Spin 1 Micrcondensates
Everything you ever wanted to know about the single mode
approximation (and more)
Austen Lamacraft
University of Virginia
May 15, 2011
Technion Quantum Magnetism Workshop
Phys. Rev. A 83, 033605 (2011)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
1 / 42
What is Quantum Magnetism in Ultracold Atoms?
Is it this?
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
2 / 42
What is Quantum Magnetism in Ultracold Atoms?
Or this?
Cs Fe8
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
2 / 42
Where are we? (rash overgeneralizations)
Equilibrium states take a backseat to Dynamics
Theory is only now coming to grips with real experiments
We don’t understand what happens when you cool the simplest spinor
system (spin 1 Bose gas)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
3 / 42
Outline
1
Spin 1 Microcondensates
2
The (semi-)classical limit
3
Some interesting features of integrable classical motion
4
Quantizing the Microcondensate
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
4 / 42
Outline
1
Spin 1 Microcondensates
2
The (semi-)classical limit
3
Some interesting features of integrable classical motion
4
Quantizing the Microcondensate
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
5 / 42
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)
Spin 1 Microcondensates
May 15th, 2011
6 / 42
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
Spin 1 Microcondensates
May 15th, 2011
7 / 42
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)
Spin 1 Microcondensates
May 15th, 2011
7 / 42
Microcondensate Hamiltonian
For N particles is
HSMA =
N̂ =
1
X
e0
e2
: N̂ 2 : +
: Ŝ·Ŝ : +HZ
2N
2N
A∗m Am
Ŝ =
A∗m Smm0 Am0 ,
m,m0
m=−1
HZ =
1
X
m=−1
Austen Lamacraft (University of Virginia)
X
A∗m pm + qm2 Am
Spin 1 Microcondensates
May 15th, 2011
8 / 42
In gory detail...
Dropping the density-density and linear Zeeman pieces, we have
Hred =
e2 2
Sz + 2 A∗1 A∗−1 (A0 )2 + (A∗0 )2 A1 A−1 +
2N 1
∗
∗
∗
A1 A1 + A−1 A−1 + q A∗1 A−1 + A∗−1 A−1 .
2 A0 A0 −
2
... and probably you are none the wiser!
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
9 / 42
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
Spin 1 Microcondensates
1 2
May 15th, 2011
10 / 42
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
Spin 1 Microcondensates
May 15th, 2011
11 / 42
Outline
1
Spin 1 Microcondensates
2
The (semi-)classical limit
3
Some interesting features of integrable classical motion
4
Quantizing the Microcondensate
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
12 / 42
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
shown 0.8 fundamental spin mixing process. b) Absorptive images of the
0.2 data
0.4are
Austen
Lamacraft
of Virginia)
Spin0.6
1with
Microcondensates
May 15th, 2011
13 / 42
in the
inset of(University
Fig. 1c. and
show good agreement
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.
backactionSpin
in the
present experiments.
Austen Lamacraft (University of other
Virginia)
1 Microcondensates
In the absence
May 15th, 2011
13 / 42
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)
Spin 1 Microcondensates
May 15th, 2011
14 / 42
A strange choice of variables: Hyperbolic Spins
1 ∗
A1 A1 + A∗−1 A−1 + 1
2
K+ = A∗1 A∗−1 ,
K− = A1 A−1 ,
1
1
B0 = A∗0 A0 + ,
2
4
1 ∗ ∗
1
B+ = − (A0 A0 ) , B− = − (A0 A0 ) .
2
2
K0 =
Invariant under the rotations about the z-axis generated by the
conserved quantity Sz
Compare K ’s with Anderson spins in the fermionic case
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
15 / 42
Hyperbolic (or SU(1, 1) spins)
K ’s and B’s are two reps of the non-compact group SU(1, 1)
[K0 , K± ] = ±K±
[K+ , K− ] = −2K0
and likewise for the {B0 , B+ , B− }
Quadratic Casimir operators
1
1 2
(K− K+ + K+ K− ) =
Sz − 1
2
4
1
3
CB = B02 − (B− B+ + B+ B− ) = − .
2
16
CK = K02 −
Classical limit is a hyperboloid instead of a sphere!
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
16 / 42
Return on investment in strange variables3
Hred =
e2 2
Sz + 8B0 K0 − 4B− K+ − 4B+ K− + 2qK0 .
2N
Pair of coupled Hyperbolic spins
Note ‘Minkowski’ scalar product in coupling
K0 + B0 is just (half) total number of particles
3
N.M. Bogoliubov, J. Math. Sci. 136, 1, 3552-3559 (2006)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
17 / 42
Classical limit
Treat Am , A∗m as classical amplitudes Am , A∗m of ‘order
√
N’
Sz2
4
CB ∼ 0.
