ultracold atoms slides

QSimQuantum simulation with ultracold atoms

Lecture 1: Introduction to quantum simulation with ultracold atoms J. H. Thywissen

Lecture 2: Hubbard physics with optical lattices B. DeMarco

Lecture 3: Ultracold bosons in optical lattices: an overview A.-M. Rey

Lecture 4: Quantum simulation & quantum information I. Deutsch

APS Tutorial 7

Quantum simulation with ultracold atoms

Joseph H. ThywissenUniversity of Toronto

20 March 2011APS March meeting

Dallas, TX

Quantumsimulator?

An introduction to

Problem: what is the minimalsurface given fixed edges?

Soap filmsa simulation to find minimal surfaces.

Problem: what is the minimalsurface given fixed edges?

Answer: construct a wire grid and dip it in soap!

minimal surface for tetrahedral edges.

Answer precedes the explanation

• Lagrange: calculus of variations1760: poses minimal surface problem

• Plateau: soap film simulations, c.1840

• Initiates a “Golden Age” of mathematical study of minimal surfaces.

• Riemann, Weierstraß, Schwarz, others: fail to find answer to surfaces of least area.

• Douglas: solves in 1930. (Fields Medal ‘36)

J. A. F. Plateau1801-1883

Answer precedes the explanation

• Lagrange: calculus of variations1760: poses minimal surface problem

• Plateau: soap film simulations, c.1840

• Initiates a “Golden Age” of mathematical study of minimal surfaces.

• Riemann, Weierstraß, Schwarz, others: fail to find answer to surfaces of least area.

• Douglas: solves in 1930. (Fields Medal ‘36)

J. A. F. Plateau1801-1883

Plateau’s laws1. Smooth surfaces

2. Constant curvature

3. Soap films always meet in threes, and they do so at an angle of 120o, forming an edge (“Plateau Border”).

4. These Plateau Borders meet in fours at an angle of arccos(-1/3) to form a vertex.

What is simulation?

• Provides the answer to a mathematical problem or model

• Typically done (today) on a classical digital computer.

• Does not “solve” the model -- does not tell us why. (unlike calculation?)

• Empirical rules might be learned; and further simulations (various initial conditions, etc) could address questions.

• Experiment? Yes, but we know Hamiltonian.

QSimQuantum simulation with ultracold atoms

Lecture 1: Introduction to quantum simulation with ultracold atoms J. H. Thywissen

Lecture 2: Hubbard physics with optical lattices B. DeMarco

Lecture 3: Ultracold bosons in optical lattices: an overview A.-M. Rey

Lecture 4: Quantum simulation & quantum information I. Deutsch

APS Tutorial 7

OutlineI. What is Simulation?

II. Length scales

III.Example - strongly interacting fermions

Tutorial 7 slides online:http://ultracold.physics.utoronto.ca/QSim.html

Classical simulation of a quantum system

• Typically on a computer...a device that cannot be in a superposition or entangled state. {more about this in Lecture 4.}

• Methods typically used are• numerical integration of the Schrödinger Eq.

(or mean field extension, such as GP Eq.)• Monte Carlo (QMC) simulations

• However QMC fails* for many-body fermion problems, or excited states of bose systems, due to ‘sign problem.’

• Feynman: Use a quantum system to simulate another quantum system [1981]

*or has exponential scaling

Quantum simulation (QSim)

• When classical simulation is inefficient, using a quantum system may be the only option.

• Not universal quantum computing...eg, couldn’t factor a number.

• Certain models “natural” fits for atoms

★ Hubbard Model: optical lattices★ 1D models: extremely elongated traps★ 2D models: pancake traps★ Universal interactions:

unitarity-limited Fermi gas

Neutral atom Hamiltonian

• V: Inter-atomic potential is deep, complex, and unique to each atom pair)

• U: Trapping potential not reminiscent of textbooks, where we typically worked “in a box” (U=0)

H =!

dr !†(r)"! !2

2m"2 + U(r)

#!(r) +

12

!drdr! !†(r)!†(r!)V (r ! r!)!(r!)!(r)

How could this Hamiltonian be useful to simulate other systems?

