The AB Effect in Neutral Liquids

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The Aharonov–Bohm effect in neutral liquids

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2010 J. Phys. A: Math. Theor. 43 354003

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IOP P UBLISHING JOURNAL OF PHYSICS A: M ATHEMATICAL AND THEORETICAL

J. Phys. A: Math. Theor. 43 (2010) 354003 (13pp) doi:10.1088/1751-8113/43/35/354003

The Aharonov–Bohm effect in neutral liquids

E B Sonin

Racah Institute of Physics Hebrew University of Jerusalem Givat Ram, Jerusalem 91904 Israel

E-mail: [email protected]

Received 13 January 2010, in nal form 8 March 2010Published 12 August 2010Online at stacks.iop.org/JPhysA/43/354003

AbstractThe Aharonov–Bohm effect was discovered as a quantum-mechanical effectfor charged particles, but it has its counterpart in classical wave mechanics.The Aharonov–Bohm interference arises at the scattering of a sound wave bya vortex in classical and quantum hydrodynamics. This interference leadsto a transverse force between quasiparticles and vortices in superuids andsuperconductors. The Aharonov–Bohm effect was also generalized to neutralparticles with magnetic or electric dipole momenta. The Aharonov–Bohmeffect for charge particles and its modication for magnetic momenta (theAharonov–Casher effect) have already been experimentally observed, andthe efforts to detect the Aharonov–Bohm effect for electrically polarizedneutral particles are on the way. A possible system for this detection is aBose-condensate of excitons in a double quantum well. Observation of theAharonov–Bohm effect in this system would provide direct evidence of Bose–Einstein condensation.

PACS numbers: 03.65.Vf, 67.25.dk, 71.35.Lk, 73.21.Fg

(Some gures in this article are in colour only in the electronic version)

1. Introduction

The 50 year history of the Aharonov–Bohm (AB) effect [ 1] continues to demonstrate afundamental importance of the effect, and more and more implications and analogs of the

effect are being found in various area of physics. This effect clearly manifests the key role of the topological phase in classical andquantum physics [ 2]. Though the AB effect hasappearedin the quantum-mechanical theory, generally speaking it is a wave-mechanical effect, whichhas analogs in classical wave mechanics.

The AB interference arises at the scattering of a sound wave by a vortex in classicaland quantum (superuid) hydrodynamics. In a sense, the AB effect entered superuidhydrodynamics even earlier than the effect itself was discovered though it was not realized atthat time. The interference leads to a transverse force on a vortex, which was rst found by

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J. Phys. A: Math. Theor. 43 (2010) 354003 E B Sonin

Lifshitz and Pitaevskii [ 3] for rotons in 1957, i.e., two years before the paper by Aharonovand Bohm. But only much later it was demonstrated [ 4] that this roton transverse force, aswell as the similar force produced by phonons, results from the AB interference.

From the very beginning the transverse force from quasiparticles scattered by a vortexwas a subject of controversy and of vivid debates. Pitaevskii [ 5] and later Fetter [6], whostudied scattering of phonons by a vortex in the Born approximation, did not reveal anytransverse force. Later Iordanskii [7] revisited the problem of phonon scattering and foundthat there is a transverse force though it cannot be obtained from the differential cross-section,which was analyzed by Pitaevskii and Fetter. Iordanskii suggested that his force and theLifshitz–Pitaevskii force were of different origins and for rotons they should be summed.Since the force calculated by Lifshitz and Pitaevskii and the Iordanskii force were equal inamplitude but opposite in sign, he concluded that the transverse force from rotons vanished.But the analysis done in [ 4] demonstrated that the Iordanskii force for rotons is identical tothe Lifshitz–Pitaevskii force and they must not be added. In addition, the Lifshitz–Pitaevskiiforce from rotons was calculated in the original paper [ 3] with a wrong sign. After itscorrection the transverse force on the vortex was of the same sign and value both for rotons

