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

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

After one more discussion with Prof J. Walraven

The continuum argument only holds in 3D (three spatial dimensions)
 1D Tonks Girardeau gas is an example of a system with increasingly strong
correlations as the density is reduced
 In 2D the situation depends on the details of the interaction
Analogue quantum simulation
THE BOSE HUBBARD MODEL
The Bose-Hubbard Hamiltonian
Occupy lowest band only
Substitute into Hamiltonian
With parameters
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Dominant contributions
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Hopping and interaction terms
2
-1
10
U ag
10
J/E R
ER aS
-2
1
10
10
-3
0
10 5
15
V0 /ER
10
25
E R ... recoil energy
ag ... ground state size
a S ... scattering length
V0 ... depth of the
optical potential
Recoil energy: 𝐸𝑅 = ℏ2 𝑘 2 /2𝑚
Na: 𝐸𝑅 ≈ 25 kHz
Rb: 𝐸𝑅 ≈ 3.8 kHz
Validity:
only lowest Bloch band occupied
𝑛 𝑎𝑠3 ≪ 1, i.e. low density, weak interactions
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Tight binding Hubbard model
Keeping dominant terms
Where 𝜖𝑖 arises from a background trap
The chemical potential 𝜇 fixes the average particle number
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Changing the lattice potential
Shallow lattice: 𝐽 ≫ 𝑈
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).
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
U
M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and
I. Bloch, Nature 415, 39 (2002).
Limit 𝐽 → 0
The Hamiltonian is diagonal in the particle number basis of each lattice site
We can consider a single site only
𝑈 † †
𝐻 = 𝑎 𝑎 𝑎𝑎 − 𝜇 𝑎† 𝑎
2
We now minimize the energy expectation value
𝐸 𝑛 = 𝑛𝐻𝑛
and find
𝑈
𝑈
𝐸 𝑛 = 𝑛2 −
+𝜇 𝑛
2
2
by evaluating when Δ𝐸𝑛 = 𝐸 𝑛 + 1 − 𝐸 𝑛 becomes negative we obtain the
value of 𝜇/𝑈 above which it becomes energetically favourable to add another
atom to a site
Δ𝐸 𝑛 = 𝑛𝑈 − 𝜇
Thus the lattice site occupation increases by one for each unit of 𝜇/𝑈
Ψ𝑀𝐼 = | 𝑛, 𝑛, 𝑛, 𝑛, 𝑛, 𝑛, ⋯ ⟩ where 𝑛 = 𝜇/𝑈
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Limit 𝑈 → 0
We consider a 1D lattice with 𝑀sites and periodic
boundaries
We rewrite the Hamiltonian
𝐻 = −𝐽
†
†
(𝑎
𝑎
+
𝑎
𝑗 𝑗 𝑗+1
𝑗 𝑎𝑗−1 )
using the operators
†
𝑎𝑞
=
1
𝑀
4𝐽
i 𝑞𝑗 †
𝑎𝑗
𝑗𝑒
and find
𝐻 = −𝐽
𝑞
†
cos
𝑞
𝑎
𝑞 𝑎𝑞
𝑗
with energies
−2 𝐽 cos 𝑞 = 𝐸𝑞
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max 𝐸𝑞 − min(𝐸𝑞 )
𝐽=
4
Limit 𝑈 → 0
Where 𝑀 is the number of sites in the lattice
The 𝑁 particle ground state is then given by
𝑁
†
Ψ0 ∝ 𝑎𝑞=0 |vac⟩
4𝐽
with mean occupation per site
𝑛𝑖 =
𝑁
𝑀
and particle number fluctuations
𝑁
†
†
2
Δ𝑛𝑖 = Ψ0 𝑎𝑖 𝑎𝑖 𝑎𝑖 𝑎𝑖 Ψ0 −
𝑀
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𝑞
2
𝑁 𝑀−1
𝑁
=
≈
2
𝑀
𝑀
Gutzwiller ansatz
The symmetry broken version of the 𝑈 = 0 ground state
is
Ψ𝛼 =
𝛼2
†
− 2 𝛼 𝑎𝑞=0
𝑒
𝑒
𝑣𝑎𝑐 =
𝛼2
†
− 2 𝛼 𝑗 𝑎𝑗
𝑒
𝑒
𝑣𝑎𝑐
This is equivalent to a product of coherent states
𝛼
Ψ𝛼 =
𝑀
𝑗
𝑗
Gutzwiller ansatz extends this by allowing non-coherent
states in each site
𝐺 =
𝑗
𝑗
𝑛 𝑓𝑛 𝑛
There are no correlations
between sites
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
𝑗
𝑗
Superfluid phase 𝑈 ≪ 𝐽
Minimization of the energy with the Gutzwiller wave
function
𝐺 𝐻 𝐺 → min
yields parameters close to the coherent state wave
function
(𝑗)
𝑓𝑛
𝑎𝑗 ≠ 0 superfluid parameter
0
1
2
3
n
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Mott insulator 𝑈 ≫ 𝐽, commensurate filling
(𝛼)
𝑓𝑛
0
1
2
3
𝑎𝑗 = 0 superfluid parameter
n
Variation around the Mott state, 𝑧 nearest neighbous:
𝐻
𝐺 ∝
𝜖
𝜖0𝑗+ 1𝑗+ 𝜖2
𝑗
𝑗
> 0 Mott phase
𝜖
minimum
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
= 0 critical point
𝑈
𝑧𝐽
≈ 5.8
crit
The Mott insulator– loading from a BEC
Theory: D.J, et al. 1998, Experiment: M. Greiner, et al., 2002
quantum
freezing
BEC phase 𝐽 ≫ 𝑈:
super
fluid
Mott
melting
/U
†
Ψ ∝ 𝑎𝑞=0
𝑁
|vac⟩
Mott insulator 𝐽 ≪ 𝑈 (commensurate):
n=3
n=2
superfluid
Mott n=1
𝑧𝐽
𝑈
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
𝑧𝐽/𝑈
𝑐𝑟𝑖𝑡
Analogue quantum simulation
ARTIFICIAL GAUGE FIELDS
Introduction: Huge magnetic fields

