## Spin squeezing in a driven Bose-Einstein condensate

Stewart D. Jenkins*and T. A. Brian Kennedy†

*School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430*

共Received 1 August 2002; published 30 October 2002兲

We consider a collection of atoms prepared in a Bose-Einstein condensate which interact via two-body elastic collisions. A resonant driving field couples two internal states which together constitute an effective spin 1/2 system for each atom, and from which a total spin for the gas can be defined. It is shown that in the limit of strong driving, the system dynamics reduces to a mechanism for spin squeezing similar to that of a recent proposal for undriven condensates. However, we find that the conditions for spatial stability in our driven system are complementary to those of the undriven system. Reasons for this difference, associated with the physics of dressed atom collisions, are discussed along with conditions for preparing and observing the spin squeezed state.

DOI: 10.1103/PhysRevA.66.043621 PACS number共s兲: 03.75.Fi, 03.65.Ud, 03.67.⫺a

## I. INTRODUCTION

*A system of N*Ⰷ1 distinguishable two state quantum
sys-tems has a Hilbert space of dimension 2*N*. Is it practically
possible to prepare arbitrary quantum states, by accessing the
recesses of this large space? The preparation of such
mas-sively entangled states is a major focus of current research in
quantum information physics. By definition, entangled states
cannot be represented as product states, or statistical
mix-tures of product states.

In recent years, a manifestation of a particular form of massive entanglement known as spin squeezing关1兴has been of great interest. Such correlated atomic wave functions could be used to enhance the accuracy of population mea-surements in spectroscopy and also have the potential to be used in atomic clocks关2– 4兴. Several methods for producing spin squeezed states have been proposed in the last decade 关3,5–11兴. Experimental realizations of these states have been made by mapping the quantum state of squeezed light onto a gas of cold uncondensed atoms 关12兴, as well as through quantum nondemolition measurements 关13,14兴.

*In a recent paper, Sorensen et al.*关15兴have suggested that
an entangled wave function for the internal state of a system
*of N atoms can be produced by means of low-energy elastic*
collisions within a vapor prepared in a special initial state.
The initial state is a zero-temperature Bose-Einstein
conden-sate 共BEC兲. A short controlled electromagnetic pulse is
ap-plied and, at the atomic level, coherently mixes two
single-particle atomic states, labeled by 兩⫹

典

and兩⫺典

, which form*an effective spin 1/2 system. The sum of N such atomic spins*

*defines a resultant N-particle spin angular momentum for the*

*gas. The effect of low-energy s-wave atomic collisions is to*

*squeeze, hence entangle, the associated N-particle spin wave*function of the condensate mixture 关1兴. The special initial condition, and the stability of a common spatial mode for condensates in either of the single-particle atomic states,

*leads to a squeezing which scales with N. This very attractive*feature, by contrast with some of the earlier proposals, means

that entanglement could be enhanced just by increasing the initial condensate particle number.

The stability of the common spatial mode, the so-called
‘‘breathe-together’’ mode is a nontrivial issue. A detailed
sta-bility analysis was reported in Ref.关16兴, and, in the
Thomas-Fermi limit, boils down to a condition on the relative
*strength of interstate and intrastate s-wave atomic collisions,*
*i.e., g*_{⫹⫺}⬍*g*_{⫹⫹}*,g*_{⫺⫺}*. Here g*_{} is proportional to the
scat-tering length for the collision of atoms in the stationary states

兩

典

and 兩典

, respectively, and is defined precisely later. In short, for a pair of overlapping condensates, interacting only via elastic collisions, a common spatial mode is stable for atomic states which have relatively weak interstate colli-sions.In this paper we present a scenario for squeezing the
*N-particle spin wave function in the opposite limit g*_{⫹⫺}
⬎*(g*_{⫹⫹}⫹*g*_{⫺⫺})/2, when the interstate collisions are
rela-tively strong. The procedure is similar to that of Ref. 关15兴,
and again relies on the existence of a common
breathe-together spatial mode, except that the initial BEC is strongly
driven by an external field, so that the atomic single-particle
states execute resonant Rabi oscillations at a frequency that
exceeds other characteristic time scales of the problem. This
type of driving between hyperfine ground states was
origi-nally demonstrated experimentally in a rubidium BEC by
*Matthews et al.*关17兴, and has since been used in the creation
of vortices关18,19兴. Spin squeezing is again caused by elastic
atomic collisions between condensate atoms which undergo
rapid Rabi oscillations, i.e., collisions of dressed atoms. The
reasons for the complementary stability criteria are, as will
be discussed, a result of the different collision properties of
bare atom and dressed atom condensates 关20兴.

The remainder of this paper is organized as follows. In Sec. II we present the basic theoretical model. An analysis of the model follows in Sec. III, where the limit of large Rabi frequency is used to simplify the equations. This section also includes a discussion of the breathe-together mode and its stability, an analytical calculation of the spin squeezing, and a comparison with the physics of Ref. 关15兴. Following our conclusion in Sec. IV, two appendixes provide further details on the breathe-together mode stability共Appendix A兲and the *Electronic address: stewart.jenkins@physics.gatech.edu

†_{Electronic address: brian.kennedy@physics.gatech.edu}

collisional interaction energy of bare and dressed conden-sates共Appendix B兲.

## II. THEORETICAL MODEL

We consider an ultracold BEC atomic gas with two single-particle states labeled by兩⫹

典

and兩⫺典

, which form a closed two state system. In an alkali atom these states should be hyperfine states belonging to the ground electronic term2_{S}

1/2*, although their total angular momenta F, and hence*
their energy, may be different. The possibility of collision
induced couplings to other degenerate hyperfine states means
that the two state system will not, in general, be closed. This
problem may be circumvented if one chooses the two
*inter-nal states with different F values such that both states have*
the minimum共or maximum兲*possible values for MF* in their
respective manifolds. Doing this will forbid spin-exchange
collisions between the two states, thus forming a closed
two-state system. Alternatively, if this choice cannot be made, the
unwanted degenerate states may be coupled via an off
reso-nant laser field to higher energy levels, thereby raising the
energy of the unwanted levels through the ac Stark shift, and
making spin-exchange collisions energetically unfavorable
关15兴. It should be noted that with the former option, it is
necessary to use an optical trap关21兴to confine the gas; since
these states generally have opposite magnetic moments, it
would not be possible to simultaneously trap them
magneti-cally. By assuming that a careful choice of the states has
been made, we will exclude from our model the influence of
other hyperfine states. The role of atomic losses in degrading
the squeezing has been discussed in Ref.关15兴, and we expect
similar qualitative features with the current scheme.