(the Lifshitz–Pitaevskii force) and for phonons (the Iordanskii force) [ 8].Later on the debates on the transverse force (especially on its phonon version, i.e.Iordanskii force) continued with a number of publications rejecting its existence. In particular,Ao and Thouless [ 9] suggested that the Berry phase provides an universal exact value for thetotal transverse force on the vortex which does not depend on the presence of quasiparticles orimpurities. This ruled out the transverse force on thevortex from quasiparticles and impurities.The arguments about this issue continued for a number of years [10–12], but eventually someconsensus was reached that the original calculation of the Berry phase missed the contributionfrom the normal-uid circulation. After its addition the calculations of the transverse forcefrom the scattering theory and from the Berry phase gave the same result [13–15].

The AB interference was also studied in classical hydrodynamics for water surface waves(the acoustic AB effect [ 16, 17]). Scattering of the light by a vortex also results in the ABinterference (the optical AB effect [ 18]). The original AB effect for electrons also leads to atransverse force on a uxon [10, 19]. The force was derived by various methods, in particular,by analyzing the scattering of electron wavepackets in the paraxial approximation [ 20–23]. Inclean superconductors the BCS quasiparticles also produce an additional transverse force onthe vortex [ 24–26] analogous to the Lifshitz–Pitaevskii–Iordanskii force in superuids. Theforce affects the Hall effect in the mixed state of a superconductor.

TheAB effect wasdiscoveredforcharged particles [1], but later wasgeneralizedon neutralparticles with magnetic (the Aharonov–Casher effect [ 27]) and electric [28, 29] dipoles. TheAB effect for charge particles and its modication for magnetic moments have already beenexperimentally observed (see bibliography in [28]). On the other hand, the AB effect forelectrically polarized neutral particles is still waiting its clear experimental detection, andsome serious steps on the way to this goal have been reported for superuid 4 He [30] and forexcitons [31].

One may expect that any Bose-condensed medium, where the coherent phase exists, is agood candidate for detection of this effect. This refers also to Bose condensates of excitons,which have already been discussed for more than four decades. The interest to this idea wasrevived after experimental observation of excitons in coupled doubled quantum wells, whichstimulated intensive theoretical and experimental investigations [ 32–38]. Recently it wasshown that the AB effect can result in rotation of the electrically polarized Bose condensate[39]. Thedetectionof theAB effect for theBose condensate of excitons is especially importantsince thequestion whether Bose condensationwasreallyobserved there remains unsettled [ 40]

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despite some experimental evidences of condensation being reported. For excitons in doubledquantum wells the electric dipole moment of an exciton is provided by spatial separation of electrons and holes in two different wells. Since the AB effect and vortices, which can appeardue to it [ 36–38], are a clear manifestation of phase coherent states, this experiment wouldprovide a ‘smoking gun’ for Bose–condensation.

The present paper will address these two manifestations of the AB phenomenon.

2. Transverse force from quasiparticles on a vortex

2.1. Scattering of phonons by the vortex

In superuid hydrodynamics the equations of motion for the superuid component areequivalent to the hydrodynamic equations for an ideal inviscid liquid:

∂ρ∂t

+ ∇ (ρ v ) =0, (1)

∂v∂t + (v ·∇ )v = −∇ µ. (2)Here we restrict ourselves to zero temperature; ρ and v are the liquid density and the velocity,respectively, and µ is the chemical potential.

We consider a sound wave propagating in the plane xy normal to a vortex line (the axisz). Then the liquid density and velocity are functions of the time t and the position vector r inthe plane xy :

ρ( r , t ) =ρ0 + ρ (1) (r , t ) , v(r , t ) = v v + v (1) (r , t) , (3)where ρ0 is the average density and

v v (r ) =κẑ ×r2πr 2

(4)

is the velocity induced by the vortex lament, κ =h/m is the circulation quantum, ρ (1) (r , t )and v (1) (r , t ) = κ2π ∇ φ areperiodical variations of thedensity andthevelocity due to the soundwave ( ρ (1) =0, v (1) =0). They should be determined from the linearized hydrodynamicequations, which after some algebra [15] are reduced to the linear sound equation for thephase:

