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Non-equilibrium in cold atom systems
K. Sengupta
Indian
Association
for the Cultivation
of Science, Kolkata
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Collaborators: Brian Clark, Michael Kobodurez, Steven Girvin, David Pekker
Stephen Powell, Subir Sachdev, Christian Trezger
1/12/12
Outline
•
Introduction
•
Atoms as quantum emulators: the Bose Hubbard model
•
Emulating the Ising model: tilted lattices
•
Non-equilibrium Dynamics: A perspective
•
Dynamics of cold atom systems
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Laser and Cold atoms in a trap
Apply counter propagating laser:
standing wave of light.
The atoms feel a potential V = -a |E|2
Unit of energy
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Energy Scales
dEn = 5Er ~ 20 U
U ~10-300 t
20 Er
The atoms can hop from one lattice
site to the next: kinetic energy
The neutral atoms interact by short-range
potential: effectively on-site interaction.
Ignore higher
bands
Model Hamiltonian
For a deep enough potential, the atoms
are localized : Mott insulator described by
single band Bose-Hubbard model.
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Coherence Pattern
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Mott-Superfluid transition: preliminary analysis using Bose-Hubbard model
Mott state with 1 boson per site
Stable ground state for 0 < m < U
Adding a particle to the Mott state
Mott state is destabilized when
the excitation energy touches 0.
Removing a particle from the Mott state
z
Destabilization of the Mott state via addition of particles/hole: onset of
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superfluidity
Beyond this simple picture
Higher order energy calculation
by Freericks and Monien: Inclusion
of up to O(t3/U3) virtual processes.
Mean-field theory (Fisher 89)
Quantum Monte Carlo studies for
2D systems: Trivedi and Krauth.
Phase diagram for n=1 and d=3
O(t2/U2) theories
MFT
Superfluid
Predicts a quantum phase
transition with z=2 (except at
the tip of the Mott lobe where
z=1).
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Mot
t
Tilted optical lattice: Emulating the Ising model
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Applying an electric field to the Mott state
Shift the center of the trap. This generates
effective electric field for the atoms
1.
2. If the atoms have a net spin, apply a linearly
varying magnetic field.
Generation of an effective tilt for the lattice:
One can access regime where the effective
electric field has the same magnitude as U
which is impossible in standard condensed
matter systems.
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Construction of an effective model: 1D
Parent Mott state
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Neutral dipoles
Resonantly coupled to the parent Mott
state when U=E.
Neutral dipole state with
energy U-E.
Two dipoles which are not nearest neighbors
with energy 2(U-E).
Effective dipole Hamiltonian: 1D
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Weak Electric Field
For weak electric field, the ground state is dipole vaccum and the low-energy
excitations are single dipole
• The effective Hamiltonian for the dipoles for weak E:
• Lowest energy excitations: Single band of dipole excitations.
• These excitations soften as E approaches U. This is a precursor of the
appearance of Ising density wave with period 2.
• Higher excited states consists of multiparticle continuum.
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Strong Electric field
•
•
•
The ground state is a state of maximum dipoles.
Because of the constraint of not having two dipoles on
consecutive sites, we have two degenerate ground states
The ground state breaks Z2 symmetry.
The first excited state consists of band of domain walls
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between the two filled dipole states.
•
Intermediate electric field: QPT
Quantum phase transition at EU=1.853w. Ising universality.
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Recent Experimental observation of Ising order (Bakr et al Nature 2010)
First experimental realization of effective Ising model in ultracold atom system
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A digression: Readout scheme for boson occupation
Let the system equilibrate and raise the optical lattice depth so that the
boson number configuration is frozen.

Send in photons of “right” wavelength.

Even number of bosons in a lattice site undergo light assisted collisions
and move out of the lattice.

