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
The electron gets dressed by la1ice vibrations (Phonons) of the polar crystal.
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.
The electron gets dressed by la1ice vibrations (Phonons) of the polar crystal.
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
Bose Polaron in the Strongly interacting limit. Ming-‐‑Guang Hu et al , PRL 117, 055301 (2016).
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 in the Strongly interacting limit. Ming-‐‑Guang Hu et al , PRL 117, 055301 (2016).
BOSE POLARON EXPERIMENTS-‐‑JILA 17
µ = E0 ( _____ )− E0 ( ____ )
AOrac:ve Branch (AOrac:ve Polaron)
Repulsive Branch (Repulsive Polaron)
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
Polaron energy
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
Polaron energy
Many-‐Body conFnuum
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
SOMETHING WRONG!
GROUND STATE ENERGY-‐‑ Aarhus Experiment 18
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
FEW-‐‑BODY PHYSICS – EFIMOV STATES?
No larger than
10−6 !2
mR2
1. V. N. Efimov. Phys. Le1. B, 33(563), 1970. 2. D. S. Petrov and F. Werner. arXiv.org., (1502.04092v1), 2015.
(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?
FEW-‐‑BODY PHYSICS – EFIMOV STATES?
1. V. N. Efimov. Phys. Le1. B, 33(563), 1970. 2. D. S. Petrov and F. Werner. arXiv.org., (1502.04092v1), 2015.
Unbounds states appear for N=6 (Imp+5 bosons)
26
Efimov states1: N-‐‑Body bound states?
FEW-‐‑BODY PHYSICS – EFIMOV STATES?
1. V. N. Efimov. Phys. Le1. B, 33(563), 1970. 2. D. S. Petrov and F. Werner. arXiv.org., (1502.04092v1), 2015.
Unbounds states appear for N=6 (Imp+5 bosons)
26
Efimov states1: N-‐‑Body bound states?
FEW-‐‑BODY PHYSICS – EFIMOV STATES?
1. V. N. Efimov. Phys. Le1. B, 33(563), 1970. 2. D. S. Petrov and F. Werner. arXiv.org., (1502.04092v1), 2015.
Unbounds states appear for N=6 (Imp+5 bosons)
Efimov states Extremely
shallow
26
CONCLUSIONS
Very dilute gas
1/ n1/3b
AARHUS JILA