∂ 2φ∂t 2 −c

2s∇

2φ = −2v v (r ) ·∇∂φ ( r )

∂t , (5)

where cs =√ ρ∂µ/∂ρ is the sound velocity.In the long-wavelength limit k →0 onemay usethe perturbationtheory with theparameterκk/c s . The latter is on the order of the ratio of the vortex core radius rc ∼ κ/c s to thewavelength 2 π/ k . The standard scattering theory in the Born approximation yields (see [ 10]and references therein for details) the asymptotic solution at large r , which is a superpositionof the incident plane wave

∝exp(ik ·r ) and the scattered wave ∝exp(ikr ) :

φ =φ0 exp(−iωt) exp(ik · r ) + ia(ϕ)√ r exp(ikr ) . (6)

Here

a(ϕ) = k2π 1cs ei π4 [κ̂ ×k ] ·k 1q 2 1 − q2

2k2 = −12 k2π κcs ei π4 sin ϕ cos ϕ1 −cos ϕ (7)

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s c a t

t e r e

d

w

a v eincident wave

θ

Figure 1. Scattering of a sound wave by a vortex.

is the scattering amplitude, which is a function of the angle ϕ between the initial wavevectork and the wavevector k =kr /r after scattering (see gure 1), q =k −k is the momentumtransferred by the scattered phonon to the vortex, and q

2

=2k2

(1 −cos ϕ) .Scattering produces a force on the vortex,F ph =σ cs j ph −σ ⊥cs [ẑ × j ph], (8)

where j ph = ρ (1) v (1) is the mass ow related with the propagating sound wave. So theforce is determined by two effective cross-sections [ 4, 10]: the transport cross-section for thedissipative force,

σ = σ(ϕ)( 1 −cos ϕ) dϕ, (9)and the transverse cross-section for the transverse force,

σ ⊥ = σ(ϕ) sin ϕ dϕ. (10)

The differential cross-section σ(ϕ) = |a(ϕ) |2 with a(ϕ) from (7) is quadratic in the circulationκ and even in ϕ. Therefore, in theBornapproximation the transverse cross-section σ

⊥vanishes.

This explains why Pitaevskii [ 5] and Fetter [6] did not reveal the transverse force.However, due to a very slow decrease of the velocity vv ∝ 1 /r far from the vortex, thescattering amplitude is divergent at small scattering angles ϕ →0:

a(ϕ) ≈ − k2π κcs ei π4 1ϕ . (11)This divergence is integrable in the integral for the transport cross-section ( 9), and its

calculation is reliable. In contrast, the integrand in ( 10) for the transverse cross-sectionhas a pole at ϕ = 0. The principal value of the integral vanishes, but there is no justicationfor the choice of the principal value, and the contribution of the small angles requires anadditional analysis.

At small scattering angles ϕ 1/ √ kr the asymptotic expansion ( 6) is invalid, andone cannot use the differential cross-section or the scattering amplitude for description of the small-angle scattering [ 4, 7, 20]. Meanwhile, the small-angle behavior is crucial for thetransverse force as demonstrated below. A more accurate calculation, which still uses the

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perturbation theory 1 but not the asymptotic expansion leading to the expression ( 6) (see [4]and appendix B in [ 10]), yields that at ϕ 1

φ

=φ0 exp(

−iωt + ik

·r ) 1 +

iκk

2csϕ

kr

2i. (12)

Using an asymptotic expression for the error integral

(z) =2

√ π z

0e−t 2 dt −→

z

|z| −1

√ πz exp(−z2) (13)

at |z| → ∞, one obtains for angles 1 ϕ 1/ √ kr :φ =φ0 exp(−iωt) exp(ik ·r ) 1 +

iκk2cs

ϕ

|ϕ| −iκcs k2πr 1ϕ exp ikr + i π4 . (14)

The second term in square brackets coincides with scattering wave at small angles ϕ 1 withthe amplitude given by ( 11). But now one can see that the standard scattering theory missesto reveal an important non-analytical correction to the incident plane wave, which changes a

sign when the scattering angle ϕ crosses zero. Its physical meaning is discussed in the nextsection.

