Quantum Mechanics of Bose-Einstein Condensates Alexander L ...

Quantum Mechanics of Bose-Einstein Condensates
1. Brief review of superfluid 4 He
Alexander L. Fetter
Stanford University
Natal, Brazil, August 2012
• two-fluid model and quantized circulation
• quasiparticles and Landau critical velocity
2. Three-dimensional ideal Bose gas in a box and in a trap
3. Energies and length scales for interacting Bose gas
4. Gross-Pitaevskii picture for a trapped Bose gas
• static and time-dependent behavior
• Bogoliubov spectrum and collective modes
• effect of attractive interactions
5. Optical lattices and reduced dimensionality
1

1. Brief review of superfluid 4 He
• 4 He (and also 3 He) remain liquid down to T = 0 K under atmospheric
pressure
• arises from weak interatomic potentials and large quantum zero-point
motion
• 4 He has two electrons, two protons and two neutrons so it acts like a boson
under interchange of two atoms
• in contrast, 3 He has only one neutron and thus acts like a fermion under
interchange of two atoms
• this apparently small difference leads to wholly distinct behaviors at low
temperature
• below T λ ≈ 2.17 K, 4 He makes a phase transition from a viscous normal
fluid to a wholly different and remarkable superfluid form
2

1.1 two-fluid model
• superfluid 4 He can flow through fine channels with no pressure drop,
implying zero viscosity
• yet a direct measurement of viscosity (through a rotating cylinder device)
yields viscosity like that of the normal fluid above T λ
• Landau explained this with a phenomenological two-fluid model:
(1) a viscous normal component with density ρ n and velocity v n and
(2) a superfluid component with density ρ s and an irrotational velocity v s
with ∇ × v s = 0
• both ρ n and ρ s depend on temperature, with ρ n + ρ s = ρ (the total mass
density)
• ρ n vanishes at T = 0 K, and ρ s vanishes at T λ
• can measure ρ n through viscous drag on oscillating disks, and then ρ s =
ρ − ρ n
3

• superfluids exhibit persistent currents in multiply connected geometries
• rotate in normal state above T λ and then cool; superfluid continues to rotate
even if container comes to rest
• similar behavior for superconductors, where electrical charge makes direct
measurement straightforward
• measured critical velocity v c sets upper limit on size of persistent currents
• for channel of lateral dimension d, observed critical velocity v c ∼ /Md,
where M is atomic mass
• this is believed to arise from formation of quantized vortices (Feynman)
• there is very different mechanism for v c in bulk (Landau)
4

1.2 quantized circulation
• superfluid is irrotational; can describe with velocity potential v s = ∇Φ
• for slow flow, it is also incompressible with ∇ · v s = 0
• together, these imply that Φ obeys Laplace’s equation with ∇ 2 Φ = 0
• quantum mechanics shows that the velocity potential is proportional to the
phase S of a single-particle wave function with Φ = S/M (Feynman)
• originally, irrotational superfluid flow was thought to imply that superfluid
would not rotate
• experiments showed, however, that meniscus for rotating superfluid was just
like that for rotating normal fluid
• Onsager (1949) and Feynman (1955) suggested that rotating superfluid has
an array of quantized vortices
• each has quantized circulation κ = ∮ dl · v s = h/M
• for external rotation with speed Ω, vortex areal density is n v = MΩ/π
5

1.3 quasiparticles and Landau critical velocity
• Landau proposed that superfluid 4 He has a long-wavelength phonon
spectrum ω k ≈ sk with s ≈ 240 m/s the speed of compressional sound
• to fit the specific heat, he suggested that the spectrum also has a local
minimum at shorter wavelengths
ω k ≈ ∆ + (k − k 0) 2
,
2M ∗
where ∆/k B ≈ 8.7 K is known as the roton gap, k 0 ≈ 1.9 × 10 10 m −1
(k0 −1 ∼ interparticle separation), and M ∗ ≈ 0.16 M( 4 He)
• this model yields a good fit to the measured temperature dependence of ρ s
and ρ n
• it also predicts a critical velocity v c by considering the motion of a massive
macroscopic object through the superfluid
• energy/momentum conservation shows that the object emits quasiparticles
only when it exceeds a critical velocity v c given by the minimum value of
the ratio ω k /k, which here occurs for rotons with v c ≈ ∆/k 0 ≈ 50 m/s
6

