Polaron Seminar, AG Widera — AG Fleischhauer, 05/06/14

Introduction to polaron physics !in BECs

!!

Fabian Grusdt !

Physics Department and Research Center OPTIMAS, University of Kaiserslautern, Germany

Graduate School of Materials Science in Mainz, Kaiserslautern, Germany !

!

1

Motivation

2

Q: What is a polaron?

A: a long-lived quasiparticle compromised of an impurity dressed by phonons.

�

e� phonon

Motivation

3

Renormalization group analysis of mobile impurities in a BEC

F. Grusdt,1, 2, 3 A. Shashi,3, 4 and E. Demler3

1Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, Germany2Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany

3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA4Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA.

(Dated: April 30, 2014)

We discuss a mobile impurity which is immersed in a homogeneous Bose Einstein condensate(BEC) and interacts strongly with the phonon modes of the latter to form a polaron. We developa renormalization group (RG) formalism within the Lee-Low-Pines (LLP) scheme, where the totalsystem momentum q is conserved.

For small momenta q the polaron is subsonic and we calculate ground state properties, like itsenergy, mass and quasiparticle weight. Comparing our results to mean-field (MF) calculations,which are valid in the weak-coupling approximation, allows us to identify the intermediate-couplingregime. We also develop a generalization of our RG formalism to non-equilibrium situations, and inparticular calculate the impurity’s Green’s function. From the latter we obtain the radio-frequency(RF) absorption spectrum of the impurity, which can provide direct experimental signatures forpolaron formation. For experimentally accessible parameters we find that deviations from simple MFcalculations can take substantial values, highlighting the importance to take into account quantumfluctuations in the interpretation of experimental results.

For large system momenta q on the other hand, the system is supersonic and using the RG wecalculate the polaron’s phase diagram. Within MF theory we show that the transition from asubsonic polaron (with non-vanishing quasi-particle weight Z) to the supersonic side (where Z = 0)can be understood as a second order quantum phase transition in three dimensions, whereas ahigher-order transition is predicted in lower-dimensional systems.

PACS numbers: 67.85.-d,67.85.Pq,71.38.Fp,03.75.Kk

x

zy q

FIG. 1. (Color online) We consider a single impurity im-mersed in a three dimensional homogeneous BEC (a). Thetotal momentum q of the system is conserved, and the inter-action of the impurity with the Bogoliubov phonons of theBEC leads to formation of a polaron. The dispersion relationof the phonons is shown in (b) and their scattering amplitudeVk with the impurity in (c).

I. INTRODUCTION

introIn this paper we investigate a single mobile impu-

rity immersed in a homogeneous BEC, as depicted inFIG.1(a). The latter provides Bogoliubov phonons,whose dispersion relation !

k

is shown in FIG.1(b). For

not too large boson-impurity interaction and su�cientlylarge BEC density, the system can be described by aFrohlich Hamiltonian [1]. The corresponding impurity-phonon scattering amplitude V

k

is shown in in FIG.1(c).This system has been treated before using di↵erent ap-

proaches. ... Here we investigate non-equilibrium as wellas equilibrium properties of this system by means of arenormalization group (RG) approach.A powerful method to investigate the properties of mo-

bile impurities is given by the Lee-Low-Pines (or Lang-Firsov) unitary transformation [2]. It makes use of acoordinate frame with the impurity in its center. Theconservation of total momentum q, a direct consequenceof translational invariance of the combined impurity-BECsystem, is manifestly present in this scheme. Focusing ona sector of fixed total momentum, the remaining Hamil-tonian only describes phonon degrees of freedom withimpurity-mediated interactions between them.Our paper is organized as follows. In Sec. II we in-

troduce our microscopic model and apply the LLP trans-formation to treat the total system momentum q as aconserved quantity. In Sec. III we review the MF de-scription of the polaron ground state and discuss thephase diagram. This sets the stage for the ground stateRG procedure, which is formulated in terms of quantumfluctuations around the MF polaron in Sec. IV. We startthat section by a simple dimensional analysis (IVB), be-fore we present the formal concept of the RG (IVC) andcalculate corrections to the polaron ground state proper-ties (IVE). In section VI the generalization of the RG to

Polaron’s in BECs

BECs

single impurity Anderson et al., Science 269 (1995)

VOLUME 85, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 OCTOBER 2000

FIG. 2. Loading the dipole trap from the MOT with 100%efficiency. Shown are photon counts detected with the APD.Discrete signal levels correspond to an empty MOT (N ! 0,MOT stray light only), one and two atoms, respectively. Alter-nating to normal MOT operation the dipole trap is switched onand the MOT off for periods of 1 s. The signal decreases tothe lowest level which is due to the dipole trap laser stray lightonly. After that the atoms are recaptured into the MOT show-ing the same fluorescence level as before. During normal MOToperation atoms are occasionally loaded into the MOT from thebackground vapor (L) or leave the MOT (C, in this particularcase as a result of a cold collision [14]).

depending on the laser intensity. The trap size was ob-served to be independent of the atom number N up toN ! 8.

