Antiferromagnetism of spin 1 bosons :
experimental study with ultracold sodium atoms
Vincent Corre
Lingxuan Shao
Tilman Zibold
Camille Frapolli
Jean Dalibard
Fabrice Gerbier
Former members :
David Jacob
Luigi de Sarlo
Emmanuel Mimoun
jeudi 30 mai 13
Laboratoire Kastler Brossel
Ecole Normale Supérieure
Université Pierre et Marie Curie Paris 6
CNRS
•
Spin 1 condensates
•
•
•
Spin 1 electronic ground state for Na, Rb
All Zeeman components can be trapped in optical traps
mF=-1
mF=0
mF=+1
F=1
First studies at MIT (Ketterle group)
Many experiments, mostly in non-equilibrium situations.
Chapman et al. (Georgia Tech)
Sengstock et al. (Hamburg)
Klempt, Arlt, Ertmer et al. (Hannover)
Bloch et al. (Mainz-Munich)
Lett et al. (NIST-Maryland)
•
Coherent spin oscillations driven by spin-exchange collisions
•
Quenches and phase ordering kinetics across quantum phase
transitions
•
Exotic topological defects (half-vortices, skyrmions, knots, ...)
Leanhardt et al. (MIT)
Bigelow et al. (Rochester)
Shin et al.( Seoul)
Spin squeezing analogous to parametric quantum optics
Klempt et al. (Hannover)
Chapman et al. (Georgia Tech)
Oberthaler et al.(Heidelberg)
•
Stamper-Kurn et al. (Berkeley)
Raman et al.( Georgia Tech)
Equilibrium properties have been well studied theoretically, not so much experimentally
jeudi 30 mai 13
Outline
jeudi 30 mai 13
•
Antiferromagnetic spinor condensates : basic properties
•
Magnetic phase diagram : magnetization-dependent quantum phase transition
•
Anomalous fluctuations near zero field and zero magnetization
•
manifestation of collective spin fluctuations acting to restore spin
rotational symmetry
•
kinetics of relaxation towards equilibrium : «pre-thermalized» state
Single-mode approximation

Single-mode approximation : the bosons condense in the same single-particle orbital
Us : spin-interaction energy per atom
Spin exchange interaction :
m=0
Na
Us>0 :antiferromagnetic
m=+1
Only one spin-changing collisional channel

m=0
m=-1
Total angular momentum (along z) Mz = N+1-N-1 is conserved by binary collisions

Mean field approximation : |
⇥N = |⇥0 ⇥N
|
N
⇥
spin
Mean-field («spin nematic») states minimize the spin exchange energy at zero field
jeudi 30 mai 13
Nematic spin order
n: unit vector
Family of states with
zero average spin :
all possible
rotations of the
mF=0 state
eigenstate of
Nematic spin ordering :
Qab
with zero eigenvalue
1
= ⇥Ŝa Ŝb + Ŝb Ŝa ⇤
2
1
3
ab
= na nb
1
3
ab
Spins fluctuate in a plane perpendicular to a particular direction n (director)
n
|
n
jeudi 30 mai 13
⇥
0
| = ⇤1⌅
0
⌥
1
i
e
⇧ 2
=⇧
⇤ ⌥0
1 i
2e
S⇥ = 0, n = ez
⇥
⌃
⌃
⌅
“easy-axis”
S⇥ = 0, n = e = cos( )ex + sin( )ey
“easy-plane”
n and -n correspond to the same state : the order parameter behaves as the
macroscopic alignement of nematic liquid crystals (2nd rank tensor)
Magnetic energy
Total spin-dependent energy in a magnetic field :
q = B2
Magnetization is conserved :
Quadratic Zeeman energy α ~270 Hz/G2
‣ Linear Zeeman shift acts only as a constant offset
‣ Magnetic field enters only through the quadratic Zeeman shift
Note this is an approximative conservation law : there are (slow) mechanisms for spin relaxation -e.g.
due to dipole-dipole interactions, or to evaporation.
