8.514:
Many-body phenomena in condensed matter and atomic physics Last modi ed: September 29, 2003
1 Lecture 6. Vortices, superuidity. Trapped gases.
BEC at nite temperature.
To treat hydrodynamics and BEC in spatially varying background, need a more general
approach that does not assume condensation into a particular plane wave state. One
can formulate theory of BEC so that the wavefunction of the condensate is an arbitrary
function in space-time that is determined self-consistently from a classical nonlinear eld
equation, known as the Gross-Pitaevskii equation.
1.1 Gross-Pitaevskii equation
Let us start again from the Hamiltonian of weakly interacting Bose gas with short-range
interaction,
!
!
Z
2
h
H = '^+(x) ; 2m r2x + U (x) '^(x) + 2 '^+(x)'^+(x)'^(x)'^(x) dx
(1)
To formulate
the condensate
dynamics, start with the Heisenberg evolution, ih @t '^ =
'
^ H] = ; 2hm2 r2 + U (x) '^ + '^+'^2, and replace '^, '^+ by classical variables ', ',
which gives a classical eld dynamics problem,
!
2
h
2
ih@
t ' = ; 2m r + U (x) ' + ''
2
(2)
!
2
h
2
;ih @t ' = ; 2m r + U (x) ' + '2'
(3)
called the Gross-Pitaevskii equations.
It is instructive to write eqs. separately for the modulus and phase ' = j'jei .
h ('r' ; 'r') = h j'j2r
@t n + rj = 0 n = j'j2 j = 2mi
(4)
m
and
!
2
h
2
h @t = ; U (x) + n + 2m jr'j
(5)
Comparing the above expressions for the density and current, obtain the superow velocity
h
v = r
(6)
m
The ow is irrotational, r v = 0 (this is true away from singularities in ), and
2
p
h
2
m@t v = ;r ~ + mv =2 ~ = U (x) + n + 2mpn r2 n
(7)
(compare to the Euler equation).
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1.2 Superuidity. Vortices.
Let us consider the circulation of velocity in a superow. It follows from the relation
between the velocity and the phase, Eq. (6), that the circulation around any contour C
obeys
I
h 2l
; = v dr = m
(8)
C
with some integer l. We see that
The circulation is quantized in multiples of h=m
The ows in multi-connected geometries, such as a ring-shape tube, are discrete
Quantum leaps are required to change a ow.
The fact that a superow, due to the dicreteness of circulation, cannot be dissipated
gradually, but only in dicrete steps, is the origin of superuidity. The only way to eliminate
a superow is to produce excitations with discrete vorticity and then remove them (along
with the vorticity) from the system.
Also mention the Landau criterion for superuidity: The quasiparticle energy (k) =
(k) ; v k, Doppler-shifted due to the ow, should be positive, to prevent massive
production of quasiparticles. This criterion denes a critical velocity
(k)=jkj
(9)
vc = min
k
0
above which the ow without excitations is unstable, thus showing that superuidity can
be sustained only at the ow velocity below certain critical value.1 The Landau criterion
points at a neccessary condition for superuidity. However, since it does not take into
account vortices which can be generated in the ow even at velocities below vc, one cannot
use it to predict the actual value of critical velocity. The observed critical velocities are
system-dependent (non-universal) and are typically several orders of magntude below vc
estimated from quasiparticle dispersion.
Now let us consider velocity eld in a vortex with singularity on the z axis.The ow
lines are concentric circles parallel to the (x y) plane. Constant circulation requires that
the velocity falls inversely with the distance from the z axis:
h l ^
v(r) =
(10)
m 2
where ^ is the asimuthal unit vector of the cylindrical coordinate system.
Density variation is important only close to the vortex core, at distances where the
kinetic energy per particle exceeds the interaction energy, 12 mv2 > n, which gives
= ph
(11)
m
1 For
weakly interacting Bose gas, the critical velocity is equal to Bogoliubov sound velocity.
2
The so-called healing length determines the size of the vortes core where density is
depleted below its bulk value.
This qualitative picture can be conrmed by an analysis based on the Gross-Pitaevskii
equation. One can look for a solution of the equation that describes a vortes.
p For a
single-quantized vortex with unit circulation, l = 1, we expect '( ! 1) = nei , and
'( ) / ei . Thus one can take a trial function of the form
p
'(r) = n q 2 2 ei
(12)
+ r0
p
and minimize the energy H(' '), which gives r0 = 2 , in agreement with the estimate
above.
