Quantum Interference and the Trapped Bose Condensed System

a r X i v :c o n d -m a t /9802161v 2 [c o n d -m a t .s t a t -m e c h ] 25 F e

b 1998

Quantum Interference and the Trapped Bose Condensed System

Juhao Wu and A.Widom

a

a

Physics Department,Northeastern University,Boston MA 02115

In experiments involving Bose condensed atoms trapped in magnetic bottles,plugging the hole in the bottle potential with a LASER beam produces a new potential with two minima,and thus a condensate order parameter (i.e.wave function)with two maxima.When the trapping potential is removed and the condensate explodes away from the trap,the two wave function maxima act as two coherent sources which exhibit amplitude interference.A simpli?ed theoretical treatment of this experimental e?ect is provided by considering momentum distributions.PACS numbers:03.75.Fi,05.30.Jp,32.80.Pj,64.60.-i

The two ?uid[1]model has long served as the standard picture of super?uidity in liquid He 4.In spite of the great success of the model,Lan-dau warned[2]that the view was merely quasi-classical.It would be desirable to have a quan-tum ?uid mechanical model in which superposi-tion of amplitudes on a macroscopic scale played a more central role.For the dilute Bose condensed gas[3],such coherence would mean that the Bose condensate order parameter[4]Ψ(r )would be re-quired to maintain many of the properties nor-mally attributed to single particle wave functions exhibiting amplitude superposition[5].

Recent experimental probes[6],developed for observing Bose condensates[7]in atomic traps[8],have provided that which is needed to go beyond the Landau “quasi-classical”two ?uid picture.Direct observation of amplitude superposition in the Bose condensate order parameter Ψ(r )has been observed[9]in mesoscopic samples.

Consider some central features of recent ex-perimental methods.To see what is involved,let us ?rst contemplate a “thought experiment”.Suppose that at times t <0?,there is but one atom localized near the origin in the region of a trap potential U (r ).At time t =0,the trap potential is suddenly removed,and thereafter t >0+the atomic motion is described by the free particle Schr¨o dinger equation i ˉh ?ψ/?t )=?(ˉh 2/2M )?2ψ.

From the viewpoint of quantum mechanics,the

sudden[10]removal of the trap potential yields an out going wave function described using the non-relativistic propagator[11]G (r ?s ,t ).The outgoing wave function is determined by

ψ(r ,t )=

G (r ?s ,t )ψ(s ,0)d 3s,(1)where G (r ?s ,t )=

M

2ˉh t

.(2)

From the viewpoint of classical mechanics,the particle exits the trap region at a uniform veloc-ity along the path r =v t .Both the classical (measurement)and quantum (virtual)viewpoints are resolved by considering what happens after the previously trapped particle freely moves far away from the original trapping region.Employ-ing |r |>>|s |in Eqs.(1)and (2),one ?nds that ψ(r ,t )≈ M/(2πi ˉh t ) 3/2e iS (r ,t )/ˉ

h

A (M r /t ),

(3)

where S (r ,t )=(M |r |2/2t )and

A (p )=

e ?i (p ·s )/ˉ

h ψ(s ,0)d 3s.

(4)

is the momentum amplitude of the particle when it was trapped by the potential.Eq.(3)implies that

|ψ(r ,t )|2≈

|A (p )|2

δ r ?(p t/M )

d 3

p

Thus,when a particle has escaped away from the trapping region(long after the trap poten-tial has been removed),the probability density in space|ψ(r,t)|2can be computed by averaging the classical equation of motion r=(p t/M)over the momentum probability density|A(p)|2,present when the particle was in a trapped state.

To understand recent experiments on quantum interference for trapped Bose condensed meso-scopic systems,one must replace the above single particle amplitudes with their second quantized operator counterparts

?ψ(r)= e i(p·r)/ˉh?A(p)d3p

(2πˉh)3

.(9) The ballistic Eq.(9)has been the central theo-retical tool employed in the recent measurements of trapped Bose condensed systems.The mo-mentum distribution of particles within the trap is probed by the spatial distribution of particles detected away from the trap.

Let us now review the notion of superposi-tion of amplitudes as it becomes apparent in the

classic“two slit quantum interference exper-iment”[13].Quantum particle di?raction ex-periments present us with a closely analogous situation regarding initial probability distribu-tions in momentum and?nal probability distri-butions in space.In standard slit di?raction ex-periments,the momentum distribution of parti-cles just behind the slits is probed by the spa-tial distribution of the particles at the detectors very far behind from slits.Once this close anal-ogy between di?raction experiments and recent Bose condensed experiments is appreciated,it then becomes evident why these measurements have exhibited interference e?ects on the meso-scopic scale of the trapped Bose condensates. Our brief review of di?raction theory from slits is as follows:(i)A particle with high velocity v is normally incident on a screen with one or more slits.The slits have a length scaleˉd and the high incoming momentum is de?ned by the inequality Mv>>(ˉh/ˉd).(ii)After a time period

t s≈(D/v)(slit to detector time),(10) the particle hits some detector at a distance D behind the slits.(iii)Before the particle hits the slits from incidence side of the screen,the mo-mentum parallel to the screen is zero.If the par-ticle is squeezed through the slits to the emission side of the screen at time zero,then the quan-tum mechanical uncertainty principle forces the momentum amplitude

A(p)= slits e?i(px′)/ˉhψ(x′,0)dx′.(11)

The momentum component p is in the x-direction,and lives in the plane of the screen with the slits.In that plane,p is normal to the long slit direction.(iv)Far behind the slits,the spatial distribution in x near the detectors is given by a one-dimensional version of Eq.(5);It is

|ψ(x,t s)|2≈

|A(p)|2δ x?(pt s/M) dp

2πˉh D

A p=(xMv/D) 2.(13)

For example,for di?raction through one slit hav-ing a width w,Eq.(11)reads

A(1?slit)(p)=1

w

w/2

?w/2

e?i(px′)/ˉh dx′,(14)

so that Eq.(13)yields the one slit di?raction pat-tern

|ψ(1?slit)(x)|2≈ Mvw sin2(Mvwx/2ˉh D)

2cos pb

2ˉh D |ψ(1?slit)(x)|2.(17) Thus,one sees that the momentum distribution near the slits give rise to the di?raction pattern in space near the detectors,via Eq.(13).Let us now turn to the quantum interference di?raction pat-tern expected from the momentum distribution of a trapped Boson condensate.

In the zero temperature Gross[14]-Pitaevskii[15]theory for dilute Bosons in a trap potential U(r),the variational trial ground state is taken to be“quantum coherent”in a sense that is closely analogous to coherent states in quantum optics.The trial ground state is a normalized eigenstate of the Boson?eld operator,

?ψ(r)|Ψ>=Ψ(r)|Ψ>.(18) The Bose condensation order parameterΨ(r)is then chosen to minimize the energy functional E=<Ψ|?H?μ?N|Ψ>(19) whereμis the chemical potential.Eq.(19)is eval-uated using the two body scattering length a;

E=

Ψ?(h?μ)Ψ+(2πˉh2a/M)|Ψ|4 d3r,(20) where

h=? ˉh2

by

ρ0(r,t)≈ N0(p)δ r?(p t/M) d3p