R (nm)

Csinter-atomic potential,

1st simplification: low-energy limit• Dilute atoms scatter pair-wise, because their typical spacing

is much smaller than the potential range r0

• Below 0.1mK, atom pairs do not have enough E to overcome the p-wave centrifugal barrier

Two-body collision

m1 m2

R

V!(R) = V (R) + !2!(! + 1)/(2µR2)

! = 1

! = 0

R = n!1/3

For elastic scattering, must be

!!k("r ) = ei!k·!r ! a

1 + ika

eikr

r

The scattering term has an amplitude

plane wave

+

spherical wave

“scattering amplitude”fk = ![1/a + ik]!1

S-wave ( ) scattered wave function

from which you find the phase k

!1/a! ! = 4"|f!k(#n)|2 =

4"a2

1 + k2a2,

& cross-section

! = 0

For elastic scattering, must be

!!k("r ) = ei!k·!r ! a

1 + ika

eikr

r

The scattering term has an amplitude

plane wave

+

spherical wave

“scattering amplitude”fk = ![1/a + ik]!1

S-wave ( ) scattered wave function

Only one free parameter!

“scattering length” a

from which you find the phase k

!1/a! ! = 4"|f!k(#n)|2 =

4"a2

1 + k2a2,

& cross-section

! = 0

Pseudo-potential

• Two interaction potentials V and V’ are equivalent if they have the same scattering length

• So: after measuring a for the real system, we can model with a very simple potential.

g =4!!2

ma

V (!R) = g"(!R)

V (!R)Replace interaction potential with delta function!

where

V (!R) = g"(!R)#R(R ·)• Actually, to avoid divergences you need

“regularized”

Neutral atom Hamiltonian (revisited)

H =!

dr !†(r)"! !2

2m"2 + U(r)

#!(r) +

12

!drdr! !†(r)!†(r!)V (r ! r!)!(r!)!(r)

Can write V(..) as pseudopotential:

in limit of dilute ( ) and ultracold ( ).

V (!R) = g"(!R)#R(R ·)

R! r0

T ! 100µK

Neutral atom Hamiltonian (revisited)

H =!

dr !†(r)"! !2

2m"2 + U(r)

#!(r) +

12

!drdr! !†(r)!†(r!)V (r ! r!)!(r!)!(r)

What about the trap?

Can write V(..) as pseudopotential:

V (!R) = g"(!R)#R(R ·)

in limit of dilute ( ) and ultracold ( ).

R! r0

T ! 100µK

• What if a cold gas were a distribution of local creatures?

2nd simplification: Local chemical potential

snake line of ants

{scare the ant at the front of the line, and the last ant won’t rattle its tail...}

• What if a cold gas were a distribution of local creatures?

2nd simplification: Local chemical potential

snake line of ants

{scare the ant at the front of the line, and the last ant won’t rattle its tail...}

• Recipe:

µ !" µlocal = µ! U(!r)

Local chemical potential: “how to use your Stat. Mech. textbook”

• Thomas Fermi density profiles:

• ideal quantum gas functions:

• Similar Thomas Fermi expression for bosons:

n =1

6!2

!2mEF

!2

"3/2

nTF =(2m)3/2

6!2!3[EF ! U("r)]3/2

n = !!3T f3/2(z)

z = e!µ !" z = e!(µ!U("r))

for zero-temperature fermions in semiclassical limit.

at finite temperature ( ), where z=fugacity.! = 1/kBT

nTF =1g

[µ! U(!r)]µ = gn

textbook local µ

textbook local µ

textbook local µ

µlocal = µ! U(!r)

Validity of local chemical potential• A “local density approximation” (LDA).

• Not a good approximation when:-tunneling can occur through barriers-long-range order affected (eg, phase coherence)-gradients perturb states (eg, localized states [AM Rey])-long-range interactions (Coulomb etc)

• In those cases, QSim model must include trapping potential.

• However in some important cases works well: -important length scales (eg, Fermi length or lattice constant) much smaller than trap size-Far from edges, compared to healing length :

! = 1/!

8"na!2

2m!2= gn

!

such that

Cold neutral gases: length scales• inter-atomic potential range, r0: 2 nm

•

• thermal de Broglie wavelength: 100 nm

• average inter-particle spacing: 100 nm-same length scale as 1/kF

• lattice constant: 400 nm

• ground state width: 1µm @ 100Hz (typ. magnetic trap)100nm @ 10kHz (single site of optical lattice)

• cloud size: 1-100 µm

• scattering length, a-low-field (background) 5 nm-near a Feshbach resonance 100 nm to 1000 nm


•






Quantumdegeneracy



•






Quantumdegeneracy

Simulation space


Cold neutral gases: length scales (in traps)• inter-atomic potential range, r0: 2 nm

•



• ground state width: 1µm @ 100Hz (typ. magnetic trap)


Simulation space


QSim in local µ picture:en

ergy

position

U(r)

dens

ity

µ

position


ergy

position

U(r)

dens

ity

µ

position

uniform H, simulated with local µ & T.


ergy

position

U(r)

dens

ity

µ

position

uniform H, simulated with local µ & T.

bosons (for single component):

fermions (for 2-component gas):

H =!