2.2. The Iordanskii force and the Aharonov–Bohm effect

Theanalogy between thephononscatteringby a vortex andthe ABeffect for electronsscatteredby a magnetic-ux tube becomes evident if one rewrites the sound equation ( 5) in presence of the vortex as

k2φ − −i∇ + kcs

v v2

φ =0. (15)On the other hand, the stationary Schr ödinger equation for an electron in presence of themagnetic ux conned to a thin tube (the AB effect [ 1]) is

Eψ ( r ) =1

2m −ih̄∇ −ec

A2ψ( r ). (16)

Here ψ is the electron wavefunction with energy E and the electromagnetic vector potential isconnected with the magnetic ux by the relation similar to that for the velocity v v aroundthe vortex line [see ( 4)]:

A =[ẑ ×r ]2πr 2

. (17)

Let us consider the semiclassical solution of the sound equation:

φ =φ0 exp −iωt + ik · r + iδS

h̄ =φ0 exp(−iωt + ik ·r ) 1 + iκk2πc s

θ , (18)

where δS

= −(h̄k/c

s)

r v

v ·dl

=h̄θκk/ 2πc

s is the variation of the action due to interaction

with the circular velocity from the vortex along semiclassical trajectories. The angle θ is anazimuth angle for theposition vector r measured from the direction opposite to the wavevectork (see gure 1). This choiceprovides that thesemiclassical correction vanishesfor the incidentwave far from the vortex. One can check directly that ( 18) satises the sound equation (5) inthe rst order with respect to the parameter κk/c s . For the AB effect the phase δS/ h̄ arisesfrom the electromagnetic vector potential: δS = −(e/c)

r A ·dl .1 One can nd a more general expression, which does not use the perturbation theory, in [ 41].

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Figure 2. Aharonov–Bohm interference.

According to (18) the phase φ is multivalued, and one must choose a cut for an angle θ at the direction k , where θ = ±π . The jump of the phase on the cut line behind the vortex isa manifestation of the AB effect [ 4]: the sound wave after its interaction with the vortex hasdifferent phases on the left and on the right of the vortex line. This results in an interference[4, 10]. In the interference region the semiclassical solution is invalid and must be replacedby ( 12). The width of the interference region is d int ∼√ λr where λ = 2π/ k is the soundwavelength (gure 2). The interference region corresponds to very small scattering angles∼

dint /r =1/ √ kr .The next step is to consider the momentum balance in the area around the vortex lament.Surrounding thelament with a cylindric surface of theradiusmuch largerthan thewavelength,the total momentum ux is determined by the integral over this surface, dS j ⊥j =0, wheresubscript ⊥ is for a component normal to the wavevector k of the incident wave. Here

ij = P δ ij + ρv i (r )v j (r ) is the hydrodynamic momentum–ux tensor. Expanding it withrespect to perturbations ρ (1) and v (1) (see [15] for details), one obtains ( 8) with the transversecross-section

σ ⊥ = δS −−δS +h̄k = κcs . (19)

It is crucial that just the interferenceregion, which corresponds to an innitesimallysmall angleinterval, yieldsa nite contribution to the transverse force, which onecould notobtain from thestandard scattering theory using the differential cross-section. In fact the details of the solutionin the interference region are not essential: only a jump of the phase δ−−δ+ =(δS −−δS +)/ h̄across the interference region is of importance.