BEC atoms in the states 兩⫹

典

and兩⫺典

suffer low-energy elastic collisions, while a resonant external electromagnetic field induces transitions between them. The Hamiltonian for*the N-particle system may be written*

*Hˆ*共*t*兲⫽*Kˆ*⫹*Vˆext*⫹0*Sˆz*⫹*UˆC*⫹*HˆAF*共*t*兲, 共1兲
*where Kˆ is the kinetic energy, Vˆext* is the external trapping
*potential, UˆC* is the collisional interaction energy, and
*HˆAF(t) is the interaction between the atomic states and the*
external electromagnetic field. These operators are defined
by

*Kˆ*⫽⫺

兺

⫽⫹,⫺冕

*d*3*r* *ˆ*_{}†共**r**兲ប
2_{ⵜ}2

*2m* *ˆ*共**r**兲, 共2兲

*Vˆext*⫽

冕

*d*3

*r*

*ˆ*

_{⫹}†

_{共}

**r**兲*V*_{⫹}共**r**兲*ˆ*_{⫹}共**r**兲

⫹

冕

*d*3

*r*

*ˆ*

_{⫺}†共

**r**兲

*V*

_{⫺}共

**r**兲

*ˆ*

_{⫺}共

**r**兲, 共3兲

*HˆAF*共*t*兲⫽⫺ប
2

冉

冕

*d*

3_{r}_{}* _{ˆ}*
⫹

†_{共}_{r}_{兲}_{}_{共}_{r}_{兲}* _{e}*⫺

*i*

**(r)**⫺

*i*

_{L}t⫻*ˆ*

⫺共**r**兲⫹H.c.

冊

, 共4兲where*ˆ*_{}**(r) and***ˆ*_{}†**(r) are boson field annihilation and **
cre-ation operators for the bare atomic states,**(r) and****(r) are**
the effective Rabi frequency and phase of the external
*driv-ing field, respectively, and V*_{⫾}**(r) are the external trapping**
potentials for the atomic states兩⫾

典

. For transitions between two hyperfine states separated by a microwave frequency, the wavelength can be assumed large with respect to the vapor, and we can take the Rabi frequency and phase to be constant throughout the sample, i.e.,**(r)**⫽0 and

**(r)**⫽. The tran-sition frequency between the states is0 and the spin

*opera-tor Sˆ*is defined in terms of the Pauli spin matrix

_{z}*,*

_{z}*Sˆz*⫽
ប

2 _{},

兺

⫽⫹,⫺冕

*d*3*r* *ˆ*_{}†共**r**兲共*z*兲*ˆ*共**r**兲. 共5兲

*We similarly define Sˆxand Sˆy*, by replacing*z*with*x*and

*y*, respectively.

The two-body collisional interaction potential may be written

*UˆC*⫽
1
2 _{},

兺

⫽⫾*g*_{}

冕

*d*3

*r*

*ˆ*

_{}†共

**r**兲

*ˆ*†

_{}共

**r**兲

*ˆ*

_{}共

**r**兲

*ˆ*

_{}共

**r**兲,

*where the collision coefficients g*_{}⫽42ប2*a*_{}*/m with a*_{}
the scattering length between atoms in the internal states
and . Note that aside from the atomic energy and Rabi
coupling terms in the Hamiltonian, this is exactly the same
*Hamiltonian discussed by Sorensen et al.*关15兴.

## III. ANALYSIS OF THE MODEL

### A. Limit of large Rabi frequency

We consider the limit that the Rabi frequency is the
largest frequency in the problem. The analysis then proceeds
by means of two unitary transformations 关22兴. The first is
*generated by Uˆ (t)*⫽exp(⫺*i**LSˆz*/ប), and yields a time
depen-dent Schrodinger equation with a time independepen-dent
*Hamil-tonian Hˆ*

⬙

*. The latter is identical to the Hamiltonian Hˆ (t)*

*with the substitutions HˆAF(t)→HˆAF*(0)⫽⫺

*Sˆx*and 0

*→*⌬⫽0⫺*L*. Assuming atomic resonance,⌬⫽0, we may
*write Hˆ*

⬙

as*Hˆ*

⬙

⫽*Kˆ*⫹

*Vˆ*⫹

_{ext}*Uˆ*⫺

_{C}*Sˆ*. 共6兲

_{x}*A second unitary transformation is applied with Wˆ (t)*⫽

*exp(i*

*Sˆx*/ប), which yields the Schro¨dinger equation

*i*ប(/

*t)*兩

*(t)*

典

⫽*Hˆ*

⬘

*(t)*兩

*(t)*

典

*with Hˆ*

⬘

*(t)*⫽

*Kˆ*⫹

*Wˆ*†

*(t)(Vˆext*⫹

*UˆC)Wˆ (t). Here,*兩

*(t)*

典

is the state vector in the rotating frame, and is related to the Schro¨dinger picture state vector in the laboratory frame 兩⌿*(t)*

