Spinor Bose Microcondensates
and Hamiltonian monodromy
Austen Lamacraft
University of Virginia
August 17, 2011, NIST
Phys. Rev. A 83, 033605 (2011)
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
1 / 56
Manifesto
The order parameter of a BEC is a macroscopic variable
For an spinor BEC, it should correspond to some kind of pendulum
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
2 / 56
Outline
1
Spin 1 Microcondensates
2
Some features of integrable classical motion
3
Classical mechanics of the Mexican hat
4
Interlude: Hyperbolic spins
5
Monodromy in spin-1 Bose condensates
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
3 / 56
Outline
1
Spin 1 Microcondensates
2
Some features of integrable classical motion
3
Classical mechanics of the Mexican hat
4
Interlude: Hyperbolic spins
5
Monodromy in spin-1 Bose condensates
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
4 / 56
What is a Microcondensate?1
Convenient shorthand for all atoms are in the same spatial state
System size L ξd,c , the healing lengths for density and spin
If spinor condensate ←→ magnet
then microcondensate ←→ single domain
1
Coinage due to Fabrice Gerbier and Jean Dalibard, I believe.
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
5 / 56
Microcondensate Hamiltonian – Just two ingredients
Contact interactions
Hint =
X
i<j
=
X
i<j
δ(ri − rj ) [g0 P0 + g2 P2 ]
δ(ri − rj ) [c0 + c2 Si · Sj ]
c0 = (g0 + 2g2 ) /3
Austen Lamacraft (University of Virginia)
c2 = (g2 − g0 ) /3
Spinor Bose Microcondensates
August 17, 2011
6 / 56
Microcondensate Hamiltonian – Just two ingredients
Zeeman energy of atom with z-component m = −1, 0, 1: both linear
2
(∝
3 m) and quadratic (∝ m ) contributions
DEUTSCHE PHYSIKALISCHE GESELLSC
39
K atoms in a magnetic field. Levels are lab
using zero-field quantum numbers.
Figure 1. Energy levels of
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
6 / 56
Microcondensate Hamiltonian
For N particles is
HSMA =
N̂ =
1
X
e0
e2
: N̂ 2 : +
: Ŝ·Ŝ : +HZ
2N
2N
∗
am
am
Ŝ =
∗
am
Smm0 am0 ,
m,m0
m=−1
HZ =
1
X
m=−1
Austen Lamacraft (University of Virginia)
X
∗
am
pm + qm2 am
Spinor Bose Microcondensates
August 17, 2011
7 / 56
In gory detail...
Dropping the density-density and linear Zeeman pieces, we have
Hred =
e2 2
∗
Sz + 2 a1∗ a−1
(a0 )2 + (a0∗ )2 a1 a−1 +
2N
1
∗
∗
∗
∗
a−1 .
a1 a1 + a−1 a−1 + q a1∗ a−1 + a−1
2 a0 a0 −
2
... and probably you are none the wiser!
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
8 / 56
Related models
Dicke–Jaynes–Cummings
HDJC = 2Sz + ωb † b + g (bS+ + b † S− )
Lipkin–Meshkov–Glick (shape transitions in nuclei, magnetic
molecues, Bose–Hubbard dimers...)
HLMG = −
1
γx Sx2 + γy Sy2 − hSz
N
Interacting Boson Model of the nucleus
HIBM = s ns +d nd +
X
l1 ,l2 ,l10 ,l20 ,L
Austen Lamacraft (University of Virginia)
L L
vlL1 l2 l 0 l 0 bl†1 × bl†2 · b̃ l10 × b̃ l20
1 2
Spinor Bose Microcondensates
August 17, 2011
9 / 56
Intriguing features in the spectrum...
HSMA =
e2 > 0, q > 0 (e.g.
