Persistent currents in a toroidal trap
Juha Javanainen
Department of Physics, University of Connecticut, Storrs, Connecticut 06269-3046
Sun Mok Paik and Sung Mi Yoo
Department of Physics, Kangwon National University, Chunchon, Kangwon-do 200-701, Korea
Received 14 August 1997
Using elementary microscopic methods, we theoretically study persistent flow of an alkali-metal vapor
Bose-Einstein condensate around a tight toroidal trap. The angular velocity of a persistent current must be
smaller than the angular frequency of the lowest condensate excitation. A supercurrent may be excited by
rotating a perturbing potential that is strong enough to cut the toroidal condensate. S1050-29479807907-4
PACS numbers: 03.75.Fi, 05.30.-d, 32.80.Pj
Experimental studies of Bose-Einstein condensates BEC
in alkali-metal vapors are now well under way 1. While
experimental data are not yet available, the potential of su-
perfluidity and persistent currents in these systems has been
recognized from the outset. Theoretical effort 2 has focused
on vortex states of a weakly interacting Bose gas bound to a
harmonic trap, states in which the entire condensate rotates
in accordance with a quantized value of angular momentum.
Studies of excitations of such vortices 3,4 have led to ar-
guments that a vortex cannot be stable 4 in a trap with the
minimum of the confining potential at the center.
Unimpeded by the no-go rule 4, a persistent current may
be stable in other types of traps that can pin the vortex. We
consider a rather extreme case, a trap that confines a conden-
sate to a torus. With the additional assumption that trans-
verse confinement is tight, the motion of the condensate
along the torus is amenable to a simple microscopic treat-
ment. We point out that superfluid flow may be stable as
long as the angular velocity at which the condensate circu-
lates around the torus is lower than the angular frequencies
of the elementary excitations of the condensate. Creating a
persistent current also proves to be a nontrivial task. We
develop an approach whereby the condensate is stirred up by
a rotating potential strong enough to cut the torus.
Specifically, we take the radius of the condensate torus R
to be much larger than the transverse dimensions of a cut
across the condensate. Two simplifications ensue from such
an assumption. First, the frequencies of the excitations in-
volving the transverse coordinates, call them y and z , tend to
be much higher than the excitation frequencies of the motion
in the direction along the ring coordinate x . In what follows
we assume that the transverse motion is frozen to a wave
function ( y , z ). Second, as has been anticipated in our no-
tation already, in our mathematics we straighten the torus
and treat the motion along the ring as linear translation. As a
vestige of the original topology we impose periodic bound-
ary conditions over the circumference of the torus 2 R .
For atom-atom interactions we adopt the conventional
function pair potential characterized by the s wave scattering
length a . As it comes to the interactions, the only relevant
parameter of the transverse motion is the length scale l de-
fined by l
-2
= dydz | ( y , z ) |
4
. We embody atom-atom in-
teractions into the dimensionless parameter =2 aR / l
2
. It
will frequently prove convenient to discuss the fluid in a
rotating coordinate system in which a stationary condensate
would rotate like a wheel at the angular velocity - . We
employ the dimensionless parameter =mR
2
/ for the an-
gular velocity. Without restricting the generality, in the fol-
lowing we assume that 0. Finally, we use R as the unit of
length, the atomic mass m as the unit of mass, and
2
/( mR
2
)
as the unit of energy.
All told, the atoms move in the interval x - , with
periodic boundary conditions. In the basis of the plane waves
u
k
( x ) =1/ 2 e
ikx
with k =0,1, . . . , the second-
quantized many-body Hamiltonian in the rotating frame
reads
H =
k
k
2
2
- k
b
k
†
b
k
+
p , q
V
˜
p -q b
p
†
b
q
+
1
2
k , p , q
b
k +q
†
b
p -q
†
b
p
b
k
. 1
We allow for a potential V ( x ) in the direction of the torus,
and V
˜
( k ) =(1/2 )
-
dxe
-ikx
V ( x ) are the Fourier coeffi-
cients of the potential. The corresponding Gross-Pitaevskii
equation 5,6GPE for a system of N atoms is
-
1
2
2
x
2
+i
x
+V +2 N | |
2
= . 2
In the absence of the potential V , the plane waves u
k
are still
the eigenstates of the GPE, though the energy chemical po-
tential depends on both rotation and atom-atom interactions:
k
=k
2
/2- k +N . 3
It is a peculiarity of the transformation to the rotating frame
that atoms in the rotating-frame eigenstate u
k
still have the
velocity v =k with respect to the stationary frame.
Let us momentarily ignore both atom-atom interactions
and the potential V ( x ), and work in the nonrotating labora-
tory frame with =0. A state with all the N atoms in any
one-particle state u
k
is evidently an eigenstate of the Hamil-
tonian 1. By the translational symmetry, such a state should
be a good first approximation to an eigenstate of the Hamil-
tonian 1 even in the presence of atom-atom interactions.
PHYSICAL REVIEW A JULY 1998 VOLUME 58, NUMBER 1
PRA 58 1050-2947/98/581/5804/$15.00 580 © 1998 The American Physical Society