centre for quantum physics & technology, clarendon laboratory, university of oxford. dieter...

Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.

Dieter Jaksch

(University of Oxford, UK)

Bose-Einstein condensates and optical lattices

(basics)

Dieter Jaksch

University of Oxford

EU networks: OLAQUI, QIPEST


Irreversible loading of optical lattices

Motivation


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


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



Basics of many particle quantum mechanics


One species of particles and one motional state only

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

Fock states (bosons)

n


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


Number operator and commutator

The commutation relation is

Matrix representation in the Fock basis |ni

We also define the number operator


Several motional states and species

0

1

a b

+1


Number operators and commutators

We define number operators similar to before

The commutation relations are


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.


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


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 for a BEC this corresponds to assuming a coherent state of the atoms in a motional state described by destruction operator a.


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

Kinetic energy

E0

E1

+1

A particle gains energy by hopping betweendifferent states


Particle energies (II)

Interaction energy

n particles in the same state

Interactions between particles in different states

E0

E1

Each particle interactions with all particles inthe other state

E0

E1

Each particle interacts with n-1 particles in the same state


Example: Tunnelling (I)

Chain of atoms with kinetic energy and periodic boundaries

Hamiltonian

Introduce discrete Fourier transformed operators q 2 ]-,(N-1)/N, … 1]

with commutation relations

´


Example: Tunnelling (II)

Rewrite the Hamiltonian

The eigenstates are

with single particle eigenenergiesq

Blochband

excitations

4J

ground state

Eq


Example: Repulsive Interaction

A single quantum state with repulsive interaction and potential energy

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

E0

E1



Optical lattices and Hubbard models


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


Munich: I. Bloch, T. Haensch et al.


Optical lattices: Basics

AC – Stark shift

Spontaneous emission

|0i

|1i

laser

|0i

|1i

AC –Stark shift <{}

Spontaneous emission I{}¼

À 1

Spontaneous emission rates of less than 1s-1

shift:


Optical lattice: Basics

The dominant real part acts as a conservative potential V(x). For a standing wave laser configuration we obtain

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

k … laser wave vector

V0 … lattice depth / laser intensity


Lattice design

laser

square lattice

laser

triangular lattice

1D

2D

3D

different internal states


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


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)


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


Optical lattice

Described by a Hubbard model

Hopping term J and interaction U (s-wave scattering for bosons) are adjustable via the lattice depth

… destruction operator for an atom in lattice site

Aq are the momentum destruction operators

… number of particles in lattice site defined as

U

JV0


Hopping and interaction terms

10 -310 -2

5 15 25100101-110102

V /E0R

J/ERU aE aSR ER... recoil energy

a ... ground state size

a ... scattering length

V 0... depth of the optical potential

S

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


U

Hubbard picture:

¹h! t rap

trap levels

molecular levels

¹h! t rap

Microscopic picture: Two atoms in one well

scattering lengthtrap size

molecular picture:

trap

energy

0

internuclear separation


Changing the lattice potential

Shallow lattice: JÀU Deep lattice J ¿ U

U4JU

4J

D.J., C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).

M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39 (2002).


The Mott insulator– loading from a BEC

BEC phase J À U:

Mott insulator J ¿ U (commensurate):

Theory: D.J, et al. 1998, Experiment: M. Greiner, et al., 2002

J/U

Mott n=1

n=2

n=3

superfluid

/U

quantumfreezing

superfluid Mott

melting



Simulation of dynamic and static properties


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 |iUnitary evolution according to Hamiltonian (setting ~ =1)

Non-unitary evolution due to interaction with bath or collisions starting from an initial density operator


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

analyticalmean field theory (Gutzwiller)

numericalexact diagonalization

Quantum Monte Carlo

Improvements for the time-dependent case

analyticalmean field theory (Gutzwiller)

... ???

numericalexact time evolution for small systems

DMRG in 1D

... ???

standard (?) condensed matter

??


Allow for non-coherent state in each lattice site

Not number conserving ansatz

Remarks: not number preserving (i.e. the superfluid will have a phase)

number preserving version

Variational method

Time independent Gutzwiller

P N jG i

occupation

lattice site

contains the chemical potential to fix the mean number of particles hG|N|Gi.


0 1 2 3 nsuperfluid parameter

f (®)n

> 0 Mott phase

= 0 critical point²

hH i ²

minimum

µUzJ

¶

cr it¼ 5:8:::

Variation around the Mott state:

Mott insulator UÀJ


Superfluid phase U¿J

0 1 2 3 n

f (®)n

superfluid parameter

or

recover Gross-Pitaevskii equation


Time dependent Gutzwiller

Time-dependent ansatz

Variational method

Resulting equations

Only nearest neighbour hopping h,iJ,= J for h,i

J,= 0 otherwise superfluid parameter


In the superfluid limit JÀU

Gross Pitaevskii equation

coherent state

k

Blochband

BEC

excitations

Recover the GPE


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!


Time dependent DMRG (I)

System described by a state

and fix the maximum occupation as nmax.

Based on work of Vidal (2003, 2005), Verstraate & Cirac (2004), Werner (1990) for spins

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 [] n replaces fn() from Gutzwiller ansatz


Examples

A superposition state

is written as

with = 2-1/2, 1A=|1i, 2

A=|0i, 1B=|0i, 2

B=|1i

A superposition state

is written as

with = 1, 1A=|0i, 1

B=2-1/2(|0i+|1i)


State in site 2 State in site 3

CorrelationsCorrelations

State indexSite NumberSite Number

Time dependent DMRG (II)

Applying a series of Schmidt decompositions


Local quantum operations

Only the Gamma tensor for the corresponding site needs to be updated (O(2) basic operations)


Only the Gamma tensor for the corresponding sites and the lambda tensor in between need to be updated (O(3) basic operations)

Two site operations


Recovering the required form

Arrange into a matrix with row index nk,1 and column index nm,3

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 two-site 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.


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 :

― 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 grows at worst

logarithmically with the size at small energies

one and two site operations

Trotter parameters


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

L

A


Towards higher dimensions

MPS and MERA: G. VidalPEPS: F. Verstraete and J.I. CiracWGS: M. Plenio and H. Briegel


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 hi| can thus describe systems prepared in pure states (a ket |i) as well as in mixed states (kets |ii with probability pi)


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

centre for quantum physics & technology, clarendon laboratory, university of oxford. dieter...

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