Transcript BEC and optical lattices

Bose-Einstein condensates
and optical lattices
(basics)
Dieter
Dieter Jaksch
Jaksch
(University
of of
Oxford,
University
OxfordUK)
EU networks: OLAQUI, QIPEST
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Irreversible loading of optical lattices
Motivation
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
The System
Ultracold atoms
Weakly interacting BEC  GPE
Atoms in a lattice  strong correlations
Fermions & Bosons  quantum statistics
Polar interactions  long range
Main Properties
Adjustable spatial dimension
Very low temperatures pK to nK
Strong correlations possible
 no mean field approach possible
Full quantum dynamics
 no semiclassical approach
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a=1
N
Aims and Goals
Provide the physics background for better understanding current research on:
BEC
optical lattice physics
mathematical methods for strongly correlated quantum systems
Explain the physical basis of
the Gross-Pitaevskii equation
the (Bose)-Hubbard model in optical lattices
approximate descriptions of strongly correlated 1D systems
Give an overview of a selection of recent work in this field
Dynamics of the superfluid-Mott insulator transition
Excitation spectrum of the 1D Bose-Hubbard model
Loading and Cooling / mixtures of different species
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Irreversible loading of optical lattices
Basics of many particle quantum mechanics
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Fock states (bosons)
One species of particles and one motional state only
n
The Fock states are orthogonal and normalized (h .|. i is the scalar product)
Since bosons are indistinguishable these Fock states fully describe the state of
one species of bosons in a single motional state
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Creation and destruction operators (bosons)
This leads to the definition of creation and destruction operators
Creation operator
Its hermitian conjugate, the destruction operator a
Therefore we can write
Furthermore we find
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Number operator and commutator
The commutation relation is
Matrix representation in the Fock basis |ni
We also define the number operator
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Several motional states and species
a
b
1
0
a
a+1
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Number operators and commutators
We define number operators similar to before
The commutation relations are
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Note: Fermionic particles
No two fermions can occupy the same quantum state (Pauli principle). This is
reflected by properties of the fermionic creation and destruction operators.
The anticommutator relations are
so that the square of each creation operator gives zero. No two particles can
be created in a single quantum state.
These anticommutator relations extend to several species and quantum states
like the commutator relations do for bosons.
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Note: Bose-Einstein condensates
In the case of a Bose-Einstein condensate a large number N of bosonic
particles occupy the same quantum state. As a crude approximation (better
justified and mathematically more rigorous approaches yielding the same result
exist) one assumes that it does not matter physically whether N or N-1
particles exist in the condensate. Therefore
This effectively means that the destruction and the creation operators for
particles in the Bose-Einstein condensate are replaced by a number
The Bose-Einstein condensate is thus described classically by c-numbers
instead of a full quantum treatment
Note: The macroscopic wave function arises from similar arguments if the
spatial degrees of freedom are included in the treatment
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Example: Coherent state
A coherent state is a superposition of Fock states
It is an eigenstate of the destruction operator
The expected number of particles is
When replacing a  a for a BEC this corresponds to assuming a coherent
state of the atoms in a motional state described by destruction operator a.
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Particle energies (I)
The Hamiltonian H which governs the dynamics of the quantum system will be
the sum of all energies in our case. There will be several contributions
Potential energy
E1
E0
Kinetic energy
A particle gains energy by hopping between
different states
a
a+1
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Particle energies (II)
Interaction energy
n particles in the same state
Each particle interacts with n-1 particles in
the same state
E1
E0
Interactions between particles in different states
E1
E0
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Each particle interactions with all particles in
the other state
Example: Tunnelling (I)
Chain of atoms with kinetic energy and periodic boundaries
a=1
N
N+1 ´ 1
Hamiltonian
Introduce discrete Fourier transformed operators q 2  ]-1,(N-1)/N, … 1]
with commutation relations
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Example: Tunnelling (II)
Rewrite the Hamiltonian
Eq
The eigenstates are
Blochband
4J
with single particle eigenenergies
q
ground state
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excitations
Example: Repulsive Interaction
A single quantum state with repulsive interaction and potential energy
E1
E0
Apply the Hamiltonian to a Fock state
It is thus an eigenstate with eigenenergie (U (n-1)/2 + E0) n. This is the
ground state for ng particles given by
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Irreversible loading of optical lattices
Optical lattices and Hubbard models
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Optical lattices superimposed on a BEC
Interference of standing wave laser beams induces AC-Stark shifts to trap the
atoms in a periodic lattice potential
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Munich: I. Bloch, T. Haensch et al.
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Optical lattices: Basics
AC – Stark shift
a
|1i
w
laser
w01
|0i
Spontaneous emission