Let us consider The energy of the vortex, that can be estimated as the kinetic energy
of the ow, is positive. Thus vortices do not appear unless the system is driven, or stirred.
Let us consider a cylindrical jar rotating with angular velocity , and nd the critical
rotation velocity at which vortices start to appear.
For a vortex located on the symmetry axis of the jar of radius b and height L, the
kinetic energy of the ow is
!
Z LZ b
2 Z L Z b d
2
1
h
h
b
Ev =
n() 2 mv2 ()2ddz = n m
(13)
dz = n m ln L
0 0
0 (According to Eq.(12) the density can be approximated by a constant for > , while
the depletion of density in the vortex core, at , cuts the log divergence at small .)
the contribution to the energy due to the core, which
can be estimated using the trial
2
h
function (12), turns out to be approximately n m ln 1:464L, which is smaller than our
kinetic energy estimate (13).
The energy of the vortex in a jar rotating with velocity is Ev () = Ev ; M , where
M is the angular momentum of the vortex,
Z
Z LZ b
Mv = mn r v d3r =
n()mv()2ddz = b2h nL
(14)
0 0
The vortex becomes energetically favorable at
!
h
b
(1)
> c = Ev =Mv = mb2 ln (15)
Note the inverse square dependence of (1)
c on the radius b, which means that it is easier
to produce vortices in a larger jar.
If the rotation velocity is larger than (1)
c and keeps increasing, one can reach the next
critical value (2)
at
which
the
second
vortex
appears, and then, at some higher value
c
(3)
c , the third vortes enters the jar, and so on.
At high rotation speed, when there are many vortices, one can estimate their number
N from the requirement that the total circulation
due to the vortices, mh N , matches the
H
circulation of a uniformly rotating uid, v dr = b2 , which gives a linear dependence
2
N () ' m
b
(16)
h
3
Of course, since N is integer, in reality the number of vortices increases discretely, in
steps, on average following the proportionality relation (16).
1.3 Trapped gases.
Bose condensation of conned gasses diers somewhat from BEC in a uniform system that
we discussed so far. Most importantly, the BEC transition is accompanied by an abrupt
change of density distribution. This is due to the fact that the lowest energy quantum
state in which atoms condense is peaked at the trap center and has spatial extent much
less than the size of thermal cloud at temperatures slightly above TBEC .
In the experiments on BEC in trapped gasses, atoms are conned by a magnetic trap,
which can be described by a harmonic potential U (r) = 21qm!2r2. The ground state is a
.
gaussian wavepacket 0 (r) / exp(;r2=2l!2 ) of width l! = h=m!
In an ideal Bose gas, in the absence of interactions, in the BEC state at T = 0 all
the atoms populate the state 0. One notes that the density of this state, at the peak,
n ' N=l!3 , can be extremely high when the number of atoms is large. In the presence of
interactions, on can easily reach the limit when the interaction energy per particle is much
larger than the level spacing in the trap, n h !. For that, the number of atoms should
exceed Nc = h!l
!3 =. However, for realistic parameters the value Nc can be 103 ; 104,
which is much less than the typical atom numbers N in the experiments.
To understand the BEC state at a larger number of atoms, one can start with the
Gross-Pitaevskii energy functional and look for a non-uniform state '(x) that minimizes
the energy,
!
Z h 2
1
2
2
4
E (') = 2m jr'(x)j + ((U (x) ; )j'(x)j + 2 j'(x)j dx
(17)
R
with the particle number N = j'j2dx being xed by a chemical potential .
Let us argue that one can discard the gradient term in the energy functional, since the
expected condensate size is much larger than the oscillator ground state width l! , for which
the kinetic and potential energy terms are approximately equal. Indeed for a condensate
of size R, the kinetic, potential, and interaction energies can be estimated as 2hm2 N=R2,
m!2 NR2 , and N 2 =R3 , respectively (since the gas density n ' N=R3 ). Comparing the
2
potential and interaction energy, obtain the condensate size R 2' (N=m!2 2)1=5 . At large
N Nc, the value R is much larger than l! that satises 2hm l! 2 = m!2 l!2 . Hence the
kinetic energy is small, 2hm2 R 2 m!2 2 R2 , which justies ignoring it in the estimate of R.