‣ if mz=Mz/N=0, the QZE selects the spin nematic state along z (all atoms in m=0)
BUT :
‣ the system cannot minimize its interaction energy «freely» if prepared with a non-zero mz
‣ competition between spin-exchange interactions and QZE
jeudi 30 mai 13
Outline
jeudi 30 mai 13
•
Antiferromagnetic spinor condensates : basic properties
•
Magnetic phase diagram : magnetization-dependent quantum phase transition
•
Anomalous fluctuations near zero field and zero magnetization
•
manifestation of collective spin fluctuations acting to restore spin
rotational symmetry
•
kinetics of relaxation towards equilibrium : «pre-thermalized» state
Experimental system
• Spin 1 BEC in a crossed-dipole trap (CDT) (trap size~8 microns)^3
Atom number
N
3500
Temperature
T
100nk
Chemical potential
µ
200nK
Trap frequency
⇥
50nK
Detection via time of flight with SternGerlach imaging
‣
‣
Spin-exchange interaction
Us
Quadratic Zeeman energy
q
1nK
13nk/G2
Preparation of a well-defined magnetization done
at high temperature (T~10 TBEC)
0.55
0.5
time of flight with magnetic gradient
switch gradient field off, take absorption
image
D. Jacob et al., NJP 13 (2010)
0.4
Mz/N
0.3
0.2
0.1
0
mF=+1
mF=0
mF=-1
0
500
1000
Depolarization time
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1500
Evolution of Zeeman populations for mz=0.47
mF=+1 mF=0 mF=-1
n
0
0.6
0.4
asymptote
0.2
0
Bc
0
0.4
0.8
B [G]
• Fit the phase boundary
• Find Bc
as the intersection point
• Repeat for many different mz
mz=0.47
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B
• n0 = N0/N
Results for the phase diagram
mF=+1 mF=0 mF=-1
D. Jacob et al., PRA (2012)
Experiment
«almost polar»
Quadratic Zeeman
energy wins
n0
1
1
0.8
0.6
B
(a)
0.5
0.4
[Saito & Ueda, arXiv:1001.2072]
0.2
0
0
0.2
0.4
mz
0.6
0.8
0
Competition between spin-dependent interactions and Quadratic
Zeeman energy, under the constraint of fixed magnetization
Quantum phase transition analogous to «spin-flop» transition of
Néel antiferromagnets
jeudi 30 mai 13
«antiferromagnetic»
or «canted phase»
spin-dependent
interactions win
Energy
Results for the phase diagram
mF=+1 mF=0 mF=-1
D. Jacob et al., PRA (2012)
Experiment
Mean-field theory (T=0) n0
n0
1
1
1
0.8
0.8
0.6
0.4
0.6
0.5
0.4
0.2
0.2
0
0
0.2
(b)
B
B
(a)
1
0.4
mz
0.6
0.8
0
0
0
0.2
Competition between spin-dependent interactions and Quadratic
Zeeman energy, under the constraint of fixed magnetization
Quantum phase transition analogous to «spin-flop» transition of
Néel antiferromagnets
jeudi 30 mai 13
0.5
0.4
mz
0.6
0.8
0
asymptote
Measurement of the critical field
1
0.8
0.6
0.4
0.2
D. Jacob et al., PRA (2012)
Bc =
(a)
0
0.2
0.4
0.6
0.8
Bc
mz
0.5
0.4
0.3
0.2
0.1
0
r
Us ⇣
1
↵
p
Us from a numerical solution of
the Gross-Pitaevskii equation
(measured atom number and trap
frequencies as input parameters)
(b)
↵ = h ⇥ 277 Hz/G
No free parameters
0
0.2
0.4
0.6
0.8
mz
Good agreement with calculated boundary (solid)
completes measurements for mz>0.5, B>0.2 G at NIST Liu et al. ,PRL 102 (2009)
