bose polarons theory and experimentsbqmc.upc.edu/pdfs/doc904.pdf · bose polarons theory and...

Bose Polarons theory and experiments Luis A. Peña Ardila!

Barcelona, November 29 2017

WHY THE POLARON PROBLEM IS IMPORTANT? 1

Charge-‐Transfer excitons in DNA

Elementary par:cles and Higgs bosons

Electrons in solids

3He (Impuri:es)–4He mixtures (Strongly correlated Quantum liquid)

Because represents a general scenario in physics: an impurity particle! interacting with a medium.!

Solid State Physics

The electron gets dressed by la1ice vibrations (Phonons) of the polar crystal.

WHAT IS A POLARON (SOLID STATE) 2

Impurity!

Medium!

Polaron!

Solid State Physics Fröhlich solid state polaron1

Low-‐‑energy description

Electron Bath of phonons

Electron dressed by phonon Fröhlich polaron

THE FRÖHLICH HAMILTONIAN

1. Fröhlich, Adv. Phys. 3, 325 (1954)

Electron Phonons Electron-‐‑Phonon interaction

ω q =ω 0Interac:on with LO phonons (are the important contribution in P)

Solid-‐‑state Fröhlich:

H = P2

2M+ hω q

q∑ a†qaq + Vq

k,q∑ cp+q

† cp a†q +aq( )aq + a−q

†( )

3

Low-‐‑energy description

impurity dressed by phonon Fröhlich polaron

M 0 ∝1ε∞

− 1ε0

⎛⎝⎜

⎞⎠⎟

1/2

THE FRÖHLICH HAMILTONIAN

1. R. Feynman .Slow Electrons in a Polar Crystal.PRL 97 (1955) 2. R. Feynman. Statistical Mechanics.(1972)

Fixed and depends of the material

Electron Bath of phonons

3

Optical-‐‑Phonons Electron-‐‑Phonon coupling Electron

H = P2

2M+ hω 0

q∑ a†qaq +

M 0

υ1/2k,q∑ exp(iq ⋅r)

qa†q +aq( )aq + a−q

†( )

Solid State Physics


POLARONS IN ULTRACOLD QUANTUM GASES (BOSONS) 4

?

Solid State Physics

An impurity gets dressed by low energy excitations of the BEC and forms a polaron.


Ultracold Gases

POLARONS IN ULTRACOLD QUANTUM GASES (BOSONS) 4

ULTRACOLD QUANTUM GASES 5

Interactions can be controlled

Clean systems

nBa3 ≪1

Diluteness

nBR03 ≪1

Low temperatures

Λ = h2πmKBT

>> R0

Feshbach resonances in cold gases1

1. C. Chin et all. Feshbach resonances in ultracold gases

a = −∞

HOW INTERACTIONS ARE CONTROLLED?

V (R − R' ) = 4π!2

ma δ (R − R' )

S-‐wave Sca7ering length

Sca7

ering length (1

000 a0)

6

Short range interactions

Atoms interact by colliding

Low energy collisions

S-‐‑wave collisions dominate

ULTRACOLD QUANTUM GASES 7

ATTRACTIVE POLARONS REPULSIVE POLARONS

Ultracold Gases

Solid State Physics Ultracold Gases QUESTION

Interaction tunable Interaction Fixed

QUESTION: can the solid state system be

mapped onto an ultracold system?

?!

7

Bose Polaron in the Strongly interacting limit. Ming-‐‑Guang Hu et al , PRL 117, 055301 (2016).

BOSE POLARON EXPERIMENTS-‐‑JILA 8

RF spectroscopy

nBECa3 ≈ 3×10−5

Bosonic bath (Rb)

Impurity gas bath (K)

1knb

= a6π 2na3( )1/3b


POLARON ENERGY BRANCHES 9

Attractive branch!

Repulsive branch!µ/µ

0

1/ knb

µ = E0 ( _____ )− E0 ( ____ )

BOSE POLARON EXPERIMENTS-‐‑AARHUS 39KRadio Frequency Spectroscopy!

Observation of a1ractive and repulsive polarons in a Bose Einstein Condensate. Nils B. Jørgensen et al ,PRL 117, 055302 (2016).

