Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Statistical Mechanics and Dynamics
of Multicomponent Quantum Gases
Austen Lamacraft
October 26, 2011
6
2010 Fellows
9
Scialog 2010
19
Cottrell Schola
26
Additional Pro
28
Advisory Comm
30
Financials
32
Officers, Board
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Boson pairing and unusual criticality
• Yifei Shi, Austen Lamacraft and Paul Fendley
arXiv:1108.5744
• Also Andrew James and Austen Lamacraft
Phys. Rev. Lett. 106, 140402 (2011)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Why study phase transitions?
Magnetic Phase Diagram
magnetic field
first order=
jump in magnetization at zero field
temperature
near critical point
jump~ (Tc − T )β
Concepts in Critical Phenomena: Amsterdam Nov. 27 2006
€
critical point
jump=0
slide
7
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Why study phase transitions?
Magnetic Phase Diagram
magnetic field
first order=
jump in magnetization at zero field
temperature
near critical point
jump~ (Tc − T )β
Concepts in Critical Phenomena: Amsterdam Nov. 27 2006
€
critical point
jump=0
slide
7
Amazingly, there is a sense in which the two problems are the same.
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Universality: pick your battles
• Forget phase diagram and focus on phase transitions
• Continuous transitions characterized by critical exponents
• M ∝ (T − Tc )β , C ∝ (T − Tc )−α
• At T = Tc correlation functions hM(x)M(y)i =
• Behavior is characteristic of scale invariance
C
|x−y|2∆
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
From scale invariance to simple models
(Scale invariance)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
From scale invariance to simple models
Studying simple models is a really good idea!
Z=
X
e −βHIsing [σ] ,
{σ}
βHIsing [σ] = −J
σi = ±1
X
<ij>
σi σj
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Universality classes
Characterized by broken symmetry of order parameter1
e.g. XY model
βHXY [θ] = −J
1
Also spatial dimension
X
<ij>
cos(θi − θj )
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Examples of 3D universality classes
Ising class
XY class
• Liquid-gas, binary mixtures,
• Easy plane magnets,
• C ∝ (T − Tc )−0.11
• C ∝ (T − Tc )−0.01
uniaxial magnetic systems,
micellization,. . .
• ξ ∝ (T − Tc )−0.63
λ-transition in 4 He,
superconductors, BEC,. . .
• ξ ∝ (T − Tc )−0.67
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Examples of 3D universality classes
Ising class
XY class
• Liquid-gas, binary mixtures,
• Easy plane magnets,
• C ∝ (T − Tc )−0.11
• C ∝ (T − Tc )−0.01
uniaxial magnetic systems,
micellization,. . .
• ξ ∝ (T − Tc )−0.63
Beautiful classification...
λ-transition in 4 He,
superconductors, BEC,. . .
• ξ ∝ (T − Tc )−0.67
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Examples of 3D universality classes
Ising class
XY class
• Liquid-gas, binary mixtures,
• Easy plane magnets,
• C ∝ (T − Tc )−0.11
• C ∝ (T − Tc )−0.01
uniaxial magnetic systems,
micellization,. . .
• ξ ∝ (T − Tc )−0.63
λ-transition in 4 He,
superconductors, BEC,. . .
• ξ ∝ (T − Tc )−0.67
Beautiful classification...
Let’s try to break it!