jeudi 30 mai 13
m2z
1
⌘
2
Outline
jeudi 30 mai 13
•
Antiferromagnetic spinor condensates : basic properties
•
Magnetic phase diagram : magnetization-dependent quantum phase transition
•
Anomalous fluctuations near zero field and zero magnetization
•
manifestation of collective spin fluctuations acting to restore spin
rotational symmetry
•
kinetics of relaxation towards equilibrium : «pre-thermalized» state
Fluctuations and depletion
near zero applied field and magnetization
‣ if mz=0, the QZE should select the spin nematic state along z (all atoms in m=0)
Standard deviation of n0
0.4
0.6
0.3
0
0.8
n
1−n0
Depletion of m=0
0.4
0.2
0.1
−2
−1
0
B [G]
1
2
0
−3
3
0.4
0.6
0.3
−2
−1
0
B [G]
1
2
3
0
0.8
n
1−n0
0
−3
0.2
0.4
0.2
0
−4
10
0.2
0.1
−2
10
0
10
q [Hz]
2
10
4
10
0
−4
10
−2
10
0
10
q [Hz]
Near zero applied field :
• large depletion of the m=0 state
• anomalously large fluctuations of n0 (standard deviation ~N)
jeudi 30 mai 13
2
10
4
10
Quantum eigenstates for q=0
q=0
can be diagonalized
Ground state : singlet state S=0 Zeeman populations for the
singlet state are isotropic, with
super-Poissonian fluctuations
Law, Pu Bigelow, PRL 1998
Castin & Herzog, CRAS Paris 2000
Ho & Yip, PRL 2000
N
⇥n0 ⇤ = , n0
3
Fragmented condensate with 3 macroscopically occupied quantum states
0.3N
Mueller, Ueda, Baym, Ho, PRA 2006
Excited states : angular momentum eigenstates S, Mz
S=4
10 Us/N
ladder of angular momentum eigenstates S, Mz
describing collective spin fluctuations (“thin spectrum”)
S=2
3 Us/N
S=0
N+S must be even (bosonic symmetry) if N even
(assumed here)
Mz=-2 Mz=-1 Mz=0 Mz=1 Mz=2
Super-poissonian fluctuations, common to all low-energy eigenstates Stot <<N
jeudi 30 mai 13
Broken symmetry approach for q=0 and T=0
Spin nematic states :
form an over-complete basis
One can rewrite the singlet ground state as
Statistical mixture of spin nematic states
with equal weights :
Statement : measurements corresponding to a few-body operator Ok (k=1,2,...) cannot distinguish
between the singlet state and a statistical mixture of spin nematic states with equal weights
Castin & Herzog, CRAS Paris 2000
Analogy with the problem of relative phase for independent condensates : the measurement
process projects the singlet state into a random spin nematic state
Castin & Herzog, CRAS Paris 2000; Ashaab and Legget, PRA 2002
jeudi 30 mai 13
Broken symmetry approach for non-zero q and T
BS
0.6
Statistical mixture of spin nematic
states with Boltzmann weight
1−n0
0.5
0.4
0.3
exact
0.2
0.1
0 −1
10
0
10
Moments of n0 are
universal functions of
1
10
Nq/T
2
10
De Sarlo et al., in preparation
0.3
Excellent agreement with exact
numerical diagonalization of H
[here N=1000, T=10 Us]
∆ n0
0.25
0.2
0.15
0.1
0.05
−1
10
jeudi 30 mai 13
0
10
1
10
Nq/T
2
10
Connection with spontaneous symmetry breaking
• Phase transition to an ordered phase, characterized by some order parameter O that
breaks a symmetry of the hamiltonian H
