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Файл:Introduction to superfluidity and superconductivity. Учебное пособие
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3. THE NON-UNIFORM WEAKLY INTERACTING BOSE GAS
3.1. The time-independent Gross-Pitaevskii equation
Consider a weakly interacting Bose gas at low temperatures. As we have
seen above, the interaction is weak if na3 << 1, i.e. there are almost no particles
in a volume with a characteristic size a. Alternatively, one may say that the mean
interparticle distance is much greater than the scattering length. Physically this
means that the particles “do not see” much of each other. For all practical purposes
we may then take the interaction energy to be of a contact type:
x
,x
xx,
(130)
Then the Hamiltonian is
†
x
xx
(131)
We now need to write down a state vector of the gas. If the interaction
is weak enough to be neglected, so that the probability distribution for the
coordinates x1, …, xN is factorised into probability distributions for the
coordinates of individual particles, we take the ansatz that the wave function of
the condensate is a product of single-particle wave functions, each normalised to
unity:
x, ,x
x,
x
x
(132)
The next step is to construct the energy functional
x
x
x
*x, ,xx, ,x,
(133)
where the asterisk denotes complex conjugation. It is left as an exercise to
the reader to prove that inserting (132) into (133) produces
xx
x
xx
x
(134)
On physical grounds, it is more reasonable to normalise the wave
function so that the volume integral of its modulus squared would give the total
number of particles. So, we define a new wave function
xx,
(135)
which makes the number-density of the particles equal to its modulus
squared:
†
In this chapter we shall not be putting hats on operators because there will be no danger of confusing
c-numbers and q-numbers.
32
xx
xx
(136)
The energy functional written in terms of the new wave function then
becomes:
xx
x
xx
x
,
(137)
where we have neglected the difference between N and N – 1, since we
work with a macroscopic system.
To obtain an equation forxwe need to compute the variation of the
energy functional with respect to*xsubject to the constraint that the total
number of particles should be constant. We use the method of Lagrange
multipliers and construct another functional:
xx,
(138)
where μ is a Lagrange multiplier to be determined later. Variation of F with
respect to*xyields:
*xx
* x*
*
(139)
Integrating by parts the first term and using Gauss's theorem we get
*x x
x
S *
(140)
The surface integral is zero because the wave function is fixed at the
boundary, and we finally have the time-independent Gross-Pitaevskii equation
(TIGPE) for a weakly interacting Bose gas:
xxx
x
xx
(141)
Note that the structure of the TIGPE is the same as the structure of the timeindependent Schroedinger equation (TISE) for a particle moving in the
potentialx
x
The first term is just the usual external potential, while the
second is the effective potential due to the presence of the other particles.
In order to determine the meaning of the Lagrange multiplier μ,
rewrite the TIGPE in terms of x,multiply both sides by*x,and integrate over
the total volume of the gas:
x
x
xx
x
,
(142)
which is δE/δN. This means that μ is the chemical potential of the gas. From
the TIGPE we conclude that the chemical potential is equal to
,
(143)
33
where n is the number density of the uniform gas in the absence of an
external potential.
3.2. The coherence length
As an example of the application of the TIGPE, consider a Bose gas
constrained by a rigid wall to occupy the half-space x > 0. The external potential
energy is then
, ,
, ,
(144)
and the wave function only depends on one variable x.
Recall the quantum-mechanical expression for the probability
current
J
p
,
(145)
where χ is the phase of the wave function. In view of the normalisation
(136), the probability current is in fact a particle current. If the phase of the wave
function is constant in space, the current is zero. The converse is also true. Since
any wave function is defined up to a phase factor, we may put this constant equal
to zero and thus make the wave function real.
