On the rotational state of a Bose–Einsteincondensate

Tsutomu Kambe1

Higashi-yama 2-11-3, Meguro-ku, Tokyo, 153-0043, Japan

E-mail: kambe@ruby.dti.ne.jp

Received 30 June 2013, revised 9 April 2014Accepted for publication 10 April 2014Published 28 May 2014

Communicated by Y Fukumoto

AbstractA quantum liquid of an almost ideal Bose gas brought into rotation isinvestigated from a physical and mechanical point of view on the basis of theGross–Pitaevskii (GP) equation by applying a quantum-mechanical scenario,i.e. the London scenario. This scenario allows a superfluid to have rotationalstates. Considering that the equation governs an interacting Bose gas, it isproposed that the GP equation admits rotational flows of a superfluid. This iscarried out without incurring essential change of the equation. By this refor-mulation, a superfluid placed in a rotating vessel is able to have a solid bodyrotation with the same angular velocity as its container and also to have ameniscus approximated by a parabolic profile. The solid body rotation isaccompanied by a density increase proportional to the square of its angularvelocity. These are consistent with experimental observations. In addition, thisformulation allows a vortex of quantized circulation with coaxial rotationalcore whose density does not necessarily vanish at its center.

1. Introduction

In view of recent renewing studies of interacting Bose gases in rotation at extremely lowtemperatures, we consider a possible reformulation of a quantum condensate brought intorotation on the basis of the Gross–Pitaevskii equation without incurring essential change.

Some early experimental studies of rotating helium II (Osborne 1950, Andronikashviliand Kaverkin 1955) showed that the quantum liquid appeared to be rotating uniformly as awhole in a rotating container. The idea of a quantum vortex in a rotating superfluid wasproposed by Onsager (1949) and Feynman and Gorter (1955). According to them, the vortexinduces a potential flow around it with its circulation quantized and with the density vanishing

0169-5983/14/031418+16$33.00 © 2014 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK 1

| The Japan Society of Fluid Mechanics Fluid Dynamics Research

Fluid Dyn. Res. 46 (2014) 031418 (16pp) doi:10.1088/0169-5983/46/3/031418

1 Former Professor, University of Tokyo (Physics).

at its center. At sufficiently fast rotation, the quantum vortices form a lattice that imitates onaverage the solid-body rotation (Andronikashvili et al 1961, Andronikashvili and Mama-ladze 1966). This is understood as follows. The rotating superfluid is threaded by a lattice ofquantized potential vortices in a manner which rotates as a whole (Donnelly 1991). Insection 2 of this paper, we reconsider whether the vortices are characterized by potential flowsonly and propose the possibility of having a rotational core.

Although their approach is different from the present study, the same subject of possi-bility for a quantum liquid to be in a rotational state was recently studied by Greenberg andZelevinsky (2012). Our approach, however, is motivated by a recent new representation ofrotational flows of an ordinary fluid (Kambe 2013), which satisfies Eulerʼs equation of motionof a compressible ideal fluid. Note that the system of equations derived from Gross andPitaevskii (Gross 1961, Pitaevskii 1961) have analogous form to that of rotational flows of anordinary inviscid fluid.

In the present study we consider a quantum liquid, i.e, a degenerate state of an interactingBose gas of mass m, which is governed by the Gross–Pitaevskii equation. On the basis of theLagrangian formulation of particle mechanics in a rotating frame and also correspondingformulation of quantum mechanics, we find a certain type of rotational state that does notnecessarily result in the formation of potential vortices (but a solution of the potential vortexis not excluded). Section 3 investigates its formulation. Some examples of solution resultingfrom the formulation are presented for solid-body rotation of a quantum liquid in section 4and for quantum vortices with coaxial rotational core in section 5.

1.1. Superfluids

First we review past studies on the superfluids brought into rotation, and pose the questionwhether the theory of potential vortices only can explain all. We consider the possibility if arotational state is allowed for a degenerate almost ideal Bose gas at absolute zero within thecurrent framework of the Gross–Pitaevskii system. (For a degenerate almost ideal Bose gas,see Lifshitz and Pitaevskii (1980), section 25.)

(a) Regarding the solid-body rotation of superfluids, there exists some experimentalevidence. Tkachenko (1966) showed that the superfluid behaved like a fluid rotating with thevelocity of a solid body at a macroscopic level. There is a critical angular velocity Ωc of arotating vessel for vortex formation. General understanding of a superfluid in a fast rotatingvessel is that the rotating liquid forms a lattice of quantum vortices.

By their experimental study, Andronikashvili and Tsakadze (1965a, 1965b) showed thathelium II (i.e. liquid helium below the λ-point temperature) brought into motion in a rotatingvessel increases its density appreciably. This fact urges more careful study of rotatingquantum condensates because the structure of the vortex lattice with density defect at thevortex centers implies a reduction of the density of helium II. There must be an unknowneffect that compensates the density reduction at the lattice points if the theory of potentialvortices is the only valid one. In regard to the helium I of normal fluid, an appreciableincrease in the density is not observed by the increase of angular velocity. Section 4 gives aclue to this problem.