• Very general mechanism present in high energy physics and condensed matter :
➡
➡
<O> seen as the expectation value of an operator O
One imagines a fictitious «symmetry-breaking field» h coupling to O : H’ = H +h O
Some examples in condensed matter :
Physical system
Order parameter
Broken symmetry
symmetry breaking
field
crystal
center-of-mass
position
center-of-mass
translation
pinning potential
antiferromagnet
Staggered
magnetization
spin rotation
staggered magnetic
field
superfluid
macroscopic
wavefunction
global phase rotation
particle source/sink
spin 1 BEC
nematic director
spin rotation
quadratic Zeeman shift
jeudi 30 mai 13
Connection with spontaneous symmetry breaking
Thermodynamic limit :
any infinitesimally small h will mix many eigenstates and project the system into a «meanfield» state
all states collapse to the ground state in
thermodynamic limit (large enough N) :
S=4
7 Us/N
P. W. Anderson’s “tower of states” picture
of spontaneous symmetry breaking
S=2
3 Us/N
S=0
Mz=-4 Mz=-2 Mz=0 Mz=2 Mz=4
A few related works for Heisenberg
antiferromagnets :
Anderson, PR 86 (1952)
Lieb, Matthis, J. Math. Phys. 3(1962)
Neuberger & Ziman, PRB 39 (1989)
Kaplan, von der Linden, Horsch, PRB 42 (1990)
Bernu, Lhuillier, Pierre, PRL 69 (1992)
Finite systems : when h is too small, collective fluctuations of the order parameter («thin spectrum»)
will overcome its effect and restore the broken symmetry (crossover of width ~1/N)
The typical magnitude of h vanishes as 1/N and thus goes to zero in the thermodynamic limit.
jeudi 30 mai 13
Outline
jeudi 30 mai 13
•
Antiferromagnetic spinor condensates : basic properties
•
Magnetic phase diagram : magnetization-dependent quantum phase transition
•
Anomalous fluctuations near zero field and zero magnetization
•
manifestation of collective spin fluctuations acting to restore spin
rotational symmetry
•
kinetics of relaxation towards equilibrium : «pre-thermalized» state
Spin thermometry (preliminary)
Finite temperatures also implies finite condensed fraction
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
−4
10
−2
10
0
10
q [Hz]
0
−4
10
2
10
0.3
0.3
0.25
0.25
0.2
0.2
n0
n0
Higher temperature (fc~70 %)
1−n0
1−n0
Lowest temperature (fc>90 %)
0.15
0.15
0.1
0.1
0.05
0.05
0
−4
10
−2
10
0
10
q [Hz]
2
10
0
−4
10
−2
10
−2
10
0
10
q [Hz]
0
10
q [Hz]
2
10
2
10
Thermal pedestal visible in the 1-n0 plot, but has negligible impact on standard deviation
jeudi 30 mai 13
Spin thermometry (preliminary)
Finite temperatures also implies finite condensed fraction
0.35
We use the variance of n0 to extract
the relevant parameters because it
is only sensitive to the SMA part :
0.3
n0
0.25
0.2
0.15
The contribution of the thermal
component is Poissonian and
much smaller.
0.1
0.05
0 −4
10
−2
10
0
10
q [Hz]
2
10
We fit Ts (the temperature associated to collective spin fluctuations) and the condensed
fraction fc.
We use a Hartree-Fock model of the thermal cloud to extract the temperature Tk of the
thermal cloud from fc.