An atom strongly driven by resonant laser light emits anaverage fluorescence power of hvG!2 " 3 pW, where Gis the natural linewidth, G ! 2p 3 5.2 MHz for cesium.With a realistic overall detection efficiency of 1023 it isnecessary to discriminate about 3 fW fluorescence powerfrom a stray light background. Fluorescence of the atomstrapped in the MOT is observed with an avalanche photo-diode (APD) in single photon counting mode with a mea-sured photon detection efficiency of 50% at l ! 852 nm.The fluorescence light is collected by a lens mounted in-side the vacuum chamber and is sent through a telescopeonto the APD; see Fig. 1. Typical photon counting ratesare #3 20$ 3 103 s21 per atom depending on the detuningand intensity of the trapping laser. Well separated equidis-tant steps in the fluorescence signal allow us to monitor thenumber of trapped atoms in a noninvasive way and in realtime [12]; see Fig. 2. During normal MOT operation wecan thus easily choose a desired atom number to be trans-ferred into the dipole trap.

The dipole trap consists of a single tightly focusedNd:YAG laser beam which is linearly polarized and super-imposed on the MOT. We use the same lens inside thevacuum chamber both for focusing the dipole trap laserand for collecting the fluorescence. Because of the largedifference between the wavelengths of the Nd:YAG laser(1064 nm) and the D2 line of cesium (852 nm), dipoletrap laser radiation is easily blocked from the detectionby interference filters. During simultaneous operation ofboth traps the fluorescence of the trapped atoms is sub-

stantially reduced due to the light shift. Thus, the optimalgeometrical overlap of the dipole laser with the MOTtrapping volume can be achieved by minimizing thefluorescence. The dipole trap laser has a waist of about5 mm yielding a trap depth corresponding to 16 mK and amaximum photon scattering rate of 190 s21 at the centerof the trap for a typical laser power of 2.5 W.

Transfer of atoms between the two traps is accomplishedthrough suitable timing sequences. First only the MOT isoperated to collect atoms. To transfer these atoms into thedipole trap, the Nd:YAG laser is turned on a few ms be-fore the MOT lasers are turned off. To recapture the atomsinto the MOT, this procedure is reversed. By analyzingthe resonance fluorescence we measure without any uncer-tainty the number of atoms right before transferring theminto the dipole trap and directly after reloading the MOT;see Fig. 2. Note that the dipole trap provides a conserva-tive potential and can therefore not capture atoms from thebackground vapor. The probability of capturing an atomby the MOT during the detection immediately after reload-ing the MOT is less than 1% and can be neglected.

Atoms can be caught by the dipole trap only at placeswhere the dipole potential exceeds the atomic kinetic en-ergy Ekin, being of the order of kBTD (TD ! hG!2kB !125 mK for Cs) yielding the geometric loading effi-ciency P ! 1 2 #Ekin!U$w2

0 !r20 , where U . Ekin is the

dipole potential in the trap center, w0 and r0 are 1!eradii of the dipole trap and the MOT, respectively. Evenfor small MOT sizes used in our experiments (typicallyr0 ! 10 mm) P is about 70%. However, during a fewms of simultaneous operation of both traps, the MOTeffectively cools the atoms down into the dipole potential.We have indeed found that 1 s after loading N atomsfrom the MOT, the probability to find the same N atomsin the dipole trap is more than 98% for all N up to 7. Thisis consistent with the measured dipole trap lifetime (seebelow) and 100% loading efficiency.

Varying the time spent by atoms in the dipole trap wehave measured the fraction of atoms transferred back intothe MOT and hence the lifetime in the dipole trap to bealso independent of N . The results are shown in Fig. 3demonstrating again 100% loading efficiency and a stor-age time of 51 6 3 s. Each point shows an averaging over400 atoms (about 100 single observations on N atoms, Nvarying from 1 to 7). The same procedure repeated withthe dipole trap laser blocked is also presented in Fig. 3as circle symbols. In this case atoms are stored in thequadrupole magnetic field as reported previously [3]. Asexpected, roughly half of the atoms are immediately lostafter switching off the MOT lasers due to a statistical dis-tribution of the spin orientations relative to the local mag-netic field. From the fact that the atoms’ lifetime in themagnetic trap is the same, we conclude that storage in bothphysically very different traps is limited by background gaspressure only.

