Squeezing and entanglement of matter-wave gap solitons

a r X i v :q u a n t -p h /0412036v 1 6 D e c 2004

Squeezing and entanglement of matter-wave gap solitons

Ray-Kuang Lee,1,2Elena A.Ostrovskaya,1Yuri S.Kivshar,1and Yinchieh Lai 2

1

Nonlinear Physics Centre and ARC Centre of Excellence for Quantum-Atom Optics,

Research School of Physical Sciences and Engineering,

The Australian National University,Canberra,ACT 0200,Australia 2

Department of Photonics and Institute of Electro-Optical Engineering,

National Chiao-Tung University,Hsinchu 300,Taiwan

We study quantum squeezing and entanglement of gap solitons in a Bose-Einstein condensate loaded into a one-dimensional optical lattice.By employing a linearized quantum theory we ?nd that quantum noise squeezing of gap solitons,produced during their evolution,is enhanced compared with the atomic solitons in a lattice-free case due to intra-soliton structure of quantum correlations induced by the Bragg scattering in the periodic potential.We also show that nonlinear interaction of gap solitons in dynamically stable bound states can produce strong soliton entanglement.

PACS numbers:03.75.Lm,42.50.Dv

In the past two decades,the active research into quan-tum properties of nonlinear many-particle systems has enabled control and engineering of quantum noise in non-linear optics [1].Quantum noise reduction of optical signals below the shot-noise level,referred to as uncer-tainty squeezing,and quantum entanglement of inter-acting optical pulses are at the core of the applications of non-classical light in quantum interferometry,preci-sion measurement,and quantum information process-ing.Non-spreading optical collective excitations,soli-tons ,supported by dispersive nonlinear media,proved the best candidates for experiments on quantum noise reduction and entanglement due to their robust dynam-ics and scattering properties [1].

It has been recognized [2]that general methods of quantum noise squeezing and entanglement developed in quantum optics could apply to other nonlinear bosonic ?elds,such as weakly interacting ultracold atoms in a Bose-Einstein condensate (BEC).Consequently,a num-ber of theoretical proposals were put forward [3,4,5]and experiments carried out [6,7,8]on generating macro-scopic entangled number-squeezed states in BEC.Pro-duction of quantum correlations in the macroscopic quan-tum states relies on interactions between the atoms,the mechanism that is analogous to optical Kerr e?ect.

Recently,gap solitons in a repulsive condensate,sup-ported by periodic potentials of optical lattices,have attracted a great deal of attention due to their con-trollable interaction and robust evolution uninhibited by collapse [9,10,11,12,13].The existence of gap solitons is a unique property of nonlinear periodic sys-tems .Atomic gap solitons form due to combination of an inherent nonlinearity of BEC and the bandgap struc-ture of the matter-wave spectrum that can modify and ultimately reverse dispersion properties of the atomic wavepackets [14].The current techniques for gap soli-tons generation su?er greatly from “technical”noise [12].Provided this problem can eventually be overcome,gap solitons may represent an attractive high-density source for atomic interferometry,quantum measurements and quantum information processing with ultracold atoms.

Understanding and control of quantum noise associated with atomic gap solitons is therefore a fundamental is-sue which so far has not been explored.However,the studies of quantum polariton solitons in a frequency dis-persive medium [15]and the amplitude-squeezed opti-cal solitons in shallow periodic media described by the couple-mode model [16]suggest that both strong disper-sion and bandgap spectrum of a nonlinear soliton-bearing system with periodically varying parameters can dramat-ically modify the properties of the soliton squeezing.In this Letter,we study quantum ?uctuations of matter-wave gap solitons in an optical lattice and inves-tigate the band-gap e?ect on the quantum noise squeez-ing.We employ the soliton perturbation approach [17]to analyze quantum ?uctuations around the soliton so-lutions of the Gross-Pitaevskii (GP)equation with a pe-riodic https://www.sodocs.net/doc/a02725509.html,ing this approach,we demonstrate enhancement of quantum noise squeezing e?ect induced by the gap soliton evolution.Furthermore,we show that the existence of dynamically stable bound states of gap solitons provides a favorable environment for generating entangled soliton pairs.