2. Three-dimensional ideal Bose gas
Brief review of familiar uniform system and less familiar case of harmonic trap
2.1 Qualitative picture of Bose-Einstein condensation in uniform system
• assume uniform ideal gas with particle mass M, number density n and
volume per particle n −1
• one characteristic length is interparticle spacing ∼ n −1/3
• if system is in equilibrium at temperature T , then mean particle momentum
is p T ∼ √ Mk B T
• the de Broglie wavelength λ T ∼ h/p T ∼ / √ Mk B T then gives another
characteristic length
• in classical limit → 0 (or high T ), thermal wavelength λ T is small relative
to interparticle spacing n −1/3
• neglect quantum diffraction, like ray optics in electromagnetism
7

• as T falls, thermal wavelength λ T eventually becomes comparable to
interparticle spacing n −1/3
• now quantum diffraction between nearby particles is important
• define dimensionless “phase space density” nλ 3 T
(small in classical limit)
• when this parameter is of order 1, get onset of quantum degeneracy
• for fermions, this defines the familiar Fermi temperature T F , given
equivalently by k B T F ≈ 2 n 2/3 /M
• for electrons in metals with n ∼ 10 28 m −3 and M as electron mass, find
T F ∼ 10 4 K
• for liquid 3 He with similar n but much bigger M, find T F ∼ 1 K
• unlike the gradual fermion crossover, an ideal Bose gas undergoes a sharp
transition at a similar temperature k B T c ∼ 2 n 2/3 /M
• parameters for both 3 He and 4 He are similar and yield T c ∼ 1 K, comparable
with observed T λ ∼ 2.17 K
8

2.2 quantitative description of ideal Bose gas in external potential V tr
• for any specific trap potential V tr , there is a set of single-particle states with
energies ɛ j
• for an ideal Bose gas in equilibrium at temperature T and chemical potential
µ, the mean occupation of the state j is
n j =
1
exp[β(ɛ j − µ)] − 1 ≡ f(ɛ j)
where β −1 = k B T and f(ɛ) = {exp[β[ɛ − µ)] − 1} −1 is the familiar
Bose-Einstein distribution function
• the mean total particle number and mean total energy are
N(T, µ) = ∑ j
f(ɛ j ) and E(T, µ) = ∑ j
ɛ j f(ɛ j )
• formally, can invert the first to find µ(T, N), and substitution into second
yields the more physical quantity E(T, N)
9

• for large N, introduce “density of states” g(ɛ) = ∑ j δ(ɛ − ɛ j)
• previous expression for N then becomes
∫
N(T, µ) = dɛ g(ɛ)f(ɛ) =
∫ ∞
ɛ 0
g(ɛ)
dɛ
exp[β(ɛ − µ)] − 1
• for a classical system, the chemical potential is large and negative
• as T falls at fixed N, chemical potential increases toward zero and eventually
reaches the lowest single-particle energy ɛ 0
• this equality µ(T c , N) = ɛ 0 defines the critical temperature for the onset of
Bose-Einstein condensation because distribution function f(ɛ 0 ) is singular
• nevertheless, the integral that represents N continues to decrease for T < T c
10

• let N 0 be the number of particles condensed in the lowest single particle
state
• by definition, N 0 = 0 above the Bose-Einstein condensation temperature T c
• separate out N 0 from integral and write N = N 0 (T ) + N ′ (T ), with
N ′ (T ) =
∫ ∞
ɛ 0
g(ɛ)
dɛ
exp[β(ɛ − ɛ 0 )] − 1
the number of particles not in the condensate
• if g(ɛ 0 ) is finite, integral diverges and BEC cannot occur
11

2.3 uniform ideal three-dimensional Bose gas
• three-dimensional cubical box with dimension L and volume V = L 3
• periodic boundary conditions give plane waves ψ k (r) = V −1/2 exp(ik · r)
• energy eigenvalue is ɛ k = 2 k 2 /2M
• boundary conditions require k = (2π/L)(n x , n y , n z ) where n j is an integer
• lowest state is uniform with k = 0, ɛ 0 = 0 and density 1/V
• integral gives density of states
g(ɛ) = V
4π 2 ( 2M
2 ) 3/2
ɛ 1/2
• g(ɛ) vanishes at ɛ = 0, and get BEC with nλ 3 T c
≈ 2.612 that defines T c
12