Effect of a magnetic field

Resulting energy spectrum
1
® 0.5

c =1/2
c =1/3

The wave function accumulates
a phase characterized by  when
hopping around a plaquette.
 Phase proportional to enclosed
magnetic flux
0
-4
0
" /J
/J
4
Ultracold atoms in rotating lattices

Effective magnetic field via rotation





N.K. Wilkin et al. PRL 1998
B. Paredes et al. PRL 2001
Experiment: J. Dalibard, ENS
Experiment: C. Foot, Oxford
Alternative ways for realizing artificial magnetic fields, e.g.
 A.S. Sorensen et al. PRL 2005
 G. Juzeliunas et al. PRL 2004
 E.J. Mueller, PRA 2004
 D.J et al. New J. Phys. 2003
Magnetic field vs rotating system

Hamiltonian of the form
with vector potential
z-direction leading to terms



for a magnetic field along the
plus potential terms.
Compare with system of neutral particles rotating around z-axis
with angular momentum operator
In both cases a force orthogonal to the direction of motion acts on the
particle.
Quantum mechanically this leads to an energy and thus phase
difference when one and the same path is travelled in two different
directions  broken symmetry.
Artificial magnetic field on a lattice

For a lattice geometry rotation or a magnetic field leads to the following
properties

When hopping from one lattice site
to the next a phase is acquired.
 When a closed path is travelled the
wave function should get a phase
proportional to the surrounded area
(i.e. the enclosed flux).

When discretizing the Hamiltonian a
Peierl’s transformation can be used to
bring the Hamiltonian into a form which
obviously fulfils these properties
(in Landau gauge)
Energy bands

Fractal energy bands
Hofstaedter butterfly

Investigate magnetically induced
effects
® 0.5

 quantum Hall effect



1
c =1/2
fractional quantum Hall effect
Atomic systems allow detailed
study of the energy bands
Interaction effects are
controllable
c =1/3
0
-4
0
" /J
/J
The optical lattice setup allows to explore exotic parameter regimes
 novel effects?
4
Alternative methods?

Rotating the lattice creates centrifugal terms in the potential part of
the Hamiltonian
r

These need to be precisely balanced by a trapping potential which is
experimentally difficult

Use alternative methods to create an artificial magnetic field

Laser induced hopping along the x direction
DJ et al., New J. Phys. 5, 56 (2003).