jeudi 30 mai 13
Kinetic temperature saturates near chemical potential
(shown as stars)
Spin temperature (nK)
Spin temperatures are very low (as low as 15 nK=Tc/
50), much lower than «kinetic» temperatures
Kinetic temperature (nK)
Spin thermometry (preliminary)
250
200
150
100
50
0
0
5
10
Trap depth (µ K)
15
5
10
Trap depth (µ K)
15
5
10
Trap depth (µ K)
15
800
600
400
200
0
Ability to measure high condensed fractions reliably
Condensed fraction
1
0.8
0.6
0.4
0.2
0
jeudi 30 mai 13
Distribution of measured n0 (preliminary)
observed
expected
Nq/T =0.003 mz=0.017 +/−0.036
Nq/T =0.458 mz=0.038 +/−0.040
s
Nq/T =1.123 mz=0.050 +/−0.043
s
s
0.2
0.2
0.1
0
0.15
0.1
0.1
0.05
0.05
0
0
0.5
n
0
0
1
0
P (n0)
0.2
0.2
0.15
P (n )
P (n0)
0.3
0.5
n
0
0
1
0
Nq/T =4.556 mz=0.005 +/−0.036
s
0.4
0.3
P (n0)
0.4
P (n0)
Nq/T =1.701 mz=0.051 +/−0.048
s
0.15
0.1
0.1
0.05
0.5
n
0
0
1
0
0.2
0.5
n
0
0
1
0
0.5
n
1
0
0.3
0.6
0.6
0.1
0
0
0.2
0.4
0.2
0
0
0.5
n0
1
1
0
0
0.4
1
P (n0)
0.8
P (n0)
0.8
P (n )
0.4
P (n )
P (n0)
Nq/Ts=4.853 mz=0.033 +/−0.046 Nq/Ts=19.772 mz=0.040 +/−0.039 Nq/Ts=19.157 mz=0.036 +/−0.042 Nq/Ts=75.355 mz=0.018 +/−0.036 Nq/Ts=230.075 mz=0.038 +/−0.038
0.5
0.2
0.5
n0
1
0
0
0.5
n0
1
0
0
0.5
n0
Distribution calculated for N=300, rescaling the temperature accordingly
Spin temperature and condensed fraction are extracted from the data
No adjustable parameter used
Conclusion : spin fluctuations seem fairly close to thermal equilibrium
jeudi 30 mai 13
0.5
1
0
0
0.5
n0
1
Pre-thermalization : how isolated systems reach equilibrium
Why such a large difference ?
spin and kinetic degrees of freedom
separately in equilibrium but decoupled
1st «phonon mode»
• Vast differences in energy scales suggests very weak
coupling between spin collective fluctuations and other
collective modes (phonons and spin waves)
•Independent mechanisms to reach equilibrium :
‣ Dephasing of single-mode spin excitations on the
‣
thin
spectrum
one hand : «quantum thermalization»
Evaporative cooling with standard thermalization on
the other
• Example of a
«pre-thermalized» state
Ref :Polkovnikov et al, RMP 2011
Asymptotic state of an isolated system reaching a pseudo-thermal equilibrium via unitary evolution
jeudi 30 mai 13
What determines the spin temperature ? (preliminary)
In an isolated system, the temperature after thermalization is determined by the available energy
extracted from the data
Spin temperature (nK)
250
200
150
100
50
0
0
5
10
15
Trap depth (µ K)
This argument gives the correct order of magnitude for the temperature
NB: Fluctuations of q (magnetic field) can be excluded for qualitative and quantitative reasons
jeudi 30 mai 13
Conclusion
•
Experimental study of spin 1 antiferromagnetic condensates at (or near) equilibrium :
•
•
•
•
•
•
Exploration of phase diagram n0 vs B, Mz
Measurement of critical “nematic-flop” transition line
Observation of large fluctuations : symmetry-restoring fluctuations in finite samples
(still in progress) spin and kinetic degrees of freedom separately in equilibrium but
decoupled
Kinetics of relaxation towards thermal equilibrium : quenches from large to low q
Impact of collective fluctuations on quantum coherence
universal decoherence mechanism
•
van Wezel, van den Brink, Zaanen, PRL 94 (2005)
Quantum limit of collective fluctuations :
•
•
jeudi 30 mai 13
Decrease N (~100) and trap volume to keep US constant
Dynamical collapse and revival experiments
Diener & Ho, arXiv:cond-mat/0608732
L. Chang, PRL 2007
R. Barnett et al., PRA 2010, 2011
Cui et al., PRA 2008
Q. Zhai et al., PRA 2009