Image the remaining atoms and thus obtain the parity of occupation of the
original boson configuration.
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
Non-equilibrium Dynamics
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Non-equilibrium dynamics of closed quantum system
Relation with cold atoms: These experimental systems (and ion traps ) provide
perfect realization closed zero temperature quantum systems
Conceptual issues with non-eq. dynamics:
a)
Universality in out of equilibrium systems
a)
Thermalization: Nature of the steady state at long times
c)
Short time dynamics: universal signatures of quantum criticality
c)
Understanding defect production in quantum systems
c)
Study of dynamics in field theories in strongly coupled regime and
its relation to gravity via ADS/CFT
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Absence of thermalization in 1D Bose gas
Blue detuned laser used to create tightly bound
1D tubes of Rb atoms.
The depth of the lattice potential is kept large to
ensure negligible tunneling between the tubes.
The bosons are imparted a small kinetic energy
so as to place them initially at a superposition
state with momentum p0 and –p0.
The evolution of the momentum distribution of
the bosons are studied by typical time of flight
experiments for several evolution times.
The distribution of the bosons are never gaussian
within experimental time scale; clear absence of
thermalization in nearly-integrable 1D Bose gas.
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Kinoshita et al. Nature 2006
Dynamics in the Bose-Hubbard model
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Projection operator formalism for Bose-Hubbard model
Mott state
Distinguishing
between hopping
processes
Distinguish between two types of hopping processes
using a projection operator technique
Define a projection operator
Divide the hopping to classes (b) and (c)
Eliminate the “offending terms” by a cannonical transformation to get Heff
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Building fluctuations over MFT
Design a transformation which eliminate hopping
processes of class (b) perturbatively in J/U.
Obtain the effective Hamiltonian
Use the effective Hamiltonian
to compute the ground state
energy and hence the phase
diagram
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Equilibrium phase diagram
Reproduction of the phase diagram with remarkable accuracy in d=3: much better
than standard mean-field or strong coupling expansion (of the same order)
in d=2 and 3.
Allows for straightforward generalization for treatment of dynamics
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Non-equilibrium dynamics
Consider a linear ramp of J(t)=Ji +(Jf - Ji) t/t.
For dynamics, one needs to solve the Sch. Eq.
Make a time dependent transformation
to address the dynamics by projecting on
the instantaneous low-energy sector.
The method provides an accurate description
of the ramp if J(t)/U <<1 and hence can
treat slow and fast ramps at equal footing.
Takes care of particle/hole production
due to finite ramp rate
Final equation to be solved
Generic result: Defect density scales
with ramp rate for slow ramps
through critical point: Kibble-Zurek law.
To be tested for realistic experimental
systems.
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Defect production and residual energy for linear dynamics
One finds a plateau like behavior of
both P and Q instead of the expected
scaling behavior for large quench time.
Absence of critical scaling: may be
understood as the inability of the system
to access
the critical (k=0) modes.
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For typical values of h/Jc= 70ms while
the ramp time is roughly around 20-30 ms.
The dynamics is not adiabatic enough to
display scaling. This is also the regime
where the local theory works.
1-
Order parameter dynamics
Fast quench from the Mott to the SF phase; study of superfluid dynamics.
Single frequency pattern near the critical point; more complicated deeper in the SF
phase.
Strong quantum fluctuations near the QCP; justification of going beyond mft.
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Study of non-linear ramp dynamics
Protocol:
with the exponent varying between 1 and 5
Non-monotonic initial slope signifying
minimal defect production for slow
to moderate ramps around
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Experiments with ultracold bosons on a lattice: finite rate dynamics
2D BEC confined in a trap and in the presence
of an optical lattice.
Single site imaging done by light-assisted collision
which can reliably detect even/odd occupation
of a site. In the present experiment one detects
sites with n=1.
Ramp from the SF side near the QCP to deep inside
the Mott phase in a linear ramp with different
ramp rates.
The no. of sites with odd n displays plateau like
behavior and approaches the adiabatic limit
when the ramp time is increased asymptotically.
No signature of scaling behavior. Interesting
spatial patterns.
1/12/12W. Bakr et al. arXiv:1006.0754
Dynamics in the dipole model
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Quench dynamics across the quantum critical point
Tune the electric field from
Ei to Ef instantaneously
Compute the dipole order
Parameter as a function of time
The time averaged value of the order parameter is maximal near the QCP
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Dynamics with a finite rate: Kibble-Zureck scaling
Change the electric field linearly
in time with a finite rate v
Quantities of interest
Scaling laws for
finite –size systems
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Q ~ v2 (v) for slow(intermediate ) quench. These are
termed as LZ(KZ) regimes for finite-size systems.
Kibble-Zureck scaling for finite-sized system
Expected scaling laws for Q and F
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Dipole dynamics
Observation of Kibble-Zurek law for
intermediate v with Ising exponents.
Correlation function
Conclusion
1.
2.
2.
Ultracold atom systems emulates several spin and boson models
and allow us to experimentally access the properties of these
models in the strongly correlated regime.
They provide excellent examples of closed quantum systems.
Together, they provide us with the requisite test-bed for studying
non-equilibrium dynamics of strongly correlated quantum systems.
4. In the Bose-Hubbard model, experimental regime do not (yet)
correspond to the slow limit and thus do not exhibit KZ power law.
5.
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In the finite-size 1D dipole model, there is a range of ramp rate
for which one expects to observe finite-size version of KZ scaling
with universal Ising exponents.