(b) Experimental investigations of rotating helium II in early times (Osborne 1950,Andronikashvili and Kaverkin 1955, Donnelly 1956) showed that the quantum liquid He II ina rotating vessel formed a meniscus as if it was a normal fluid undergoing solid-body rotation.In fact, the meniscus, i.e. the shape of free surface, of rotating helium II was given by thesame parabolic height as that of the normal fluid with respect to the distance from the rotationaxis. This implies that the superfluid component is participating in the rotation. This was also

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

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observed in the experiments of rotating He II by Hall and Vinen (1955). Measurement of theangular momentum of rotating helium II showed also that the liquid motion corresponds tothe solid-body rotation (Reppy et al 1960). Kiknadze et al (1965) showed that two-dimen-sional vortex lattices in rotating helium II are capable of rotation together with its rotat-ing vessel.

(c) Note that the study of Kiknadze et al (1965) took analogy with the theory of type IIsuperconductors of Abrikosov (1957). A type-II superconductor is characterized by the for-mation of magnetic vortices in an applied magnetic field. This occurs above a certain criticalstrength Hc of applied magnetic field. The theory of type-II superconductors in magnetic fieldswas developed by Abrikosov, who applied the ideas of Onsager–Feynmanʼs quantum vorticesof superfluids to the lines of magnetic flux passing through the material. The region of lines ofmagnetic flux is surrounded by a circulating supercurrent of electrons. In analogy with thesuperfluid, the swirling supercurrent creates the so-called Abrikosov vortex. He found that thevortices arrange themselves into a regular array. The density distribution of superconductingelectrons coincides exactly with that of a superfluid in corresponding regular array of vortices.There is a close analogy between the Abrikosov vortex of type-II superconductor and theOnsager–Feynman vortex of superfluid.

In the case of type-II superconductor, a flux of magnetic lines of force passes through thesuperconducting material. It would be interesting to consider whether this implies that a fluxof vortex lines passes through the superfluid and forms a vortex of quantized circulation witha rotational core. In this regard, Andronikashvili and Kaverkin (1955), cited by Androni-kashvili and Mamaladze (1966), describes their experiment of helium II in a container whichwas brought into rotation from a resting state. They observed a difference between He I(ordinary fluid) and He II (superfluid) during the process of dragging the fluid into rotation. Inthe case of He II, i.e. the quantum condensate, only its peripheral layers were dragged intorotation at first, while the central part of the meniscus remained flat. Gradually the radius ofthe flat part of the meniscus became smaller and at last a parabolic free surface was formed. Itwas indistinguishable from the parabolic meniscus of a classical viscous liquid at moderatevelocities of rotation. This observation implies that vertical vortex lines gradually penetrateinward into the liquid helium from the peripheral layers. This reminds us of the penetration ofmagnetic lines in the type-II superconductor. They conclude at the end that the superfluidcomponent maintains motion for which vcurl s differs from zero (where vs denotes the velocityof the superfluid component).

Section 5 presents an example of quantum vortices with coaxial rotational core, whosedensity does not vanish at its center. Jackson et al (2009) and also Allen et al (2013) studiedthe finite-temperature effect on vortex dynamics and found non-vanishing density at thevortex core, which is a thermal cloud owing to the finite temperature. However, in the presentcase, the rotational core is obtained by solving the Gross–Pitaevskii equation at the absolutezero temperature.

1.2. Londonʼs scenario

Thus in this paper, we consider how a quantum condensate is endowed mathematically withrotational state. Early in the history of quantum mechanics, London (1927) proposed ascenario to improve Schrödingerʼs quantum mechanics and succeeded in introducing elec-tromagnetism into the Schrödinger system (see OʼRaifeartaigh (1997)). We consider ananalogous scenario to endow a superfluid with a rotational property. Although it is usuallyconsidered that the Gross and Pitaevskii equation (GP equation) implies potential flows ofsuperfluid, we investigate the possibility that a quantum liquid governed by the GP equation

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

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may support rotational motion. According to Londonʼs scenario applied to the GP equation, itis shown in section 2.3 that the same equation can describe rotational motions of superfluid.This is not surprising because the non-linearity of the Gross–Pitaevskii equation has its originin the interaction between two particles in the condensate.

2. Gross–Pitaevskii equation

We consider a quantum condensate of almost ideal Bose gas of atomic mass m. There existinteractions between atoms, represented by a short-range interatomic potential δ xU ( )0 , which

is repulsive if U0 is positive. The coefficient U0 is given by π= U a m402 with a the s-wave

scattering length and π= h 2 with h the Planck constant. In such a slightly non-ideal Bosegas at absolute zero temperature, almost all the particles, but not all, are in the condensate. A

wave function operator is written as Ψ Ψ Ψˆ = + ′x x xt t t( , ) ( , ) ( , ) where ⟨ ⟩Ψ Ψ= ˆ is the

order parameter (the symbol ⟨ ⟩· denoting ensemble average). The condensate wave func-

tion Ψ describes the mode that has macroscopic occupation with Ψ = xn ( )2 giving thecondensate particle number density at x.