Since everything depends on x, the current can only be along the xaxis. However, on physical grounds we expect that there can be no current in the
case we are considering, so we may choose the wave function real, and the TIGPE
takes the form:
,
(146)
where primes denote differentiation with respect to x. The reader is invited
to show that the solution for x > 0 with the boundary conditions ψ(0) = 0 and
ψ(∞) < ∞ is
,
(147)
where we have defined the coherence length
(148)
3.3. The time-dependent Gross-Pitaevskii equation
Drawing on the analogy between the TIGPE and the TISE, we make
an educated guess that the time-dependent phenomena may be described by a
34
time-dependent Gross-Pitaevskii equation (TDGPE):
x
x
x
x
(149)
If the external potential does not depend on time, we can separate the
variables in the TDGPE and obtain a solution:
xx
(150)
Just as in the case of the TDSE, one can obtain a continuity equation,
which expresses on the probability conservation:
J ,
(151)
where the probability current is (cf. (145))
J
p
(152)
Writing x
x
x
,where n(x, t) is the particle number-
density, we obtain an expression for the current:
J
v,
(153)
where we have defined the superfluid velocity
v
(154)
Putting the expression for the condensate wave function in terms of
the number density and phase into the TDGPI produces two equivalent equations:
,
v
(155)
The second of Eqns. (155) is just the continuity equation, in
agreement with (151). Taking the gradient of the fist produces an equation
v
(156)
Equation (156) is similar in structure to the Euler equation in the
hydrodynamics of an ideal irrotational liquid. Thus we see that the existence of
the wave function of the condensate leads to superfluidity. The only crucial
difference from classical hydrodynamics is the “quantum” term containing
Plank's constant. Let's compare it with the U0n term assuming that the external
potential, and consequently n, varies on a length-scale l:
(157)
Thus, if the external potential remains almost constant over distances
35
comparable to the coherence length, then the quantum term in (156) may be
omitted. In the opposite case the quantum term becomes dominant.
3.4. Bogoliubov equations
Suppose the system is driven out of equilibrium, but the departure of the
local properties from their equilibrium values is small, so we may put
,
,
(158)
whereis the equilibrium value of the condensate wave function, and
is
a small fluctuation. Let's put (158) into the TDGPE and its complex conjugate,
and only keep the terms linear in the fluctuation. Thus we have
*
,
*
*
*
*
*
(159)
Let's look for solutions of the form
x
x
*x
(160)
Here u(x) and v(x) are arbitrary functions of coordinates, and the
overall factor
is included to cancel the same factor present in the
equilibrium solution of the TDGPE
If we do not include this
factor, the equations resulting from (159) will not be satisfied for an arbitrary
moment of time. As will be seen below the inclusion of both positive and negative
frequency components ensures a non-trivial solution of (159), and the choice of
the complex conjugate of v(x) makes things come out nice.
Putting (160) into (159) and dropping the subscript “0” on n(x) we
get the Bogoliubov equations for u(x) and v(x):
xxxxx,
xxxxx
(161)
Consider the case of a constant external potential. Then
and we
solutions in the form of plane waves:
x
qx
,x
qx
,
(162)
where we have taken the amplitudes to depend only on the magnitude of
the wave vector q. The result is
36
,
(163)
These equations have a non-trivial solution if the determinant is zero.
This condition gives us the relation between the frequency and the wave vector:
,
(164)
The limiting cases are
,
,
,
,
(165)
where
The above results are precisely the results that have been
obtained within the Bogoliubov theory of the weakly interacting Bose gas. The
first limiting case corresponds to excitations with a wavelength much greater than
the coherence length, which makes them ordinary sound waves. In the opposite
case the excitations behave like free particles.
3.5. Vortex states
The Gross-Pitaevskii theory connects the superfluid velocity and the phase
of the wave function:
v
s
(166)
It follows then that the superfluid flow is irrotational, meaning that v
s
Using Stokes's theorem we could say that the circulation of the superfluid
velocity along an arbitrary contour was equal to zero. This would indeed be true
if the region occupied by the superfluid were simply-connected. If, however, the
region is multiply-connected, i.e. if there are "holes" in it, one can set up a
supercurrent around such a hole (see Fig. 4). Therefore the circulation of the
velocity along an arbitrary contour drawn round such a hole is certainly not zero.
After traversing the contour, the phase of the wave function must get back to its
initial value modulo 2π, since physically nothing has changed:
v
s
x
x
,
(167)
37
The circulation of the superfluid velocity is thus quantised, and the integer
l is called the number of quanta of circulation.
Equation (167) is reminiscent of Ampere's law of electrodynamics in
integral form:
Bx
(168)
In the special case of a thin current-carrying wire, the curl of B is zero
everywhere except the wire. We therefore make a guess that in the case of a Bose
gas we have a similar situation: there are vortex lines such that the curl of vs is
zero everywhere except the line.
Suppose we have an infinite straight vortex line in an infinite
medium. Then the system is invariant under rotations about the axis directed along
the line (call it z), and the superfluid velocity only depends on the distantce from
the z-axis. Then recalling a similar problem in electrodynamics, we can write for
the superfluid velocity:
v
s
(169)
What is the structure of the vortex line? Now that we know the
dependence of the superfluid velocity on the distance from the line, we should
like to know how the modulus of the wave function depends on this distance.