Austen Lamacraft (University of Virginia)
23
e2
: Ŝ · Ŝ : +q [N1 + N−1 ]
2N
Na)
e2 > 0, q < 0
Spinor Bose Microcondensates
August 17, 2011
10 / 56
These coupled equations are nonlinear Josephson-type
3(b)
equations and point to the equivalency of spin mix0 ms
ing in a spin-1 condensate to Josephson systems re(a)in superconductors [27] and other superfluids
alized
[11, 14, 39, 40, 41, 42, 43, 44]. The non-linearity of these
70 ms
equations provides a rich manifold of dynamical trajectories that can be accessed experimentally by choice of
0 components
+1 of the spin
-1
initial populations and phases
140 ms
and the strength of the applied magnetic field.
To investigate the coherent dynamics of this system, we
87
87
begin with Rb condensates created using an improved
mf 1
0
-1
version of the all-optical trapping technique we have pre1.0
(c)
viously reported [16, 19]. Using a dynamical compression
0.25
technique
(b) and just a single focused laser beam, conden0.20
sates with up
to
300,
000
atoms
are
created
after
2
s
of
0.8
0 ms
0.15
evaporative cooling. The condensates created in this op0.10
tical trap are generally in a mixture of all F = 1 spin
0.6
0.2 0.4 0.6 0.8
states and reveal complicated spatial domains. To create
!"
a well-characterized initial condition, we first prepare a
70
ms
0.4
condensate in the |F = 1, mF = −1" state by applying a
magnetic field gradient during the evaporative cooling.
To initiate spin dynamics, a coherent superposition of
0.2
spin states with non-equilibrium spin populations is cre140
ms
ated by applying a sequence of phase-coherent microwave
0.0
pulses tuned to F = 1 ↔ F = 2 transitions. Follow0.0
0.1
0.2
0.3
0.4
0.5
0.6
ing this state preparation, the condensate is allowed to
m
1
0
-1
f
Time
(s)
freely evolve in the optical trap. A typical evolution
is shown1.0
in Fig. 1c for an initial spin configuration of
(c)
ρ(1,0,−1) $ (0, 3/4, 1/4). Up to0.25
four distinct oscillations
FIG. 1: Coherent spin mixing of spin-1 Bose condensate in
an optical trap. Coherent spin mixing producing oscillations
are observed in this example before the spin populations
0.20
in the populations of the F = 1, mF = 0, ±1 spin states of
damp to0.8
a steady state. These oscillations demonstrate
87
Rb condensates confined in an optical trap starting from a
0.15
the coherence of the spin mixing
process.
superposition of condensate spin components at t = 0 that is
We have measured the spin0.10
oscillation frequency for
subsequently allowed to evolve freely. a) Schematic indicates
different0.6
initial spin populations. These
are shown 0.8 fundamental spin mixing process. b) Absorptive images of the
0.2 data
0.4
Austen
Lamacraft
of Virginia)
Spinor 0.6
Bose
August 17, 2011
11 / 56
in the
inset of(University
Fig. 1c. and
show good agreement
withMicrocondensates
Experiments in the single mode regime
Period (s)
lations
Rb
Period (s)
Spin populations
Chapman group (GA Tech, 2005) working with
Experiments in the single mode regime
(NIST,
P H Y SLett
I C A group
L REV
I E W L2009)
E T T Eworking
RS
23 Na
week ending
27 MARCH 2009
1.0
Faraday Signal (arb. units)
d state populations as a
edicted phase transition
spinor BEC.
ntaining up to 6 ! 109
cal dipole trap derived
nm is then loaded, folalization. A weak magg 6 s forced evaporation
1:5 ! 105 atoms in the
trap frequencies are
d the mean Thomasp up the magnetic field
the field gradient. The
3 and 60:7 "T with an
d uncertainties are estined statistical and sysstate with any desired m
atomic spin followed by
n mF state. The rf pulse
splitting, and its ampli-
with
B = 26 µT < Bc
0.8
B = 40 µT > Bc
(a)
(b)
0.6
0.4
0.2
0.0
0
20
40
60
80 100 0
Hold Time (ms)
20
40
60
80
100
Hold Time (ms)
FIG. 1 (color online). Faraday signal (proportional to hFx i2 )
taken from a single measurement for m ¼ 0 at two magnetic
fields 26 and 40 "T starting with #0 ¼ 0:5, $ ¼ 0. The solid
line is a fit with a damped sinusoid. The signals show (a) an
oscillating phase and (b) a running phase at B below and above
Bc , respectively, as evidenced by the signal reaching zero or not.
backaction
in the
experiments.