典

, by 兩⌿*(t)*

典

⫽*Uˆ (t)Wˆ (t)*兩

*(t)*

典

.*Wˆ*†_{V}_{ˆ}

*extWˆ*⫽

兺

⫽⫹,⫺冕

*d*3_{r}_{}* _{ˆ}*

†_{共}_{r}_{兲}_{V}_{¯}_{共}_{r}_{兲}_{}* _{ˆ}*
共

**r**兲

⫹cos*t*

冕

*d*3

*r*关

*ˆ*

_{⫹}†共

**r**兲

*Vd*共

**r**兲

*ˆ*⫹共

**r**兲 ⫺

*ˆ*

⫺
†_{共}

**r**兲*Vd*共**r**兲*ˆ*⫺共**r**兲兴

⫹*i sin**t*

冕

*d*3

*r*关

*ˆ*

_{⫹}†共

**r**兲

*Vd*共

**r**兲

*ˆ*⫺共

**r**兲 ⫺

*ˆ*

⫺
†_{共}_{r}_{兲}_{V}

*d*共**r**兲*ˆ*⫹共**r**兲兴, 共7兲
where *V ¯ (r)*⫽1

2关*V*⫹**(r)**⫹*V*⫺**(r)**兴 and *Vd***(r)**⫽

1
2关*V*⫹**(r)**

⫺*V*_{⫺}**(r)**兴. This expression is formally in agreement with the
*mean-field theory of Williams et al.* 关22兴. The two-body
in-teraction potentials transform as

*Wˆ*†*UˆCWˆ*⫽*¯g*

冕

*d*3

*r Tˆ*0

(0)_{共}_{r}_{兲}_{⫹}_{g}

⬘

冕

*3*

_{d}

_{r}_{关}

_{Tˆ}*z*

(1)_{共}_{r}_{兲}_{cos}_{}_{t}

⫹*Tˆ _{y}*(1)共

**r**兲sin

*t*兴⫺

*g*

冕

*d*3

*r*关

*e2i*

*tTˆ*

_{⫺}(2)

_{2}共

**r**兲 ⫹

*e*⫺

*2i*

*tTˆ*

_{2}(2)共

**r**兲兴

⫹*g*

冕

*d*3

*r*

*ˆ*†

_{x}_{⫹}共

**r**兲

*ˆ*†

_{x}_{⫺}共

**r**兲

*ˆx*⫺共

**r**兲

*ˆx*⫹共

**r**兲, 共8兲

where*ˆ _{x}*†

_{⫹}

**(r)**关

*ˆ*†

_{x}_{⫺}

**(r)**兴 are bosonic field creation operators for a spin one-half oriented in the positive 关negative兴

*x*di-rection, i.e.,

*ˆ*†

_{x}_{⫹}

**(r)**⫽关

*ˆ*

_{⫹}†

**(r)**⫹

*ˆ*

_{⫺}†

**(r)**兴/

冑

2 and *ˆ*†

_{x}_{⫺}

**(r)**⫽ ⫺

*i*关

*ˆ*

_{⫹}†

**(r)**⫺

*ˆ*

_{⫺}†

**(r)**兴/

冑

2 关26兴. Alternatively, it is useful to re-gard these as field operators for dressed atomic single-particle states, arising as a result of the Rabi coupling of the bare atom to the external electromagnetic field关20兴.The collision coupling coefficients are given by

*g*
*¯*_{⫽}1

4 _{},

兺

⫽⫹,⫺*g*_{}, 共9兲

*g*

⬘

⫽14共*g*⫹⫹⫺*g*⫺⫺兲, 共10兲

*g*⫽1

4共*g*⫹⫹⫹*g*⫺⫺⫺*2g*⫹⫺兲. 共11兲

*We have also defined the collision kernels Tˆ _{q}(k)*

**(r) as**fol-lows:

*Tˆ*_{0}(0)共**r**兲⫽

兺

,⫽⫾
*ˆ*
†_{共}

**r**兲*ˆ*_{}†共**r**兲*ˆ*_{}共**r**兲*ˆ*_{}共**r**兲, 共12兲

*Tˆ*_{⫾}(1)_{1}共**r**兲⫽⫿ 1

冑

2关*ˆ*
*x*⫹
† _{共}

**r**兲*ˆ _{x}*†

_{⫾}共

**r**兲

*ˆx*⫿共

**r**兲

*ˆx*⫹共

**r**兲

⫹*ˆ*
*x*_{⫺}
† _{共}_{r}_{兲}_{}_{ˆ}

*x*_{⫾}
† _{共}_{r}_{兲}_{}_{ˆ}

*x*⫿共**r**兲*ˆx*⫺共**r**兲兴, 共13兲

*Tˆ*_{⫾}(2)_{2}⫽1
2*ˆx*⫾

† _{共}_{r}_{兲}_{}_{ˆ}*x*⫾
† _{共}_{r}_{兲}_{}_{ˆ}

*x*_{⫿}共**r**兲*ˆx*_{⫿}共**r**兲, 共14兲

and⫿

冑

*2Tˆ*

_{⫾}(1)

_{1}

**(r)**⫽

*Tˆ*(1)

_{y}**(r)**⫾

*iTˆ*(1)

_{z}**(r).**

*The notation Tˆq*

*(k)*

*indicates the q component of a rank k*
spherical tensor with respect to the effective spin, as defined
by the commutation relations

关*Sˆ _{x},Tˆ_{q}(k)*共

**r**兲兴⫽ប

*qTˆ*共

_{q}(k)**r**兲, 共15兲 关

*Sˆy*⫾

*iSˆz,Tˆq*

*(k)*

共**r**兲兴⫽ប

冑

共*k*⫿

*q*兲共

*k*⫾

*q*⫹1兲

*Tˆ*

_{q}(k)_{⫾}

_{1}共

**r**兲. 共16兲 Rewriting the collision interaction in terms of these spherical tensors is useful for the purpose of calculating the unitary transformation, since it follows from their definition that

*Wˆ*†共*t*兲*Tˆ _{q}(k)*共

**r**兲

*Wˆ*共

*t*兲⫽

*e*⫺

*iq*

*tTˆ*共

_{q}(k)**r**兲. 共17兲 The one collision kernel that has not been explicitly defined in terms of tensors, is a linear combination of tensors of rank

*0 and 2, each with component q*⫽0. As a consequence, it is unaffected by the second unitary transformation, and thus we leave it in the explicit form given.

