The Bose polaron- theory and experiments Georg M. Bruun Aarhus - - PowerPoint PPT Presentation

The Bose polaron- theory and experiments

Georg M. Bruun Aarhus University

• R. S. Christensen, J. Levinsen & GMB, PRL 115, 160401 (2015)
• J. Levinsen, M. M. Parish & GMB, PRL 115, 125302 (2015)
• N. B. Jørgensen et al, arXiv:1604.07883

Bose Polaron

Mobile impurity interacting with bosonic reservoir

• Electrons coupled to phonons
• Helium mixtures
• High Tc superconductors
• Elementary particles coupled

to the Higgs boson

Nice to have experimental realisation in cold atoms

Very recently two independent experimental realisations of the Bose polaron:

• 1. Schirotzek et al., Phys. Rev. Lett. 102,

230402 (2009)

• 2. Kohstall et al., Nature 485, 615 (2012)
• 3. Koschourek et al., Nature 485, 619 (2012)

❶ N. B. Jørgensen et al., arXiv:1604.07883 ❷ Ming-Guang Hu et al., arXiv:1605.00729

c signal

−3.53

−1 1 0.5 1 1.5

Fermi polaron gave lots of new insights

This Talk

• 1. Theory

Good understanding, both at weak and strong coupling

• 2. Experiment

First observation of long lived Bose polaron using RF spectroscopy

People

Jesper Levinsen Meera Parish Jan Arlt Nils Jørgensen Lars Wacker Kristoffer T. Skalmstang

Experiment Theory

Rasmus S. Christensen

Aarhus University

Monash University

Mean-field:

Astrakharchik & Pitaevskii, Phys. Rev. A 70, 013608 (2004)
 Cucchietti & Timmermans, Phys. Rev. Lett. 96, 210401 (2006)
 Kalas & Blume, Phys. Rev. A 73, 043608 (2006) Bruderer, Bao & Jaksch, Eu. Phys. Lett. 82, 30004 (2008)

Theory

Fröhlich:

Huang & Wan, Chin. Phys. Lett. 26, 080302 (2009)
 Tempere et al., Phys. Rev. B 80, 184504 (2009)
 Castels & Wouters, Phys. Rev. A 90, 043602 (2014) Grust et al., Sci. Rep. 5, 12124 (2015) Vlietinck et al., New J. Phys. 17, 033023 (2015)

Field theory: Variational:

Rath & Schmidt, Phys. Rev. A 88, 053632 (2013) Li & Das Sarma, Phys. Rev. A 90, 013618 (2014) Schhaddilova, Schmidt, Grusdt & Demler, arXiv:1604.06469

H = X

k

✏B

k a† kak + 1

2V X

k,k0,q

VB(q)a†

k+qa† k0−qak0ak

+ X

k

✏kc†

kck + 1

V X

k,k0,q

V (q)c†

k+qc† k0−qak0ck

BEC Impurity Impurity-BEC interaction

BEC weakly interacting naB3≪1 ⇒ Bogoliubov theory

Perturbation theory V(q):

• 1. order
• 2. order

Perturbation theory

a aB

Replace V (q) → Tv = 2πa/mr Diagrams like comes from expanding

T (p) = Tv 1 − TvΠ11(p) = Tv + T 2

v Π11(p) + . . .

Self-energy in powers of a: in a consistent way

Σ(p, ω) = Σ1(p, ω) + Σ2(p, ω) + Σ3(p, ω) + . . .

(a) (b) 1 2 2 1 with 1 = + 2 = + (c) 1 2 2 1 1 2 2 1

28 third order diagrams

Energy

E(0) Ω = a ξ + A(α)a2 ξ2 + B(α)a3 ξ3 ln(a∗/ξ)

A(1) = 8 √ 2/3π B(1) = 2/3 − √ 3/π

α = m/mB a∗ = max(a, aB) Ω = 2πnξ/mr

E N = 4πna m  1 + 32 3√π (na3)1/2 + 4(2 3π − √ 3)na3 ln(na3)

• E

N = 2πna m  1 + 128 15√π (na3)1/2 + 8(4 3π − √ 3)na3 ln(na3)

• a=aB:

Same structure as Lee-Huang-Yang + Wu-Hugenholz-Pines-Sawada

Weakly interacting BEC

Residue & Effective Mass

Z−1 = 1 + C(α) a2 aBξ + D(α) a3 aBξ2

m∗ m = 1 + F(α) a2 aBξ + G(α) a3 aBξ2

C(1) = 2 p 2/3π D(1) ⇡ 0.64 F(1) = 16 p 2/45π G(1) ' 0.37

Condition for Z≃1:

a2 aBξ ⌧ 1 Breaks down for ideal BEC

Variational Theory

Multichannel model

ˆ H = X

k

h Ek†

kk + ✏kc† kck +

• ✏d

k + ⌫0

• d†

kdk

i

+g√n0 X

k

⇣ d†

kck + h.c.