2.3. The transverse force from the Berry phase

Now let us consider a connection between the transverse force and the Berry phase. The Berryphase can be found from the hydrodynamic Lagrangian

L =κρ2π

∂φ∂t −

κ 2ρ8π

(∇ φ) 2 −

V 2

ρ 2 . (20)

The rst term with the rst time derivative of the phase φ (Wess–Zumino term) is responsiblefor theBerry phase = S B / h̄ , which is thevariation of thephaseof thequantum-mechanicalwavefunction for an adiabatic motion of the vortex around a closed loop [2]. Here

S B = dr dt κρ2π ∂φ∂t = − dr dt κρ2π (v L ·∇ L )φ. (21)6


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is theclassical actionvariation aroundthe loop and ∇ L φ is thegradientof thephase φ [r −r L (t ) ]with respect to the vortex position vector r L (t ) . However, ∇ L φ = −∇ φ , where ∇ φ is thegradient with respect to r . Then the loop integral

d l =

vL dt yields the circulation of total

current j

= (κ/ 2π) ρ ∇ φ for points inside the loop, but vanishes for points outside. As a

result, the Berry-phase action is given by

S B =V κ

2π (dl · j ). (22)where V is the volume inside the loop (a product of the loop area and the liquid height alonga vortex). Here we considered the zero temperature when the superuid density ρ s is equal tothe total density ρ . But more general analysis [ 42] showed that the same expression is validfor nite temperatures when j =ρ s v s + ρn v n .Since the Berry phase is proportional to the current circulation, which determinesthe transverse force, there is a direct connection between the Berry phase and the amplitudeof the transverse force on a vortex, as was shown by Ao and Thouless [ 9]. Thus, in orderto nd the transverse force, one should calculate the current circulation. If the circulation of

the normal velocity at large distances vanished (as assumed in [ 9, 42]), the current circulationwould be (dl · j ) = ρs κ , and the Berry phase (as well as the transport force) would beproportional to ρ s . This would mean that there is no contribution to the force from normalquasiparticles. However, at very large distances the normal circulation κn = (dl ·v n ) doesnot vanish (see [ 13–15] for details) and (dl · j ) = ρs κ + ρ n κn . Using a proper value of the asymptotic normal circulation κn the Berry phase yields the same transverse force asdetermined from the momentum balance.

2.4. The transverse force in the semiclassical theory

Up to now the transverse force was derived for phonons. But one can suggest a more generalsemiclassical theory, which demonstrates the existence of the transverse force also for rotons.

Let us consider a quasiparticle with an arbitrary spectrum ε(p) . If the quasiparticle movesin the velocity eld induced by the vortex, its energy is E( p) = ε(p) + p ·v v which may betreated as a Hamiltonian for the classic equations of motion:

drdt =

∂E∂ p =vG

p

p+ v v ,

d pdt = −

∂E∂ r = −

∂( p ·v v )∂ r

, (23)

where vG (p) = dε/ dp is the quasiparticle group velocity. As before, we assume that thequasiparticle moves in the plane xy normal to the vortex. From these equations one can ndthe classical trajectory of the quasiparticle moving past the vortex line. Usually it is closeto a straight line, and a distance of the straight trajectory from the vortex line is the impactparameter b. If we are looking for the transverse force, we need only the variation δp

⊥ of the

momentum component normal to the initial momentum p which results from quasiparticlemotion past the vortex. The momentum of the quasiparticle is connected to the classical

action: p = ∂S/∂ r . Then δp⊥(b) = ∂δS(b)/∂b where the total variation of the classicalaction δS(b) along the trajectory is a function of the impact parameter b. The solution of ( 23)in the rst order with respect to vv yields that

δS(b) = − ∞−∞dlp

2πv G (l)κb

b2 + l2, (24)

where l is the coordinate along the trajectory. The scattering angle of the quasiparticle isϕ ≈ −δp⊥(b)/p and the transverse cross-section is

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σ ⊥ = ∞−∞db sin ϕ ≈

∞

−∞db ϕ =

δS(−∞) −δS( +∞)p

=1

2πv G

∞

−∞

db∂

∂b

∞

−∞

dlκb

b2 + l2 =κ

vG. (25)

This expression can be used for rotons with the spectrum ε(p) = + (p −p R )2/ 2µ R and thegroup velocity vG =(p −p R )/µ R . For phonons the group velocity vG is equal to the soundvelocity cs and does not depend on l. Therefore, the action variation δS given by (24) does notdepend on the impact parameter b at b = 0. This means that there is no phonon scattering inthe semiclassical approximation: ϕ ∝∂δS(b)/ db =0. A nonzero transverse cross-section σ ⊥for phonons arises from a nite jump of δS at b =0. In fact, one may not use the semiclassicaltheory to calculate this contribution. Nevertheless, ( 25) yields the correct value σ

⊥ = κ/c sfor phonons.