shift:
|1i
G
|0i
AC –Stark shift <{}
¼
Spontaneous emission I{}

G
À1
 Spontaneous emission rates of less than 1s-1
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w
Optical lattice: Basics
The dominant real part acts as a conservative potential V(x). For a standing
wave laser configuration we obtain
k … laser wave vector
V0 … lattice depth / laser intensity
Spatially periodic potential  realization of a lattice model
Very little spontaneous processes  motion described by Schroedinger
equation
Shape and properties of the potential adjustable by varying laser parameters!
Additional background potential by magnetic or optical fields
Superlattice potentials by superimposing additional lattice potentials
Creation of quasi random patterns using additional incommensurate lasers
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Lattice design
1D
laser
2D
different internal states
3D
square lattice
laser
triangular lattice
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Ultracold atoms in an optical lattice
Only two particle interactions for as ¿ a0 and few particles per lattice sites, i.e. a
dilute gas ¿1
V0 is varying quickly on the length scale of optical wave lengths  ¼ 500nm
and cannot be treated as a small perturbation like the trap potential VT and
interactions
Solve the one particle problem including kinetic term and optical potential
Treat trap potential and interaction term as a perturbation
Restrict calculations to small temperatures T
Hamiltonian of trapped interacting particle
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Single particle problem in 1D
Mathieu equation for the mode functions (~ dimensionless parameters)
Bloch bands with normalizable Bloch wave functions in the stable regions
Stable regions
a) V0 = 5 ER
b) V0 = 10 ER
c) V0 = 25 ER
Lowest band:
E(0)q = -2 J cos(q)
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Wannier functions
These are mode functions pertaining to a certain Bloch band and localized at a
lattice site
Note: This definition is not unique because of the arbitrary phase in the Bloch
wave functions. The degree of localization depends strongly on their choice.
At small temperatures only the lowest Bloch band n=(0,0,0) will be occupied
Wannier
functions
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Optical lattice
… destruction operator
for an atom in lattice
site a
V0
J
U
a
Aq are the momentum
destruction operators
… number of particles in
lattice site a defined as
Described by a Hubbard model
Hopping term J and interaction U (s-wave scattering for bosons) are
adjustable via the lattice depth
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Hopping and interaction terms
2
-1
10
Ua
10
J/E R
ER aS
-2
1
10
10
-3
0
10 5
15
V0 /ER
10
25
E R ... recoil energy
a ... ground state size
a S ... scattering length
V0 ... depth of the
optical potential
Recoil energy: ER = ~2k2/2m
Na: ER ¼ 25 kHz
Rb: ER ¼ 3.8 kHz
Validity:
only lowest Bloch band occupied
n as3 ¿ 1, i.e. low density, weak interactions
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Microscopic picture: Two atoms in one well
¹h!
molecular picture:
U
t r ap
energy
Hubbard picture:
trap