Spectral Response

10

2 = F = 1,mF = 0

1 = F = 1,mF = −1

1/ knb1knb

= a6π 2na3( )1/3b

Bosonic bath (K) nBECa

3 = 3×10−8

Impurity gas bath (K)

AOrac:ve Branch

Repulsive Branch

µ/µ

0

RECOVERING THE LOW-‐‑ENERGY HAMILTONIAN

Hamiltonian impurity coupled to the bosonic bath:

Impurity Bosons Boson-‐‑ boson interaction Impurity-‐‑bosons interaction

Bosonic bath

Impurity

a: s-‐‑wave sca1ering length between bosons

Dilute bath bosons

H = P2

2mI

+ εkk∑ a†kak +12

VBB(q)k,k ',q∑ a†k '−qak+qa†kak ' + VIB(q)k,q∑ ρ I (q)a

†k−qak

The Bogoliubov approximation weakly interacting BEC quasiparticles system.

na3 ≪1

H = E0 + Ekk≠0∑ bk

†bk +P2

2mI

+ nBVIB(0)+ε k0nBε kk≠0

∑ VIB(k) bk + b−k†( )

12

..and now write this Hamiltonian with respect to the BEC…

SINGLE IMPURITY PROBLEM

Impurity Bogoliubov excitations

Impurity-‐‑Bogoliubov exc. interactions

Ultracold gases

Solid state physics

Phonons Electron-‐‑phonon interaction Electron

Fröhlich Hamiltonian in

HElectron−phonon =P2

2me

+ !ω qa†q

q∑ aq + Vqc

†p+qc

†p a†q + aq( )

q,p∑

HPolaron =P2

2mI

+ Ekb†k

k≠0∑ bk + Vk exp(ik ⋅rI ) b

†−k + bk( )

k≠0∑

map?

YES In the weak coupling

F R Ö H L I C H ?

aq + a−q†( )

Impurity-‐Boson Sca7ering length

13

PERTURBATION THEORY-‐‑WEAK COUPLING

Total Energy:

H0 H ' (Perturbation)

Effective mass:

Perturbation theory works for:

nBa3 ba

⎛⎝⎜

⎞⎠⎟2

<<1

m*

m= 1+ 64

45 πnBa

3 ba

⎛⎝⎜

⎞⎠⎟2

E(NB ,1) = E0 (NB )+ 8π nBa3( ) b

a⎛⎝⎜

⎞⎠⎟ +

323 π

nBa3( )1/2 b

a⎛⎝⎜

⎞⎠⎟2⎡

⎣⎢

⎤

⎦⎥!2

2ma2

H = E0 + Ekk≠0∑ bk

†bk +P2

2mI

+ nBVIB(0)+ε k0nBε kk≠0

∑ VIB(k) bk + b−k†( )

Involving up to one Bogoliubov excitaFon on top of the BEC

COMPLETE ENERGY SPECTRUM NEEDS TO INCLUDE MORE THAN one Bogoliubov excitaFon on top of the BEC L. A. and S. Giorgini. Phys. Rev. A 92 033612 (2015)

14

Is there a method that investigate this regime? YES: Monte-‐‑Carlo methods

THE METHODS

Bosonic bath

Bosonic bath

16

Strong interacting regime:

Weak interacting regime: Fröhlich Hamiltonian

QUANTUM MONTE-‐‑CARLO METHOD-‐‑ ALL COUPLING

Bosonic bath

Physical system Diffusion Monte-‐‑Carlo

64 bosons* 1 Impurity

Trial wave function Local energy

ψ T (R) = fI (riα )i=1

NB

∏ f (rij )i< j∏

V (R) Hard Sphere potential

V (R) = +∞ R ≤ a0 R > a

⎧⎨⎩

R

Impurity

V (R)

R

b = R0 1−tan(k0R0 )k0R0

⎡

⎣⎢

⎤

⎦⎥

ASW potential

V (R) =−V0 R ≤ R00 R > R0

⎧⎨⎪

⎩⎪

R0 a

EL (R) = − !2

2m i=1

N

∑ ∇2ψ T (R)ψ T (R)

+V (R)

*SIZE EFFECTS

Periodic Boundary conditions

15


BOSE POLARON EXPERIMENTS-‐‑JILA 17

µ = E0 ( _____ )− E0 ( ____ )

AOrac:ve Branch (AOrac:ve Polaron)

Repulsive Branch (Repulsive Polaron)

GROUND STATE ENERGY-‐‑ Aarhus Experiment 18


Polaron energy


Polaron energy

Many-‐Body conFnuum


SOMETHING WRONG!