3. Gauge-eld rotation of Bose-condensed electric dipoles by magnetic eld

Now letus switch to another manifestation of theAB physics: theAB effect for neutral electricdipoles. In fact, the Hamiltonian for electrically polarized excitons taking into account thegauge eld, which leads to the AB effect, was known long ago [43], though without anyconnection to the AB effect:

H =1

2m p −

ec

Ã2

=1

2m p −

1c

[H ×d ]2

. (26)

The Hamiltonian takes into account that a dipole d = ea formed by two charges ±e connectedwith the position vector a feels the electromagnetic vector potential A (r + a ) −A (r + a ) ≈(a ·∇ )A (r ) = ∇ (a ·A ) + [H ×a ]. The rst gradient term can be removed with a gaugetransformation. The second term yields the ‘effective’ vector potential Ã = [d ×H ]/e . ForBose-condensed excitons in electron–hole bilayers (double quantum wells) this Hamiltonianwas used in [ 36–38].

The velocity of a dipole is given by the standard expression v = p − ec Ã m , so thegauge eld Ã can produce a neutral current of dipoles. However, in a cloud of uncorrelatedexcitons the gauge-eld effect will be compensated by the canonical momentum p so thatto minimize the kinetic energy. For Bose-condensed particles the situation is different. Thegauge-eld velocity vd =e Ã /mc is not curl-free in general, its vorticity being

ω = [∇ ×v d ] =1

mc[∇ ×[H ×d ]]. (27)

The canonical momentum p = h̄∇ φ is curl-free and therefore cannot compensate the gauge-eld velocity if vortices are absent. Thus, the gauge eld makes a Bose condensate of dipolesrotate.

Excitons in a double well have a z component of the dipole moment d z =ea , where a ison the order of the distance between two wells. For an axisymmetric 2D exciton cloud in anaxisymmetric magnetic eld (gure 3) the z component of the gauge-eld vorticity is

ωz =dzmc

∂H r∂r

+ H r

r, (28)

where H r is the radial in-plane component of the magnetic eld produced by a long thinsolenoid (see gure 3). The aring magnetic eld around its end is similar to that of themagnetic monopole. If the distance L of the exciton cloud from the solenoid end is muchlarger than the cloud radius R the radial magnetic eld is linear with respect to r : H r =H 0r/R ,8


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+ + + ++ + + + + + + ++ + + + + + + +

+ + + + + ++ + + +- --- -- --

L

a

z R

Figure 3. The exciton cloud in the electron-hole bilayer in the aring magnetic eld.

where H 0 = H zR/L is the radial magnetic eld at the cloud boundary r = R and H z is theaxial magnetic eld in the cloud center. Then the gauge eld fully simulates the solid-bodyrotation with the angular velocity = ωz / 2 =H 0d/mcR =eH 0a/mcR . Later on we shallignore the axial eld H z . Its effect can be really removed by putting a second solenoid abovethe cloud mirroring the rst one.