• below T c , fraction of particles in excited states with finite momentum is less
than 1
N ′ (T )
N = ( T
T c
) 3/2
• correspondingly, fraction of particles in condensate is
N 0 (T )
N = 1 − N ′ (T )
N = 1 − ( T
T c
) 3/2
,
which increases from 0 at T c to 1 at T = 0 K
• for uniform ideal Bose gas in d dimensions, same analysis gives density of
states g(ɛ) ∝ ɛ d/2−1
• note that g(0) is finite for two dimensions, so two-dimensional Bose gas has
no BEC
• but this case is subtle since it depends logarithmically on order of limits
N → ∞ or L 2 → ∞
13

2.4 ideal three-dimensional Bose gas in harmonic trap
• general three-dimensional harmonic trap has potential
• familiar single-particle energies are
V tr (r) = 1 2 M ( ω 2 xx 2 + ω 2 yy 2 + ω 2 zz 2)
ɛ nx ,n y ,n z
= (n x ω x + n y ω y + n z ω z ) + ɛ 0
where n j is a non-negative integer and ɛ 0 = 1 2 (ω x+ω y +ω z ) is ground-state
(zero-point) energy
• ground-state wave function ψ 0 (r) is product of three Gaussians with
characteristic dimensions d j = √ /Mω j (oscillator length)
• if sums are replaced by integrals (holds for ɛ ≫ ɛ 0 ), density of states becomes
g(ɛ) =
ɛ2
2 3 ω0
3 ,
where ω 0 = (ω x ω y ω z ) 1/3 is geometric mean trap frequency
14

• onset of BEC in this harmonic trap occurs for µ(T c , N) = ɛ 0 , with transition
temperature
k B T c ≈ 0.94 ω 0 N 1/3
• below T c , number of particles in excited states falls like N ′ (T )/N = (T/T c ) 3
• compare power 3/2 for uniform BEC
• remaining particles occupy the non-uniform single-particle ground state
ψ 0 (r) (product of Gaussians) with
N 0 (T )
N = 1 − ( T
T c
) 3
• like uniform BEC, condensate fraction grows from 0 at T c to 1 at T = 0
• zero-temperature condensate density has Gaussian profile: n 0 (r) = N|ψ 0 (r)| 2
• note trap potential introduces new characteristic length d 0 = √ /Mω 0 ,
unlike case of uniform ideal BEC
15

• images of this three-dimensional Gaussian condensate rising out of thermal
cloud provided clear evidence of BEC
• typical atomic traps have d 0 ∼ a few µm and N ∼ 10 6
• leads to transition temperature T c ∼ 100-1000 nK, depending on atomic
mass (varies from 7 Li to 87 Rb)
16

• for isotropic harmonic trap in general dimension d, find density of states
g(ɛ) ∝ ɛ d−1
• note two dimensions now yields g(0) = 0, so two-dimensional harmonic trap
will have BEC for ideal gas
• but even weak interactions lead to different physical interpretation related
to Berezinskii-Kosterlitz-Thouless (BKT) transition
• here, case d = 1 is similar to two-dimensional uniform ideal BEC; both have
finite value for g(0)
17

3. Energies and length scales for interacting Bose gases
Inclusion of two-body interactions leads to several new features for realistic
BECs
• Interaction potential v(r − r ′ ) has short range, much less than interparticle
spacing
• at low temperatures, only s-wave scattering is important
• introduce a simple model with contact interactions: v(r) = gδ (3) (r),
where g has the dimension of energy times volume
• standard scattering theory for two identical particles with mass M yields
the identification
g = 4π2 a
M ,
where a is the s-wave scattering length that relates the phase of the scattered
wave to that of the incoming wave
18

• for common bosonic alkali-metal atoms ( 23 Na, 87 Rb), s-wave scattering
length is positive and a few nm, so interaction is repulsive
• for 7 Li, scattering length is negative (because of accident of bound states)
• this effective attraction leads to very different behavior for 7 Li (discussed
later)
• for positive a, dimensionless parameter na 3 characterizes “diluteness” of gas
• usually na 3 is very small ∼ 10 −6
• in some cases, magnetic field can enhance a with “Feshbach resonance”
leading to large values of na 3 19