2.1. Equation of a condensate of almost ideal Bose gas

If only pair interactions are taken into account, the equation governing Ψ is given by theGross–Pitaevskii equation. In fact, in a trap with confining external potential Ve, the Gross–Pitaevskii energy functional for the condensate (Fetter 2009) is defined by

∫ Ψ Ψ Ψ= + +⎛⎝⎜

⎞⎠⎟

xm

V U2

1

2d . (1)GP e

22 2

0

4 3

The total particle number is defined by ∫ Ψ= xN d2 3 . Minimizing μ− NGP under the

constraint condition =N const, we obtain the following Gross–Pitaevskii equation:

Ψ Ψ μ Ψ− + + − = ( )( )m U V2 0, (2)e2 2

0

2

(Ginzburg and Pitaevskii 1958, Gross 1961, Pitaevskii 1961), where μ is a chemical potential.In the uniform stationary state of Ψ = =n const0 in the constant negative potential

η= −Ve (where η > 0), in which Ψ = 0, it is immediately seen from (2) that we haveμ η= +U n0 0 . The GP equation assumes: (i) there exist a large macroscopic number of Bose

atoms in the ground state (i.e. in the condensate) at absolute zero temperature, (ii) Ψ ′ x t( , )

is negligible in comparison with ⟨ ⟩Ψ Ψ= ˆ , and (iii) the inter-particle spacing (∼ −n01 3) is much

larger than a. The conditions (i) and (ii) are referred to as those for a slightly non-ideal oralmost ideal Bose gas.

2.2. Time-dependent Gross–Pitaevskii equation

The equation (2) can be generalized to the following time-dependent GP equation:

Ψ Ψ Ψ Ψ∂ = − + + ( )( )i m U V2 , (3)t e2 2

02

where ∂ ≡ ∂ ∂tt , and Ψ now depends on t as well as on x.2

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2 Comparison of (2) and (3) implies that a stationary solution has the time factor μ− [ ]texp i .

The non-idealness of the Bose gas causes the presence of particles with non-zeromomentum even at absolute zero because of the interaction potential δ xU ( )0 (see e.g. Lif-shitz and Pitaevskii (1980), section 25).3 This GP picture fits remarkably well to the system ofdilute ultracold trapped alkali-metal gases, developed since 1995.

Bogoliubov (1947) introduced an essential mechanism of such a system of a dilute Bosegas of repulsive interactions (see Lifshitz and Pitaevskii (1980), ch 3). It is particularlyremarkable that the existence of superfluidity in a uniform dilute Bose gas arises indeed from

the repulsive interactions, since the Landau critical velocity is given by =v U n mc 0 0 , which

vanishes for =U 00 (Fetter 2009).

Expressing Ψ with a complex polar form Ψ Ψ φ= ( )iexp , substituting it in (3), and

dividing the resulting expression into imaginary and real parts, we have

| |Ψ Ψ φ Ψ φ∂ + · + = m m

2 2 0, (4)t2

φ φ Ψ

ΨΨ∂ + − + + = ( )

m mV U

2 20. (5)t e

22

2 2

0

2

The mass probability density ρ is defined by ρ Ψ= m 2. Introducing a function Φ by

φ Φ= ( )m , a current vector of velocity u can be defined by

Φ Φ φ= ≡ ( )u m, . (6)

Then the first equation (multiplied by Ψm ) can be transformed to

ρ ρ∂ + · =u( ) 0, (7)t

i.e. the continuity equation. The second (divided by m) can be rewritten as

Φ∂ + + + + =u h h V m1

20, (8)t e

2nl qm

ρ Ψ Ψ= = − −( ) ( ) ( )h U m h m, 2 . (9)nl 02

qm2 2 1 2

2.3. Non-integrable factor to represent rotational flows

In quantum mechanics, the electromagnetism is introduced into the Schrödinger system. Thiswas first proposed by London (1927) by applying an ingenious mathematics to reformulatethe Schrödinger equation. In this subsection we attempt to introduce a rotational property byapplying an analogous mathematical scenario to the GP system of superfluid. An alternativeformulation is given in section 3.

In quantum mechanics, a momentum p is represented by the operator − i . However,for the motion of a charged particle (of electric charge e) in an electromagnetic field(represented by a vector potential A x t( , )), the momentum should be replaced by

− = − − = − −⎜ ⎟⎛⎝

⎞⎠

p A A A

e

ci

e

ci i

e

c, (10)

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3 By collision of two particles of momenta =p 01

and =p 02

in the condensate of non-ideal Bose gas, there existsnon-zero probability of transition to new states of ′ = ≠p p 01 and ′ = −p p2 respectively.

(Landau and Lifsitz 1977, section 111). This is equivalent to the following replacement:

→ ≡ − ( )Aie

ca covariant derivative . (11)A

Namely, the space derivative should be replaced by the covariant derivative

− ( ) Ae ci , together with the time derivative ∂t replaced by ϕ∂ + ( )eit (where ϕ is

an electric scalar potential). For the Schrödinger system represented by the equation such as(3), the above replacement is equivalent to the replacement of Ψ :

Ψ Φ Ψ Φ→ = −* *σ

σ⎡⎣⎢

⎤⎦⎥

xe

cx A x( ) exp i ( ), d d , (12)

where four vectors are introduced by ϕ= −σ ( )AA , and =σ xx ct( , ) with A and x three

vectors. In fact, the factor *Φ( )e c is called Londonʼs phase factor introducing

electromagnetism into the Schrödinger system (OʼRaifeartaigh 1997).Using a structural analogy of the present system with the Schrödinger system, we follow

the formulation of quantum mechanics. It is observed that the current field would not besingly connected when there exists a potential vortex, because its axis is characterized as aline singularity. In order to accommodate vortices, it is proposed that the wave function