Consider an isolated vortex line. The external potential is zero. We
take Eqn. (156) as our starting point. The situation is stationary, so the left-hand
side is zero, and we have
const.
(170)
The constant on the right-hand side is fixed if we go far from the line. Then
the first and the third term will vanish and the second will be equal to U0n0, with
n0 being the superfluid particle density far from the line. The result is a non-linear
equation for n that can only be solved numerically:
Fig. 4. If the region occupied by the superfluid is multiply connected, a current can be set up
round each domain where the wave function is zero. In a simply connected region this cannot
be done because there are not sources and drains of the current.
ψ=0
J≠0
38
(171)
We shall be interested in the behaviour of the superfluid particle
density at distances from the axis greater than the coherence length. In this case
the third term may be neglected and we have
,
(172)
Thus we can construct the following approximate solution of (171):
, ;
,
(173)
The solution (173) is presented in Fig. 5.
The energy associated with the presence of a single vortex is the
kinetic energy of the flow. For a cylindrically symmetric vortex flow with l quanta
of circulation, the energy per unit length is
v
,
(174)
where R is the characteristic size of the region occupied by the superfluid
flow, and we have neglected the variation of the superfluid particle density with
distance wishing only to calculate the vortex energy with logarithmic accuracy.
It is left as an exercise to the reader to show that the energy of
interaction of two vortices separated by a distance a , per unit length, is equal to
int
(175)
The total energy due to the presence of two vortices is is then equal
to
2v
,
(176)
Fig. 5. The superfluid density as a function of the distance from the vortex line. At distances
less than the coherence length the superfluid density is zero.
n(r⊥)
r
⊥
∼ξ
39
where we have approximated log(lξ) with log(ξ). The second term in the
brackets is negative because ξ < a, so it is energetically more favourable to have
two vortices each carrying one quantum of circulation than one vortex carrying
two quanta of circulation.
Consider now a rotating condensate. Experiment shows that when
the angular velocity exceeds a certain value (called critical), vortices begin to
appear. In order to find the critical angular velocity we should go over to the frame
of reference in which the condensate is at rest. Recall the transformation from the
laboratory (stationary) frame of reference to a frame rotating with an angular
velocity Ω about the common origin:
x x,
v vx,
(177)
where the primes refer to the rotating frame. The Lagrangian in the rotating
frame is
vxx
xvx x
(178)
Differentiating with respect to the velocity we obtain the canonical
momentum:
p
v
vxv p,
(179)
which shows that not only the linear momentum remains unchanged as we
go over from one frame to the other, but also the angular momentum:
Mxpx p M
(180)
The energy of the particle in the rotating frame is
vp
xx
xv x M
(181)
With a single vortex present, the angular momentum of the vortex
per unit length is
M
v
,
(182)
where we have also neglected the variation of the superfluid density with
distance from the vortex core. Thus the energy per unit length of the vortex line
in the rotating frame becomes
vv
(183)
When the angular velocity reaches its critical value, this energy will become
zero. Thus
c
(184)
It is interesting to estimate the angular velocity at which the number
40
of vortices is so large that their cores begin to overlap. When this happens the
relative motion in the bulk of the fluid ceases owing to the cancellation of the
equal and opposite flows between adjacent vortices. Therefore the liquid rotates
as a solid so that the velocity of each point is
v x
(185)
Taking a contour surrounding the axis of rotation we get for the circulation
vl vS S ,
(186)
where we have made use of Stokes' theorem, and where A is the projected
area of the surface subtended by the contour. On the other hand, since each vortex
carries one quantum of circulation, the total circulation is equal to the number of
vortices enclosed by the contour multiplied by the circulation carried by each.
The former is approximately equal to the ratio of the projected area A to the crosssection area of the core. Thus
c
(187)
Both situations are presented in Fig. 6
Exercises
1. Solve the one-dimensional TIGPE for the potential (144) and thus obtain the
solution (147).
2. The equations (157) for the amplitude and phase of the wave function can be
obtained from the principle of least action:
Fig. 6. (a) When the angular velocity reaches the value given by (184), a vortex carrying a single
quantum of circulation appears. (b) As the velocity increases, more and more vortices with a
single quantum of circulation enter the system until their density is such that the cores begin to
touch. This situation corresponds to the velcity given by (187).
Ω≥Ω
C2
∼ξ
Ω=Ω
C1
∼ξ
(a)
(b)
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