If the magnetic eld is strong enough it creates vortices [ 36]. Let us estimate the criticalmagnetic eld. The kinetic energy of the Bose condensate is given by

E =m2

n2(r)

h̄m∇ φ −v d (r )

2

dr , (29)

where φ is the phase of the Bose-condensate wavefunction, and integration is over the whole2D condensate with the 2D density distribution n2(r ). For low magnetic elds the phase-gradient velocity is absent, and the kinetic energy of the axisymmetric cloud of the radius R isgiven by

E 0 =πm R

0n2(r)v d (r) 2r dr. (30)

This energy increases with the magnetic eld and at some critical magnetic eld becomeslarger than the energy of the state with a single-quantum vortex at the cloud center:

E v =πm R

0n2(r) vd (r) −

h̄mr

2

r dr. (31)

In an electrostatic trap for excitons with steep walls [33] one may neglect spatial variation of the exciton density n2, and the condition E v = E 0 yields the critical value of the magneticeld H 0 [36]:

H cr =h̄cdR

ln Rr c

, (32)

where r c is the vortex core radius used as a lower cutoff in the integral for the vortex energy.Assuming r c of the order of the interparticle distance n−1/ 22 (this yields an upper bound on thecritical magnetic eld) and using the values n2 =1010 cm−2 and R =10 µ m for the density

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and the cloud radius, ( 32) yields H cr =293 G. Thus, magnetic elds of about a few hundredG are enough for vortex creation. At further growth of the magnetic eld more and morevortices should penetrate into the cloud eventually forming a vortex array similar to that inmechanically rotated superuids. However, the role of vortices is opposite in the two cases.At the mechanical rotation of the container the superuid does not rotate without vortices,but does rotate when vortices appear. At the gauge-eld rotation the superuid rotates in thevortex-free state, whereas vortices tend to suppress rotation eventually stopping it when thereis a large number of them.

One can detect the gauge-eld rotation by observation of photoluminescence. Bosecondensation of excitons in the double quantum well should lead to emission of coherentlight [ 44–46]. The dipole interaction of excitons with the electromagnetic eld is linear withrespect to exciton creation and annihilation operators. Their averages vanish in the Fock stateswith xed numbers of excitons, but must be replaced by the Bose condensate wavefunctionin the coherent state resulting from Bose condensation. This leads to an oscillating classicalpolarization component

P (r ) =Re dψ( r )

√ S e−iω0t

, (33)where ψ( r ) is the Bose condensate wavefunction normalized to the total number N of excitons( |ψ( r )|

2 d2r = n2(r ) d2r =N ), S is the area of the exciton cloud, and ω0 is the frequencydetermined by the energy of the exciton annihilation E b = h̄ω 0. Thus, the total oscillatingpolarization component P (r ) d2r is proportional to √ N , whereas the constant average totalpolarization d N is proportional to the total number of particles. In general the exciton dipolemoment d may have three complex components d x , dy , dz . For example, the dipole rotating inthe plane corresponds to d y = ±idx . A coherent light emitted by an oscillating macroscopicdipole can be treated classically [ 47]. The light power emitted into the elementary solid bodyangle d = sin θ dθ dφ is given by

dW(θ,φ)

d =

ck 4

8π

|d⊥|2S

S

ψ( r ) eik in r d2r2

, (34)

where θ and φ are the polar and azimuthal angle of the wavevector k of light with the inplanecomponent k in (kin = k sin θ , k = ω0 /c ), d⊥ is the projection of the dipole moment onto theplane normal to k , which determines the polarization of light, and|d⊥|2 = |dx |2(1 −sin2 θ cos2 ϕ) + |dy |2(1 −sin2 θ sin2 ϕ)

+ |dz|2 sin2 θ −(d∗x dy + dx d∗y ) sin2 θ sin ϕ cos ϕ+ sin θ cos θ [(d∗x dz + dx d∗z ) cos ϕ + (d∗y dz + dy d∗z ) sin ϕ]. (35)

This resultfullyagreeswithearlier calculationsof theradiativedecayrate usingFermi’s GoldenRule [ 48, 49 ], if one denes the rate of emission of a photon with the inplane wavevectork in as = 4π 2(dW/ d )/(Sk 2h̄ω 0 cos θ) . This justies a purely classical treatment of luminescence.