Austen Lamacraft (University of other
Virginia)
Spinor
Bosepresent
Microcondensates
In the absence
August 17, 2011
11 / 56
Dynamics in the classical limit
Gross-Pitaevskii dynamics2
i
da±1
∗
= e2 ±sz a±1 + (a∓1
+ a±1 )(1 − |a1 |2 − |a−1 |2 )−
dt
∗
(a1∗ a−1
+ a1 a−1 + |a1 |2 + |a−1 |2 )a±1 + qa±1 .
...can be solved in terms of elliptic functions, but what do you learn?
√
Lower case throughout means ‘per particle’ quantities e.g. a1 = a1 / N,
sz = Sz /N, etc.
2
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
12 / 56
Outline
1
Spin 1 Microcondensates
2
Some features of integrable classical motion
3
Classical mechanics of the Mexican hat
4
Interlude: Hyperbolic spins
5
Monodromy in spin-1 Bose condensates
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
13 / 56
A very simple system
(Loading Asteroids)
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
14 / 56
Periodic boundary conditions = motion on a torus
Quasiperiodicity: asteroid always hits spaceship!
(Loading torus)
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
15 / 56
Description of phase space
Phase space is a product T 2 × R2
Not as special as it seems!
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
16 / 56
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
This is the simplest example of action (p1 L1 , p2 L2 ) angle (θ1 , θ2 ) variables
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
17 / 56
Quantizing the system
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
18 / 56
Other choices are possible
a(p1 L1 ) + b(p2 L2 )
(1, 0)
Austen Lamacraft (University of Virginia)
a, b ∈ Z
(0, 1)
Spinor Bose Microcondensates
(1, 1)
August 17, 2011
19 / 56
Different actions = different unit cells
I1
2 1
p1 L1
=
I2
1 1
p2 L2
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
20 / 56
Outline
1
Spin 1 Microcondensates
2
Some features of integrable classical motion
3
Classical mechanics of the Mexican hat
4
Interlude: Hyperbolic spins
5
Monodromy in spin-1 Bose condensates
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
21 / 56
The Mexican hat
Consider the Hamiltonian for two dimensional motion
H=
Austen Lamacraft (University of Virginia)
p2 r2
− + r4
2
2
Spinor Bose Microcondensates
August 17, 2011
22 / 56
A natural approach – separate angular motion
p2 r2
− + r4
2
2
pr2
`2
r2
=
+ 2−
+ r4
2
2r
2
px x + py y
` = xpy − ypx
pr =
r
H=
Defines potential for radial motion V (r ) =
`2
2r 2
−
r2
2
+ r4
VHrL
100
80
60
40
20
0.5
Austen Lamacraft (University of Virginia)
1.0
1.5
2.0
2.5
Spinor Bose Microcondensates
3.0
r
August 17, 2011
23 / 56
Phase plane for reduced motion
VHrL
100
80
60
40
20
0.5
Austen Lamacraft (University of Virginia)
1.0
1.5
2.0
2.5
Spinor Bose Microcondensates
3.0
r
August 17, 2011
24 / 56
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
=− 2
θ̇ = −
∂`
r
(Note that θ motion is not trivial)
The motion at fixed (E , `) lies on a torus.