By immersing the lattice into a rotating BEC
A. Klein and DJ, EuroPhys. Lett. 85, 13001 (2009).
Laser induced magnetic field

Two component optical lattice


trapping two internal states in different columns
The polarization of the lasers determines the position of the lattice sites
U
J

eg
…
…
…
…
onsite interactions
hopping rate
trap potential
atomic energy difference
Acceleration, laser induced hopping



Acceleration or inhomogeneous electric field yielding offset 
Apply two Raman lasers with detunings  and Rabi frequency 1,2
which induces hopping along x direction
The phases 1,2= eiqy of the lasers determine the phase
imprinted on the atoms
Raman lasers
Resulting setup
Laser imprinted hopping phases
Φ
-Φ
𝑡
Ω1
Ω2
M Aidelsburger et al., PRL. 2011
z
𝜃
𝑦
y
𝑥
Proposal: DJ et al., New J. Phys. 2003
x
K. Jimenez-Garca et al., PRL. 2012
Realizing flux 𝜋/2 per unit cell
M. Aidelsburger et al.,
arXiv:1407.4205
M. Aidelsburger et al.,
PRL 111, 185301 (2013)
H. Miyake et al.,
PRL 111, 185302 (2013)
Digital quantum simulation
GATE OPERATIONS
Two-level atoms and their manipulation
Single atom as a two level
system
|0⟩
|1⟩
Single qubit manipulation
laser
qubit in long-lived
internal states
|0⟩
|1⟩
addressing
qubits
Use hyperfine states e.g. 87Rb
|1⟩
F=2
|0⟩
F=1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
laser
Focussing the laser to a single
atom position is challenging
Controlled interactions
DJ et al. PRL 82, 1975 (1999); DJ et al. PRL 85, 2208 (2000)
V(R)
atoms
Cold controlled collisions
design 𝐻 = Δ𝐸 𝑡 𝑔
to obtain 𝑔
1
𝑒
2
1
𝑔 ⊗ 𝑒 2 ⟨𝑒|
→ 𝑒 𝑖𝜙 𝑔
1
𝑒
2
Rydberg atoms
𝐸 ≈ 1kV/cm, 𝑅 = 𝜆𝑜𝑝𝑡 2 ≈ 300nm
Δ𝐸 ≈ 60GHz
Udip ≈ 4GHz
Exp: Bloch, Greiner, 2002/03
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Exp: Grangier, Saffman, 2008/09
Engineering a Cluster-state
0i 0
i
i+1
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i1
Engineering a Cluster-state
1
2
i
0
i+1
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i
1i
 0
i 1
 1 i 1

Engineering a Cluster-state
i
1
2
0
i
1
2
0
i
i+1
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0
i 1
1i
 0 i 1 i 2  1 i 1 0
 0
i 1
 1 i 1

i 1
 1 i 1 1 i 2

Engineering a Cluster-state
i
1
2
0
i
1
2
0
i
i+1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
0
i

0
1

e
1 i 1 0
i 1
i
i2
1i
 0
i 1
 1 i 1

i 1
 1 i 1 1 i 2

Engineering a Cluster-state
i
1
2

1
2
0
i
1
2
0
i
i+1
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
0i 0
0
i

0
1

e
1i 0
i 1
i
i 1
i 1
i

0
1

e
1 i 1 0
i 1
i
i2
1i
 0
i 1
 1 i 1

 1 i 1 i 1
i 1

 1 i 1 1 i 2

Engineering a Cluster-state
i
1
2
(1  ei ) Bell  12 (1  ei ) 1 i 1 i 1
1
2
0
i
1
2
0
i
i+1
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0
i

0
1

e
1 i 1 0
i 1
i
i2
1i
 0
i 1
 1 i 1

i 1
 1 i 1 1 i 2

Dipole-dipole interactions
Excite atoms to high lying states with large electron orbit
Electric Field
Apply electric field to induce large dipoles
Dipole-dipole interaction potential (atomic units)
Large molecules bound by this interaction can be formed for large 𝑛
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Rydberg atoms: Internal states
|𝑟⟩
|𝑟⟩
1
|𝑒⟩
|𝑔⟩
Atom 1
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2
|𝑒⟩
|𝑔⟩
Atom 2
Fast phase gate - excitation
|𝑟𝑟⟩
U
|𝑟𝑒⟩
|𝑒𝑟⟩
|𝑟𝑔⟩
1
|𝑒𝑒⟩
1
|𝑔𝑒⟩
Laser pulse:
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|𝑔𝑟⟩
|𝑒𝑔⟩
|𝑔𝑔⟩
1
2
1