We are interested in the limit of large Rabi frequency
compared to any other characteristic frequency. In this limit
we make a form of rotating wave approximation and cycle
*average the rotating frame Hamiltonian Hˆ*

⬘

*(t) over the Rabi*period to get

### HˆRWA

⬘

⫽*Kˆ*⫹

兺

⫽⫾冕

*d*

3_{r}_{}* _{ˆ}*
†

_{共}

**r**兲*V¯*共**r**兲*ˆ*_{}共**r**兲

⫹1

2*¯g*,

兺

_{}

_{⫽⫾}

冕

*d*3*r* *ˆ*_{}†共**r**兲*ˆ*_{}†共**r**兲*ˆ*_{}共**r**兲*ˆ*_{}共**r**兲

⫹*g*

冕

*d*3

*r*

*ˆx*

_{⫹}†

_{共}

**r**兲*ˆx*_{⫺}
† _{共}

**r**兲*ˆx*⫺共**r**兲*ˆx*⫹共**r**兲, 共18兲

*in which all collision tensors with q*⫽0 have disappeared. In
the limit of strong driving the vapor experiences only the
*average external potential V ¯ (r) of the two strongly coupled*
single-particle states关22兴. In a far off-resonance optical trap
关21兴, unlike a magnetic trap, it is reasonable to assume that

*the latter potentials are identical, and thus equal to V*

**¯ (r).**### B. Single-mode approximation: the breathe-together mode

*In the rotating frame the Hamiltonian H _{RWA}*

⬘

is indepen-dent of the driving field. This enables us to make use of the analysis of Ref. 关16兴, and assume the existence of a stable single common ‘‘breathe-together’’ mode

**¯ (r,t). Perhaps***surprisingly, the condition for stability of this mode, g*

_{⫹⫺}⬎

*(g*

_{⫹⫹}⫹

*g*

_{⫺⫺})/2, is opposite to that required for its stability in the BEC squeezing scheme of Ref. 关15兴. In the latter scheme squeezing results from the collisional interaction of two bare atomic condensates without any Rabi coupling.

of the mean interaction energy of two condensates in which
the macroscopically occupied single-particle states are,
*re-spectively, bare atomic states E _{int}b* and dressed atomic states

*E*, is revealing共Appendix B兲. These energies are given by

_{int}d*E _{int}b* ⫽

*g*

_{⫹⫺}

冕

*d*3

*r*

_{⫹}共

**r**兲

_{⫺}共

**r**兲, 共19兲 where

_{⫾}

**(r)**⬅

*N*

_{⫾}兩

_{⫾}

**(r)**兩2 are the spatial densities of the bare states, and

*Eint*
*d* _{⫽}1

2共*g*⫹⫹⫹*g*⫺⫺兲

冕

*d*3

_{r}_{}

*x*⫹共**r**兲*x*⫺共**r**兲, 共20兲

where*x*⫾**(r)**⬅*Nx*⫾兩*x*⫾**(r)**兩2are the spatial densities of the
dressed states. The interaction energy is proportional to the
overlap integral of the corresponding condensate spatial
*den-sity profiles, with the expected prefactor g*_{⫹⫺} in the bare
*atom case, but with a prefactor (g*_{⫹⫹}⫹*g*_{⫺⫺})/2 for dressed
*condensates. Indeed the coefficient g*_{⫹⫺} only contributes to
the self-interaction energy of dressed condensates. These
un-usual features of elastic collisions in dressed condensates are
reflected in the different stability criterion for the
breathe-together mode.

Assuming the atomic states are chosen such that the sta-bility condition for the breathe-together mode is satisfied, we may define annihilation operators for this mode by 关20,23,24兴

*aˆ _{x}*

_{⫾}共

*t*兲⫽

冕

*d*3

*r*

*¯**

_{共}

*兲*

**r,t***ˆ*

_{x}_{⫾}共

**r**兲, 共21兲 with corresponding definitions for creation operators, while

*the number operator aˆ*†

_{x}_{⫾}

*(t)aˆx*⫾

*(t)*⫽

*aˆx*⫾

†

*aˆx*⫾ is time
inde-pendent.

*Hence, in Hˆ _{RWA}*

⬘

we replace the field operators *ˆx*⫹

**(r)**

*→** ¯ (r,t)aˆx*⫹

*(t),*

*ˆx*⫺

**(r)**

*→*

*⫺*

**¯ (r,t)aˆ**x*(t) to give*

### HˆRWA

⬘

*→NE*共

*t*兲⫹ប⍀共

*t*兲

*aˆ*†

_{x}_{⫹}

*aˆ*†

_{x}_{⫺}

*aˆx*⫺

*aˆx*⫹ ⫽

*NE*共

*t*兲⫺ប⍀共

*t*兲

冉

*Sˆx*

2 ប2⫺

1
4*N*

2

冊

_{,}

_{共}

_{22}

_{兲}

where

*E*⫽

冕

*d*3

*r*

*¯**共

*兲共*

**r,t***K*⫹

*V¯*兲

*¯*共

*兲*

**r,t**⫹1_{2} *¯g*

冉

冕

*d*3

*r*兩

*¯*共

*兲兩4*

**r,t**冊

共*N*⫺1兲

and

ប⍀共*t*兲⫽*g*

冉

冕

*d*3

*r*兩

*¯*共

*兲兩4*

**r,t**冊

.*Thus in the breathe-together mode approximation H _{RWA}*

⬘

re-duces to the one-axis twist Hamiltonian originally discussed in the context of spin squeezing by Kitagawa and Ueda关1兴.*The nomenclature arises since Sˆx*generates a rotation about

*the x axis, and Sˆx*

2

generates a rotation in which the
*eigen-states of Sˆx*are rotated in opposite directions.