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
25 / 56
Motion on the torus
Hradial =
Austen Lamacraft (University of Virginia)
pr2
`2
r2
+ 2−
+ r4
2
2r
2
Spinor Bose Microcondensates
August 17, 2011
26 / 56
Motion on the torus
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
26 / 56
Quasiperiodic motion
(Loading hat)
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
27 / 56
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
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
28 / 56
At a pinch...
In the plane of (`, E ), there is a special point (0, 0) where the torus pinches
EFSTATHIOU, JOYEUX, AND SADOVSKIÍ
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
29 / 56
At a pinch...
In the plane of (`, E ), there is a special point (0, 0) where the torus pinches
EFSTATHIOU, JOYEUX, AND SADOVSKIÍ
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
29 / 56
Rotation angle
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
30 / 56
Rotation number in the Mexican hat
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
31 / 56
Hamiltonian monodromy in a nutshell
Rotation angle increases by 2π as we circumnavigate the pinched torus
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
32 / 56
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 (more later...)
1997 Molecular physicists become interested. Candidate systems are
flexible triatomic molecules HAB, such as HCN, HCP, HClO.
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
33 / 56
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,
HE
YR
S ISC Athat
L Rthe
E Vsingular
I E W L E T T17
E RJULY
S
PHY
CThere
A034301
L isRthus
E(2009)
V Ithe
EW
L E TPpossibility
T
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
;
$It¼
:
in the CO2 molecule, whose monodromatic
number of surprises. Chief among these is a phenomena
is a simple
distin
3=2features have
H2
"H
2
stepwise precession ad
known as Hamiltonian monodromy, which was introduced
been thoroughly investigated theoretically [14,16]. Despite
behavior
by Duistermaat in 1980 as a topological obstruction to the
the large number of systems in which monodromy
is along a loop
With this scaling, the thread pierces
the 2D lem
the key to our first
existence of global action-angle variables [1]. In the resotheoretically predicted, there have been no previousYet,
clas∼3
ð$;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
to build the intuition
ne
rotationtonumber
of the
integrable
approxim
a smooth, but multivalued function of the constants of
useful to have acauses
classicala example
guide one’s
intuition.
by
no
means
rare,
motion. This functional form results in loops of valuesFIG.
of 4. Thus
we designed
experiment
on abetween
readily realized
to beour
multivalued.
This
rotation
number correspondsin
Experimental
measurements
of the step angle
FIG.
3.
Measured
mass
positions
as
projected
onto
the
XY
foundly influences
Austen Lamacraft
(University
of Virginia)
Spinor
Bose
Microcondensates
August
17, 2011
34 /plane
56 thd
successive
swinging
motions
loops
#) the
space
are mapped
the constants
of motion
having differing overall
behavior,
classical
system
inaswhich
the(",consequences
of monodromy
step
size in
!%
of
stepwise-precessing
swing
Outline
1
Spin 1 Microcondensates
2
Some features of integrable classical motion
3
Classical mechanics of the Mexican hat
4
Interlude: Hyperbolic spins
5
Monodromy in spin-1 Bose condensates
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
35 / 56
Singular reduction – a different view of phase space
1
x± = √ (x ± iy )
2
1
p± = √ (px ± ipy )
2
[x± , p∓ ] = i
1
a± = √ (x± + ip± )
2
∗
∗
a± , a∓ = 0
a± , a± = i
Classical version of QM oscillator variables with ± circular polarization
|a+ |2 − |a− |2 = xpy − ypx = `
Note that under rotations in the plane a± → e ±iθ a±
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
36 / 56
Hyperbolic spins hidden in harmonic oscillators
Let’s consider the variables
1 ∗
∗
a+ a+ + a−
a−
2
∗ ∗
K+ = a+
a−
K − = a+ a− ,
K0 =
{K0 , K+ } = −iK+
{K0 , K− } = iK−