2

Fast phase gate - blockade
|𝑟𝑟⟩
No excitation!
|𝑟𝑒⟩
|𝑒𝑟⟩
2
|𝑒𝑒⟩
|𝑔𝑒⟩
Laser pulse:
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U
|𝑟𝑔⟩
2
„-“
|𝑒𝑔⟩
|𝑔𝑔⟩
1
2
1

2

|𝑔𝑟⟩
Fast phase gate – de-excitation
𝑔𝑔
𝑔𝑒
𝑒𝑔
𝑒𝑒
|𝑟𝑒⟩
1
|𝑒𝑒⟩
|𝑒𝑟⟩
1
|𝑔𝑒⟩
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
U
|𝑟𝑔⟩
„-“
Laserpuls:
|𝑟𝑟⟩
→ −|𝑔𝑔⟩
→ −|𝑔𝑒⟩
→ −|𝑒𝑔⟩
→ +|𝑒𝑒⟩
|𝑒𝑔⟩
|𝑔𝑟⟩
„-“
|𝑔𝑔⟩
1
2
1

2

Adiabatic gate – no addressing
|𝑟𝑟⟩
detuned by
large interaction
|𝑟𝑒⟩
|𝑒𝑒⟩
|𝑒𝑟⟩
U
𝑟𝑔 + |𝑔𝑟⟩
𝛿 𝑡
𝛿 𝑡
𝛿 𝑡
Ω 𝑡
Ω 𝑡
2Ω 𝑡
|𝑔𝑒⟩
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|𝑒𝑔⟩
|𝑔𝑔⟩
Dressed states picture
Ω ≪ 𝑈  adiabatically eliminate the state |𝑟𝑟⟩
dressed states:
resulting interaction phase:
𝑡
𝑑𝑡 ′ (𝜖𝑔𝑔 𝑡 ′ − 2 𝜖𝑒𝑔 𝑡 ′ )
𝜑 𝑡 =
0
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Optical addressing
High resolution optical imaging systems
Strathclyde: S Kuhr et al.
Oxford: C Foot et al.
Bonn: D Meschede et al.
Harvard: Greiner et al.
Munich: Bloch et al.
Greiner Lab, Science 2010
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Bloch Lab, Science 2011
Single site addressing
Scanning electron microscopy
Mainz: H Ott et al.
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Quantum simulator criteria
J. Ignacio Cirac and Peter Zoller, Nature Physics 8, 264 (2010)
Quantum system
Large number of degrees of freedom, lattice system or confined in space
Initialization
Prepare a known quantum state, pure or mixed, e.g. thermal
Hamiltonian engineering
Set of interactions with external fields or between different particles
Interactions either local or of longer range
Detection
Perform measurement on the system, particles individually or collectively.
Single shot which can be repeated several times.
Verification
Increase confidence about result, benchmark by running known limiting
cases, run backward and forward, adjust time in adiabatic simulations.
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Quantum simulation
DETAILS (IF TIME PERMITS ONLY)
State selective potential
Lin angle Lin laser configuration
Electrical field
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Atomic Level Structure
Alkali atoms
Qubit states
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Stark shift
Fine structure shift
Hyperfine structure (Clebsch Gordon)
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Moving harmonic potentials
Retain motional ground state
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Accumulated phases
Kinematic (dynamic) phase
Interaction (entangling) energy shift
Entanglement phase
𝜙 𝑎𝑏
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𝑖
=−
ℏ
𝑡
Δ𝐸 𝑎𝑏 𝑡 ′ 𝑑𝑡 ′
0
Evolution truth table
Resulting evolution
Ignore 𝜙 𝑎 and 𝜙 𝑏 as trivial single particle phases
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
Gate fidelity
Consider entanglement between motional and internal
degrees of freedom as source of infidelity
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.