Because of the large detuning of the dipole trap laserfrom atomic resonances the light shifts of both hyperfine

3778

Frese et al, PRL 85 (2000)

Outline

Fröhlich Hamiltonian in a BEC derivation from first principles

!

conditions for the approach !

!

Polaron properties (MF polaron theory) ground state energy

4

5

Bogoliubov-Fröhlich Hamiltonian !— derivation from first principles —

see e.g. Tempere et al., PRB 80 (2009) Bruderer et al., PRA 76 (2007)

Fröhlich Hamiltonian

6

Microscopic model:

H =

Zd3~r�†(~r)

� r2

2mB+

gBB

2�†(~r)�(~r)

��(~r)

+

Zd3~r †(~r)

� r2

2M+ gIB�

†(~r)�(~r)

� (~r).

�(~r)

(~r)

Bose field

impurity field

87Rb

133Cs

pseudo-potentials: VIB(~r) = gIB�(~r)

+traps or lattices

Fröhlich Hamiltonian

7

BEC - Bogoliubov theory

�(~r) = L�3/2X

~k

ei~k·~r�~k

discrete set of modes

[�~k, �†~k0 ] = �~k,~k0

HB =X

~k

�†~k�~k

~k2

2mB+

1

L3

X

~k,~k0,�~k

gBB

2�†~k+�~k

�†~k0��~k

�~k0 �~k

Hamiltonian

BEC �0 =p

N0

HB = E0 +X

~k

�†~k�~k

~k2

2mB+

gBBN0

2L3

X

~k 6=0

⇣2�†

~k�~k + �†

~k�†�~k

+ �~k��~k

⌘

weakly interacting

BEC

Fröhlich Hamiltonian

8

ˆ�~k = cosh ✓~ka~k � sinh ✓~ka†�~k

,

Bogoliubov transformation

HB = E00 +

X

~k

!ka†~ka~k

Bogoliubov phonons:

!k = ckp1 + (⇠k)2/2

BEC characterization:

⇠ ⇠ 1µm

⇢BEC ⇠ 1018...1021m�3

c ⇠ 1mm/s

phonons: a~k, a†~k

0 1 2 30

2

4

6

8

Fröhlich Hamiltonian

9

HIB =gIBL3

Zd3~r †(~r)

X

~k,~k0

ei(~k�~k0)·~r�†~k�~k0 (~r)

Boson - impurity interaction

HIB =gIBL3

2

4N0 +X

~k 6=0

⇣ei

~k·~rp

N0�†~k+ h.c.

⌘+

X

~k,~k0 6=0

ei(~k�~k0)·~r�†

~k�~k0

3

5

BEC �0 =p

N0

BEC mean field shiftphonon-phonon

scatteringphonon-impurity scattering

Fröhlich Hamiltonian

Phonon-impurity scattering

Ha�� =

Zd3~r †(~r)

X

~k 6=0

V disck e�i~k·~r

⇣a~k + a†

�~k

⌘ (~r)

ˆ�~k = cosh ✓~ka~k � sinh ✓~ka†�~k

,

V disck =

gIBL3

pN0

✓k2/2mB

2gBB⇢BEC + k2/2mB

◆1/4

scattering amplitude

modified by Bogoliubov mixing angles ✓~k

HIB =gIBL3

2

4N0 +X

~k 6=0

⇣ei

~k·~rp

N0�†~k+ h.c.

⌘+

X

~k,~k0 6=0

ei(~k�~k0)·~r�†

~k�~k0

3

5

Fröhlich Hamiltonian

11

Vk = gIBp⇢BEC(2⇡)

�3/2

(⇠k)2

2 + (⇠k)2

!1/4

Renormalization group analysis of mobile impurities in a BEC

F. Grusdt,1, 2, 3 A. Shashi,3, 4 and E. Demler3

1Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, Germany2Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany

3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA4Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA.

(Dated: May 1, 2014)

We discuss a mobile impurity which is immersed in a homogeneous Bose Einstein condensate(BEC) and interacts strongly with the phonon modes of the latter to form a polaron. We developa renormalization group (RG) formalism within the Lee-Low-Pines (LLP) scheme, where the totalsystem momentum q is conserved.