The degree of spin squeezing that results from time
evo-lution under this Hamiltonian can be quantified by the
pa-rameter 2⫽inf兵*N(*⌬*Sˆ*_{⬜})2/兩

具

**Sˆ**

典

兩2*:Sˆ*

_{⬜}⫽

**Sˆ**•

**n, where n**⬜

具

**Sˆ**

典

其. This definition for the degree of entanglement is consistent*with those discussed by Wineland et al.*关3,4兴 and Sorensen

*et al.*关15兴. It has been shown that if2⬍1, then the conden-sate exhibits massive entanglement and spin squeezing关15兴. It can also be demonstrated that we attain maximal squeezing when our initial state vector is a coherent spin state 共as de-fined by Kitagawa and Ueda 关1兴兲whose mean spin vector is perpendicular to the axis about which the one axis twisting

*occurs, in this case the x axis. A convenient realization of this*initial condition would be a zero-temperature BEC with all atoms in the internal state,兩⫺

典

.With the coherent state initial condition, the squeezing parameter as a function of time can be computed analytically 关1兴, and is given by

2_{共}_{t}_{兲}_{⫽}

## 再冋

1⫹14共*N*⫺1兲*A*共*t*兲

册

⫺ 1### 4共N⫺1兲

冑

*A*共

*t*兲

2_{⫹}_{B}_{共}_{t}_{兲}2

冎

cos*2N*⫺2

冉

冕

0*t*

⍀共*t*

⬘

兲*dt*

⬘

冊

,

共23兲

where *A(t)*⫽1⫺cos*N*⫺2关2兰0
*t*_{⍀}

*(t*

⬘

*)dt*

⬘

兴, and*B(t)*⫽ ⫺4 sin关兰

_{0}

*t*⍀

*(t*

⬘

*)dt*

⬘

兴cos*N*⫺2关兰

_{0}

*t*⍀

*(t*

⬘

*)dt*

⬘

兴 共Fig. 1兲*. For large N,*it has been shown that the minimum attainable squeezing parameter scales as2⬃

*N*⫺2/3关1兴.

*For spin squeezing to occur the collision coefficient g*
must be nonzero. Furthermore, for our approximations to be
valid, the characteristic frequency ⍀ must be small
com-pared to the Rabi frequency . To illustrate the time scales,
we give some estimates of the characteristic parameters. We
assume, for simplicity, that the vapor is trapped in a
spheri-cally symmetric potential *V(r)*⫽*V*_{⫺}*(r)*⫽*V*_{⫹}*(r)*
⫽1

2*m**tra p*
2

*r*2. The initial wave function 0**(r) can be **
deter-mined using the Thomas-Fermi approximation to the time
FIG. 1. The squeezing parameter 2 as a function of time,

⫽兰0

*t*

⍀*(t*⬘*)dt*⬘, for the evolution of the system under the
Hamil-tonian⫺⍀*(t)Sˆx*

2

independent Gross-Pitaevski equation关25兴. Using this proce-dure we numerically determine the characteristic frequency ⍀ to be

⍀关s⫺1兴⫽1.4⫻10⫺4*m*共*a*_{⫹⫹}⫹*a*_{⫺⫺}⫺*2a*_{⫹⫺}兲

⫻共*tra p*关s⫺1兴兲6/5

冉

*ainit*

*N*

冊

3/5### , 共24兲

where the scattering lengths are given in Bohr radii, and the
*atomic mass m is measured in atomic mass units. The *
*scat-tering length ainit*corresponds to the initial condensate, prior
to the application of the external electromagnetic field. For
example, if the system condenses in the single-particle state

兩⫺

典

*, then ainit*⫽

*a*⫺⫺.

We further note, using the results of Ref.关1兴, that the time taken to attain the minimum squeezing parameter is given by

*t*min关s兴⬇

3.7⫻104
*a _{init}*3/5

*m*⫺1/5共*tra p*关s⫺1兴兲⫺6/5*N*⫺1/15
共*a*_{⫹⫹}⫹*a*_{⫺⫺}⫺*2a*_{⫹⫺}兲 , 共25兲

which for*tra p*/2⬇1 kHz, is typically a fraction of a
*sec-ond, virtually independent of N for practical purposes.*

### C. Discussion of BEC spin squeezing scenarios

In the scheme of Ref.关15兴, spin squeezing is produced by coherent elastic collisions in a BEC mixture of atoms which individually constitute effective spin one-half bosons. A fast

/2 pulse is applied to the condensate with all atoms initially prepared in the internal state兩⫺

典

共for ease of comparison we will use as close a notation as is possible to that employed in the rest of our paper兲. The /2 pulse produces a bare atom condensate mixture with all of the ‘‘spins’’ oriented in the*positive x direction. Elastic collisions then act to squeeze the*initial coherent spin state. By contrast, in our scheme, elastic collisions take place during the resonant Rabi oscillations induced by a strong applied field, squeezing the dressed atom condensate mixture.

The Hamiltonian of Ref. 关15兴is given in our notation by
*Hˆsor*⬅*Hˆ (t)*⫺*HˆAF(t)* 共with0⬅0),