{K+ , K− } = 2iK0
which have the Casimir
2
KK
≡ K02 − K+ K− =
`2
4
Surface of constant ` is hyperboloid
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
37 / 56
Hyperbolic spins are invariant variables
1 ∗
∗
a+ a+ + a−
a−
2
∗ ∗
K+ = a+
a−
K − = a+ a− ,
K0 =
Invariant under rotations in the plane a± → e ±iθ a±
Useful to characterize reduced dynamics: motion on hyperboloid
H=
p2 r2
− + r4
2
2
1
= −(K+ + K− ) + K0 + (K+ + K− )
2
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
2
August 17, 2011
38 / 56
Geometric construction
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
39 / 56
Outline
1
Spin 1 Microcondensates
2
Some features of integrable classical motion
3
Classical mechanics of the Mexican hat
4
Interlude: Hyperbolic spins
5
Monodromy in spin-1 Bose condensates
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
40 / 56
Spin 1 Bose condensates
Bose condensation involves macroscopic occupancy of single-particle state
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
41 / 56
Spin 1 Bose condensates
Bose condensation involves macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
41 / 56
Spin 1 Bose condensates
Bose condensation involves macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
A: Allow macroscopic occupancy of different states, and their interaction
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
41 / 56
Spin 1 Bose condensates
Bose condensation involves macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
A: Allow macroscopic occupancy of different states, and their interaction
∗ , a , m = −1, 0, 1
Bosons just oscillator quanta: degrees of freedom are am
m
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
41 / 56
Spin 1 Bose condensates
Bose condensation involves macroscopic occupancy of single-particle state
Q: But what if Bosons have spin?
A: Allow macroscopic occupancy of different states, and their interaction
∗ , a , m = −1, 0, 1
Bosons just oscillator quanta: degrees of freedom are am
m
Macroscopic occupancy =⇒ Oscillators are (close to) classical
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
41 / 56
Spin-1 gas in the single mode approximation
Hred =
e2 2
∗
Sz + 2 a1∗ a−1
(a0 )2 + (a0∗ )2 a1 a−1 +
2N 1
∗
∗
∗
∗
a1 a1 + a−1 a−1 + q a1∗ a−1 + a−1
2 a0 a0 −
a−1 .
2
Again can introduce hyperbolic spins and arrive at reduced Hamiltonian
e2 2
Hclass =
Sz + 8 (N/2 − K0 ) (K0 − Kx ) + 2qK0 .
2N
where again
2
KK
≡ K02 − K+ K− =
`2
4
Surface of constant ` is hyperboloid
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
42 / 56
Geometric construction3
1
h = sz2 + 2(1 − 2k0 ) (k0 − kx ) + 2q̃k0
2
h = const. fixes a hyperbola with asymptotes k0 =
Austen 3Lamacraft (University of Virginia)
Spinor Bose Microcondensates
1
2,
q̃ ≡ q/e2
k0 = kx + q̃2
August 17, 2011
43 / 56
Geometric construction3
1
h = sz2 + 2(1 − 2k0 ) (k0 − kx ) + 2q̃k0
2
h = const. fixes a hyperbola with asymptotes k0 =
3
1
2,
q̃ ≡ q/e2
k0 = kx +
q̃
2
Lower case denotes per particle quantities
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
43 / 56
Two regimes
q̃ > 0
Separatrix at h = 12 sz2 + q̃
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
44 / 56
Two regimes
Separatrix in the spectrum
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
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Two regimes
q̃ > 0 : topologically distinct trajectories at sz = 0
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
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Two regimes
Signature in the spectrum. What’s happening at the origin?
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
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Two regimes
For q̃ < −2 origin moves through top boundary
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
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Evidence for separatrix in
23
Na (q̃ < 0)
week ending
17 AUGUST 2007
PHYSICAL REVIEW LETTERS
m equilibrium, the full
q. (1) are revealed. We
e rf pulse as above, but
cond time scales.
ibed by the Hamilton
2 @E
"_ !