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Bose condensates and superfluids are XY systems
Bose condensation: macroscopic occupancy of single-particle state
• Wavefunction Ψ(r) is condensate order parameter
• Free to choose phase: XY symmetry breaking
~
• Superfluid velocity v = m
∇θ
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Vortices give a twist in 2D
Quantized vortices: phase increases by 2π × q (Integer q)
v=
~
~ êθ
∇θ =
m
m r
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Vortices give a twist in 2D
Logarithmic interaction between vortices of charge q1 , q2
Vq1 ,q2 (x − y) = −q1 q2
πn~2
ln |x − y|
2m
2D density n
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
The Kosterlitz–Thouless transition
Consider contribution to the partition function from a q = ±1 pair
Zpair =
=
Z
Z
dxdy exp [−βV1,−1 (x − y)]
dxdy
|x − y|
βπn~2
2m
Pair found at separation r with probability ∝ r 1−
• Pair dissociates for
kB T > kB TKT ≡
• Pair bound for
T < TKT
πn~2
2m
βπn~2
2m
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Pair condensates: an Ising transition in an XY system
Two Bose condensates with a definite phase
|∆θi =
N
X
n=0
Detect phase by interference
e in∆θ |niL |N − niR
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Pair condensates: an Ising transition in an XY system
Take a condensate of molecules and split it
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Pair condensates: an Ising transition in an XY system
Dissociate pairs
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Pair condensates: an Ising transition in an XY system
What is the resulting state?
Superposition involves only even numbers of atoms
N/2
X
n=0
|2niL |N − 2niR =
N
N
n=0
n=0
1X
1X
|niL |N − niR +
(−1)N |niL |N − niR
2
2
= |∆θ = 0i + |∆θ = πi
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Pair condensates: an Ising transition in an XY system
Pair condensate −→ condensate breaks an Ising symmetry!2
2
Romans et al (2004), Radzihovsky et al. (2004)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
A simple model
HGXY = −
X
hiji
[(1 − ∆) cos(θi − θj ) + ∆ cos (2θi − 2θj )]
Korshunov (1985), Lee & Grinstein (1985)
• ∆ = 0 is usual XY; ∆ = 1 is π-periodic XY
• ∆ < 1 has metastable minimum
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Schematic phase diagram
What is the nature of phase transition along dotted line?
action parameter !, which is defined below!. Second
which have fourfold-rotational symmetry, they can form a
Statistical Mechanics of Boson
Pairssymmetric tetratic phase.
of Spinor
Condensates
phase diagramDynamics
is quite similar
to that
found by Mart
fourfold
Ratón et al. for particles with excluded-volume repu
An interesting question is what happens if the symmetry
of the particles is slightly broken. Will the symmetry of the
This similarity indicates that the phase behavior is ge
phase also be broken, or can the particles still form a higherfor particles with almost-fourfold symmetry, independe
symmetry phase? For example, we can consider particles
the specific interparticle interaction.
with approximate fourfold-rotational symmetry that is
The plan of this paper is as follows. In Sec. II, we pr
slightly broken down to twofold, as in Fig. 1. Can these
our model and calculate the mean-field free energy. We
particles still form a tetratic phase, or will they only form a
examine the phase behavior and calculate the phase dia
less symmetric nematic phase?
Recently, several experimental
theoretical
studiesP H Y S I C A L R E V I E W L E T T E R S
VOLUME 92, and
N UMBER
16
θ2
have addressed this problem. Narayan et al. #1$ performed
with c ! 1 $ 2%d$2&=2 , a1 ! 2%d$4
experiments on a vibrated-rod monolayer
and found that
ν
a2 ! 2)d a1 , b1 ! 1, b2 ! 2)%d
twofold symmetric rods can form a fourfold symmetric tet%h2 =2'm1 kB &"n=c(%d=2&'2=d the tr
ratic phase over some range of packing fraction and aspect
at the tricritical point ! ! 0.
ratio. Zhao et al. #2$ studied experimentally the ASF
phase behavIn the neighborhood of Tc1 the ‘
ior of colloidal rectangles and found what they called an
^ 2 decouples at low energies
Normal
T
field
c0
almost tetratic phase. Donev et al. #3$ simulated the phase
grated
out of the partition function
behavior of a hard-rectangle system withν an
aspect
ratio
of
2,
T
Tc2 Tc1
c(n,T)
tive quartic coupling g1 ! g! 1 ( g1
and showed they form a tetratic phase. Another simulation
N-ASF
transition is identical to tha
by Triplett et al. #4$ showed similar results. In further theoθ1 as g!