Without the kineticR energy term, the functional
be written in terms of the density
1
2
2
n = j'j only, E (n) = ((U (x) ; )n + 2 n dx. After taking the minimum, one has
;
;
= U (x) + n
(18)
One can arrive at this result by making a local density approximation, i.e. treating each
small part of the BEC cloud as a uniform system. For the latter, as we already know,
4
the relation between the chemical potential and density is given by Eq.(18). In addition,
the chemical potential in equilibrium must be constant throughout the system. This
condition xes the density distribution n(x) so that the term n in Eq.(18) compensates
the potential U (x) variation in space, which gives
( ; U (x)) = U < n(x) = 0 (19)
U >
We note that the argument used to nd the density distribution is similar to that of the
Thomas-Fermi theory of many-electron atoms, based on a local density approximation
for electrons moving in an eective potential that is determined selfconsistently from
an elecrostatic problem. The local density approximation in trapped BEC, along with
Eqs.(18),(19), is often referred to as Thomas-Fermi approximation.
For a central-symmetric harmonic trap
potential, by relating BEC radius parameter R
2
m!
2
with the chemical potential via = 2 R , from Eq.(19) we obtain density distribution
of the form
2
2 ; r2 r < R
n(r) = m!
R
(20)
2
The particle number can be related with R (and thus with ) as follows:
1 1 m! 2
Z R m! 2 2
2
2
5 = 4 m! 2 R5
N=
R
;
r
4
r
dr
2
;
R
(21)
15
3 5 0 2
which gives a relation between the BEC radius and the number of particles,
!1=5
15
R = 4m!2 N
(22)
p
For large N Nc, the radius R is much larger than the BEC healing length = h=
nm
estimated for typical density n = N= 43 R3, which determines the scale of spatial nonlocality in BEC correlations. This means that the Thomas-Fermi approximation is indeed
a local density approximation.
The above discussion summarizes the situation at T = 0. Let us briey discuss how
the BEC transition aects density distribution. At temperatures above the transition,
the gas in the trap forms a cloud of width RT that can be estimated from 12 m!2RT2 ' T .
At T < TBEC , condensate appears forming a much more narrow peak of radius (22) that
coexists with the broad thermal distribution. As temperature goes down and becomes
very small, the thermal component in the density distribution disappears, and one obtains
the zero-temperature state (20).
1.4 Finite T eects: quasiparticle lifetime.
Decay of quasiparticles due to elastic scattering
X
Hint = 2
a+k4 ak+3 ak2 ak1
k1 +k2 =k3 +k4
5
(23)
Golden Rule for transition rate:
X
Wi f = 2h jhf jHintjiij2 (Ef ; Ei)
(24)
f
In a normal Bose gas, at T > TBEC , the rate of scattering out of the state jii is
dfi = ; 2 X jM j2 f f (1 + f )(1 + f )( + ; ; )
out :
(25)
m
n
i
j
m
n
dt
h ijmn ijmn i j
while the rate of scattering in jii is
dfi = 2 X jM j2(1 + f )(1 + f )f f ( + ; ; )
in :
(26)
i
j m n i
j
m
n
dt h ijmn mnij
with Mijmn = Mmnij = , fi = ha+i ai i, etc. The resulting rate is the sum of the in-rate
and out-rate dfi=dt = dfi=dtjin + dfi=dtjout,
dfi = ; 2 X jM j2 (f f (1 + f )(1 + f ) ; (1 + f )(1 + f )f f ) ( + ; ; )
m
n
i
j m n
i j m n
dt
h ijmn ijmn i j
(27)
Features:
The rate dfdt vanishes in equilibrium, since 1 + fj = e fj , etc.
!
i
j
For near-equilibrium distribution, dfdt = ; 1 (fi ; fi(0) ) with 1 = a2nvT the classical
i
scattering rate
q
(recall: = 4h 2a=m, vT = 2T =m)
Despite scattering, quasiparticles are well dened: (k) 1
In the BEC state, scattering is stimulated by the presence of the condensate:
p
an a+n ! a0 a+0 = N
which gives the rate of scattering out of the state jii as
dfi = ; 2 X jM j2f f (1 + f )( + ; )
out :
m
i
j
m
dt
h ijm ijm i j
and the rate of scattering in jii,
dfi = 2 X jM j2(1 + f )(1 + f )f ( + ; )
in :
i
j m i
j
m
dt h ijm mij
p
(28)
(29)
(30)
with Mijmn = Mmnij / N .
The resulting rate, dfi=dt = dfi=dtjin + dfi=dtjout, is
dfi = ; 2 X jM j2 (f f (1 + f ) ; (1 + f )(1 + f )f ) ( + ; )
(31)
m
i
j m
i
j
m
dt
h ijm ijm i j
Due to the presence of Bose condensate, scattering rate is enhanced at T < TBEC .
