Download BEC and optical lattices

CLARENDON LABORATORY
PHYSICS DEPARTMENT
UNIVERSITY OF OXFORD
and
CENTRE FOR QUANTUM TECHNOLOGIES
NATIONAL UNIVERSITY OF SINGAPORE
Quantum Simulation
Dieter Jaksch
Outline

Lecture 1: Introduction


Lecture 2: Optical lattices


Analogue simulation: Bose-Hubbard model and artificial gauge fields.
Digital simulation: using cold collisions or Rydberg atoms.
Lecture 4: Tensor Network Theory (TNT)


Bose-Einstein condensation, adiabatic loading of an optical lattice.
Hamiltonian
Lecture 3: Quantum simulation with ultracold atoms


What defines a quantum simulator? Quantum simulator criteria. Strongly
correlated quantum systems.
Tensors and contractions, matrix product states, entanglement properties
Lecture 5: TNT applications

TNT algorithms, variational optimization and time evolution
Remarks

References to stock market models:



Melvin Lax, Wei Cai, and Min Xu, Random Processes in Physics and
Finance, Oxford University Press, 2006
Neil F. Johnson, Paul Jefferies, and Pak Ming Hui, Financial Market
Complexity, Publ. Oxford University Press, 2003
Brief introduction: T.H. Johnson, Non-equilibrium strongly-correlated
dynamics, DPhil thesis, Email: [email protected]
Remarks

Lattice systems/crystals: strong correlations can be achieved at any
density by quenching the kinetic energy
 Continuum systems/gases: for finite range interactions the system
will become weakly interacting if mean separation between particles
much larger than range of the interaction
 Thanks to Prof J. Walraven for pointing this out