MSF
nent system, continuous so long
c(n,0)
retical work, Martínez-Ratón etνal.
#5,6$ developed
a densityN-ASF
functional theory to study the effectPRL
of particle
P H Y Stransition
I C A LtakesRplace
E V atI EvaW
106,geometry
015303on(2011)
(the N phase is simply a vacuum of
phase transitions. They found a range of the phase diagram
nontrivial, is exactly soluble [7],
3 as
in which the tetratic phase
long
shape is
FIG.can
2. exist,
Phase as
diagram
forthe
a bosonic
atom-molecule
mixture
FIG. 1.
Schematic illustration of an interacting particle s
An old problem with many guises
Korshunov (1985)
Lee & Grinstein (1985)
Sluckin & Ziman (1988)
Carpenter & Chalker (1989)
buildup of atomic superfluid (with
Romans et al (2004)
pha
in d ! 3, expressed in terms of detuning ! and
T.
in temperature
the tetratic phase.
shape
particles
indicates
th
nThe !
j" of
j the
, j#j
, with
meansym
and a quantum
It illustrates a finite-T tricritical point at T rotational
symmetry offor
thedinteraction
is$broken
down
from
fo
>
2,
and
!
d=4
for
d
<
2)
Radzihovsky et al.*jvs@lci.kent.edu
(2004) critical point at ! %n; 0&. In the weakly interacting
limit
appropha
to twofold.
At T ! 0 the N-ASF transition
priate to experiments the ratio T =T ! 2 .
lec
d-dimensional XY universality class
Geng & Selinger 1539-3755/2009/80"1!/011707"6!
(2009)
2
011707-1
©2009 The
American
PhysicalofS
Similarly,
in the
neighborhood
wh
or bound into molecules) to N, and detuning ! is related
the resulting N-MSF transitions are
James & AL (2011)
spof
to the energy of a molecule at rest, which can be experisality classes discussed above. The
mentally controlled with a magnetic field. In the dilute
illustrated in Fig. 2. In 3D itpro
ex
Ejima et al. (2011)
gas limit g ; g ; g are proportional to the two-body
singularity at the tricritical pointlon
and
c0
c
1
c2
10
2
2$
5=3
1/U
1
c1
10
1
2
12
1 atom-molecule, and molecules-wave atom-atom,
molecule scattering lengths, respectively, and " characterizes the coherent atom-molecule interconversion rate,
encoding that molecules are composed of two atoms [5].
The mean-field phase diagram as a function of
#1;2 and $ ! 1=kB0T can be worked out by minimiz-10
-8
-6
-4 action
-2
ing Rthe imaginary-time
(%) coherent-state
R
P
S ! $h! d% dd x 2 " &# h@
! % & $ H# % &# ; & &'. Simple
totes to the single-component BEC
pha
To study the MSF phase, we sepa
into classical condensate fields "&0O
MSF) and fluctuations about it. With
the
to expand H^ # to second order in flu
siti
leads to H^ # ! E%0& %"20 & $ H^ %2& with
0
2#
4
Z
g2
%0&
d
# ^
4
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
How things change on the Ising critical line
Redo KT argument accounting for string
Domain wall terminates at disorder operator µ(x)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
How things change on the Ising critical line
Disorder operators dual to σ(x) of Ising model
hσ(x)σ(y)i = hµ(x)µ(y)i =
Zpair =
=
Z
Z
1
|x − y|1/4
dxdy hµ(x)µ(y)i exp −βV1/2,−1/2 (x − y)
dxdy
1
|x − y| 4 +
βπn~2
8m
Dissociation at higher temperatures than for ‘free’ half vortices
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
How things change on the Ising critical line
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Numerical simulation using worm algorithm
Z=
YZ
c
π
−π
dθc Y
w (θa − θb ),
2π
habi
w (θ) is written in terms of the Villain potential wV (θ)
w (θ) ≡ wV (θ) + e −K wV (θ − π)
∞
∞
X
X
J∗ 2
2
J
e inθ e − 2 n
wV (θ) ≡
e − 2 (θ+2πp) ∝
n=−∞
p=−∞
VHΘijL
0.5
1
-0.5
-1.0
2
3
4
5
6
Θij
J∗ = J −1
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Numerical simulation using worm algorithm
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Boson pairing and unusual criticality: summary
• We found an Ising transition where you’d expect an XY (KT)
transition!