:
@ @!0
(2)
Oscillation Period (ms)
40
Oscillation Amplitude
ization are obviated, as
each point as the differ! !" # !# . The meathe predictions, except
n region. There the syse ground state, and the
ce. We observe equilifrom 200 ms at high
ds, by which time atom
(a)
30
20 0.6
0.4
ρ
0
10
0.2
m
0
0.30
0
0.25
(b)
0
40
Time (ms)
80
0.20
0.15
0.10
0.05
e double-well ‘‘bosonic
0
50
0
10
20
30
40
and exhibits a regime of
Magnetic Field (µT)
near a critical field Bc ,
rmonic oscillations. At
FIG. 2 (color online). Period (a) and amplitude (b) Black
of spin
oscillations
of applied
magnetic field, following a
#$B
c&$1 # !(University
Austen
Lamacraft
of
Virginia) as a function
Spinor Bose
Microcondensates
c% !
0% "
et al. PRL 2007
August 17, 2011
45 / 56
Relation to mean-field phase diagram4
e2 < 0
4
Stenger et al. Nature 396, 345-348 (1998)
Austen Lamacraft (University of Virginia)
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August 17, 2011
46 / 56
Relation to mean-field phase diagram4
e2 < 0
4
Stenger et al. Nature 396, 345-348 (1998)
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
46 / 56
Relation to mean-field phase diagram4
e2 > 0
4
Stenger et al. Nature 396, 345-348 (1998)
Austen Lamacraft (University of Virginia)
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August 17, 2011
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What is happening at the origin for q̃ < 0?
Austen Lamacraft (University of Virginia)
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August 17, 2011
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What is special about Sz = H = 0?
Tip of the cone corresponds to the state (a1 , a0 , a−1 ) = (0, e −iχ , 0),
invariant under rotations about the z-axis
Austen Lamacraft (University of Virginia)
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August 17, 2011
48 / 56
What is special about Sz = H = 0?
Thus the torus is pinched at this point: the circle corresponding to
EFSTATHIOU,
JOYEUX,
AND SADOVSKIÍ
rotations
through φ
has contracted
to nothing
M
i
c
d
FIG. 16. Two possible plots of a pinched torus; cf. Chap. IV.3,
Fig. 3.5 on p. 163 of Ref. [4]. Both representations are equivalent in
the four-dimensional phase space.
Austen Lamacraft (University of Virginia)
!!lSpinor
,hBose
" =Microcondensates
!!l ,h " + k2" .
August 17, 2011
48 / 56
m
E
sis of a pendulum system reveals the relevance to
he classical concept of ‘monodromy’ — why a falling
right way up.
Falling cats
a slightly unstable state. The vertical ‘spring
mode’ motion quickly becomes a ‘swing
mode’ oscillation, just like a clock pendulum
swinging in some vertical plane. However,
this swing state is transient and the system
returns once more to its spring mode, then
back to a swing mode, and so on indefinitely.
The surprise is that the successive planes in
which it swings are different at each stage.
Moreover, the angle through which the
swing plane turns, from one occurrence to
the next, depends sensitively on the amplitude of the original spring mode.
The apparent paradox here is that the initial state has zero angular momentum — the
this net spin about the vertical axis is zero.Yet the
es, or swing state rotates from one instance to the
attice, next. Analogously, a falling cat that starts
d, the upside down has no angular momentum
h the about its own longitudinal axis, yet it can
They invert itself, apparently spinning about that
od in axis. The resolution of the paradox, for a cat,
nom- is that the animal changes its shape by movtum- ing its paws and tail in a particular way. At
cat, each stage of the motion, angular momensition tum remains zero and is thus conserved, but
e and the overall effect of the shape changes is to
uces a invert the cat. The final upright state also
romy has zero angular momentum, so there is no
s feet contradiction of conservation. This effect
new is known as the ‘geometric phase’, or monouanti- dromy, and is important in many areas of
ple of physics and mathematics.