Hubbard models
𝑎𝑖† 𝑎𝑗
𝐻 = −𝑡
𝑖,𝑗


𝑈
+ ℎ. 𝑐. +
2
𝑎𝑖† 𝑎𝑖† 𝑎𝑖 𝑎𝑖
𝑖
Landay theory fails for too strong interactions: perturbation series,
renormalization fails to converge
Universal behaviour is often encountered near phase transitions, this
motivates the study of toy models
Atoms
Rydberg
CMP
CMP cooled
Bio
s – ms
𝜇s – ns
ps – fs
ps – fs
ps - fs
Hz - kHz
MHz
THz
THz
THz
nK
nK
300K
mK
300K
1 − 10
104 − 106
1 − 10
104 − 106
1 – 10
Coherence 𝛾, 𝜅
Hz
kHz
THz
GHz
THz
Driving frequency Ω
kHz
MHz
THz
THz
THz
Experiment Time
Energy 𝐸/ℎ
Temperature 𝑇
Ratio 𝐸/𝑘𝐵 𝑇
200Hz ⟺ 10nK ⟺ 1peV
Quantum simulation
TWO PARTICLES IN TWO BOXES
2nd quantization formalism for bosons
Single particle Hilbert space ℋ with complete set of orthogonal functions 𝜙𝑖 .
We denote a quantum state of identical bosons with 𝑛𝑖 particles in the single
particle orbital 𝜙𝑖 by
| … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … 〉
We define annihilation operators 𝑎𝑖 which destroy a particle in the 𝑖-th orbital. It
act according to
𝑎𝑖 … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … = 𝑛𝑖 … 𝑛𝑖−1 , 𝑛𝑖 − 1, 𝑛𝑖+1 …
Similarly operators 𝑎𝑖† create a particle in the 𝑖-th mode and act according to
𝑎𝑖† … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … =
𝑛𝑖 + 1 … 𝑛𝑖−1 , 𝑛𝑖 + 1, 𝑛𝑖+1 …
Their commutation relations are those of harmonic oscillator ladder operators
𝑎𝑖 , 𝑎𝑖† = 𝛿𝑖,𝑗
The combination 𝑛𝑖 = 𝑎𝑖† 𝑎𝑖 counts the number of particles in orbital 𝑖.
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
2nd quantization formalism for bosons
It acts according to
𝑛𝑖 … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … = 𝑛𝑖 … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 …
A general pure many body state of the system is given by a coherent
superposition of configuration states
Ψ =
𝑐 𝑛𝑖 | 𝑛𝑖 〉
𝑛𝑖
Mixed states are build as incoherent mixtures of pure states as usual
𝜌=
𝑑𝑖𝑗 Ψ𝑖 〈Ψj |
𝑖,𝑗
It is also useful to introduce the vacuum state 𝑣𝑎𝑐 = 0,0,0,0, ⋯ = |0〉
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Particle energies (I)
Potential energy
E1
E0
Kinetic energy
A particle gains energy by hopping between
different states
a
a+1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
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
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Each particle interacts with all particles in
the other state
Two atoms in two sites
Hamiltonian
Basis states
L
R
|20〉
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
L
R
|11〉
L
R
|02〉
Hamiltonian and ground state
Matrix form
Ground state
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Density matrix
The density matrix corresponding to the ground state wave function is
𝜌 = Ψ 〈Ψ|
For the case 𝐽 = 0 and 𝑈 > 0 this density matrix is diagonal in the single
particle orbitals 𝜙𝐿 and 𝜙𝑅 given by
𝜌 = 1,1
𝐿𝑅
1,1 =
0 0
0 1
0 0
0
0
0
For the case 𝑈 = 0 and 𝐽 > 0 the density matrix is diagonal in a different basis
given by symmetric and antisymmetric linear combinations ∝ 𝜙𝐿 ± 𝜙𝑅
1 0 0
𝜌 = 2,0 ± 2,0 = 0 0 0
0 0 0
For general values of 𝐽 and 𝑈 no such single particle matrix exists.
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Single particle density matrix
This is defined by
𝜌1 =
𝑎𝐿† 𝑎𝐿
𝑎𝑅† 𝑎𝐿
𝑎𝐿† 𝑎𝑅
𝑎𝑅† 𝑎𝑅
For the ground state
𝜌1212
1.0
0.8
0.6
0.4
0.2
𝑈
J
𝐽
U
0
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
5
10
15
20
Particle number fluctuations
How well defined is the particle number in each site?
2
Δ𝑛
n2
0.5
0.4
0.3
0.2
0.1
𝑈
J
𝐽
U
0
5
10
This is given by Δ𝑛2 = 𝑛𝐿2 − 𝑛𝐿
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