Ψ Ψ Φ= ⎡⎣ ⎤⎦( )mexp i of (3) is extended to *Ψ with an extended phase factor *Φ( )m

instead of Φ( )m , where

Ψ Φ Ψ Ψ Θ= =* *⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

xm

x tm

exp i ( ) ( , ) exp i , (13)

Θ Φ Φ Φ= + = + +* * w x w x w xd d d , d d d d , (14)11

22

33

and Φ Φ Φ Φ= ∂ + ∂ + ∂x x xd d d d11

22

33. The expression of Θd implies that the current density

* * * *Ψ Ψ Ψ Ψ≡ ¯ − ¯ ( )( )v mi 2c is given by Ψ Θ Ψ Φ= +( )w2 2 , and the general-

ized momentum p is given by

ρ Φ= = ≡ + =p w wmv v v w w w, , ( , , ). (15)c 1 2 3

It is found that the potential velocity Φ=u of (6) is now replaced by a rotational velocity v.In place of (6)∼(8), we have an extended system of equations:

Φ ρ ρ Φ= + ∂ + · = ∂ + + + + =wv v v h h V m const, ( ) 0,1

2, (16)t t e

2nl qm

where = ( )w x x xw w w( ), ( ), ( )1 2 3 . Here, we define a differential one-form ( )1 by v xd11

+ +v x v xd d22

33. If ( )1 is an exact differential (by =w 0), i.e. if Φ= d( )1 , then it defines

the potential velocity Φ=v . However, if ( )1 is not an exact differential, namely if

∑= ∂ − ∂ ∧ ≠< ( )w w x xd d d 0,( )j k k j

j kj k

1

we have a rotational flow v of vorticity ω = × w, since its ith component

is ω ϵ= ∂ = ∂ − ∂( )w w wi ijk j k j k k j .

In this reformulation, the original momentum φ Φ− = ( ) mi i is replaced by

Θ φ Φ− = − + = +⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ( ) ( ) ( ) w wm m m mi i i i i .

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

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Equivalently, the operator of (3) is replaced by the covariant derivative w:

→ ≡ + ( )wm

i a covariant derivative . (17)w

This is analogous to (11) of the electromagnetic case, and wm corresponds to theelectromagnetic momentum − Ae c( ) . Note that the vector w denotes a current velocity ofrotational motion. Thus, a rotational component is introduced into the GP system. Thisimplies that the GP equation can support rotational motion in general.

3. Rotational state of an almost ideal Bose condensate

We consider an extension of the GP equation to that for a rotating frame with a differentapproach. Beginning with the particle mechanics of section 3.1, we extend it to its quantummechanical formulation for a rotating frame and in section 3.2 apply it to the description of arotational state of an interacting Bose gas. Here, an important remark is to be made from aphysical point of view on possible macroscopic motions. Namely, a closed system in ther-modynamic equilibrium can take only two states: uniform translation or uniform rotation(Landau and Lifsitz (1980), section 10). Therefore, we consider the possibility of solid bodyrotation of a macroscopic condensate, in addition to a static state or uniform translation.

3.1. Mechanics of a point mass in a rotating frame

Suppose that a particle of mass m is moving with a velocity v0 in a potential field V in a fixedinertial frame of reference K0. In the mechanics, the Lagrangian of the particle motion is given

by = − mv V01

2 02 .4 The momentum and energy of the particle are

= ∂ ∂ = = · − = + p pv mv v mv V,1

2,

0 0 0 0 0 0 0 0 02

in the frame K0. Concerning the same motion, let us take another frame of reference K whichis rotating relative to K0 with an angular velocity Ω around the common origin. The particlevelocity is now vK relative to K, and v0 is composed of two components:

Ω Ω= + × = + ≡ ×Ω Ωx w w xv v v , ,K K0

where Ωw is the velocity of the frame rotation at the position x. The Lagrangian in K is

= + · + −Ω Ω w wmv mv m V1

2

1

2. (18)K K K

2 2

Generalized momentum pKand energy K in the frame K are

≡∂∂

= + = ≡ · − = − +Ω Ω p w p wv

mv m mv v mv m V,1

2

1

2. (19)

KK

KK K K K K K0

2 2

Thus, the momentum pKand angular momentum = ×M x pK K

are the same as p0and

= ×M x p0 0, while K is different from 0. Substituting = − Ωwv vK 0 , we obtain

Ω Ω= + − · × = − · ( )x Mmv V mv1

2.K 0

20 0 0

Thus the energy is transformed according to Ω= − · MK 0 0.

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4 By the particle trajectory q t( )i , the velocity and momentum are given by = ˙v qi i0 and = ∂ ∂ ˙p qi i

0 0 .