For small momenta q the polaron is subsonic and we calculate ground state properties, like itsenergy, mass and quasiparticle weight. Comparing our results to mean-field (MF) calculations,which are valid in the weak-coupling approximation, allows us to identify the intermediate-couplingregime. We also develop a generalization of our RG formalism to non-equilibrium situations, and inparticular calculate the impurity’s Green’s function. From the latter we obtain the radio-frequency(RF) absorption spectrum of the impurity, which can provide direct experimental signatures forpolaron formation. For experimentally accessible parameters we find that deviations from simple MFcalculations can take substantial values, highlighting the importance to take into account quantumfluctuations in the interpretation of experimental results.

For large system momenta q on the other hand, the system is supersonic and using the RG wecalculate the polaron’s phase diagram. Within MF theory we show that the transition from asubsonic polaron (with non-vanishing quasi-particle weight Z) to the supersonic side (where Z = 0)can be understood as a second order quantum phase transition in three dimensions, whereas ahigher-order transition is predicted in lower-dimensional systems.

PACS numbers: 67.85.-d,67.85.Pq,71.38.Fp,03.75.Kk

0 1 2 30

0.5

1(c)

FIG. 1. (Color online) We consider a single impurity im-mersed in a three dimensional homogeneous BEC (a). Thetotal momentum q of the system is conserved, and the inter-action of the impurity with the Bogoliubov phonons of theBEC leads to formation of a polaron. The dispersion relationof the phonons is shown in (b) and their scattering amplitudeVk with the impurity in (c).

I. INTRODUCTION

introIn this paper we investigate a single mobile impu-

rity immersed in a homogeneous BEC, as depicted inFIG.1(a). The latter provides Bogoliubov phonons,whose dispersion relation !

k

is shown in FIG.1(b). For

not too large boson-impurity interaction and su�cientlylarge BEC density, the system can be described by aFrohlich Hamiltonian [1]. The corresponding impurity-phonon scattering amplitude V

k

is shown in in FIG.1(c).This system has been treated before using di↵erent ap-

proaches. ... Here we investigate non-equilibrium as wellas equilibrium properties of this system by means of arenormalization group (RG) approach.A powerful method to investigate the properties of mo-

bile impurities is given by the Lee-Low-Pines (or Lang-Firsov) unitary transformation [2]. It makes use of acoordinate frame with the impurity in its center. Theconservation of total momentum q, a direct consequenceof translational invariance of the combined impurity-BECsystem, is manifestly present in this scheme. Focusing ona sector of fixed total momentum, the remaining Hamil-tonian only describes phonon degrees of freedom withimpurity-mediated interactions between them.Our paper is organized as follows. In Sec. II we in-

troduce our microscopic model and apply the LLP trans-formation to treat the total system momentum q as aconserved quantity. In Sec. III we review the MF de-scription of the polaron ground state and discuss thephase diagram. This sets the stage for the ground stateRG procedure, which is formulated in terms of quantumfluctuations around the MF polaron in Sec. IV. We startthat section by a simple dimensional analysis (IVB), be-fore we present the formal concept of the RG (IVC) andcalculate corrections to the polaron ground state proper-ties (IVE). In section VI the generalization of the RG to

scattering amplitude

continuum limit L ! 1N0

L3= ⇢BEC a~k !

✓2⇡

L

◆3/2

a(~k) [a(~k), a†(~k0)] = �(~k � ~k0)

V disck =

gIBL3

pN0

✓k2/2mB

2gBB⇢BEC + k2/2mB

◆1/4

HIB =gIBL3

2

4N0 +X

~k 6=0

⇣ei

~k·~rp

N0�†~k+ h.c.

⌘+

X

~k,~k0 6=0

ei(~k�~k0)·~r�†

~k�~k0

3

5

Fröhlich Hamiltonian

12

HIB =gIBL3

2

4N0 +X

~k 6=0

⇣ei

~k·~rp

N0�†~k+ h.c.

⌘+

X

~k,~k0 6=0

ei(~k�~k0)·~r�†

~k�~k0

3

5

Phonon-phonon scatteringX

~k

�†~k�~k ⇠ Nph

1

L3

X

k<⇤

=1

(2⇡)3(2⇡)3

L3

X

k<⇤

! 1

(2⇡)3

Z ⇠�3

d3~k ⇠ ⇠�3

hHa-aihHa��i

!⌧ 1

hHa-ai ⇠ gIBNph

⇠3hHa��i ⇠ gIB

p⇢BEC

sNph

⇠3

condition for Fröhlich model

⇢BEC � Nph

⇠3

Fröhlich Hamiltonian

13

H =

Zd3~k !ka

†(~k)a(~k) + gIB⇢BEC+

+

Zd3~r †(~r)

✓� r2

2M+

Zd3~k Vke

�i~k·~rha(~k) + a†(�~k)

i◆ (~r)