15
2
20
Compare: Mean field approximation
Assume the particles are not correlated
Ψ = 𝑐𝐿 𝐿 1 + 𝑐𝑅 𝑅 1 ⊗ 𝑐𝐿 𝐿 2 + 𝑐𝑅 𝑅 2
Ψ = 𝑐𝐿2 𝐿𝐿 + 𝑐𝐿 𝑐𝑅 ( 𝐿𝑅 + 𝑅𝐿 ) + 𝑐𝑅2 |𝑅𝑅〉
With this ansatz the Schrödinger equation (obtained from the principle of least
action) becomes non-linear in the remaining coefficients
𝐻=
𝑐𝐿 2 𝑈
−𝐽
−𝐽
𝑐𝑅
2𝑈
Ψ𝑀𝐹
𝑐𝐿
= 𝑐
𝑅
The resulting approximate ground state is not correlated and describes one
particle moving in a mean background field created by the other particle
Dimension of vectors increases linearly with size.
|𝐿〉
|𝑅〉
Reduction of degrees of freedom is achieved by
ignoring correlations and introducing non-linearity.
Often valid for sufficiently weak interactions
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Particle number fluctuations
Δ𝑛𝐿
1
Mean field approach
0.5
Exact solution
0
5
10
𝑈/𝐽
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Limit of large interactions U
Mean field theory gives (independently of 𝑈)
1
Ψ =
𝐿
2
1
+ 𝑅
1
⊗ 𝐿
2
+ 𝑅
2
Compared to the exact solution
Ψ ∝ 𝐿
1
⊗ 𝑅
2
+ 𝑅
1
⊗ 𝐿
2
Mean field theory predicts wrongly
on-site particle number fluctuations
Particle-particle correlations
Coherence
Superfluidity
We need to account for correlations when interactions are strong.
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Quantum simulation
BOSE-EINSTEIN CONDENSATION
Bose-Einstein condensation (BEC)
Prediction 1924 by Bose and Einstein
Helium
Strong interactions shield BEC
Alkali gases
Weak interactions
Small temperature
Mean field description
99% of particles in same orbital
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Single particle orbitals and Hamiltonian
Single particle Hamiltonian
𝒑2
𝐻1 =
+𝑉 𝒙
2𝑚
Single particle orbitals (ordered by ascending energy)
𝐻1 𝜙𝑖 𝒙 = 𝜖𝑖 𝜙𝑖 𝒙
Many body Hamiltonian using operators defined through 𝜙𝑖
𝜖𝑖 𝑎𝑖† 𝑎𝑖
𝐻𝑖𝑑 =
𝑖
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Ideal Bose gas
Define field operator
𝜓 𝑥 =
𝜙𝑖 𝑥 𝑎𝑖
𝑖
Hamiltonian
𝐻𝑖𝑑
2
𝒑
= ∫ d𝒙 𝜓 † 𝒙
+ 𝑉 𝒙 𝜓(𝒙)
2𝑚
Field operator commutation relation
𝜓 𝒙 , 𝜓 † 𝒙′
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
= 𝛿(𝒙 − 𝒙′ )
Eigenstates
A general eigenstate
And the ground state in this notation is
With 𝑁 the total number of particles in the system
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Symmetry breaking
Particle number conserving ground state
Ψ0 = 𝑁, 0,0, ⋯ =
† 𝑁
𝑎0
𝑁!
|𝑣𝑎𝑐〉
Symmetry breaking coherent state by 𝑎0 → 𝑁 = 𝛼
𝛼 =
𝛼2
− 2 𝛼 𝑎0†
𝑒
𝑒
𝑣𝑎𝑐 =
𝛼𝑛
𝛼2
𝑒− 2
𝑛
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
𝑛!
|𝑛, 0,0, ⋯ 〉
Symmetry breaking
There is no physical reason for choosing 𝛼 real or to
believe that each realization gives the same phase
1
𝜌=
2𝜋
2𝜋
𝑑𝜑 |𝛼 𝑒 𝑖𝜑 〉〈𝛼𝑒 𝑖𝜑 |
0
This density matrix is identical to the mixture
𝜌=
𝑝 𝑛 𝑛, 0,0,0, ⋯ 〈𝑛, 0,0,0, ⋯ |
𝑛
Here 𝑝 𝑛 is a Poisson distribution with mean 𝛼
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
2
BEC?
Need to consider single particle density matrix
〈𝑎0† 𝑎0 〉
𝜌1 = Tr2,…,N 𝜌 = 〈𝑎† 𝑎 〉
1 0
⋮
〈𝑎0† 𝑎1 〉
…
〈𝑎1† 𝑎1 〉
⋮
…
⋱
Does 𝜌1 have an eigenvalue of order N in a thermal state
with finite temperature  BEC
Depends on the density of single particle orbitals
Interactions deplete the BEC
Photons are usually not considered a BEC
Laser: not a thermal state
Black body radiation: photons emitted and absorbed
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Isotropic harmonic trap - recap
Energy
Degeneracy
𝑗ℏ𝜔
𝑗+1 𝑗+2
2
⋮
⋮
2ℏ𝜔
6
ℏ𝜔
3
0
1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Thermal properties – harmonic trap
Total number of particles