In the long-wavelength limit k inr → 0 (34) yields a smooth distribution of the emittedpower over the solid angle sphere: ∝ sin2 θ for the dipole moment normal to the plane(dx =dy =0) and ∝(1+cos 2 θ) for the circularly polarized inplane dipole moment (d z =0,dy = ±idx ). But if the inplane dipole moment is linearly or elliptically polarized the axialsymmetry of the distribution is broken, and the emitted power depends on the azimuthal angleφ . The total emitted power does not depend on orientation of the dipole moment:

W = S dW(θ,φ)d d = ck4

3|d |2

S S ψ( r ) d2r2

. (36)

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0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0d W̃

d Ω

θ

1

2

3

Figure 4. Angular dependence of the reduced luminescence power W̃ =8πW/ck 4|dz|2N for theaxial dipole d z . (1) The vortex-free state in the long wavelength limit kR →0. ( 2) The vortex-freestate, kR =

3. (3) The central-vortex state, kR

=3. The luminescence in the central-vortex state

at kR →0 vanishes.

In the vortex-free state, where the phase φ of ψ ( r ) is constant, the power W = ck4|d |2N/ 3is proportional to the total number of Bose-condensed excitons. However, if a vortex appearsin the cloud center, emission becomes impossible because of destructive phase interference inthe condensate ( ψ( r ) d2r =0). This is a strong effect, which can be used for experimentalidentication of Bose condensation and vortices.

In the short wavelength limit kR → ∞ the effect of vortices is also present, but thecentral vortex is not able to fully suppress radiation. The vortex effect can be evaluated withcomparison of the coherence factor in ( 34) for the vortex-free state,

1S S ψ( r ) eik in r d2r = 2√ n

2

R 2 R

0r dr

2π

0

dϕ2π

eikin r cos ϕ = 2√ n2

kinRJ 1(k inR), (37)

and for the state with a vortex in the center:

1S S ψ( r ) eik in r d2r = 2√ n2R 2

R

0r dr

2π

0

dϕ2π

ei(ϕ+kin r cos ϕ)

=i√ n2kinR

3 1F 2

32

, 2,52

, −k2inR

2

4, (38)

where 1F 2({a}, {b, c }, z) is the hypergeometric function and spatial variation of the excitondensity n2 = |

ψ

|2 was neglected. The effect of interference on the luminescence by the axial

dipole d z is demonstrated in gure 4.Though thepresent analysis is focused on the exciton Bose-condensate in double quantum

wells, the gauge-eld rotation is as general phenomenon as the AB effect itself. One canobserve the gauge-eld rotation replacing the exciton cloud in gure 1 by a helium layer ora cloud of Bose-condensed cold atoms and adding an axial electric eld necessary for theirelectrical polarization. Note that this geometry (axial electric and radial magnetic eld) isdifferentfrom thegeometry planned by Sato andPackard [30] for their experiment in superuid4He (radial electric and axial magnetic eld).

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4. Conclusions

Two examples of the AB phenomenon in neutral liquids were analyzed. The rst one is thetransverse force from quasiparticles scattered by a vortex in superuids. In the present paperthe phonon scattering was investigated in the congurational space. But one can also derivethe phonon transverse force using the partial-wave expansion [10, 15]. In the latter methodthe problem of small-angle divergence manifests itself in ambiguity of treating the partial-wave sums divergent at large angular momenta. But a proper treatment of this expansioneventually gives the same result as that in the congurational space. These two methodsare supplemented by the analysis of the wavepacket deection in the paraxial approximation[20, 21]. Full agreement between the three methods provides a solid basis for the well denedtransverse force on a vortex or a uxon despite numerous attempts to refute its existence.

The second example is the AB effect in the Bose condensate of electrical dipoles. It wasshown that a radial magnetic eld is able to rotate a Bose condensate of electrically polarizedneutral particles. This leads to penetration of vortices at some critical eld. In an exciton Bosecondensate in a double quantum well vortices strongly affect the photoluminescence from the

exciton cloud. This can be used for their experimental detection. The existence of vortices isa direct consequence and evidence of exciton Bose condensation.These two examples illustrate how important are the Aharonov–Bohm effect and its

analogs in modern condensed matter physics.

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The AB Effect in Neutral Liquids

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