Quantum Fluctuations of Polaronic Cloud in a BEC of ultracold atoms - - PowerPoint PPT Presentation

Yulia Shchadilova

Quantum Fluctuations

• f Polaronic Cloud

in a BEC of ultracold atoms

mail to: yes@rqc.ru

Russian Quantum Center

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In collaboration with: Fabian Grusdt (University of Kaiserslautern), Eugene Demler (Harvard University), Alexey Rubtsov (RQC, Moscow State University)

Tuesday, September 16, 14

Plan of the talk

• I. Introduction
• II. Model and observables
• III. Mean field approach
• IV. Gaussian approach
• V. Results
• VI. Summary and Outlook

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Tuesday, September 16, 14

Polaron

Lev Landau 1933

polar semiconductors ionic crystal doped quantum magnets QFT high-Tc superconductors ultra-cold atoms e− a fermion interacting with a scalar boson field

impurity + self-induced polarization

Tuesday, September 16, 14

Ultracold atoms

Two-component mixtures

α = a2

IB

aBBξ

RF

• Adjustable interaction parameter
• High mobility

P

Tuesday, September 16, 14

Ultracold atoms. Observables

RF spectroscopy:

E0

ω I(ω) Z

Oscillations in a trap:

• polaron binding energy
• polaron mass

Tuesday, September 16, 14

Frölich Model

Polaronic frame strongly interacting bosons

ˆ H = P2 2M + X

k

Vk ⇣ ˆ ak + ˆ a†

−k

⌘ eik·R + X

k

ωkˆ a†

kˆ

ak

Impurity problem fermion in effective media

P

Tuesday, September 16, 14

Polaron frame

ˆ H = 1 2M ~ P − X

k

~ kˆ b†

kˆ

bk !2 + X

k

Vk ⇣ ˆ b†

k + ˆ

b−k ⌘ + X

k

!kˆ b†

kˆ

bk

1 2 3 4 0.0 0.2 0.4 0.6 0.8 k VkêV• 1 2 3 4 2 4 6 8 10 12 k Ωk

• T. D. Lee, F. E. Low, and D. Pines,

P Review 90, 297 (1953).

ˆ ULLP = ei ~

R P

k ~

kˆ a†

kˆ

ak

ˆ H → ˆ U −1

LLPH ˆ

ULLP

Tuesday, September 16, 14

Limits

ˆ H = 1 2M ~ P − X

k

~ kˆ b†

kˆ

bk !2 + X

k

Vk ⇣ ˆ b†

k + ˆ

b−k ⌘ + X

k

!kˆ b†

kˆ

bk

1) Heavy impurity,M → ∞ 2) Weak interactions,

ˆ bk → ˆ bk − Vk/ωk Vk → 0 ˆ nk = 0 nonlinearity polarization

Tuesday, September 16, 14

Polarization of phonon modes

Basis of coherent states: ˆ b† + ˆ b i(ˆ b† − ˆ b)

α

• A. Shashi, et al. PRA 89, 053617 (2014)

|MFi = ˆ D(β) |0i ˆ D(β) = Y

k

eβk(ˆ

bk−ˆ b†

k)

D H(ˆ b,ˆ b†) E

MF = h0| H(ˆ

b β,ˆ b† β) |0i

Tuesday, September 16, 14

Polarization of phonon modes

Basis of coherent states: ˆ b† + ˆ b i(ˆ b† − ˆ b)

α

• A. Shashi, et al. PRA 89, 053617 (2014)

|MFi = ˆ D(β) |0i ˆ D(β) = Y

k

eβk(ˆ

bk−ˆ b†

k)

Minimization gives the mean field self-consistent condition:

D H(ˆ b,ˆ b†) E

MF = h0| H(ˆ

b β,ˆ b† β) |0i

k = − Vk !k + k2

2M − ~ k M

⇣ ~ P − ~ Pph ⌘ ~ Pph = X

k

~ k |k|2 Ωk k

Tuesday, September 16, 14

Phonon Zero-Point Fluctuations

ˆ b† + ˆ b i(ˆ b† − ˆ b)

α

Q

Basis of squeezed coherent states:

|GSCi = ˆ D(β) ˆ S(Q) |0i ˆ S(Q) = e

1 2

P

kk0 Qkk0ˆ

b†

kˆ

b†

k0−H.c.

Tuesday, September 16, 14

Phonon Zero-Point Fluctuations

ˆ b† + ˆ b i(ˆ b† − ˆ b)

α

Q

Polarization + Bogoliubov transformation Basis of squeezed coherent states:

|GSCi = ˆ D(β) ˆ S(Q) |0i ˆ S(Q) = e

1 2

P

kk0 Qkk0ˆ

b†

kˆ

b†

k0−H.c.

ˆ S†(Q) ˆ D†(β)ˆ bk ˆ D(β) ˆ S(Q) = βk + X

k0

[cosh Q]kk0 ˆ bk0 + X

k0

[sinh Q]kk0 ˆ b†

k0

Tuesday, September 16, 14

Phonon Zero-Point Fluctuations

ˆ b† + ˆ b i(ˆ b† − ˆ b)

α

Q

Polarization + Bogoliubov transformation Basis of squeezed coherent states:

|GSCi = ˆ D(β) ˆ S(Q) |0i ˆ S(Q) = e

1 2

P

kk0 Qkk0ˆ

b†

kˆ

b†

k0−H.c.