Fröhlich Hamiltonian:

condition for Fröhlich model

gIB⇠�3 ⌧ 2c/⇠

typical numbers (priv.comm., Farina)gIB⇠

�2/c = 0.2⇠ = 1.3µm, c = 0.4mm/s, aIB = 34nm

(alternative derivation: Bruderer et.al., PRA 2007)

14

Model parameters!— polaron energy —

see Tempere et al., PRB 80 (2009) Rath & Schmidt, PRA 88 (2013) Shashi et al., PRA — in press (2014)

Model parameters

15

parameter typical value weak coupling strong coupling

BEC density small large

impurity-boson scattering length small large

impurity mass large small

van-der-Waals length large small

length scale healing length:

time scale via speed of sound:

energy scale

mass scale boson mass

units

free parameters

⇠

⇠/c

~c/⇠mB

⇢BEC 1...103

aIB 10�2...10�3

M 10�1...101

`vdW 10�3

~ = 1

Polaron energy

16

Infinite mass limit classical impurity

H = gIB⇢BEC +

Zd3k !ka

†~ka†~k + Vk

⇣a~k + a†

�~k

⌘M ! 1

integrable model ˆU =

Y

~k

exp

⇣↵~ka

†~k� ↵⇤

~ka~k

⌘

U†a~kU = a~k + ↵~k

U†HU = gIB⇢BEC +

Zd3k !ka

†~ka~k � V 2

k

!k

Polaron energy

17

UV-scaling!k ⇠ k2 Vk ⇠ 1

UV divergent !?Z ⇤

dk k2V 2k

!k⇠

Z ⇤

dk 1 = ⇤ ! 1Z ⇤

dk k2V 2k

!k⇠

Z ⇤

dk 1

E0 = gIB⇢BEC � 4⇡

Z ⇤

0dk k2

V 2k

!k

ground state energy momentum cut-off

Polaron energy

18

Lippmann-Schwinger equation

remember — pseudo potential!

k(~r) = (0)k (~r) +

eikr

rf(k) +O(r�2)

f(k) = � aIB1 + ikaIB

low-energies: universal k . 1/`vdW =: ⇤for

see e.g. Bloch et al., RMP (2008)

strategy: use pseudo potential with same scattering length!

1

aIB=

2⇡

gIB

✓1

mB+

1

M

◆+

2

⇡⇤

V (~r) = gIB�(~r)

see e.g. Rath & Schmidt, PRA (2013)

Polaron energy

19

E0 = gIB⇢BEC � 4⇡

Z ⇤

0dk k2

V 2k

!kground state energy

E0 = 2⇡aIB⇢BEC

mB�

Z ⇤

d3~kV 2k

!k+ 4

a2IBmB

⇤

gIB =2⇡aIBmB

1 + aIB

2⇤

⇡+O(a2IB)

�

UV convergent

SummaryFröhlich Hamiltonian in a BEC

!!!

condition: !

!

20

H =

Zd3~k !ka

†(~k)a(~k) + gIB⇢BEC+

+

Zd3~r †(~r)

✓� r2

2M+

Zd3~k Vke

�i~k·~rha(~k) + a†(�~k)

i◆ (~r)

, � , a~k�0 =

pN0

gIB⇠�3 ⌧ 2c/⇠

1

aIB=

2⇡

gIB

✓1

mB+

1

M

◆+

2

⇡⇤

Polaron properties, localized impurity ground state energy

Related work

21

experiments:! !

Schirotzek et al., PRL 102 (2009) Observation of the Fermi polaron !Scelle et al., PRL 111 (2013) Deep lattice Bose polaron !Fukuhara et al., Nature Phys. 9 (2013) Spin-impurity polarons in a BEC !!theory:!

Tempere et al., PRB 80 (2009) Feynman path integral treatment !Bruderer et al., PRA 76 (2007) Strong coupling treatment (lattice) NJP 10 (2008) !Cucchietti & Timmermans, PRL 96 (2006) Strong coupling treatment (continuum) Casteels et al., Laser Phys. 21 (2011) !Shashi et al., to appear in PRA (2014) RF spectra of Fröhlich polaron in BEC

22

Thanks to…

Eugene Demler Aditya Shashi Dmitry Abanin

Shashi et al., arXiv:1401.0952 (2014) Radio frequency spectroscopy of polarons in ultracold Bose gases

Grusdt et al., in preparation Bosonic lattice polaron Bloch oscillations

Grusdt et al., in preparation Renormalization group treatment of polarons in ultracold Bose gases

… and thanks for your attention!

23