3.1 interaction energy for a uniform Bose gas
In the present model, each particle in an N-particle interacting gas experiences
an effective local mean-field (Hartree) potential
V H (r i ) = g ∑ j≠i
δ (3) (r i − r j ) ≈ gn(r i ),
where sum is over all other particles
• takes energy E(N + 1) − E(N) = gn to add one particle to uniform dilute
Bose gas in box
• this quantity is just the chemical potential µ = (∂E/∂N) V
• hence here µ = gn
• from thermodynamic identity, find E(V, N) = 1 2 gN 2 /V
• pressure is p = −(∂E/∂V ) N = 1 2 gn2
20

3.2 healing length for uniform gas
• compare kinetic energy − 2 ∇ 2 /2M with Hartree energy V H = gn
• leads to characteristic squared length
ξ 2 =
2
2Mgn = 1
8πna ,
making use of relation between coupling constant g and s-wave scattering
length a
• if gas is perturbed locally, it takes a distance ∼ ξ to heal back to bulk
density n
• in dilute limit na 3 ≪ 1, healing length ξ is large compared to interparticle
spacing n −1/3 , confirming mean-field picture
21

4. Gross-Pitaevskii picture for trapped Bose gas
At zero temperature, a uniform interacting Bose gas has two microscopic length
scales:
• the s-wave scattering length a
• the interparticle spacing n −1/3 (equivalently, the healing length ξ)
In contrast, an interacting trapped Bose gas has an additional length associated
with the trap potential
• for harmonic trap, this is just the oscillator length d 0 = √ /Mω 0
• this additional length leads to many interesting new effects and an additional
dimensionless parameter
22

4.1 static behavior
Gross and Pitaevskii (1961) independently started from time-independent Schrödinger
equation for condensate wave function Ψ in an ideal gas and added nonlinear
Hartree potential V H = g|Ψ| 2
• including trap potential V tr leads to Gross-Pitaevskii (GP) equation for a
trapped Bose gas
(− 2 ∇ 2
)
2M + V tr + g|Ψ| 2 Ψ = µΨ,
where µ is chemical potential
• in dilute Bose gas at low temperature with na 3 ≪ 1, usually normalize to
the total N with ∫ dV |Ψ| 2 = N
23

For deeper understanding of this time-independent GP equation, it is helpful
to start from the corresponding GP energy functional
⎛
⎞
∫
⎜
E GP [Ψ] = dV ⎝ 2 |∇Ψ| 2
+ V tr |Ψ| 2 + 1
} 2M {{ }
} {{ } 2 g ⎟
|Ψ|4 ⎠
trap
} {{ }
kinetic
interaction
where the three terms are the kinetic energy, the trap energy (proportional
to the condensate density), and the interaction energy (proportional to the
condensate density squared), respectively
• variation of E GP [Ψ] with respect to Ψ ∗ at fixed normalization reproduces the
time-independent GP equation, with the chemical potential µ as a Lagrange
multiplier (
− 2 ∇ 2
)
2M + V tr + g|Ψ| 2 Ψ = µΨ,
• note that this GP equation looks like a nonlinear eigenvalue equation, but
the right-hand side contains µ rather than the energy per particle
24

For harmonic trap, use oscillator length d 0 and trap frequency ω 0 times as
units of length and energy
• resulting energy functional has one-particle kinetic energy and trap potential
energy of order unity
• two-particle interaction term contains new dimensionless parameter Na/d 0
that characterizes importance of interaction energy relative to that of ideal
gas in harmonic trap
• if Na/d 0 is small then BEC acts like ideal gas
• situation is very different if Na/d 0 is large
• note that d 0 ∼ a few µm and a ∼ a few nm, so ratio a/d 0 ∼ 10 −3
• but for typical N ∼ 10 6 , interaction parameter can be large ∼ 10 3
• in this case, repulsive interactions act to expand condensate to larger radius
R 0 (typically R 0 /d 0 ∼ 10)
25