• The same phenomenon in 3D would be truly remarkable (true
long-range XY order developing at an Ising transition)
Work underway
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Spin 1 microcondensates
AL, Phys. Rev. A 83, 033605 (2011)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Manifesto
• The order parameter of a BEC is a macroscopic variable
• For a BEC with spin, it should be some kind of pendulum
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
A very simple system
(Loading Asteroids)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Periodic boundary conditions = motion on a torus
Quasiperiodicity: asteroid always hits spaceship!
(Loading torus)
Statistical Mechanics of Boson Pairs
Description of phase space
Phase space is a product T 2 × R2
Not as special as it seems!
Dynamics of Spinor Condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Action angle variables
Hamilton’s equations for H = E (p1 , p2 )
∂H
= v1
∂p1
∂H
ẋ2 =
= v2
∂p2
ẋ1 =
θi =
2πxi
Li
are angles on the torus obeying
θi =
2πvi
t + consti
Li
Simplest example of action (p1 L1 , p2 L2 ) angle (θ1 , θ2 ) variables
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Quantizing the system
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Other choices are possible
a(p1 L1 ) + b(p2 L2 )
(1, 0)
(0, 1)
a, b ∈ Z
(1, 1)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Different actions = different unit cells
I1
2 1
p1 L1
=
I2
1 1
p2 L2
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
The Mexican hat
Consider the Hamiltonian for two dimensional motion
H=
p2 r2
− + r4
2
2
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
A natural approach – separate angular motion
p2 r2
− + r4
2
2
2
p
`2
r2
= r + 2−
+ r4
2
2r
2
px x + py y
` = xpy − ypx
pr =
r
2
`2
Defines potential for radial motion V (r ) = 2r 2 − r2 + r 4
H=
VHrL
100
80
60
40
20
0.5
1.0
1.5
2.0
2.5
3.0
r
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Phase plane for reduced motion
VHrL
100
80
60
40
20
0.5
1.0
1.5
2.0
2.5
3.0
r
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Phase space of integrable systems
This is an integrable system:
• 2 degrees of freedom and two integrals of motion (energy E,
angular momentum `). Motion lies on two dimension
submanifold of four dimensional phase space.
• Closed trajectories for reduced motion in (r , pr ) plane, and
angle θ in the (real) plane is cyclic coordinate
ṗθ = 0 −→ pθ = `, const
∂H
`
θ̇ = −
=− 2
∂`
r
(Note that θ motion is not trivial)
• The motion at fixed (E , `) lies on a torus.
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Motion on the torus
Hradial =
pr2
`2
r2
+ 2−
+ r4
2
2r
2
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Motion on the torus
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Quasiperiodic motion
(Loading hat)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Action angle variables, and the Liouville–Arnold theorem
Liouville–Arnold theorem
• For a system integrable in the above sense, can find N
conjugate pairs of action-angle variables (Ii , φi ), such that
evolution of angles is trivial φi = ωt + φi,0 φ̇i = ∂H
∂Ii
• Submanifold of phase space at fixed {Ii } is N-Torus T N
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
At a pinch...
In the (`, E ) plane, there is a special point (0, 0) where torus
pinches
EFSTATHIOU, JOYEUX, AND SADOVSKIÍ
Ma
in
co
do
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
At a pinch...
