M. Shahriar1,2

1 Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

2 Department of ECE, Northwestern University, Evanston, IL 60208, USA

Abstract

We show that using the process of one-axis-twist squeezing in an echo configuration, it is possible

to control the orientation of the macroscopic magnetic moment of a large number of atoms by

manipulating the quantum state of a single atom that is physically isolated from the ensemble.

With this control technique, it is also possible to entangle an ensemble with a single atom

deterministically, which mimics the thought experiment known as the Schrödinger’s cat. In

addition, this technique would make it possible to generate a mesoscopic Schrödinger cat state for

a large number of atoms far more rapidly that the conventional process for generating such a state,

with an orientation that is independent of the parity of the number of atoms. Apart from the echo

configuration, we have also investigated the behavior of one-axis-twist squeezing for some special

values of the squeezing parameter. We find that the squeezing propagator can be expressed as the

sum of n rotation operators if the product of n and the squeezing parameter equals pi, where

a non-zero integer. A direct consequence of this property of one-axis-twist squeezing is that there

is a hidden order in a squeezed state generated under this condition even if its Husimi quasi-

probability distribution looks irregular.

1. Introduction

Reconciliation of quantum mechanics (QM) or, more generally, the standard model, with general

relativity has not yet been achieved. Aside from this challenge, QM is viewed as the undisputed

theory of how nature works. The law of QM applies on all scales. Just as the center of mass of an

electron or an atom is expected to obey the Schrödinger equation (in the non-relativistic limit), so

should the center of mass of a large object such as a glass marble with a cm-scale radius. Modern

experiments routinely produce quantum superposition of different states of microscopic objects.

For example, a recent experiment has produced a quantum superposition of a Rb atom separated

by 54 cm [1]. Given that an atom is very small at the scale of human perception, such a

superposition does not seem to elicit a sense of wonder at how QM differs from common sense

understanding of the laws of nature. On the other hand, if it were possible to create such a

superposition of a cm-scale marble, it would fundamentally alter our perspective on nature. Such

a superposition of a macroscopic object is so radical that in his famous paper [2] Schrödinger, after

describing a plausible scenario for creating a superposition of a cat being alive and dead, implied

that quantum theory is not complete and fully understood. In fact, there are many current theories

[3, 4, 5, 6, 7] that imply that QM, as formulated, is incomplete, and additional modification thereof

is warranted when applying to macroscopic objects. It is generally understood that a spatially

separated superposition of a macroscopic object would decay to one of the states extremely rapidly.

From the point of view of microscopic quantum theory, such a collapse is expected to result from

the decoherence of internal constituent particles that are at a finite temperature. However, other

theories, such as the one posed independently by Diósi and Penrose [3, 4, 5, 6, 7] imply that such

a collapse may occur even if the decoherence of the internal particles is not taken into account. In

order to address the question of whether QM indeed allows the creation of a spatially separated

superposition of a macroscopic object, one must study such a process experimentally. One way to

create spatially separated superpositions of macroscopic objects is to produce a quantum

superposition of a cavity mirror on a cantilever with the idea of optomechanics [8, 9, 10]. However,

the two quantum states can be only separated by a small distance with this method, and thermal

noise is a big challenge for such a method. Recently, we had proposed a scheme [11, 12] that can,

in principle, produce a spatially separated superposition of as many as 140 billion atoms (limited

by the maximum number of atoms caught in a magneto-optic trap to date [13]). However, this

process is probabilistic since the nature of the superposition depends on the parity of the number

of atoms involved. Besides, a relatively long atom-cavity interaction time is required for this

scheme, significantly lowering the success rate, especially for a large number of atoms. In this

paper, we propose a method using one-axis-twist squeezing in an echo configuration to produce a

Schrödinger cat state in a deterministic manner and with a much shorter atom-cavity interaction

time.