⌘ + g X

k,q

• d†

qcq−kbk + h.c.

• Bog. modes

Impurity Molecule

Introduces effective range r0 Regularises 3-body problem

Variational wave function

|ψi = α0c†

0 +

X

k

αkc†

−kβ† k + 1

2 X

k1k2

αk1k2c†

−k1−k2β† k1β† k2 + γ0d† 0 +

X

k

γkd†

−kβ† k

! |BECi

BEC BEC

k −k

BEC

−k1 − k2

k1 k2

Recovers Efimov spectrum for 1+2 bosons for n0→0

a−

a− ' 9000r0 Can always keep |a/r0|≫1
 Even when a- large

• 1 bog. approx.

(ladder) Efimov state 2 bog. approx.

• 3. order pert.

Mean field

Residue small close to unitarity Avoided crossing with Efimov state.

n1/3 a− = −1

a = a− Efimov physics suppressed for a-≫n0-1/3 Efimov state “too large”

Spectral functions

−0.4 −0.3 −0.2 −0.1 0.1 0.2 2 4 6 ω/En EnI0(ω) TBM1 TBM2

• Pert. Th.

−0.45 −0.4 −0.35 −0.3 −0.25 50 100

−0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 2 4 6 ω/En EnI0(ω) TBM1 TBM2

• Pert. Th.

−0.1 −0.08 −0.06 100 200 300

1/kna = −5 1/kna = −1

Many-body continuum

BEC k −k

Polaron peak Weak coupling: Variational theory agrees with pert. theory Strong coupling: Pert. theory breaks

• down. Many-body continuum significant

Theory - bottom lines

• Analytical perturbation theory to 3.order in a/ξ
• Polaron well-defined for weak coupling
• Strong coupling: Variational ansatz including 3-

body Efimov correlations

• Significant many-body continuum for strong

coupling

• Impurity atoms in BEC not the Fröhlich model

Experiment bottom lines

• First realisation of the Bose polaron (See also

JILA group)

• Well-defined polaron both for repulsive and

attractive interaction

• Many-body continuum dominates at strong

coupling

• Excellent agreement with theory
• 3-body decay has no significant effects

Jan Arlt

Spethman et al. Phys. Rev. Lett. 109, 235301 (2012)

Charged or fixed impurities in BEC:

Ospelkaus et al., Phys. Rev. Lett. 96, 180403 (2006) Zipkes et al., Nature 464, 388 (2010) Schmid et al., Phys. Rev. Lett. 105, 133202 (2010) Balewski et al., Nature 502, 664 (2013) Scelle et al., Phys. Rev. Lett. 111, 070401 (2013)

Earlier experiments

Impurity in thermal bose gas: Impurities in lattice: Magnons:

Marti et al., Phys. Rev. Lett. 113, 155302 (2014)

Experimental procedure

BEC of 39K in |1⟩

|1i = |F = 1, mF = 1i |2i = |F = 1, mF = 0i

RF flip ≤10% to |2⟩ Wait for a while TOF Count # |1⟩ remaining as fn

• f detuning Δ=ω0-ωRF

∆/En Remaining atoms kna = −0.84 ωRF ω0 ∆

}3-body loss

Independent of wait time ⇒ loose 100% of |2⟩ atoms ⇒ lost |1⟩ atoms = 3×created |2⟩ atoms

Scattering lengths

a (in units of a0) B (G) 50 100 150 200 −400 −200 200 400

|1⟩ - |1⟩:aB |1⟩ - |2⟩ resonance B0 = 113.83G ∆B = −15.93G abg = −45.24a0 R∗ ' 60a0

Lysebo and Veseth, PRA 81, 032702 (2010)

Experiment takes place here. aB≃9a0

Advantages of RF flipping out of BEC

❶ Perfect spatial overlap between impurities and BEC ❷ Selectively probe only k=0 polarons ❸ Simple theoretical interpretation ˙ N2 = −2Ω2ImD(ω)

D(t t0) = iθ(t t0)h[ X

k

a†

k1(t)ak2(t),

X

k0

a†

k02(t0)ak01(t0)]i

D(ω) = n0G2(k = 0, ω) Bogoliubov theory:

+ +

RF probes k=0 impurity spectral function:

˙ N2 ∝ A(k = 0, ω) = −2ImG2(k = 0, ω)

Contrast with Fermi gas or thermal Bose gas

vertex corrections

kn = (6π2n)1/3 En = k2

n

2m

Generic Physics

h∆/E ¯ n 1/k a n a −4 −2 2 4 −1.5 −1 −0.5 0.5 1 1.5 1/k a n b −4 −2 2 4 0.5 1 c signal

−3.53

−1 1 0.5 1 1.5 d

−0.62

−1 1 h∆/E ¯ n e

−0.04

−2 2 f

1.6

−2 2 g

4.39

−1 1

Experiment Theory

1 bog.