We consider an elongated cigar-shape BEC loaded into a one-dimensional optical lattice and described by the GP equation for the macroscopic wave function [11],

i ˉh ?Ψ

2?2Ψ

2

X

-1

1

2

(a)

-1

1

(b)

-50

-2502550

-101

(c)

P

FIG.1:(Color online)Left:Band-gap diagram for Bloch waves (bands are shaded),and the family of the gap solitons in the ?rst ?nite gap for V 0=4.0.Right:Pro?les of gap solitons with di?erent chemical potentials;μ=1.91,3.0,and 3.85,corresponding to the points a ,b ,and c ,respectively.

Since the degree of localization of gap solitons varies across the gap [see Fig.1(a-c)],near the bottom edge of the gap,μ≈μ0,the weakly localized soliton pro?le is well described by the “envelope”approximation [18],

ψ(x ;μ)=AF (x )Φ(x ;μ0),

(2)

where Φ(x ;μ0)is the periodic Bloch state at the corre-sponding band edge,and F (x )is a slowly varying func-tion.The inset in Fig.2shows the oscillating wavefunc-tion ψ(x )of a gap soliton near the band edge,at μ=1.91,together with the corresponding Bloch-wave envelope F (x ).The envelope function F (x )is the solution of the lattice-free GP equation or nonlinear Schr¨o dinger (NLS)equation with the e?ective anomalous di?raction and in-teraction energy modi?ed by the lattice [18].

To study the quantum ?uctuations of the gap soli-tons,we replace the ‘classical’mean ?eld described by

Eq.(1)by the bosonic ?eld operator ?Ψ.

Then,we use the linearization approach with the perturbed quan-tum ?eld operator ?ψ

around the mean-?eld solution Ψ0.By using the back-propagation method [19],we calculate the optimal squeezing ratio for gap solitons as a func-tion of the evolution time.The optimal squeezing ratio of the quadrature ?eld for gap solitons,as a result of the homodyne detection scheme,can be de?ned as [19]:

R (t )=min[var ΨL (t )|?ψ

(t ) /var ΨL (t )|?ψ(0) ],where the inner product between the operator ?eld,?ψ

,and the local oscillator pro?le,ΨL ,is de?ned as:

ΨL |?ψ

=1

3

t

μ

FIG.3:(Color online)(a)Optimal squeezing ratio,R ,for the gap solitons with di?erent chemical potentials,μ,at di?erent time.(b)Time evolution of the phase in the homodyne de-tection for the optimal squeezing ratios at di?erent values of the chemical potential within the gap.

FIG.4:(Color online)Quantum correlation spectra in the spatial (x )-domain for gap solitons at di?erent points within the gap and at the values of θcorresponding to optimal quadrature squeezing:(a)θ=41.4?,(b)θ=9?,(c)θ=5.4?,and (d)θ=5.4?.Insets show the corresponding soliton com-ponents in the x -domain.

?uctuation of matter-wave gap solitons,we analyze the quadrature correlations between di?erent spectral com-ponents of the gap soliton induced by its nonlinear evo-lution.The intra-soliton correlation coe?cients,C ij ,are found by calculating the normally-ordered covariance,

C ij ≡

:??n i ??n j : ??n 2i ??

n 2j ,

(3)

where ??n j is the atom-number ?uctuation in the j -th slot ?s j in the spatial (s =x )or momentum (s =k )domain:

??n j =

?s j

d s [ΨL (t,s )?ψ?(t,s )+Ψ?L

(t,s )?ψ(t,s )],The corresponding correlation spectra in the spatial (x )-domain for gap solitons at di?erent points within the

gap and for values of θcorresponding to the maximum squeezing are shown in Figs.4(a-d),respectively.The near-band-edge gap soliton has a discrete correlation pat-tern “shaped”by the periodicity of the lattice,with the spatial components being weakly anti-correlated or uncorrelated [Fig.4(a)].The anti-correlation of the gap soliton components explains its enhanced quadrature squeezing compared to the NLS soliton without a lattice.In the middle of the gap,the soliton is strongly spa-tially localized,and its spatial components are strongly anti-correlated [Fig.4(b,c)],which enhances the quadra-ture noise squeezing.However,near the top edge of the gap,the localization of the gap soliton degrades due to resonance with the Bloch state at the corresponding edge,and strongly correlated,noisy components asso-ciated with the periodic Bloch wave structure start to dominate in the correlation pattern [Fig.4(d)].The squeezing is reduced near that edge of the spectral gap.The possibility of number squeezing for solitons can be assessed by analyzing the intra-soliton number corre-lations at θ=0.In the momentum domain NLS soliton displays the well-known symmetric correlation pattern with the noisy,strongly correlated outer regions of the soliton spectrum [20].For this reason,an e?cient num-ber squeezing of the NLS solitons can be produced by spectral ?ltering that removes the noisy spectral com-ponents [21].On the contrary,the number correlation spectrum of a near-band-edge gap soliton displays a pe-riodic pattern with regions of strong correlations at the Bragg condition,k i =k j =±(2m +1)(m is an integer)and uncorrelated o?-diagonal regions.This correlation pattern persists through the entire gap until the noisy spectral components due to resonance with a Bloch state appear near the opposite band edge.Therefore the en-hancement of number squeezing of the BEC gap soliton with additional spectral ?ltering seems unfeasible due to spectral selectivity of the Bragg scattering.

The possibility to achieve squeezing of atomic solitons paves the way to creating entangled macroscopic coher-ent states in BEC interference experiments.However,optical lattices also o?er an alternative possibility for entanglement production.It is known that the optical lattice supports stationary states in the form of bound soliton pairs [11].The nonlinear interaction between solitons in a pair could induce entanglement and hence nonseparability of the bound state.This entanglement can potentially be exploited in BEC atom number de-tection by implementing analogs of quantum nondemo-lition measurements based on entangled pairs of optical solitons [22].Potential advantage of gap solitons com-

4

1

2

FIG.5:(Color online)(a)Optimal squeezing ratio for in-phase(inset,solid)and out-of-phase(inset,dashed)bound soliton states(V0=4.0,μ=2.5);the two curves overlap.(b) Atom number correlation C12for in-phase(solid)and out-of-phase(dashed)soliton pairs.

pared to the lattice-free atomic solitons is the long-lived nature of the bound states which are pinned by the lat-tice and do not break-up as a result of evolution.Hence strong interaction-induced atom-number correlations can develop between the atomic wavepackets.

Typical in-phase and out-of-phase bound states of two gap solitons in the middle of spectral gap are shown in Fig.5.These states contain the same number of atoms for the sameμ,which equals to exactly twice the num-ber of atoms in a single soliton.Remarkably,the internal noise correlation properties are the same for both pairs, which results in their identical optimal number squeezing ratio,see Fig.5(a).Soliton entanglement can be quanti-?ed by inter-soliton atom number correlations which can be calculated according to Eq.(3),with??n i,j being per-turbations of the atom number operator of two solitons, and indices{i,j}={1,2}numbering individual solitons.The evolution of the correlation parameter C12shows dramatic di?erence between the in-and out-of-phase soli-ton pairs.As follows from Fig.5(b),the atom number correlation parameter for the in-phase soliton pair is neg-ative while it is positive for the out-of-phase pair.The di?erence in the entanglement behavior can be under-stood by examining the interaction energy of the solitons in the pair E12= E12(x)dx,with the energy density: E12= i=j(1dx dψ?j