3.2. Equation of order parameter in a rotating frame

We try to extend the energy functional (1) of the superfluid to that in a rotating frame. Asuperfluid in the rotating frame is described by an order parameter ψ determined by mini-mizing an energy functional corresponding to the energy Ω− · M0 0 (Lifshitz andPitaevskii (1980), section 29). The mechanical energy Ω− · M0 0 can be rewritten as

Ω+ − · × = − − +Ω Ωx w wmv V mvm

mv m m V1

2( )

1

2( )

1

2. (20)0

20 0

2 2

It is noted that the term in the parentheses on the right hand side is the linear momentum mvK

in the rotating frame K:

− =Ωwmv m mv . (21)K0

In quantum mechanics, the linear momentum =p mv0 and angular momentum = ×M x mv0

are expressed by = − p i and = − ×M xi respectively. Therefore, the above relation(21) may be written quantum-mechanically as

− − = − = −Ω Ω

w wmm

i i , equivalently i . (22)K K0 0

The symbols 0 and K are the nablas with respect to the frames K0 and K. Let us extend theenergy GP of (1) to that of a rotating frame. Corresponding to the right-hand side of (20), aquantum-mechanical energy GPr in a rotating frame is given by

∫ ψ ψ ψ= − + + − + +Ω Ω⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥ ( ) ( )x w w

mm m V Ud

1

21 i

1

2

1

2, (23)GPr e

3 2

0

2 2 20

4

where Ve is used in place of V, and the self-interaction term ψU1

2 04 is added. This form is

used in the studies of bosons in rapid rotation by Fischer and Baym (2003), and by Correggiet al (2007). On the other hand, corresponding to the left-hand side of (20), we have analternative expression ′ GPr for the quantum-mechanical energy:

∫ ∫Ωψ ψ ψ ψ ψ′ = − + + − · ˜⎡⎣⎢

⎤⎦⎥ x M x

mV Ud

1

2i

1

2d , (24)GPr e

30

2 2

0

4 3

where ψ̃ is the complex conjugate of ψ .In the non-rotating frame, the expression (23) reduces to (1) since =Ωw 0. Variation of

μ− NGPr under the subsidiary condition of fixed ∫ ψ= xN d2 3 yields

ψ ψ ψ ψ μψ+ + − + + =Ω Ω⎜ ⎟⎛⎝

⎞⎠( )w w

mm m V U

1

2i

1

2. (25)e0

2 20

2

This is the GP equation in a rotating frame with the angular velocity Ω. An equivalentequation was given by Yngvason (2008). This reduces to (2) in the rest frame when =Ωw 0.In view of (22), the equation (25) can be written as

ψ ψ ψ ψ μψ− − + + =Ω ( )( ) wm m V U21

2. (26)K e

2 2 20

2

Since K2 is a scalar product of the K operator, we have invariance of the scalar =K

202 by

rotational transformation between two frames K0 and K, which can be written simply as 2.Thus, we have the equation governing a quantum condensate in the frame rotating with

Ω= ×Ωw x:

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ψ ψ ψ ψ μψ− − + + =Ω ( )( ) wm m V U21

2. (27)e

2 2 20

2

In section 4, this equation will be applied to two problems related to the issues (a) and (b) ofsection 1.1, i.e. density increase and parabolic meniscus of a condensate in a rotating vessel.

3.3. Extension to a rotational quantum liquid

Now, we attempt to generalize the formulation to rotational states in general, whose velocity vis composed of a potential component Φ and rotational component wr: i.e. Φ= + wv r. Itsvorticity ω = ×x w x( ) ( )r represents local rotation of the fluid with the angular velocity

*Ω ω=x x( ) ( )1

2at a point x. We apply the principle of local gauge symmetry to the

rotational flow. The idea is as follows.

Consider a collection of local frames *K rotating with local angular velocity *Ω x( ) ateach point x of the flow field. By the principle of local gauge symmetry5, the expression of

the first two terms of (26), defined by Q2, would be ψ ψ= − − Ω* *( ) wQ m m2 ( )K22 2 1

22

where Ω*w is the velocity of local rotation. To transform it back to the non-rotating frame, we

note that, relative to the frame *K , the non-rotating frame is rotating with the velocity − Ω*w .

Hence the transformed Q2 is expressed by ψ− + Ω* ( ) ( )( ) wm m2 i2 2(by the second of

(22) where 0 is written simply as and Ωw is replaced by − Ω*w ). The last expression can be

written as ψ− + ( ) ( )( ) wm m2 i r2 2

, because the difference − Ω*w wr is irrotational and

can be absorbed in . Thus, instead of the operator of (3), we propose the covariantderivative w of (17) for rotational states under the principle of local gauge symmetry:

→ = +

wm

i . (28)w r

Thus for rotational states, the GP equation (3) is transformed to

Ψ Ψ Ψ Ψ∂ = − + + + ( ) ( )( ) wmm

V Ui 2 i , (29)t r e2

2

02

under an external potential Ve. Instead of (2), its time-independent version is given by

ψ ψ μ ψ− + + + − =( ) ( )( ) wmm

V U2 i 0. (30)r e2

2

02

In section 5, this equation will be applied to a problem related to the issue (c) of section 1.1,i.e. a quantized vortex with coaxial rotational core.

4. Density change by rotation

By their experiment, Andronikashvili and Tsakadze (1965a, b) showed that the quantumliquid He II brought into motion in a rotating vessel increases its density in proportion toalmost the square of the angular velocity Ω of the rotation. This is not recognized sufficiently

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5 The equation (26) is valid in the rotating frame K where all the points of K are related to those of K0 by rotationaltransformation as a whole under the common angular velocity Ω. This is called the global rotational transformation.The equation is said to have global gauge symmetry. The gauge principle requires local gauge symmetry as well forlocal rotational transformation under local angular velocity Ω* x( ). The local means point-wise in this case.

by most texts in general. The equation (27) gives us some insight into this observed propertybecause it is the equation with respect to a rotating frame.