The central topic of the paper is this: how
amics
Austen
Lamacraft
(University
of Virginia)
does
monodromy
show up when
the system
NHPA
hysics
s for
systions
ntum
is the
ntum
ssical
R. H.
ntum
whose
nrico
el for
Rev.
Spinor Bose Microcondensates
August 17, 2011
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Evolution of rotation angle
Rotation angle could be extracted from Faraday rotation spectroscopy
Austen Lamacraft (University of Virginia)
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August 17, 2011
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Quantizing the system
Recall the quantization of Asteroids
pi Li are the actions
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August 17, 2011
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Different actions are possible
a(p1 L1 ) + b(p2 L2 )
(1, 0)
Austen Lamacraft (University of Virginia)
a, b ∈ Z
(0, 1)
Spinor Bose Microcondensates
(1, 1)
August 17, 2011
52 / 56
Different actions = different unit cells
I1
2 1
p1 L1
=
I2
1 1
p2 L2
Austen Lamacraft (University of Virginia)
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August 17, 2011
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Hamiltonian Monodromy
Note
riod l
corres
4π
scaled
T
Wh
−Φ
2π
Recal
the re
Φ
jector
From
2T t
T
state
rotati
FIG.
13. (Color
online)lattice
(Left)toAsitself
we circle
origin
in Szpinch
,
Non trivial
mapping
of period
as wethe
move
around
this p
H space for q̃ < 0 the period lattice is deformed continuously,
φ has
returning to its original form, but after shifting the lattice
there
vector corresponding to I3 by 2π in the φ direction. (Right)
Schematic illustration of the rotation angle. While executing
the or
a (University
single period
on the reduced phase space
the
Austen Lamacraft
of Virginia) T of motion
Spinor Bose Microcondensates
August 17,
2011
54
56
of/ non
φ
Semiclassical (EBK) quantization
Ii = (ni + µi ) ~,
ni ∈ Z
µi are Maslov indices.
Austen Lamacraft (University of Virginia)
Spinor Bose Microcondensates
August 17, 2011
55 / 56
e to Josephson systems re[27] and other superfluids
4]. The non-linearity of these
70 ms
manifold of dynamical trajecexperimentally by choice of
ases of the spin components
Single
of ms
a Microcondensate holds a few surprises
lied magnetic
field. mode dynamics 140
t dynamics of this system, we
Stillandon’t
understand real experiment.
What
is-1 the origin of damping?
es created using
improved
mf 1
0
pping technique we have pre(c) 1.0
ing a dynamical compression
0.25
focused laser beam, conden0.20
toms are created after 2 s of
0.8
0.15
ndensates created in this op0.10
a mixture of all F = 1 spin
0.6
0.2 0.4 0.6 0.8
d spatial domains. To create
!"
condition, we first prepare a
0.4
F = −1" state by applying a
ng the evaporative cooling.
, a coherent superposition of
0.2
rium spin populations is creof phase-coherent microwave
0.0
F = 2 transitions. Follow0.0
0.1
0.2
0.3
0.4
0.5
0.6
the condensate is allowed to
Time (s)
l trap. A typical evolution
initial spin configuration of
p to four distinct oscillations
FIG. 1: Coherent spin mixing of spin-1 Bose condensate in
Small size necessary
but probably insufficient condition for validity of
an optical trap. Coherent spin mixing producing oscillations
e before the spin populations
in the populations(doesn’t
of the F = 1,refer
mF =to
0, ±1
spin states of
single
mode approximation
temperature)
hese oscillations
demonstrate
87
Rb condensates confined in an optical trap starting from a
ixing process.
superposition of condensate spin components at t = 0 that is
pin oscillation frequency for
subsequently allowed to evolve freely. a) Schematic indicates
Austen Lamacraft (University of Virginia)
Period (s)
Spin populations
Summary
Spinor Bose Microcondensates
August 17, 2011
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