*Hˆsor*⫽*Kˆ*⫹*Vˆ*⫹
1

2

兺

_{}

*g*

冕

*d*3

_{r}_{}

_{ˆ}
†_{共}

**r**兲*ˆ*_{}†共**r**兲*ˆ*_{}共**r**兲*ˆ*_{}共**r**兲⫹*g*_{⫹⫺}

冕

*d*3

*r*

*ˆ*

_{⫹}†共

**r**兲

*ˆ*

_{⫺}†共

**r**兲

*ˆ*

_{⫺}共

**r**兲

*ˆ*

_{⫹}共

**r**兲

⫽*Kˆ*⫹*Vˆ*⫹1

4共*g*⫹⫹⫹*g*⫺⫺兲

兺

_{},

冕

*d*3*r* *ˆ*_{}†共**r**兲*ˆ*_{}†共**r**兲*ˆ*_{}共**r**兲*ˆ*_{}⫹1

4共*g*⫹⫹⫺*g*⫺⫺兲

⫻

冕

*d*3

*r*

*ˆ*

_{⫹}†共

**r**兲

*ˆ*

_{⫹}†共

**r**兲

*ˆ*

_{⫹}共

**r**兲

*ˆ*

_{⫹}共

**r**兲⫺1

4共*g*⫹⫹⫺*g*⫺⫺兲

冕

*d*3

_{r}_{}

_{ˆ}⫺
†_{共}_{r}_{兲}_{}_{ˆ}

⫺
†_{共}_{r}_{兲}_{}_{ˆ}

⫺共**r**兲*ˆ*⫺共**r**兲

⫺*2g*

冕

*d*3

*r*

*ˆ*†

_{⫹}共

**r**兲

*ˆ*

_{⫺}†共

**r**兲

*ˆ*

_{⫺}共

**r**兲

*ˆ*

_{⫹}共

**r**兲. 共26兲

*For g*_{⫹⫹}⫽*g*_{⫺⫺}, which is satisfied in the case of sodium,
*Hˆsorreduces to a form very similar to HˆRWA*

⬘

in Eq.共18兲. The*last term in Hˆsor, causes spin squeezing by means of a z axis*

*twist. The operator Sˆz*is diagonal for states with a definite occupancy of the single-particle states兩⫾

典

*. With Hˆ*

_{RWA}⬘

, the*wave function in the rotating frame is squeezed by an x-axis*

*twist; Sˆx*is diagonal for states with definite occupancy of the single-particle dressed states 兩

*x*⫾

典

. However, in both cases squeezing relies on an effective single-mode approximation. As discussed in the last subsection, and in more detail in Appendix A, the stability criteria for the existence of the breathe-together mode on the intraparticle and interparticle scattering lengths are opposite inequalities in the two*sce-narios, i.e., g*

_{⫹⫺}⬍

*g*

_{⫹⫹}

*,g*

_{⫺⫺}for bare condensates, and

*g*

_{⫹⫺}⬎

*(g*

_{⫹⫹}⫹

*g*

_{⫺⫺})/2 for dressed condensates. The two sce-narios are thus complementary rather than alternatives. In the

*single-mode approximation, Hˆ*

_{sor}*contains the z-axis twist*operator 2⍀

*Sˆ*2/ប

_{z}*, while Hˆ*

_{RWA}⬘

*contains the x-axis twist*

⫺⍀*Sˆ _{x}*2/ប. The factor two difference arises because we

*aver-age over fast Rabi oscillations in the derivation of Hˆ*

_{RWA}⬘

. In Ref. 关15兴, factors such as multiple modes and atomic losses, which reduce or destroy squeezing are analyzed. We will not discuss these further, as we expect their effect to be qualita-tively similar here.mean spin vector maintains the same orientation, along the
negative *z* axis, so that

具

*(t)*兩

*Sˆz*兩

*(t)*

典

⬍0 and具

*(t)*兩

*Sˆx,y*兩

*(t)*

典

⫽0, although its magnitude shrinks with time. In the laboratory frame, relevant spin observablesshould be computed with the lab frame state vector兩⌿*(t)*

典

⫽*Uˆ (t)Wˆ (t)*兩

*(t)*

典

and the similarly transformed spin**opera-tor Sˆ**

*R(t)*⫽

*†*

**U(t)W(t)SˆW***(t)U*†

*(t). The latter can be written*explicitly as

冉

*SˆxR*共

*t*兲

*Sˆy R*共

*t*兲

*SˆzR*共*t*兲

冊

⫽冉

1 0 0

0 cos*t* sin*t*
0 ⫺sin*t* cos*t*

## 冊冉

cos*Lt* sin*Lt* 0
⫺sin*Lt* cos*Lt* 0

0 0 1

冊

冉

*Sˆx*

*Sˆy*

*Sˆz*

冊

### . 共27兲

In terms of the laboratory frame Schro¨dinger picture state
vector 兩⌿*(t)*

典

, we have, for example,具

*(t)*兩

**Sˆ**兩

*(t)*

典

⫽具

⌿*(t)*兩

**Sˆ**

*兩⌿*

_{R}(t)*(t)*

典

.*The system is subject to two rotations: one about the z*
axis at a frequency*L*, and another rotation about the
*nega-tive x axis at frequency* . In the lab frame, one would see
*the mean spin vector rotating about the negative x axis at*
frequency ; this behavior is then superimposed on a
*rota-tion about the z axis at frequency* *L*. The spin squeezing
manifests itself as a modulation on the rotational motion of
the mean spin, and occurs along axes that are perpendicular
to this vector.

The spin squeezing could, in principle, be observed using
*a method similar to that described by Wineland et al.*关4兴. For
the purpose of measuring the spin components orthogonal to
the mean spin, it is necessary to have the mean spin oriented
*in a known direction, say along the z axis. Since the direction*
of the mean spin vector oscillates rapidly, making such a
measurement may be difficult at an arbitrary time. This
ob-stacle may be overcome in the following way. Near the time
*when maximum squeezing is attained, t*_{min}*, a time T is *
cho-sen such that *T*⫽*2n* *for some integer n. The coupling*
field would then be turned off leaving the Schro¨dinger
picture and rotating frame state vectors related
by 兩⌿*(T)*

典

⫽*Uˆ (T)Wˆ (T)*兩

*(T)*

典

⫽*Uˆ (T)exp*关

*iSˆx(2n*/ប)兴兩

*(T)*

典

⫽*Uˆ (T)*兩

*(T)*

典

. Since the mean spin in the rotating frame is oriented along the ⫺*z axis for all times*关1兴

*, and Uˆ (T) is*nothing more than a rotation of 0

*T about the z axis, the*mean spin as measured in the laboratory will be oriented along the ⫺

*z direction. The same Schro¨dinger state vector*

兩⌿*(T)*

典

may also be obtained by a variation of Ramsey spec-troscopy, by employing two resonant pulses, each of time*duration T/2, and with Rabi frequencies* and⫺,

*respec-tively. The net effect is that Wˆ (T)*⫽

*exp(i*

*TSˆx*/2ប)exp (⫺

*i*

*TSˆx*/2ប)⫽

*1ˆ. We have already shown that the state*vec-tor in the rotating frame,兩

*(T)*

典

, is squeezed independent of*, since Hˆ _{RWA}*

⬘

is independent of.Once the final state has been produced, the standard
de-viation of the spin along an arbitrary axis perpendicular to
the mean spin vector is measured 关4,15兴. To do this, the
**system should be rotated about the zˆ axis by an appropriate**
amount, and then a**/2 rotation about the yˆ axis should be**

applied. The population of either the兩⫹

典

or兩⫺典

state is then*measured, thereby also measuring Sˆz*. A sub-shot-noise mea-surement of the projection noise indicates spin squeezing 关3,4兴.