6
1.5 Finite T eects: two-uid hydrodynamics, I & II sound.
Momentum can be carried both by the condensate and excitations:
Z
d3p
j = vs + jex = vs + pfp
(2h )3
Normal uid is described by quasiparticle distribution
fp = exp ( ( ; p 1(v ; v ))) ; 1
p
n
s
with the quasiparticle energy Doppler-shifted due to relative motion of the normal gas
and superuid. The momentum due to the normal component is
Z
d3p = (jv ; v j) (v ; v )
jex = pfp
(2h )3 n n s n s
At small velocities, have
( 2
Z p2
3p
d
=45h3c5 )T 4 T T
n = 3 (;@fp =@p ) (2h )3 = (2
(T =TBEC )3=2 T T < TBEC
q
where T = n is the Bogoliubov sound { free particle crossover energy, and c = n=m
is the sound velocity.
With s = ; n, the total momentum density can be expressed as
j = s vs + n vn
To describe dynamics, one needs separate equations for s , vs and n , vn. We will not
discuss the two-uid hydrodynamics in full generality. Instead, we describe a particular
phenomenon, the II sound, a collective mode that appears in the two-uid regime. In this
mode, the relative fraction of the normal and superuid component oscillates and can
propagate in a sound-like fashion.
We consider a uniform system, in the absence of external potential. The total mass
and mass current density obey the continuity relation
@t + r j = 0
(32)
@t j = ;rp
(33)
and the momentum conservation law
After eliminating j from (32) and (33), one obtains
@t2 ; r2 p = 0
7
(34)
For superuid velocity, one can write
m@t vs = ;r
(35)
This relation, derived above from the Gross-Pitaevskii equation, is in fact very general,
and is true for any superuid. It follows from the relation between the phase of superuid
order parameter and the chemical potential, h @t = ;, discussed above. In this form it
was rst introduced by Josephson in the theory of superconductivity.
Use the thermodynamic Gibbs relation N d = V dp ; S dT with = N m=V , =
S=N m, to relate the gradient r with the pressure and temperature gradients:
r = m rp ; mrT
(36)
From Eqs.(36),(35),(33), combined with j = s vs + nvn, obtain
@t (vn ; vs) = ; rT
(37)
n
In a sound wave, gas compression is adiabatic, with entropy conserved:
@t () + r (vn ) = 0
(38)
(The entropy is carried by the normal component only!) After linearizing and combining
with the mass conservation Eq.(32), have
@t = s r (vs ; vn)
(39)
Combined with Eq.(37), this yields
@t2 = s 2r2 T
(40)
n
Collective modes are obtained by considering small oscillations of density, pressure, temperature and entropy, of the form exp (iqr ; i!t). It is convenient to choose density and
temperature as independent variables. Linearizing Eqs. (34), (40) in , T , obtain
2 !
! 3
!2 ; q2 4 @@ p + @@Tp T 5 = 0
(41)
T
2
! 3
!
@
@
p
!2 4 @ + @ T T 5 ; q2 s 2 T = 0
(42)
T
n
In terms of sound velocity u = !=q, the equations have a solution if
(u2 ; c21 )(u2 ; c22 ) ; u2c32 = 0
8
(43)
with
!
!
@p c2 = s2T c2 = @p 2 T
(44)
2
3
@ T
@T C~
nC~
The constants c1 and c2 are the isothermal sound velocity and the velocity of temperature
waves at constant density, while C~ = T (@=@T )
is the specic heat at constant volume,
per unit mass. The expression for constant c3 of the form (44) was obtained from Eqs.
(41),(42) by using Maxwell relation
c21 =
@p
@T
!
@S
= @V
!
T
= ;2
@
@
!
(45)
T
The I and II sound velocities, obtained from Eq.(43), are
2 2 2 1 q 2 2 2 2
1
2
uIII = 2 c1 + c2 + c3 2 (c1 + c2 + c3) ; 4c21 c22
(46)
So far, the treatment was completely general, applicable to any Bose system, irrespective
the interaction strength and form.
Using the result, one can look at several regimes. In a weakly nonideal gas, at low
temperatures, T T = n, the I sound velocity coincides with that of Bogoliubov
quasiparticles at low energies,
q
uI = n=m
(47)
p
while the II sound velocity is 3 times lower,
p q
uII = uI = 3 = n=3m
(48)
The velocity uII decreases as a function of temperature.
In 4He the II sound represents mostly a temperature wave, and to excite/detect it
people had to use oscillatory thermal sources and heat sensors.
9