BEC-BCS Crossover in Cold Atoms · Outline Theory Previous Work Our method (Results) Future work -...

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BEC-BCS Crossover in Cold Atoms Andrew Morris Theory of Condensed Matter Cavendish Laboratory University of Cambridge

Transcript of BEC-BCS Crossover in Cold Atoms · Outline Theory Previous Work Our method (Results) Future work -...

Page 1: BEC-BCS Crossover in Cold Atoms · Outline Theory Previous Work Our method (Results) Future work - Cold Atoms - BEC-BCS Crossover - Feshbach Resonance - Universal number, ξ - Modelling

BEC-BCS Crossover in Cold Atoms

Andrew Morris

Theory of Condensed MatterCavendish Laboratory

University of Cambridge

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Outline

● Theory

● Previous Work

● Our method

● (Results)

● Future work

- Cold Atoms

- BEC-BCS Crossover

- Feshbach Resonance

- Universal number, ξ

- Modelling the interaction

- Pairing wavefunctions

- Total Energy

- Condensate fraction

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Theory

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Cold Atoms

● Bose Gas

● Fermi Gas

● Vary the interaction strength between fermionic atoms...

- BEC (1995)

- Quantised Vortices

- Propagation of solitons

- e.g. 6Li, 40K, 2H

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BEC-BCS crossover

● Strong pairing :

● Weak pairing :

● Interesting point at unitarity :

● How would this occur?

- Atoms form molecules of up and down spin

- These molecules are bosonic

- Bosonic molecules condense into BEC

- Atoms interact over a long range

- BCS theory

- Dilute : Interatomic potential range << Interparticle distance

- Strongly interacting : Scattering length >> Interparticle distance

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Feshbach Resonance

● 2 channels corresponding to different spin states

● Open channel (scattering process)

● Closed channel (bound state)

● Resonance occurs when Open and Closed channel energies are close

● Channel energies are tuned by a magnetic field

From Giorgini et al eprint cond-mat 0706.3360v1

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Feshbach Resonance (2)

Resonances in 6Li from Bourdel et al PRL 93 050401

● The s-wave scattering length, a, diverges at resonance

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Universal Number, ξ

● At resonance the only relevant energy scale is that of a non-interacting gas

● This value, ξ, is believed to be universal when kFR0 << 1 where R0 is the effective range of interaction

● Throughout we measure the interaction strength in units of 1/kFa

E I= EFG=3k F

2

10m

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Previous Work

● 2 previous studies using QMC,

● Other methods,

- J. Carlson et al, PRL 91 050401: = 0.44(1)ξ

- G.E. Astrakharchik et al, PRL 93 200404: = 0.42(1)ξ

- Nishida et al: eprint cond-mat/0607835 = 0.38(1)ξ

Symbols - QMC(Dot - Bogoliubov

Dot-Dashed BCS theory

Black line SC-MF)

From Astrakharchik et al PRL 95 230405.

BEC BCS

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This Work

● Unequal particle numbers / unequal masses

r = m↓/m↑n = density of particlesm = magnetisation

From Parish et al, PRL 98 160402 (2007)

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Normal Phase

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Phase Separated

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Magnetised Superfluid

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This Work

● Unequal particle numbers / unequal masses

r = m↓/m↑n = density of particlesm = magnetisation

From Parish et al, PRL 98 160402 (2007)

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The Model

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Modelling the Feshbach Resonance

● Pauli-Exclusion Principle for parallel spin

●As interatomic potential << atom spacing, the exact form of the interaction is unimportant

● 2 types of interaction normally used...

● We use the Pöschl-Teller

Pöschl-TellerSquare Well

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Pairing Wavefunctions

● In QMC for a spin-independent operator we normally use a product of Slater determinants, one containing n up-spin and one m, down-spin one-particle orbitals, φ.

● However, we want a wave function that explicitly describes pairing

=eJ∣1r1 ⋯ nr1

⋮ ⋱ ⋮1rn ⋯ nrn

∣∣1r1 ⋯ mr1

⋮ ⋱ ⋮1rm ⋯ mrm

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Pairing Wavefunctions (2)

=eJ∣r1 −r1 ⋯ r1 −r n

⋮ ⋱ ⋮rn −r1 ⋯ r n −rn

∣● We now use only one Slater determinant. It contains only one type of orbital, φ, which is a function of the distance between up and down particles

● 3-types of φ have been tried

● And combinations of the above

=∑i=1

C i expi r −r

=∑i=1

g i exp i r −r 2

=∑i=0

i r −r i

- Pairing Plane-waves

- Pairing Gaussians

- Pairing Polynomials

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Jastrow factor + Backflow

● Jastrow factor of Drummond et al PRB 70 235119 (2004) (CASINO users, that's a Jastrow U + P)

● Backflow corrections of López Ríos et al PRE 74 066701 (2006)

● We optimise the Jastrow, Backflow and orbital parameters using VMC and energy minimisation (Umrigar et al PRL 98 110201 (2007))

● Conclude using DMC

J=∑l=1

L

l r ijl∑

A

aA∑GA

cosGA⋅r ij

BF R =eJ R s X

x i=r ii R

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Pairing Wavefunctions (3)

● Polynomial of order 20 + Jastrow U (Polynomial) EVMC = 0.650(4)

● Gaussian EVMC = 0.664(2)

● Gaussian + Jastrow U EVMC = 0.5195(6)

● Gaussian + Jastrow U + Backflow (Eta) EVMC = 0.4745(1)

● Gaussian + Jastrow U + Backflow (Eta) + Jastrow P EVMC = 0.4605(2)

EDMC ~ 4% lower

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Results

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Present state

● Testing our ideas and CASINO with reproducing the value of ξ

● Condensate fraction: ~ 0.4 – not right yet!

- J. Carlson et al PRL 91 050401: = 0.44(1)ξ

- G.E. Astrakharchik et al PRL 93 200404: = 0.42(1)ξ

- This work (to date) VMC ~ 0.45ξ

DMC ~ 0.44 ( and still ξ ⇓ )

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Where next?

● Verify / improve on ξ

● Calculate 1- and 2-body density matrices

● Calculate condensate fraction

● Move on to varied mass system

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Acknowledgements

● Pablo Lopez-Rios ● Richard Needs

● Ben Simons

● Neil Drummond, John Trail, Alexander Badinski & Matt Brown

● EPSRC