Suppose that the condensate is at rest in a rotating cylinder of radius R without formingvortices. Then the wave function ψ of (27) should be real and approximated by a linearbehavior b r1 , where r is the radial distance from the rotation axis with b1 a constant (this ischecked just below). Then from the equation (27), we obtain

ψ Ω μ ξ+ ≈ + + ( )V U m r r U n1

2, (31)e 0

2 2 2 2

0 0

ξ* ≡ mU n2

, (32)0 0

where ξ is the healing length. The last term of (31) came from ψ ψ( )m22 2 , where

ψ ψ2 is replaced with =−( ) ( )b r b r r11

1 21

2, and the coefficient ( )m22 is replaced with

ξ U n20 0 by using (32). The last term may be neglected if ξ≫r . Then, the right-hand side tends

to be proportional to r2 as r becomes much larger than μ Ω−m2 1. This is consistent with the

original assumption ψ ∝ r when the external potential Ve is a constant and ≫( )b r V Ue1

2

0.

The term ψ 2 on the left-hand side is the particle number density n. Hence, ψ Ω= ∝n r2 2 2

for large r, which is consistent with the observation of Andronikashvili and Tsakadze,noted above.

Next, we consider that Ve is the gravitational potential =V mgze . On the free surface, i.e.on the meniscus =z zm of the condensate, we have ψ = =z z( ) 0m . Then we obtain

Ω≈ +( )z g r const2m2 2 from (31). This is consistent with the experiment of Androni-

kashvili and Kaverkin (1955), observing that the depth of the meniscus did not depart fromthe parabolic depth of normal liquids.

5. A quantized vortex with rotational core

The next example is a quantized vortex with a coaxial rotational core. The wave function is

expressed by ψ ψ= κφr( ) ei in the cylindrical frame φz r( , , ). In the case of a single

quantization (i.e. κ = 1), we have ψ ψ= φr( ) ei for a quantized vortex (having a rotational

core of a size δ). Namely, far from the vortex core (for δ ≫r 1), the flow tends to a potentialflow asymptotically, expressed by a velocity potential Φ γφ= of a quantized circulationπγ = h m2 (i.e. Φ→v ). Its rotational core is assumed to have a velocity wr represented by aGaussian profile:

γδ

γ= = − − =φ φ

⎡⎣⎢

⎤⎦⎥ ( )w w w

r

rm0, 0, , exp , ,r

2

2

where δ is a scale of the rotational core. Total velocity and vorticity are

Φ γ η δ= + = = − ≡* *η−( )wv W r W r

r rr0, 0, ( ) , ( )

1 1e , , (33)r

2

ω ω ω γ γδ

= = ∂ =*η−( ) ( )r r

rrW( ), 0, 0 , ( )

1 2e . (34)c c r 2

2

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

10

This represents a quantized vortex with a rotational core of strength πγ2 . Close to the center,

total velocity is expressed as *γ γ δ= +⎡⎣ ⎤⎦( ) ( )W r r r( ) 2 3 , which tends to 0 as →r 0. Thus,

the velocity is regularized at the center, while the velocity of a potential vortex diverges like−r 1 as →r 0.

5.1. Governing equation

The governing equation of ψ is given by (30). Substituting ψ ψ= φei , the imaginary part (of

the coefficient of φei ) describes the continuity equation as before, and the real part is

ψ ψμ

ψ ψ− + − =**

W r

m mU( )

2 20, (35)2 2

20

23

where ψ ψ= r( ) , and *μ μ= − Ve. The external confining potential Ve is assumed to be a

negative constant, such that *μ > 0. Let us define *μ≡n U0 0 which denotes the particle

number density in uniform state at rest (when >U 00 ). In fact, in a uniform state at rest (where

ψ = 02 and * =W r( ) 0), we have *ψ μ= =U const20 from (35).

The length is normalized by the healing length ξ of (32), and the velocity by ξ¯ = v m .The normalized amplitude A and radial coordinate ζ are defined by

ψ ζ ξ= =A n r, , (36)0

respectively. By these normalizations, equation (35) reduces to

ζ− + − =ζ δ ( )A W A A A 0, (37)2 2 3

ζζ ζ ζ

η ζ δ δ δ ξ≡ − = − = ¯ ¯ =δη η− −( )( )W e e

1 1 11 , , . (38)

2 2

In the present axisymmetric case, this is written as

ζζ

ζ ζ″ + ′ + − − =δ( ) ( ) ( )A A A W A A1

0. (39)2 3

The equation for the amplitude ζ( )Ap of a potential vortex (without rotational core) is

immediately obtained from (39) by simply replacing ζδ ( )W with ζ1 , which is given by

ζζ

ζζ

+ + − − =″ ′( ) ( )A A A A A1 1

0, (40)p p p p p23

(Ginzburg and Pitaevskii 1958, Lifshitz and Pitaevskii 1980, section 30).

5.2. Power series solution for ζ ≪ 1

We seek a power series solution to equation (39) by using the series expansion of ζ( )W :

∑ζζ

η βζ β ζ β ζ= −!