## IV. CONCLUSION

We have considered the possibility of spin squeezing in a resonantly driven mixture of Bose-Einstein condensates. In the limit that the Rabi frequency is the largest frequency scale in the problem, we have shown that elastic collisions squeeze quantum fluctuations of the spin component trans-verse to the average spin vector, according to the single-axis twist mechanism of Ref. 关1兴.

We have further shown that the proposed scheme is
*complementary to the proposal of Sorensen et al.* 关15兴 for
spin squeezing of a BEC. While both scenarios depend on
the existence of a mutually stable spatial mode for the
inter-acting condensates, the stability criteria are mutually
*exclu-sive: g*_{⫹⫺}⬍*g*_{⫹⫹}*,g*_{⫺⫺} for Ref. 关15兴*, but g*_{⫹⫺}⬎*(g*_{⫹⫹}
⫹*g*_{⫺⫺})/2 here. The new stability criterion was attributed to
the different collision properties of bare and dressed
conden-sate mixtures, and is discussed in the appendixes. The
present proposal is thus limited to atoms with relatively
strong interstate collisions, just as Ref. 关15兴 is limited to
those with strong intrastate collisions.

**ACKNOWLEDGMENTS**

We wish to thank M. Chapman, C. Raman, and A. Kuzmich for useful discussions, and the NSF, Grant No. 9803180, and NSA, under Grant No. ARO DAA55-98-1-0370, for support. One of us 共T.A.B.K.兲 would also like to thank the Aspen Center for its hospitality.

## APPENDIX A: STABILITY OF THE BREATHE-TOGETHER MODE

### HˆRWA

⬘

⫽*Kˆ*⫹

兺

⫽*a,b*

冕

*d*3_{r}_{}* _{ˆ}*

†_{共}_{r}_{兲}_{V}_{¯}_{共}_{r}_{兲}_{}* _{ˆ}*
共

**r**兲

⫹1_{2}

兺

⫽*a,b*

*g*_{}

冕

*d*3

*r*

*ˆ*

_{}†共

**r**兲

*ˆ*†

_{}共

**r**兲

*ˆ*

_{}共

**r**兲

*ˆ*

_{}共

**r**兲

⫹*gab*

冕

*d*3

*r*

*ˆa*†

_{共}

_{r}_{兲}

_{}

_{ˆ}*b*
†_{共}_{r}_{兲}_{}_{ˆ}

*b*共**r**兲*ˆa*共**r**兲, 共A1兲

where, for ease of our subsequent comparison, we use the
notation 兩*x*⫹

典

⫽兩*a*

典

and 兩*x*⫺

典

⫽兩*b*

典

to label the single-particle dressed states. The effective scattering coefficients are given by*gaa*⫽*gbb*⫽
1

4共*g*⫹⫹⫹*g*⫺⫺⫹*2g*⫹⫺兲,

*gab*⫽
1

2共*g*⫹⫹⫹*g*⫺⫺兲. 共A2兲

We can now adapt the results of Sinatra and Castin 关16兴, to
identify an effective single mode or ‘‘breathe-together’’
so-lution. These authors considered the collisional interaction of
a pair of condensates with a Hamiltonian identical in form to
Eq. 共A1兲*, but with a and b corresponding to bare atomic*
single-particle states, rather than dressed states. We briefly
summarize the relevant theory in order to make the
argu-ments clear.

By preparing an initial condensate in a self-consistent
mode 0**(r) and in the bare atomic state** 兩⫺

典

⫽(兩*a*

典

⫺*i*兩

*b*

典

)/冑

2⬅*ca*兩

*a*

典

⫹*cb*兩

*b*

典

关26兴, we have the corresponding*N particle initial state in the rotating frame*

### 兩共0兲

典

⫽兺

*N*0

_{a⫽}*N*

冑

*N!*

*N _{a}*!共

*N*⫺

*N*!兲

_{a}*ca*

*N*

_{a}*c _{b}N*⫺

*Na*兩

_{N}*a*:0*,Nb*:0典,
共A3兲

*where Naand Nb* *are the occupancies of the dressed states a*
*and b. The time evolution of this state according to H _{RWA}*

⬘

is given by兩共*t*兲

典

⫽兺

*N*⫽0

_{a}*N*

冑

*N!*

*Na*!共*N*⫺*Na*兲!
*c _{a}Na_{c}*

*b*
*N*⫺*N _{a}*

*e*⫺*iA(Na,Nb;t)*
⫻兩*N _{a}*:

*共*

_{a}*N*兲

_{a},N_{b};t*,N*:

_{b}*共*

_{b}*N*兲

_{a},N_{b};t典

, 共A4兲where *a***(r,0)**⫽*b***(r,0)**⫽0* (r) for all Na*. The dressed
state mode functions