• expanded condensate has much smaller radial density gradient
• hence ignore radial kinetic energy in GP energy functional
• leads to simpler and approximate “Thomas-Fermi” (TF) energy functional
∫
E TF [Ψ] = dV ( V tr |Ψ| 2 + 1 2 g|Ψ|4) ,
which involves only |Ψ| 2 and |Ψ| 4
• variation of E TF [Ψ] with respect to density |Ψ| 2 yields the TF approximation
V tr (r) + g|Ψ(r)| 2 = µ,
which also follows by omitting the kinetic energy term in GP equation
• solution is simple TF condensate density
n(r) = µ − V tr(r)
g
where θ(x) is unit positive step function
θ[µ − V tr (r)],
26

• for general three-dimensional harmonic trap, TF density is inverted parabola
)
n(r) = n(0)
(1 − x2
− y2
− z2
Rx
2 Ry
2 Rz
2
where right side is positive and zero otherwise
• here, n(0) = µ/g is central density and Rj
2 = 2µ/Mω2 j
condensate radii in three orthogonal directions
are the squared
• normalization condition on |Ψ| 2 yields N = 8πn(0)R 3 0/15 where R 3 0 =
R x R y R z depends on chemical potential µ
• leads to important dimensionless relation
(
R0
d 0
) 5
= 15 Na
d 0
which is large in present TF limit
27

• similarly, the TF chemical potential is also large
µ TF = 1 ( ) 2
2 ω R0
0
d 0
compared to characteristic oscillator energy ω 0
• note that µ TF is proportional to N 2/5
• thermodynamic identity µ = ∂E/∂N yields TF energy E TF = 5 7 Nµ TF
• use central density n(0) to define healing length in nonuniform condensate
• leads to simple relation
ξ
= d 0
d 0 R 0
• right side is small in TF limit, so ξ is small compared to oscillator length
• clear separation of TF length scales ξ ≪ d 0 ≪ R 0 is crucial in physics of
trapped TF condensates
28

4.2 time-dependent Gross-Pitaevskii equation
The time-independent GP equation has an intuitive time-dependent
generalization
∂Ψ(r, t)
i =
[− 2 ∇ 2
]
∂t 2M + V tr(r) + g|Ψ(r, t)| 2 Ψ(r, t),
where Ψ(r, t) now depends on t as well as on r
• comparison with time-independent GP equation shows that a stationary
solution has the time dependence exp(−iµt/).
• this nonlinear field equation can be recast in an intuitive hydrodynamic
form by writing the condensate wave function as
Ψ(r, t) = |Ψ(r, t)| exp[iS(r, t)]
in terms of the magnitude |Ψ| and the phase S
29

• the condensate (particle) density is simply n(r, t) = |Ψ(r, t)| 2
• usual one-body definition of the particle current density for the Schrödinger
equation shows that j = |Ψ| 2 ∇S/M
• hydrodynamic definition j = nv then gives the local superfluid velocity as
v(r, t) = ∇S(r, t) = ∇Φ(r, t),
M
where Φ = S/M is velocity potential for this irrotational flow (Feynman)
• relation between the velocity v and the phase S implies that the circulation
κ = ∮ C
dl · v around any closed path C in the fluid is quantized in units of
2π/M
κ = ∮
dl · ∇S = M C M ∆S C,
because ∆S C must be an integral multiple of 2π to make the condensate
wave function Ψ single valued
30

Substitute the representation Ψ = |Ψ|e iS into the time-dependent GP equation
• imaginary part gives the expected conservation of particles
∂n
∂t + ∇ · (nv) = 0
• in contrast the real part gives a generalized Bernoulli equation
1
2 Mv2 + V tr − 2
2M √ n ∇2√ n + gn + M ∂Φ
∂t = 0 (1)
• explicitly quantum-mechanical term involving 2 is called the “quantum
pressure” (but it really has dimension of an energy)
• usually negligible for low-lying collective modes of a trapped condensate in
the TF limit, since the density is slowly varying
• but the quantum pressure term is crucial for short-wavelength Bogoliubov
excitation spectrum in uniform gas
31

4.3 Bogoliubov spectrum: linearized hydrodynamics for uniform Bose gas
For uniform Bose gas, study small perturbations n ′ in density and Φ ′ in velocity
potential: n = n 0 + n ′ and Φ = Φ 0 + Φ ′
• equation of continuity becomes
∂n ′
∂t + n 0∇ 2 Φ ′ = 0
• similarly, perturbed Bernoulli equation becomes
− 2 ∇ 2 n ′
4n 0
+ gn ′ + M ∂Φ′
∂t = 0
• these coupled Bogoliubov equations for density perturbations n ′ and velocity
perturbations Φ ′ have celebrated plane-wave solutions
32