Temperature?

Polaron interacFon?

Other?

EFFECTIVE MASS-‐‑ Our Model

DMC PerturbaFon theory Self-‐localizaFon

m*

m= 1.63± 0.03

T-‐Matrix approach1

1. S. P. Rath and R. Schmidt. Field-‐theoreFcal study of the Bose Polaron. Phys Rev A 88 (2013)

Strongest interacting limit a/b=0

Results at the unitary limit for gas parameter nBa3 = 10−5

No measured in experiments yet

Similar to JILA

Akractive branch

Repulsive branch

19

1 impurity!!64 Bosons!

na3 = 10−5

a /b

Boson-‐‑boson pair correlation function

Impurity-‐‑boson pair correlation function

n(r) = nB

dr '0

r

∫ 4πr '2 gIB(r ')

4πr3 / 3

PAIR CORRELATION FUNCTIONS-‐‑ Our Model

Enhancement for the a1ractive branch

Depletion for the repulsive branch z

log n(r)nB

⎡

⎣⎢

⎤

⎦⎥

log n(r)nB

⎡

⎣⎢

⎤

⎦⎥

r /ξ

r /ξ

Very repulsive Very akractive Impurity absent

L. A. and S. Giorgini. Phys. Rev. A 92 033612 (2015)

gBB

20

Beyond the Fröhlich regime

Strongest interaction that two particles can afford

Bosonic bath

No small parameter describing properties

(universal limit)

UNITARY LIMIT

nBa3•  Energy

•  Effective mass •  Correlation pair function

Goal:

V (R)

R

b = R0 1−tan(k0R0 )k0R0

⎡

⎣⎢

⎤

⎦⎥

V (R) =−V0 R ≤ R00 R > R0

⎧⎨⎪

⎩⎪R0 a k0R0 = π / 2

b→ ±∞

WEAKLY INTERACTING

LIMIT nBa3 ba

⎛⎝⎜

⎞⎠⎟2

>>1

21

Binding energy-‐‑ UNITARY LIMIT

mB = mI

mB = mIAarhus Experiment

JILA predicFon

mB

mI

= 1

mB

mI

= 0

L. A and S. Giorgini. Phys. Rev. A 94 063640 (2016)

L. A. and S. Giorgini. Phys. Rev. A 92 033612 (2015)

22

Weakly Interacting Bose gas must exist

EFFECTIVE MASS-‐‑ UNITARY LIMIT

DMC

Linear fit

Non self-‐‑localization of the impurity

Aarhus Experiment

23

Boson-‐boson pair correlaFon funcFon Density profile

n(r) = nB

dr '0

r

∫ 4πr '2 gIB(r ')

4πr3 / 3

C = lim gIB(r)r2

a2nBa

3( )2/3

PAIR CORRELATION FUNCTION-‐‑ UNITARY LIMIT

Bosonic bath slightly modified Bosonic bath is significantly modified in the neighborhood of the impurity

r→ 0

Narrower peak as the gas parameter decreases

gBB(r)

24

Efimov states1: 3-‐‑Body bound states

Lowest Efimov state Polaron energy

FEW-‐‑BODY PHYSICS – EFIMOV STATES?

No larger than

10−6 !2

mR2

Efimov’s scaling factor

1. V. N. Efimov. Phys. Le1. B, 33(563), 1970. 2. D. S. Petrov and F. Werner. arXiv.org., (1502.04092v1), 2015.

(R ∼ a)

25

=

Efimov states1: 3-‐‑Body bound states

exp π / s0( ) ∼1986with


No larger than

10−6 !2

mR2


(R ∼ a)

THE TRIMER EFIMOV STATE IS PREDICTED TO OCCUR, BUT WE DO NOT OBSERVE IT BECAUSE THE ENERGY SCALE

NO SIGNIFICANT EFFECTS OF THREE-‐BODY DACAY

25

Efimov states1: N-‐‑Body bound states?



Unbounds states appear for N=6 (Imp+5 bosons)

26

Efimov states1: N-‐‑Body bound states?



Unbounds states appear for N=6 (Imp+5 bosons)

Efimov states Extremely

shallow

26

CONCLUSIONS

Very dilute gas

1/ n1/3b

AARHUS JILA

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