It should also be noted that the scheme proposed by Schrödinger for creating a

superposition of a dead cat and a live cat is highly misleading in this context, because the scheme

involves the detection of a quantum state, inducing a collapse of the superposition. To illustrate

more explicitly, let us consider a modern and simpler version of the experiment that would mimic

the idea presented by Schrödinger. Imagine the creation of a photonic quantum bit that is in a

linear superposition of two polarization states. We then send this photon through a polarizing

beam splitter, and place a single photon detector in one port of the beam splitter. If the detector

sees a photon, it produces a signal that moves a cat sitting on a cart on a rail to a different location,

B. If it does not, then the cat remains in the original position A. The implication in Schrödinger’s

paper (modified for the scenario presented here) is that this creates a superposition of the cat in

two different positions, A and B, at the same time. However, this is not at all correct. The detection

of the photon constitutes a measurement process, which collapses the quantum state of the photon.

As such, the process does not lead to a superposition of the cat in two different places. All we can

see from this thought experiment is that if we repeat the process many times, we will, with equal

probability, find the cat either in position A or position B. This is hardly radical, since a classical

probabilistic model would lead to the same conclusion. Thus, in order to create the scenario that

Schrödinger viewed to be problematic, implying incompleteness of quantum mechanics, what is

needed is a situation that would produce a superposition of the cat being in position A or B, without

any intervening measurement process. In this paper, we describe a technique that can indeed a

achieve this goal in a deterministic manner, for an ensemble of a large number of atoms, by

manipulating the quantum state of a single atom that is not physically close to this ensemble.

Experimental realization of such a scheme would provide a true test of whether it is indeed possible

to create a spatially separated superposition of a macroscopic object by entangling it to a

microscopic particle, or what the limit of such a process might be.

In addition to the echo configuration, we have also investigated the behavior of one-axis-

twist squeezing for some special values of squeezing parameter

. We find that the squeezing

propagator can be expressed as the sum of

rotation operators if

µπ

, where

is a non-zero

integer. A direct consequence of this property of one-axis-twist squeezing is that there is a hidden

order in a squeezed state when

µπ

even if its Husimi quasi-probability distribution looks

unregular.

The rest of this paper is organized as follows. In Sec. 2, we demonstrate the technique to

control the orientation of a coherent spins state with one or a few atoms. In Sec. 3, we show the

application of this technique to the generation of Schrödinger cat states. In Sec. 4, we prove that a

OATS propagator can be expressed as a sum of

rotation operators if

µπ

2. Control of the orientation of a coherent spin state with one or a few

atoms

To facilitate the exposition of the schemes presented here, it is convenient to summarize

first the relevant notations. A two-level atom can be modeled as a spin-1/2 spinor, with the spin

operator denoted as

( )

xyz

sss=s

, and the two eigenstates of

denoted as {

↑

↓

}, with the

eigenvalues of {

12−

}. The spin operator for an ensemble of atoms can be expressed as

=∑

, where

is the number of atoms and

is the spin operator of the j-th atom. The state

of this atom can be described by a point on the Bloch sphere. A point on the Bloch sphere can be

characterized by the polar angle

and the azimuthal angle

. The jth atom in the state

corresponding to such a point on the Bloch sphere is defined as

( ) ( )

, cos 2 e sin 2

jjj

θφ θ θ

≡ ↑+ ↓

. A coherent spin state (CSS) [14, 15] characterized by

the parameters

and

is defined as a state of

atoms with each atom in the state

θφ

, that

θφ θφ

≡⊗

The primary technology that enables this process is one-axis-twist squeezing (OATS) [16,

17, 18, 19, 20]. It has been shown previously that OATS can enhance the sensitivity of quantum

sensors by either suppressing the quantum noise [16, 21], or magnifying the quantum phase shift

[11, 22, 23, 24]. It has been found that that OATS with the squeezing parameter

µπ

can

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1RapidGenerationofaMacroscopicSchrödingerCatStateofAtomswithParity-IndependentOrientationJinyangLi1,GregórioR.M.daSilva1,SchuylerKain1,SelimM.Shahriar1,21DepartmentofPhysicsandAstronomy,NorthwesternUniversity,Evanston,IL60208,USA2DepartmentofECE,NorthwesternUniversity,Evanston,IL60208,USAAbstractWes...

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1 Rapid Generation of a Macroscopic Schrödinger Cat State of Atoms with Parity -Independent Orientation

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