CK ∼
N ∼ 2 (B0 + K0 )
B degrees of freedom are restricted to a cone B0 = |B+ |
Note that K0 , B0 ≤
Austen Lamacraft (University of Virginia)
N
2
Spin 1 Microcondensates
May 15th, 2011
18 / 42
Final stage of reduction
Hred =
e2 2
Sz + 8B0 K0 − 4B− K+ − 4B+ K− + 2qK0 .
2N
Exploit the symmetry under rotations about the z-axis in hyperbolic
spin space, B+ to be real and positive.
Then B+ = B0 = N/2 − K0 and we can eliminate the B’s
Hclass =
e2 2
Sz + 8 (N/2 − K0 ) (K0 − Kx ) + 2qK0 .
2N
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
19 / 42
Geometric construction
1
h = sz2 + 2(1 − 2k0 ) (k0 − kx ) + 2q̃k0
2
h = const. fixes a hyperbola with asymptotes k0 =
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
1
2,
q̃ ≡ q/e2
k0 = kx + q̃2
May 15th, 2011
20 / 42
Geometric construction
1
h = sz2 + 2(1 − 2k0 ) (k0 − kx ) + 2q̃k0
2
q̃ ≡ q/e2
h = const. fixes a hyperbola with asymptotes k0 = 12 , k0 = kx +
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
q̃
2
20 / 42
Two regimes
q̃ > 0
Separatrix at h = 12 sz2 + q̃
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
21 / 42
Two regimes
Separatrix in the spectrum
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
21 / 42
Two regimes
q̃ > 0 : topologically distinct trajectories at sz = 0
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
21 / 42
Two regimes
Signature in the spectrum. What’s happening at the origin?
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
21 / 42
Two regimes
For q̃ < −2 origin moves through top boundary
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
21 / 42
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
applied magnetic field, following a
#$B
c&$1 # !(University
Austen
Lamacraft
of
Virginia) as a function
Spin 1ofMicrocondensates
c% !
0% "
et al. PRL 2007
May 15th, 2011
22 / 42
Relation to mean-field phase diagram4
e2 < 0
4
Stenger et al. Nature 396, 345-348 (1998)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
23 / 42
Relation to mean-field phase diagram4
e2 < 0
4
Stenger et al. Nature 396, 345-348 (1998)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
23 / 42
Relation to mean-field phase diagram4
e2 > 0
4
Stenger et al. Nature 396, 345-348 (1998)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
23 / 42
Outline
1
Spin 1 Microcondensates
2
The (semi-)classical limit
3
Some interesting features of integrable classical motion
4
Quantizing the Microcondensate
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
24 / 42
Classical mechanics of the spin-1 gas
System of three degrees of freedom
{A∗m , Am0 } = iδmm0 .
m = −1, 0, +1
There are three (commuting) conserved quantities
1
2
3
The energy Nh
The angular momentum S z
The particle number N
The system is integrable in the sense of Liouville–Arnol’d
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
25 / 42
The Liouville–Arnol’d theorem
Liouville–Arnol’d theorem
For a system of N degrees of freedom 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)
Spin 1 Microcondensates
May 15th, 2011
26 / 42
The angles
Can use each of the N commuting conserved quantities Fi in place of
H in Hamilton’s equation to generate ‘flow’ x(t1 , . . . , tN )
Motion in general quasiperiodic: N incommensurate frequencies
Set of values of t = (t1 , . . . , tN ) for which x(t1 , . . . , tN ) = x(0, . . . , 0)
is a lattice (the period lattice) with vectors e1 , . . . , eN
Define angular variables in terms of the reciprocal lattice vectors
1 , . . . , N satisfying i · ej = 2πδij . Then
ϕi ≡ i · t,
i = 1, . . . , N,
in particular
ϕi (t1 ) = ωi t1 + ϕi (0)
ωi ≡ (i )1 ,
Austen Lamacraft (University of Virginia)
i = 1, . . . , N.
Spin 1 Microcondensates
May 15th, 2011
27 / 42
The actions
Ii ≡
1
2π
I
γi
p · dqi ,
γi around the i th circle of the torus.
Actions are conjugate to the angles
∂H
ωi =
∂Ii Ik 6=Ii
.
fixed
Matrix of angular ‘velocities’
(i )j =
∂Fj
∂Ii
or equivalently period lattice vectors
∂Ii
(ei )j = 2π
∂Fj Fk 6=Fj
Austen Lamacraft (University of Virginia)
,
Ik 6=Ii fixed
Spin 1 Microcondensates
.