In the (`, E ) plane, there is a special point (0, 0) where torus
pinches
EFSTATHIOU, JOYEUX, AND SADOVSKIÍ
Ma
in
co
do
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Rotation angle in the Mexican hat
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Rotation angle
φ
4π
T
−Φ
2π
T
Φ
2T t
FIG. 13. (Color online) (Left) As we circle the origin in Sz ,
H space for q̃ < 0 the period lattice is deformed continuously,
returning to its original form, but after shifting the lattice
N
ri
co
sc
R
th
je
F
st
ro
th
φ
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Hamiltonian monodromy in a nutshell
Rotation angle increases by 2π as we circle the pinched torus
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Some history
• 1673 Huygens finds period of spherical pendulum (20 years
before Newton!)
• Classical mechanics: Newton, Euler, .... Hamilton ...
• 1980 Duistermaat discovers Hamiltonian monodromy, with
the spherical pendulum a prominent example.
• 1988 Cushman and Duistermaat discuss signatures in
quantum mechanics (no time today...)
• 1997 Molecular physicists become interested. Candidate
systems are flexible triatomic molecules HAB, such as HCN,
HCP, HClO.
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
From another cold atom lab...
PRL 103, 034301 (2009)
PHYSICAL REVIEW LETTERS
week ending
17 JULY 2009
Experimental Demonstration of Classical Hamiltonian Monodromy
in the 1:1:2 Resonant Elastic Pendulum
N. J. Fitch,1 C. A. Weidner,1 L. P. Parazzoli,1 H. R. Dullin,2 and H. J. Lewandowski1
2
1
JILA and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia
(Received 7 April 2009; published 15 July 2009)
The 1:1:2 resonant elastic pendulum is a simple classical system that displays the phenomenon known
as Hamiltonian monodromy. With suitable initial conditions, the system oscillates between nearly pure
springing and nearly pure elliptical-swinging motions, with sequential major axes displaying a stepwise
precession. The physical consequence of monodromy is that this stepwise precession is given by a smooth
but multivalued function of the constants of motion. We experimentally explore this multivalued behavior.
To our knowledge, this is the first experimental demonstration of classical monodromy.
DOI: 10.1103/PhysRevLett.103.034301
PACS numbers: 45.05.+x, 02.30.Ik, 45.50.!j
week e
intriguing
After more than 300 years since the formulation
ofS I103,
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S ISC Athat
L Rthe
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I E W L E T T17
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A034301
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PRL
PRL 103, 034301 (2009)
behavior of monodromy may be a common feature of
Newton’s laws of motion, one would expect that a system
∼ 5 thresholds [17].
dynamics near chemical isomerization
as simple as a mass on a spring would have been fully
enclosing
loop
($3)
motion are
A quantum analog of the resonant elastic pendulum
understood for some time. In fact, an in-depth investigation
former loop does not c
under consideration here (Fig. 1) is the
Fermi
of even a subset of its possible dynamics produces a
returning
H%
H2 resonance
Lzto the initial
#¼
;
$It¼
:
in the CO2 molecule, whose monodromatic
number of surprises. Chief among these is a phenomena
features
have
is a simple
distin
3=2
H2
"H[14,16].
2
stepwise precession
ad
known as Hamiltonian monodromy, which was introduced
been thoroughly investigated theoretically
Despite
behavior
by Duistermaat in 1980 as a topological obstruction to the
the large number of systems in which monodromy
is along a loop
With this scaling, the thread pierces
the 2D lem
the key to our first
existence of global action-angle variables [1]. In the resotheoretically predicted, there have been no previousYet,
clas∼3
ð$;and
#Þ ¼
ð0;a0Þ,
producing
singularity
[Fig.