ˆ S†(Q) ˆ D†(β)ˆ bk ˆ D(β) ˆ S(Q) = βk + X

k0

[cosh Q]kk0 ˆ bk0 + X

k0

[sinh Q]kk0 ˆ b†

k0

Gaussian statistics:

D ˆ b E = β D ˆ b · ˆ b E = β · β + cosh Q sinh Q D ˆ b† · ˆ b E = β · β + sinh2 Q

Tuesday, September 16, 14

∂ hHi ∂αk = 0 ∂ hHi ∂Qkk0 = 0

Phonon Zero-Point Fluctuations

An approximate ground state solution:

• use Taylor series for averages
• considering only the terms up to in this average

Q2

Tuesday, September 16, 14

∂ hHi ∂αk = 0 ∂ hHi ∂Qkk0 = 0

✓ Ωk + kk0 M + Ωk0 ◆ Qkk0 + kk0 M βkβk0 + X

q

✓k0q M βqβk0Qkq + kq M βqβkQqk0 ◆ = 0

Phonon Zero-Point Fluctuations

An approximate ground state solution:

• use Taylor series for averages
• considering only the terms up to in this average

Q2 ~ Pph = X

kk0

~ k ⇣ |k|2 kk0 + Q2

kk0

⌘ k = − Vk !k + kνM−1

νλ kλ

2

−

~ k M

⇣ ~ P − ~ Pph ⌘ Numerical solution is without further approximations

Tuesday, September 16, 14

Phonon Zero-Point Fluctuations

✓ Ωk + kk0 M + Ωk0 ◆ Qkk0 + kk0 M βkβk0 + X

q

✓k0q M βqβk0Qkq + kq M βqβkQqk0 ◆ = 0

Solution without approximations:

⌘(k, k0) = ⇣ ~ k · ~ k0 ⌘ − Z d3q ↵2

q

MΩk,q ⌘(k, q) ⇣ ~ q · ~ k0 ⌘ − Z d3q ↵2

q

MΩq,k0 ⇣ ~ k · ~ q ⌘ ⌘(q, k0) Qkk0 = − 1 M αkαk0 Ωk +

~ k ~ k0 2M + Ωk0

ηk,k0 ~ F(k) = Z d3q ↵2

q

MΩk,q ⌘k,q~ q Fλ(k) = X

µ

(kµ − Fµ(k)) Z d3k0 α2

k0

MΩk,k0 k0

µk0 λ−

X

µ

kµ Z d3k0 α2

k0

MΩk,k0 Fµ(k0)k0

λ

Denote: Rewrite: Introduce: An equation for F to be solved iteratively:

Tuesday, September 16, 14

Polaron ground state energy

           diagMC MF Gaussian RG 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 Α E p

23Na 6Li

M/mB ≈ 0.26

diagMC: Vlietinck et. al, arXiv:1406.6506 (2014) Gaussian (this work): Y.E.S., Grusdt, Demler, Rubtsov, in preparation (2014) RG: Grusdt, Y.E.S., Rubtsov, Abanin, Demler., in preparation (2014)

Tuesday, September 16, 14

Total energy

23Na 6Li

M/mB ≈ 0.26

E = gIBn0 + hHi

ω

0.2 0.4 0.6 0.8 1 2 4 6 8 10

• E0 [c/]

M/mB=0.26316, q=0, 0=2000/, n0=−3 diagMC, Vlietinck et al. RG variational MF Feynman, Vlietinck et al.

E0

I(ω) Z

Tuesday, September 16, 14

Polaron mass

23Na 6Li

M/mB ≈ 0.26

M Mp ≡ M δvp δP = 1 − Pph P .

MF Gaussian RG 2d order PT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Α MpêM

Tuesday, September 16, 14

Coherence factor

23Na 6Li

M/mB ≈ 0.26

g(2)(k, k0) = D ˆ b†

kˆ

b†

k0ˆ

bkˆ bk0 E nknk0

q=p q=0 MF 2 4 6 8 10 5 10 15 20 kêx g2Hk,kcosqL

g(2) = 1 g(2) > 1 g(2) < 1

Bunching: Coherent: Antibunching (squeezing):

α = 1

Tuesday, September 16, 14

Light/heavy impurities

10 20 30 40 −100 −50 50

M/mB

23Na 6Li 0.26 87Rb 40K 0.46 6Li 6Li2

0.5 0.86

7Li 6Li

1.53

87Rb 133Cs

1.74

23Na 40K

2.12

40K 87Rb

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Tuesday, September 16, 14

Summary and outlook

Quantum fluctuations are significant in case of light or/and strongly interacting impurities. Signatures of entanglement can be captured experimentally.

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• Subsonic -- supersonic transition
• Non-homogeneous BEC
• Anisotropic interactions of dipolar gases
• Real-time dynamics

Tuesday, September 16, 14