• for plane-waves ∝ exp[i(k · r − ωt)], Bogoliobov dispersion relation is
Ek 2 = gn 0 2 k 2 ( 2
M + k 2 ) 2
= 2gn 0 ɛ 0 k + ( )
ɛ 0 2
k
2M
(k)/2 (kHz)
14
12
10
8
6
4
2
2 R -1
0
0 1 2 3 4
k
• solid line is Bogoliubov spectrum and dashed line is free particle spectrum
• black dots are experimental values determined from the dynamic structure
factor S(k, ω) measured by two-photon Bragg scattering
33

• for long wavelengths (k → 0), get linear spectrum E k ≈ sk with speed of
sound s = √ n 0 g/M
• need g > 0 for stable sound waves, namely repulsive interactions
• for short wavelengths, in contrast, spectrum is quadratic (free particle) with
E k ≈ 2 k 2 /2M
• crossover between two regimes is at k cr ≈ 1/ξ = √ 2Mn 0 g/
• if g → 0 (no repulsive interaction), then k cr → 0, and spectrum reduces to
free particle for all k
• Landau critical velocity here becomes v c = s = √ n 0 g/M
• note that ideal gas (g = 0) has v c = 0, so ideal uniform gas has BEC but is
not true superfluid
34

4.4 linearized hydrodynamic equations for collective modes in trap
Trapped static condensate with nonuniform density n 0 (r) has small-amplitude
collective oscillations
• assume n = n 0 + n ′ and v ′ = ∇Φ ′
• combination of linearized conservation equation and linearized Bernoulli
equation leads to
∂ 2 n ′
∂t 2
= ∇ · [
n0 (r)g
M ∇n′ ]
• this is generalized wave equation in inhomogeneous medium characterized
by nonuniform density n 0 (r)
• note that gn 0 (r)/M is local squared speed of sound s 2 (r)
• hence rewrite previous equation as
∂ 2 n ′
∂t 2 = ∇ · [s 2 (r)∇n ′]
• important in all experimental studies of collective modes
35

4.5 effect of attractive interactions for trapped condensates
Effective interaction is repulsive for 4 He because of the weak van der Waals
attraction and the strong hard-core repulsion, leading to positive s-wave
scattering length
• situation is different for alkali-metals like Li, Na, and Rb because of large
atomic polarizability
• note alkali metal atoms have odd Z, and total number of fermions is 2Z+N,
where N is neutron number
• bosonic species require even N, leading to odd total atomic number
A = Z + N such as 7 Li
• unlike He, these atoms have many bound states
• sign of scattering length determined by relative position of last bound state
36

• 7 Li has large negative scattering length a ≈ −55.4 nm, whereas 87 Rb has
a ≈ 5.77 nm
• 7 Li BEC would be unstable in bulk, but positive trap kinetic energy allows
for small condensates with small negative interaction energy
• for quantitative analysis, assume a spherical trap with frequency ω 0 ,
oscillator length d 0 , and Gaussian trial function
)
Ψ(r) ∝ exp
(− r2
2β 2 d 2 ,
0
where β is variational parameter (condensate radius is βd 0 )
• GP ground-state energy is readily evaluated
[
E g (β) = 1 ( ) √
2 Nω 3 1 2
0
2 β + 2 β2 +
π
Na
d 0
1
β 3 ]
• here three terms arise from kinetic energy, trap energy, and interaction
energy, respectively
• for positive a, expect expanded condensate with β > 1
• but for negative a, expect reduced condensate with β < 1
37

E g (β) = 1 2 Nω 0
[
3
2
( 1
β 2 + β2 )
+
√
2
π
Na
d 0
1
β 3 ]
It is clear from inspection that E g (β) becomes large for β → ∞
• detailed behavior for small β depends crucially on sign of a
• for noninteracting gas (a = 0), minimum occurs for β = 1, which is familiar
oscillator solution
• for positive a, E g (β) diverges to ∞ for small β, expanding the condensate
• for negative a, however, E g (β) diverges to −∞ for small β, so any solution
is globally unstable
38