Thermal population with 𝛽 = 1/𝑘𝑇
Ground state population
𝑍
𝑛0 =
1−𝑍
Limits on the values of the fugacity 0 ≤ 𝑍 = exp 𝛽𝜇 ≤ 1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Thermal properties – harmonic trap
Take out 𝑛0 and use continuum approximation for rest
∞
𝑁 = 𝑛0 +
0
𝑗 2 𝑛𝑗
𝑑𝑗
2
Integrals gives the Bose function (PolyLog in Mathematica)
𝑔3 𝑍
𝑁 = 𝑛0 +
𝛽ℏ𝜔
𝜁 3/2 ≈ 2.61
3
𝑔1/2
𝑔3/2
𝑔3 1 = 𝜁 3
𝜁 3 ≈ 1.202
𝑔3
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Thermal properties – harmonic trap
Critical temperature
Fraction of particles in the ground state
Appropriate statistical ensemble?
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Two particle interactions
Molecular potential
‘Approximation’ by a contact potential
Define
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Physical interpretation contact interaction
Molecular interaction potential
𝑎𝑠 > 0
𝑎𝑠
𝑎𝑠
Valid for
𝑛𝑎𝑠3
≪1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
𝑎𝑠 < 0
The Gross-Pitaevskii equation
Hamiltonian
Assume ground state to be still of the form
And minimize the expected energy to find orbital
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
The Gross-Pitaevskii equation
Write
Minimize
To find
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Thomas-Fermi approximation
Neglect the kinetic energy
𝜑0 𝒙 =
𝜇−𝑉 𝒙
𝑁𝑔
0
for 𝜇 > 𝑉(𝒙)
for 𝜇 < 𝑉(𝒙)
1
Valid if the so-called healing length 𝜉 =
8𝜋𝑎𝑠 𝑁 −2
𝑅3
is much
smaller than the size of the condensate 𝑅, i.e. 𝑅 ≫ 𝜉.
For harmonic trap an inverted parabola. The condensate size is
given by 𝑉 𝑅 = 𝜇.
𝑉(𝒙)
𝑛 𝒙 = 𝜑0 𝒙
2
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
𝑅
Time dependent GPE
Carry out minimization over action
Note:
No chemical potential
Mean field nonlinear Schroedinger equation
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Quantum simulation
OPTICAL LATTICES
Recap: Trapped bosonic atoms, Hamiltonians
Atoms in a given internal state |𝑎〉 are described by a bosonic field operator
𝜓 𝒙 which fulfils canonical commutation relations 𝜓 𝒙′ , 𝜓 † (𝒙) = 𝛿 𝒙 − 𝒙′ .
The single particle Hamiltonian contains a kinetic and the trapping energies
𝐻0 = ∫ d𝒙
𝜓†
𝒑2
𝒙
+ 𝑉 𝒙 𝜓(𝒙)
2𝑚
The particles interact via a contact type potential whose effect (at low
temperatures) can be described by a Dirac Delta function
𝑉
𝒙, 𝒙′
4𝜋𝑎𝑠 ℏ2
=
𝛿(𝒙 − 𝒙′ )
𝑚
This leads to the interaction Hamiltonian of the form
𝐻int =
1
∫ d𝒙d𝒙′ 𝜓 † 𝒙 𝜓 † 𝒙′ 𝑉 𝒙, 𝒙′ 𝜓 𝒙 𝜓 𝒙′
2
2𝜋𝑎𝑠 ℏ2
=
∫ d𝒙 𝜓 † 𝒙 𝜓 † 𝒙 𝜓 𝒙 𝜓 𝒙
𝑚
Here 𝑎𝑠 is the s-wave scattering length. The total Hamiltonian is 𝐻 = 𝐻0 + 𝐻int
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
The optical lattice potential
Periodic Potential created by a standing wave
Near the potential minimum
Period of half a wave length
𝜆
𝑎=
2
See Lecture Course by Prof Helene Perrin
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Bloch functions
Bloch wave functions
Where 𝑢𝑞𝑛 are periodic eigenfunctions of
With energies of
See Lecture course by Prof Pierre Clade
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
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 ∝ cos(q)
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
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.
See e.g. R. Walters et al., Phys. Rev. A 87, 043613 (2013) and reference
therein.
At small temperatures only the lowest Bloch band n=(0,0,0) will be occupied
Wannier
functions
See Lecture course by Prof Pierre Clade
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Wannier functions leaking into other sites
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.