= − + + ⋯⋯=

∞+( ) ( )W

k

11

2 6,

k

kk

1

12 2

33

5

where β ξ δ= ( )2, and expressing A by the form: ζ ζ ζ= + + + + ⋯A b b b b .0 2

24

46

6

Substituting these in (39) and replacing b0 by A0, the coefficients are determined by

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

11

β

β β

= − − = + − +

= − + − + + ⋯⋯

⎜ ⎟⎛⎝

⎞⎠( )

( )

b A A b A A A

b b b A A b A b

1

41 ,

1

16

1

4

3

4,

1

363 3 ,

2 0 02

4 02

02

04

6 42

23

0 0 22

02

4

If = =( )A A 0 00 , we have the zero solution ζ ≡( )A 0. Hence in order to have a sensible

solution, A0 must be non-zero. This is remarkable, since it is well known that the amplitude

ζ( )Ap of the potential vortex tends to zero like ζb1 as ζ → 0 with b1 a constant (see

section 5.3 (a)). This implies that the extended form of GP equation (30) permits a solution of

a quantized vortex with a rotational core with non-zero number density ψ ≠( )0 02 .

5.3. Numerical solutions

(a) Potential vortex with quantized circulation. The amplitude ζ( )Ap is found numerically by

solving the second-order differential equation (40) with the boundary conditions: =( )A 0 0p

and ζ →( )A 1p as ζ → ∞. In this solution, it is found that ζ≈A bp 1 as ζ → 0 and

ζ≈ − −( )A 1 1 2p2 as ζ → ∞ (Lifshitz and Pitaevskii (1980), section 30). The constant b1 is

determined numerically as 0.583190 to the first six digits.(b) Quantum vortex with rotational core. By giving the value =( )A A0 0 at ζ = 0

initially, the ordinary second-order differential equation (39) was solved. The solution wassought by varying the numerical value of δ δ ξ¯ = (the width of rotational core) with the trial-

and-error method for a fixed A0 so as to satisfy the condition at infinity: ζ →( )A 1 as ζ → ∞(by the Mathematica). Corresponding to each particular value of A0, the solution has beendetermined. Numerical parameters of some of the numerical solutions are listed in the table 1.

Figure 1 shows the amplitude ζ( )A of the case δ= ¯ =( )A 0.25 , 0.208940 by a solid

curve a, which is compared with the potential vortex plotted by a broken curve b. In this

figure, the vorticity ω ζ( )c is also plotted by a dotted curve c, with a different vertical scale. It

is seen that far from the core ( ξ>r ), the two solutions a and b tend to be indistinguishable.

5.4. Energy of vortices with rotational core

Thus, we have found a family of solutions depending on δ̄ (or A0) and having different values

of total energy δ . Using (24), we take its variation with keeping the total angular momentum

∫ ψ ψ= ˜M M xd3fixed with a fixed Ω. Hence we consider minimization of the first integral of

′ GPr only with 0 replaced by = + ( ) wmiw r of (28):

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

12

Table 1. Parameters of numerical solutions, where Δ ≡ −δ δ 0 and δ is the totalenergy of a vortex of core size δ).

A0 0 0.01 0.10 0.20 0.25 0.30 0.40

δ δ ξ¯ = 0 0.009 0.086 0.169 0.209 0.249 0.330

Δ ×δ 103 0 −0.003 −0.213 −0.528 −0.539 −0.363 +0.710

∫ ψ ψ′ = − + +⎡⎣⎢

⎤⎦⎥

( )x wm

mUd

1

2i i

1

2,GPr r

32

0

4

where the integration is taken over the plane perpendicular to the vortex axis, and the Ve termis neglected. Using the healing length ξ and the particle number density n0 in the uniform stateat rest, we normalize the wave function ψ and radial coordinate r like ψ ϕ= n0 and ξ ζ=r .

Substituting these, we have

∫ ζξ ϕ ϕ′ = + +⎛⎝⎜

⎞⎠⎟

( )wn

m

m

2i

1

2d . (41)GPr r

02

22

4 2

In non-uniform states in motion, the first term of the integrand is a dimensionless kinetic

energy, denoted by k. Writing as ϕ φ= [ ]A r( ) exp i , the term ξ ϕ+⎡⎣ ⎤⎦( ) wmi r is

ξ ϕ φ ξ φ φ+ = + +ζ⎜ ⎟⎡⎣ ⎤⎦ ⎛⎝

⎞⎠

( ) [ ] [ ]( ) w wm A

m

mA ri exp i i ( ) exp i .r r

Defining Φ φ= ( )m and Φ= + wv r, the second term is expressed as φ¯ [ ]v v Ai ( ) exp iwhere ξ¯ ≡ v m . Using ≡ ¯u v v, we have

ξ ϕ ζ≡ + = + = ′ +ζ( ) ( )w um

A A A u Ai , (42)k r2

22 2 2 2 2

where ζ ζ′ =( )A Ad d , and ζ= δ ( )u W defined by (38). Denoting the integrand of (41) as δfor a quantized vortex of rotational core of size δ, we have

ζ ζ ζ ζ= ′ + + = −δ δ δη− ( )( ) ( ) ( ) ( )A W A A W

1

2, 1 1 e . (43)

2 2 2 4 2

If we replace ζδ ( )W of the second term with ζ= −u 1, we obtain the energy density 0 of the

potential vortex of zero core-size:

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

13

Figure 1. Amplitude ζ( )A of a quantum vortex with a rotational coreδ= ¯ =( )A 0.25, 0.208 940 is shown by the solid curve a, whereas the potential

vortex ζ( )Ap is shown by the broken curve b. The curve c at the bottom shows the

vorticity ω ζ( )c with a different vertical scale. Two dotted vertical lines d show the size

δ̄ of coaxial rotational core.