*⬅*

_{a,b}(N_{a},N_{b}**;r,t)***coupled Gross-Pitaevskii equations,*

_{a,b}**(r,t) satisfy the**共* r,t*兲

*t*⫽

1
*i*ប

冉

⫺ប2
*2m*ⵜ

2_{⫹}_{V}_{¯}_{共}_{r}_{兲}_{⫹}

兺

⫽*a,b*

*g*_{}*N*_{}兩_{}共* r,t*兲兩2

冊

⫻共*兲, 共A5兲*

**r,t***and A(Na,Nb;t) is given by the equation*

*A˙*共*Na,Nb;t*兲⫽⫺
1
2

兺

⫽*a,b*

*N*_{}2*g*_{}

冕

*兩*

**dr**_{}共

*兲兩4*

**r,t**⫺*NaNb*

冕

*兩*

**dr***a*共

*兲兩2兩*

**r,t***b*共

*兲兩2. 共A6兲*

**r,t**Using the fact that the variance of occupied states is much
less than the mean ⌬*N _{a,b}*Ⰶ

*N¯*, one can approximate

_{a,b}*(Na,Nb ;r,t)→*

*(N¯a,N¯b*⬅

**;r,t)***¯*

**(r,t).**The conditions for the existence of a common spatial
mode for the two dressed states *¯a (r,t)*⫽

*¯b*⬅

**(r,t)***requires that the total effective potentials are equal关16兴, i.e.,*

**¯ (r,t),***N*

*¯ _{a}_{g}_{aa}*

_{⫹}

_{N}¯_{b}_{g}_{ab}_{⫽}

_{N}¯_{b}_{g}_{bb}_{⫹}

_{N}¯_{a}_{g}_{ab}_{⬅}

_{Ng}_{e f f}_{,}

_{共}

_{A7}

_{兲}

which implies that

*N*
*¯*

*a*

*N*
*¯ _{b}*⫽

*gaa*⫺*gab*
*gbb*⫺*gab*

### . 共A8兲

For this result to have physical solutions, it is necessary that either

*gab*⬍*gaa,gbb* 共A9兲

or

*g _{ab}*⬎

*g*. 共A10兲

_{aa},g_{bb}The linearized stability analysis of Ref. 关16兴 indicates that
the former case is stable, whereas the latter is unstable to a
demixing instability of the condensates. We note that in our
*case g _{aa}*⫽

*g*⫽

_{bb}*g¯ , so that the average particle number in*

*each dressed state is equal N¯a*⫽

*N¯b*. The condition for

*sta-bility of the common mode gab*⬍

*gaa,gbb*reduces to

1

2共*g*⫹⫹⫹*g*⫺⫺兲⬍*g*⫹⫺, 共A11兲

which is opposite to the stability criterion in the
*entangle-ment scheme of Sorenson et al.*关15兴.

Assuming that two bare atomic states which satisfy the
stability criterion have been identified, then the
breathe-together solution is*¯a (r,t)*⫽

*¯b*⬅

**(r,t)****

**¯ (r,t), where***¯*sat-isfies the equation

*t**¯*共* r,t*兲⫽
1

*i*ប

冉

⫺ប2
*2m*ⵜ

2_{⫹}_{V}_{¯}_{共}_{r}_{兲}_{⫹}_{Ng}

*e f f*兩*¯*共* r,t*兲兩2

冊

*¯*共

*兲, 共A12兲*

**r,t***with g, the effective scattering coefficient, given by*

*ge f f*⫽
3

8共*g*⫹⫹⫹*g*⫺⫺兲⫹
1

## APPENDIX B: MEAN COLLISION ENERGY IN BARE AND DRESSED CONDENSATES

A straightforward calculation shows that the mean
colli-sion energy in a pure Fock state兩*Na*:*a;Nb*:*b*

典

, with defi-nite occupancies of the dressed states 兩*a*

典

and兩*b*

典

, is given by*E _{M F}d* ⫽1
2

*g¯*

_{}

兺

_{⫽}

*a,b*

*N*_{}⫺1
*N*_{}

冕

*d*

3_{r}_{}
2_{共}_{r}_{兲}_{⫹}1

2共*g*⫹⫹⫹*g*⫺⫺兲

⫻

冕

*d*3

*r*

*a*共

**r**兲

*b*共

**r**兲, 共B1兲

where _{}**(r)**⬅*N*_{}兩_{}**(r)**兩2 is the spatial density of dressed
state ⫽*a,b. The last result may be contrasted with the*
mean-field energy of two condensates in a Fock state

兩*N*_{⫹}:_{⫹}*;N*_{⫺}:_{⫺}

典

, with definite occupancies of the bare atomic states兩⫹典

and兩⫺典

,*Eb _{M F}*⫽1
2

_{}

_{⫽⫹}

兺

,⫺*g*_{}*N*⫺1
*N*_{}

冕

*d*

3_{r}_{}
2_{共}

**r**兲⫹*g*_{⫹⫺}

⫻

冕

*d*3

*r*

_{⫹}共

**r**兲

_{⫺}共

**r**兲. 共B2兲

Comparison of these results indicates that the interstate
*scat-tering length g*_{⫹⫺} plays a completely different role for a
mixture of bare condensates and a mixture of dressed
con-densates. In the former it contributes to the interaction
en-ergy in proportion to the overlap of the condensate density
profiles, whereas for the latter it influences only the
self-interaction energy of each dressed state component.

关1兴**M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138**共1993兲.

关2兴D.J. Wineland, J.J. Bollinger, W.M. Itano, F.L. Moore, and D.J.
**Heinzen, Phys. Rev. A 46, R6797**共1992兲.

关3兴D.J. Wineland, J.J. Bollinger, W.M. Itano, and D.J. Heinzen,
**Phys. Rev. A 50, 67**共1994兲.

关4兴V. Meyer, M.A. Rowe, D. Kielpinski, C.A. Sackett, W.M.
**Itano, C. Monroe, and D.J. Wineland, Phys. Rev. Lett. 86,**
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关26兴For the calculations performed in this paper, it is convenient to
write the eigenstates of*ˆ _{x}*in terms of

*ˆ*eigenstates,兩⫹典and

_{z}兩⫺典, using a nonstandard normalization, i.e., 兩*x*⫹典⫽(兩⫹典⫹