trap
levels
¹h!
0
t r ap

molecular
levels
internuclear separation
scattering length
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trap size
Changing the lattice potential
Shallow lattice: JÀU
4J
Deep lattice J ¿ U
U
4J
D.J., C. Bruder, J. I. Cirac, C. W. Gardiner, and P.
Zoller, Phys. Rev. Lett. 81, 3108 (1998).
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U
M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and
I. Bloch, Nature 415, 39 (2002).
The Mott insulator– loading from a BEC
Theory: D.J, et al. 1998, Experiment: M. Greiner, et al., 2002
BEC phase J À U:
quantum
freezing
super
fluid
Mott
melting
/U
n=3
Mott insulator J ¿ U (commensurate):
n=2
superfluid
Mott n=1
J/U
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Irreversible loading of optical lattices
Simulation of dynamic and static properties
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Questions
Static properties of the ground state |Gi
Long range correlations
On-site fluctuations
(Linear) response to external perturbations
Dynamic properties for a given initial state |i
Unitary evolution according to Hamiltonian (setting ~ =1)
Non-unitary evolution due to interaction with bath or collisions starting from
an initial density operator 
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Theoretical Methods
The GPE cannot describe the MI in the deep optical lattice. It also fails to
include correlations between distant particles beyond mean field.
Improvements for the time-independent case
analytical
mean field theory (Gutzwiller)
numerical
exact diagonalization
Quantum Monte Carlo
standard (?)
condensed matter
Improvements for the time-dependent case
analytical
mean field theory (Gutzwiller)
... ???
numerical
exact time evolution for small systems
DMRG in 1D
... ???
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?
Time independent Gutzwiller
Allow for non-coherent state in each lattice site
Not number conserving ansatz
lattice site
occupation
a
Remarks:
not number preserving (i.e. the superfluid will have a phase)
number preserving version PN jGi
Variational method
a contains the chemical potential to fix the mean number of particles hG|N|Gi.
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Mott insulator UÀJ
f n( ®)
superfluid parameter
0
1
2
3
n
Variation around the Mott state:
hH i ²
> 0 Mott phase
²
= 0 critical point
µ
U
zJ
minimum
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¶
¼ 5:8:::
cr i t
Superfluid phase U¿J
f n( ®)
superfluid parameter
0
1
2
3
n
recover Gross-Pitaevskii equation
or
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Time dependent Gutzwiller
Time-dependent ansatz
Variational method
Resulting equations
Only nearest neighbour hopping ha,i
Ja,= J for ha,i
Ja,= 0 otherwise
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superfluid parameter
Recover the GPE
In the superfluid limit JÀU
coherent state
Gross Pitaevskii equation
Blochband
excitations
k
BEC
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Strong quantum correlations (?)
The Gutzwiller ansatz describes
Limit of small number of particles in many localized weakly coupled modes
Suppressed onsite particle fluctuations in the MI regime
System in terms of non-number conserving quantum states
Still missing
Still product state of different lattice sites (similar to GPE)  no
correlations beyond mean field
Nucleation of the superfluid
Critical region
Time scale required for build up of coherence
Local versus distant coherences
What is not described is of particular interest!
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Time dependent DMRG (I)
Based on work of Vidal (2003, 2005), Verstraate & Cirac (2004), Werner (1990) for spins
System described by a state
and fix the maximum occupation as nmax.
Perform successive SD of the system
Truncate these to a maximum rank
Use the SDs to form tensors
and
This gives an expansion in matrix product states
The tensor G[a] n replaces fn(a) from Gutzwiller ansatz
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Examples
A superposition state
is written as
with  = 2-1/2, 1A=|1i, 2A=|0i, 1B=|0i, 2B=|1i
A superposition state
is written as
with  = 1, 1A=|0i, 1B=2-1/2(|0i+|1i)
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Time dependent DMRG (II)
Site Number
Site Number
Correlations
Correlations
State in site 2
Applying a series of Schmidt decompositions
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State index
State in site 3
Local quantum operations
Only the Gamma tensor for the corresponding site
needs to be updated (O(2) basic operations)
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Two site operations
Only the Gamma tensor for the corresponding sites and the lambda tensor in
between need to be updated (O(3) basic operations)
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Recovering the required form
Arrange  into a matrix with row index nk,a1 and column index nm,a3
Perform a singular value decomposition
and identify the ~ variables with the new Schmidt decomposition
This procedure can be extended to higher numbers of involved sites but the
efficiency goes down.
Instead we will decompose the evolution of the system into single-site and twosite operations
Extensions to operations involving distant sites are possible but not necessary
in our case because of the local nature of the interactions and hopping terms.
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Trotter expansion
The unitary time evolution U according to the Schrödinger equation can be applied via a
Trotter expansion (we use a 4th–order expansion in all calculations)
Here we have defined
Overview of key advantages of TEBD :
―
―
―
―
Trotter parameters
one and two site operations
Efficient in storing a state :
Efficient update for 1 and 2-local unitaries :
Inaccuracies grow slowly :
For 1D systems with 2-local Hamiltonians the maximum
logarithmically with the size
at small energies
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grows at worst
Strong correlations (?)
This approach allows to describe systems in 1D where the correlations at long
distances are mean field like or scale like
Systems at criticality with long range strong correlations of the form
require   1 and are thus not appropriately described
In higher than one dimension  scales badly with the size of the system
The amount of entanglement and thus  scales with the size of the
boundary of the system. In 1D this is constant leading to  / log(L) while in
2D and 3D the boundary increases with the system size
A
L
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Towards higher dimensions
MPS and MERA: G. Vidal
PEPS: F. Verstraete and J.I. Cirac
WGS: M. Plenio and H. Briegel
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Density operators
Every expectation value is obtained by an average of the form
Therefore a density operator  = |ih| contains all the physical information
If a system is not fully prepared (e.g. in a thermal state or in the presence of
decoherence) classical uncertainty about the state of the system is present in
addition to the quantum nature contained in |i.
In these situations only the (classical) probability pi for the system occupying
the state |ii is known. The expectation value needs to be weighted accordingly
The density operator  = i pi |ii hi| can thus describe systems prepared in
pure states (a ket |i) as well as in mixed states (kets |ii with probability pi)
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Simulation of mixed states
Arrange the NxN matrix  as a vector with N2 elements
Introduce superoperators L on these matrices of dimension N2 x N2
The evolution equation is then formally equivalent to the Schroedinger
equation.
For a typical master equation of Lindblad type
If L decomposes into single site and two site operations the same techniques
as discussed for pure states and unitary evolution can be applied
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