ζ ζ= ′ + +− ( )A A A1

2.p p p0

2 2 2 4

where Ap denotes the amplitude of the potential vortex. Total angular momentum M is given

by δ( )M , 0, 0 , and δM by ∫ πζ ζδn 2 d0 , where

ζ= =δ uA A, .p2

02

Using the numerical solutions obtained in the previous section section 5.3, we can estimate

the total energy ∫ πζ ζ=δ δ 2 d . In this numerical estimate, the upper end of integration ζm

is chosen so as to fix the integration value ∫ ζ ζζ

δ d0

mof total angular momentum. The

ζm-value depends on δ and was determined so as to take the same total angular momentum asthat of the potential vortex of δ = 0 for which ζ = 5m was chosen.

Figure 2 shows the variation Δ = −δ δ 0 versus the normalized size δ ξ of rotational

core. It is interesting to find that there exists a minimum of Δ δ at δ ξ ≈ 0.2, which is lower

than Δ =δ 0 of the potential vortex. This property is kept unchanged qualitatively even if alarger value of ζ = 10m is taken. Thus, from the energy consideration, it is probable that avortex with a rotational core of δ ξ ≈ 0.2 is generated in reality. Note that, for ζ ξ> (i.e. outof the inner core), difference of the two solutions of δ ξ≈ 0.2 and the potential vortex ofδ = 0 is very small to such a degree that challenges experimental resolution. Relative var-

iation of Δ δ δ is less than 0.001 when Δ <δ 0, where ≈δ 5.816.

6. Summary and discussions

It is proposed that the Gross–Pitaevskii equation governing a non-ideal (interacting) Bose gasadmits rotational flows. This is verified by applying a quantum-mechanical scenario to theequation without incurring essential change in the equation, or alternatively by introducing alocal gauge transformation to the GP equation, by replacing the standard momentum operatorwith a covariant derivative including a gauge field.

By this reformulation, a superfluid placed in a rotating vessel is able to have a solid bodyrotation and also to have a meniscus approximated by a parabolic profile. The solid body

Fluid Dyn. Res. 46 (2014) 031418 T Kambe

14

4

2

0

-1

-2

0.1 0.2 0.3 0.4 0.5

Δεδ 103

�/�

×

Figure 2. Δ ×δ 103 (vertical) plotted vs. the core size parameter δ ξ (horizontal axis).

rotation is accompanied by a density increase proportional to the square of its angularvelocity. This density increase is supported by the experimental observation of Androni-kashvili and Tsakadze (1965a, 1965b). The parabolic meniscus was also observed by variousexperiments cited in item (b) of section 1.1.

In addition, the reformulation allows a vortex of quantized circulation to have a coaxialrotational core. It is shown by numerical analysis that a vortex of core size δ ξ≈ 0.2 has thelowest total energy which is lower than that of a potential vortex without rotational core.

If there is a rotational core in the vortex, the density does not vanish at its center. Mostobservations report that dips certainly exist in the density profiles. However, to the authorʼsknowledge, no experiment reports that the density vanishes exactly at the vortex center. Ifthere is a rotational core, there may be observable consequences on the dispersion relation ofKelvin waves (helical perturbations of the vortex), or on vortex reconnection in superfluid.6

This is considered in some detail next.The vortex with a rotational core obtained in the present study has a core of size

δ ξ≈ 0.2 . The Gaussian profile (34) of the core vorticity is approximated by a constantvorticity circular core of radius δ of the same strength in the following order estimate.Suppose that the vortex filament is deformed into a helix of a wave number k. Then,according to Thomson (1880), the dispersion relation is given by

ω ω ω ω γ δ α− = = − = +δ ( ) ( )L k L L k( ) 1 4 ,1

2, ln 10 0 0

2

for very small δk and α = 0.1159, where ω0 and ωδ are the angular frequency of a vortex ofhollow core and that of a rotational core respectively (both being roughly assumed to be thesame size δ for reasons of simplicity). The negative sign of the ω0-expression means rotationin the reverse direction to that of fluid swirling around the filament.

According to the data of recent observation of Fonda et al (2012) in superfluid helium,we estimate ≈L 14. Hence the relative difference is estimated as ω ω ω− ≈δ( ) 0.020 0 . Thisdifference is challengingly small, i.e. it may be too small to be detected by currently availableexperimental data. Another observable phenomenon may be the vortex reconnection. This isimportant, but is out of our scope, because the vortex reconnection is regarded as caused bydiffusion effect, or by finite-temperature effect.

Acknowledgments

The author benefited greatly from discussions with Professor Makoto Tsubota (Osaka CityUniversity). The manuscript has been improved significantly by the review comments fromtwo referees on the description of external differential and experimental consequences.

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