• for small and moderate negative values of a, the energy E g (β) has a local
minimum, and system remains locally stable, as seen in upper curve (a)
• straightforward analysis shows that variational local minimum disappears
at critical value N c |a|/d 0 ≈ 0.671 (middle curve b)
• corresponding critical condensate radius is reduced by β c = 5 −1/4 ≈ 0.669
• numerical study of GP equation yields more accurate value N c |a|/d 0
0.575
• this differs from variational estimate by ≈ 17%
• lower curve (c) has lost even local stability
≈
39

5. Optical lattices and reduced dimensionality
Response of an atom to electromagnetic fields is an old subject that has many
applications for BECs
• place an electric dipole p in an electric field E
• interaction energy is U int = −p · E
• minus sign means that p orients itself along the direction of E for lowest
energy configuration
• this holds for a fixed dipole moment p, but typically atom is polarizable
and acquires p induced by presence of E
• the dipole moment is proportional to E, with p = αɛ 0 E in SI units and α
is atomic polarizability
• simple classical model is conducting sphere of radius b, where α = 4πb 3
(three times the volume)
40

• for polarizable atom, interaction energy becomes U int = − 1 2 αɛ 0|E| 2
• if electric field is static and nonuniform, then interaction energy also becomes
spatially varying U int (r) = − 1 2 αɛ 0|E(r)| 2
• hence atom experiences a force that seeks to minimize the interaction energy,
placing the atom at the local maximum of |E| 2 because static polarizability
α is positive
• same result holds for low and intermediate frequencies of the applied electric
field
• situation eventually changes because of induced transitions among atomic
energy levels
41

• Drude model of a bound electron with resonant frequency ω 0 and damping
time τ gives simple and familiar result for frequency-dependent
polarizability
α(0)ω0
2 α(ω) =
ω0 2 − ω2 − iω/τ
where ω is frequency of applied field and α(0) is (positive) static
polarizability
• note that Re α(ω) changes sign from positive to negative for ω > ω 0
• combination of these results yields frequency-dependent complex interaction
energy
U int (r, ω) = − 1 α(0)ɛ 0 ω0
2
2 ω0 2 − |E(r, ω2 ω)|2
− iω/τ
• this result is often called the ac Stark effect
42

• these ideas apply directly to optical trapping of an atom by a focused laser
beam
• if the frequency of laser is below the resonance (“red detuning”), then
Re α(ω) is positive and atom is attracted to maximum of the intensity
• but if the frequency of laser is above the resonance (“blue detuning”), then
Re α(ω) is negative and atom is repelled from maximum of the intensity
• in practice, use far-off-resonance lasers to avoid dissipation
• for alkali-metal atoms (Li, Na, K, Rb, Cs), the lowest s to p transition of
the single valence electron dominates, as in familiar yellow lines of Na
• Ketterle’s group (MIT) successfully trapped a pre-existing 23 Na BEC in a
focused infrared laser beam (1998)
• unlike magnetic trap that can confine only certain magnetic hyperfine states,
laser trap confines all magnetic states (I’ll come back to this topic later)
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Laser trapping has important application in what are known as “optical lattices”
• consider laser beam propagating along the z axis with electric field
E(z, t) = E 0 cos(kz − ωt)
• reflect the beam off a mirror at z = 0 and form a standing wave
E 0 [cos(kz − ωt) − cos(kz + ωt)] = 2E 0 sin kz sin ωt
• corresponding intensity is |E(z, t)| 2 = 4|E 0 | 2 sin 2 kz sin 2 ωt
• this periodic array has spatial period 1 2λ, where λ = 2π/k is wavelength of
laser beam
• red detuning traps atoms at maxima; blue detuning traps atoms at nodes
• for either sign of detuning, atoms assume a periodic array in this
one-dimensional lattice
• wide application in study of cold dilute atoms
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• combination of more laser beams can lead to two-dimensional array of onedimensional
tubes and three-dimensional array of zero-dimensional dots
• strength of laser field determines the height of the barrier for tunneling
between adjacent sites
• has been used to study transition from superfluid to Mott insulator in a
three-dimensional optical lattice
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• interesting experiment at Stanford (1998) trapped atoms in a vertical
one-dimensional lattice
• adjacent sites have different gravitational potential Mgλ/2
• associated interlayer tunneling produced ac Josephson current at frequency
ω J = Mgλ/(2)
• led to observable periodic particle flux from lower end of the array at this
Josephson frequency
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