2(a)].
monodromy
is our
abi
nant elastic pendulum, monodromy has easily observable
sical experiments
only
single
quantumaexperiment
is developing
responsiblea more
for the
existence
of monod
resonant elastic
pendu
physical consequences. Specifically, the observed stepwise
[18] of which wesingularity
are aware. In
heuristic
be p
The presence
the monodromy-producing
precession of the elliptical swinging major axis is given by
understanding of monodromy
in of
quantum
systems,sical
it isexample cansingu
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Outline
Statistical Mechanics of Boson Pairs
Phase transitions and universality
Boson pair condensates
Interplay of strings and vortices
Dynamics of Spinor Condensates
Geometry of phase space
Mechanics of the Mexican hat
Connection to spinor condensates
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Spin 1 Bose condensates
Bose condensation: macroscopic occupancy of single-particle state
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Spin 1 Bose condensates
Bose condensation: macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Spin 1 Bose condensates
Bose condensation: macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
A: Macroscopic occupancy and interaction of different states
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Spin 1 Bose condensates
Bose condensation: macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
A: Macroscopic occupancy and interaction of different states
Bosons just oscillator quanta
Macroscopic occupancy =⇒ Oscillators are (close to) classical
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Spin-1 gas in the single mode approximation
HSMA =
N =
1
X
c0
c2
: N2 : +
: S · S : +HZ .
2V
2V
∗
am
am
X
∗
am
Smm0 am0
m,m0
m=−1
Smn spin-1 matrices, and HZ =
h≡
S=
P
∗
m am
pm + qm2 am
1 2
∗
∗
Sz + 2 a1∗ a−1
(a0 )2 + (a0∗ )2 a1 a−1 + 2a0∗ a0 a1∗ a1 + a−1
a−1
2
2N
q̃ ∗
∗
+
a1 a1 + a−1
a−1 .
N
q̃ = q/c2 n. h is energy per particle in units of c2 n
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Classical mechanics of the spin-1 gas
h≡
1 2
∗
∗
Sz + 2 a1∗ a−1
(a0 )2 + (a0∗ )2 a1 a−1 + 2a0∗ a0 a1∗ a1 + a−1
a−1
2
2N
q̃ ∗
∗
a1 a1 + a−1
a−1 .
+
N
2 | 0⟩ → |+⟩ + | –⟩
a
+1
0
|+⟩ + | –⟩ → 2 | 0⟩
–1
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Classical mechanics of the spin-1 gas
There are three conserved quantities
1. The energy Nh
2. The angular momentum S z
3. The particle number N
For a range of parameters this systems displays monodromy!
Mechanics of Boson Pairs
ns.Statistical
In the case
lable collision
140 ms
ith total spin
ttering lengths
re, and, in the
mF
ergy including
s 12,13, where
1.0
c
een two atoms
)/3m, where h̄
and a0,2 are the
0.8
nnels. For 87 Rb
engths a0,2 are
n-field energy
0.6
pical densities
field, c0 n, and
onetheless, the
0.4
-negligible and
res depending
87
Rb (refs 18,
0.2
a (refs 15,34).
ety of coherent
pinor mixing,
0.0
0.0
formation and
e with a large
field treatment
overned by the
. For an F = 1
Dynamics of Spinor Condensates
Single mode
dynamics
in experiment
1
0
–1
Period (s)
0.25
0.20
0.15
0.10
Spin populations
0.2
0.1
0.2
0.3
Time (s)
0.4
0.4
ρo
0.6
0.8
0.5
Chapman group (GA Tech) with
0.6
87 Rb
Figure 1 Coherent spin mixing of spin-1 Bose condensate in an optical trap.
23
87
Lett
group (NIST)
The F = 1, m F = 1,0,−1 spinAlso
states of
Rb condensates
confined in with
an optical Na
(2005)
(2007)
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Rotation angle in spinor condensates
Monitor evolution of perpendicular magnetization
Can be measured by Faraday rotation spectroscopy
Statistical Mechanics of Boson Pairs
Dynamics of Spinor Condensates
Summary
In multicomponent quantum gases find unusual phase transitions
Yifei Shi, AL & Paul Fendley arXiv:1108.5744
Andrew James & AL PRL 106, 140402 (2011)
...and unusual dynamics
AL, Phys. Rev. A 83, 033605 (2011)