✮Clear shift away from ω0 ✮Excellent agreement between experiment and 2 bog. theory

(trap averaging important!)

✮Well-defined polaron for weak coupling ✮Many-body continuum dominates for strong coupling

Fourier width =0.15 En

Trap averaging & Fourier broadening

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 5 10 ω/En EnI(ω)

1/kna = −5 1/kna = −2 BEC

1/k a n E/E ¯ n −4 −2 2 4 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

1/k a n σ/En −2 2 0.5 1 1.5 −4 −2 2 4

Energy Width

1 bog. approx. 2 bog. approx.

• Pert. theory

✮Remarkable agreement between experiment and theory (some problems at strong repulsion) ✮Pert. theory explains data for weak coupling ⇒ well defined polaron ✮3-body decay not needed to explain width Γ ∝ n2

0a4

Γ ∝ En unitarity

weak coupling

Makotyn et al., Nat. Phys. 10, 116 (2014)

Conclusions

❶ Good theoretical understanding of Bose polaron both for weak and for strong coupling ❷ Experimental observation of Bose polaron for the first time

• R. S. Christensen, J. Levinsen & GMB, PRL 115, 160401 (2015)
• J. Levinsen, M. M. Parish & GMB, PRL 115, 125302 (2015)
• N. B. Jørgensen et al, arXiv:1604.07883

23RD EUROPEAN CONFERENCE ON FEW-BODY PROBLEMS IN PHYSICS

DEPARTMENT OF MATHEMATICS, AARHUS UNIVERSITY, DENMARK

8TH-12TH AUGUST 2016

AARHUS UNIVERSITY

EFB23

International Advisory Committee: Alejandro Kievsky, Chris Greene, Christian Forssen, Craig Roberts, Doerte Blume, Eduardo Garrido, Francisco Fernandez Gonzalez, Hans-Werner Hammer, Henryk Witala, Jaume Carbonell, Jean-Marc Richard, Johann Haidenbauer, Kalman Varga, Lauro Tomio, Mantile Leslie Lekala, Nasser Kalantar-Nayestanaki, Nina Shevchenko, Nir Barnea, Peter Schmelcher, Peter Zoller, Pierre Descouvemont, Stanisław Kistryn, Teresa Pena, Victor Mandelzweig, Werner Tornow, Willibald Plessas, Xiaoling Cui Local Organisers: Dmitri Fedorov (Chair), Georg Bruun, Hans Fynbo, Jan Arlt, Michael Drewsen, Nikolaj Thomas Zinner Conference Secretary: Karin

http://conferences.au.dk/efb23

Arnoldas Deltuvas, Vilnius University Artem Volosniev, TU Darmstadt Brian Lester, JILA Boulder Chen Ji, ECT* Trento Chris Greene, Purdue University Dorte Blume, Washington State University Elzbieta Stephan, University of Silesia Evgeny Epelbaum, Bohum Univesrity Francesca Sammarruca, University of Idaho Frank Deuretzbacher, ITP University of Hannover Laura Marcucci, Pisa University Lucas Platter, University of Tenessee Mohammad Ahmed, Duke Iniversity Nicholas Zachariou, JLAB Nir Barnea, Racah Institute of Physics HUJI Or Hen, MIT Patrick Achenbach, University of Mainz Selim Jochim, University of Heidelberg Susumo Shimoura, University of Tokyo Valery Nesvizhevsky, Institut Laue-Langevin Yusuke Nishida, Tokyo Institute of Technology

INVITED SPEAKERS

−0.3 0.3 E/Ω 0.5 0.75 1 Z −0.3 −0.2 −0.1 0.1 0.2 0.3 1 1.2 1.4 a/ξ m∗/m

5 10 0.5 1.5 A(α) B(α) 5 10 0.5 1 C(α) D(α) 5 10 0.5 1 F (α) G(α)

aB/ξ = 0.1

• 1. order
• 2. order
• 3. order

−1 1 0.5 1 −1 1 2 0.5 1 relative atom number −1 1 −1 1 2 −1 1 −1 1 2 h∆/E ¯ n −1 1 −1 1 2 −1 1 −1 1 2

0.1 0.2 0.3 −0.5 0.5 1 E/E ¯ n transferred fraction 0.1 0.2 0.3 0.2 0.4 0.6 σ/En transferred fraction

Linear Response regime

1/kna = −0.84 1/kna = 1.6 Increasing RF power Position and width stable

h∆/E ¯ n 1/k a n a −4 −2 2 4 −1.5 −1 −0.5 0.5 1 1.5 1/k a n b −4 −2 2 4 −1 1 2 3 4