fixed
May 15th, 2011
28 / 42
Actions for the spin 1 Microcondensate
I1 = N
I2 = Sz
A
I3 =
πSz
dA =
Sz
2
2
sinh θ dθ dψ
Sz iψ
e sinh θ
2
Sz
K0 =
cosh θ
2
S2
K02 − |K+ |2 = z ,
4
Area A on hyperboloid induced the ‘Minkowski’ metric
2
Austen Lamacraft (University of Virginia)
Spin 1
Microcondensates
K+ =
May 15th, 2011
29 / 42
Trajectories on the half-plane
y=
Austen Lamacraft (University of Virginia)
2ky
,
k0 − kx
z=
Spin 1 Microcondensates
sz
.
k0 − kz
May 15th, 2011
30 / 42
q̃ > 0: Separatrix on the half-plane
Circles project to circles −→ phase space is a disc
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
31 / 42
Properties of the action: q̃ < 0
Topologically distinct trajectories at sz = 0
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
32 / 42
Properties of the action: q̃ < 0
A
πSz
2
dydz
Sz
dA =
2
z2
I3 =
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
32 / 42
What is going on at the origin?
Recall period lattice
(ei )j = 2π
∂Ii
∂Fj
Fk 6=Fj fixed

1
= 2π  0
∂I3
∂N
0
1
∂I3
∂Sz

0
0
∂I3
∂H
∂I3
∂I3
∂I3
e3 = 2π ∂N
tells us how to execute a closed orbit around the
∂Sz
∂H
third circle of the three-torus
1
∂I3
Evolve for a time 2π ∂H
2
∂I3
Change overall phase of the spinor by 2π ∂N
3
∂I3
rotate about the z-axis by 2π ∂S
z
Rotation angle
Φ(Sz , H) ≡ −2π
∂I3
∂Sz
not single-valued for q̃ < 0: increases by 2π around the origin H = Sz = 0
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
33 / 42
Hamiltonian Monodromy
φ
4π
T
−Φ
2π
T
Φ
2T t
Non trivial
mapping
of period
ei → the
e0i =origin
Mij ejin Sz ,
FIG.
13. (Color
online)lattice
(Left)toAsitself
we circle
M is monodromy
matrix
H space for q̃ < 0 the period lattice is deformed continuously,
 but after
 shifting the lattice
returning to its original form,
1 0 0
vector corresponding to I3 by 2π in the φ direction. (Right)
 1 0
Schematic illustrationMof=the 0rotation
angle. While executing
1 1reduced phase space the
a single period T of motion 0
on the
system matrix
rotatesofbyunit
an angle
Φ
integer-valued
determinant
(an element of the group
SL(3, Z))
Austen Lamacraft (University of Virginia)
May 15th, 2011
�Spin 1 Microcondensates�
Note
riod l
corres
scaled
Wh
Recal
the re
jector
From
state
rotati
this p
φ has
there
the or
of non
singu
mathe
34 / 42
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
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)
Spin 1 Microcondensates
May 15th, 2011
35 / 42
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
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)
Spin 1 Microcondensates
May 15th, 2011
35 / 42
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.
Spin 1 Microcondensates
May 15th, 2011
36 / 42
Evolution of rotation angle
Rotation angle could be extracted from Faraday rotation spectroscopy
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
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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
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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)
Spin
1 Microcondensates
May
2011
38 /plane
42 thd
successive
swinging
motions
loops
#) the
space
are
mapped
the constants
of motion
having differing overall
behavior,
classical
system
inaswhich
the(",consequences
of15th,
monodromy
step
size in
!%
of
stepwise-precessing
swing
Outline
1
Spin 1 Microcondensates
2
The (semi-)classical limit
3
Some interesting features of integrable classical motion
4
Quantizing the Microcondensate
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
39 / 42
Semiclassical (EBK) quantization
Ii = (ni + µi ) ~,
ni ∈ Z
µi are Maslov indices.
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
40 / 42
Monodromy as a Quantum Defect5
EBK gives lattice in space of conserved quantities with vectors
∂Fj
~
= ~ (i )j
∂Ii Ik 6=Ii fixed
Just reciprocal lattice of period lattice
Accompanying monodromy period lattice is the corresponding map on
the ‘quantum’ lattice
i → 0i = M −1 ji j ,
5
Due to Boris Zhilinskii
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
41 / 42
Exact solution using Algebraic Bethe Ansatz6
An eigenstate of the Hamiltonian is written as
|Ψi =
NR +
Y
B
j
K+
+
λj
λj + N q̃/2
|0, νK iK ⊗ |0, νB iB
The λj j = 1, . . . NR satisfy the equations
N
1−
R
X
νK
1
νB
−
=
.
λj
λj + N q̃/2
λj − λl
l6=j
With energy
NR
e2
4e2 X
E ({λj }) =
N(N − 1) + q|Sz | −
λj
2N
N
j=1
6
N.M. Bogoliubov, J. Math. Sci. 136, 1, 3552-3559 (2006)
Austen Lamacraft (University of Virginia)
Spin 1 Microcondensates
May 15th, 2011
42 / 42
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
Spin 1 Microcondensates
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