!

!†!

"! !2

2m"2

#!! + g n!n"

H = !†!! !2

2m"2

"! +

g

2n2

Feshbach resonances

How can we tune the scattering length a?

We can tune a molecular bound state into resonance with the free atoms, and affect net phase acquired during the collision.

Result is indistinguishable from tuning the single-channel square well: it’s only the phase that matters.

Tune the square well potential & calculate a: V

b

Rpote

ntia

l

0 2 4 6 8 10!2

!1

0

1

2

3

b V

a!R

2. Mostly a>0. Near a resonance when a<0 (eg, Li.)

bV = (n + 1/2)!1. Resonances at

when each new bound state appears.

We find:

Feshbach resonancessingle-channel model

Example: 6Li

Feshbach resonances

a(B) = abg

!1! !

B !B0

"

Near resonance the scattering length can be described as

Eb =!2

2µa2

For a>0, a bound state exists with binding energy

!0 =4"

k2sin2 #0

s-wave cross section is

Length scales (in traps, @Feshbach res.)• inter-atomic potential range, r0: 2 nm



• scattering length, a-at Feshbach resonance: divergent



Simulation space

Length scales (in traps, @Feshbach, T=0)• inter-atomic potential range, r0: 2 nm





Simulation space

Length scales (in traps, @Feshbach, T=0)• inter-atomic potential range, r0: 2 nm





Simulation space

Only one length scale left in the problem! “Universal”

Unitarity limit: a >> R

• If the scattering length far exceed any physical length scale of the problem, it cannot be important.

• Inter-particle spacing d only length scale left: must determine all interaction energies!

• In fact, EF is the energy scale associated with d-for both fermions *and* bosons! [Ho 2004]-so restate this condition as

where for both bosons and fermions

a! k!1F

kF ! (6!n)1/3

You may be more familiar with the resonant atom-photon cross section (which has different constants because it is a vector instead of scalar field):

Cross section at unitarity

Near a Feshbach resonance, |a| diverges. The scattering cross section departs from its low-ka form:

! =4"a2

1 + k2a2! 4"

k2

!res = "2dB/#

!res =32"

#2L

This is just a manifestation of the optical theorem, which says that complete reflection corresponds to a finite scattering length. In terms of the de Broglie wavelength,

Quantum simulation at unitarity

For a many-body system, resonant interactions also saturate but are less easy to quantify. Certainly it is the case that a divergent a can no longer be a relevant physical quantity to the problem.

where has been measured in various experiments.

For fermions, the only remaining length scale is .k!1F

This means that interaction energies must scale with the Fermi E. In particular, for resonant attractive interactions,

µLocal = (1 + !)"F

! ! "0.58

µU =!

1 + !EF

! 0.65EF

for a! "#

Using the LDA to integrate over the profile, we find

Perspective: What can cold atoms teach us?

Traditional CM approach:

see phenomenon(eg, superconductivity)

search for theory(eg, BCS model)

Ultracold atoms:

knowHamiltonian

quantum many-body physics (eg, BEC)

APS March meeting: 10,000 CM physicists. 100-yr-oldfield (SC observed in 1911 by Kammerling-Onnes)

Quantum simulation with neutral atomsConclusion:

• Contact interaction when dilute and ultracold

• “Universal” (no dependence on interactions) when unitarity-limited

• Simulate uniform physics when LDA valid

• Single-band model for high lattice depths {next 2 lectures}

Emulation of simple models relies on a separation of length (or energy) scales.

µlocal = µ! U(!r)

T ! 100µK

a! k!1F

R ! k!1F " r0

Postdoc position available!

Bose-Fermimixtureexperiment

Site-resolved optical lattice experiment

Quantum simulation at the University of Toronto

Thank you!

AddendumHow to cool atoms?

Laser system

sympathetic cooling on a chip

Aubin et al, Nature Phys. (2006)

ultracold atoms slides

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