Lifetime of Excitations in Atomic and Molecular Bose-Einstein Condensates

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Matteo Bellitti, Garry Goldstein, and Chris R. Laumann

Department of Physics, Boston University, Boston, MA 02215, USA

(Dated: October 13, 2022)

Recent experimental progress has produced Molecular Superﬂuids (MSF) in thermal equilibrium;

this opens the door to a new class of experiments investigating the associated thermodynamic and

dynamical responses. We review the theoretical picture of the phase diagram and quasiparticle

spectrum in the Atomic Superﬂuid (ASF) and MSF phases. We further compute the parametric

dependence of the quasiparticle lifetimes at one-loop order. In the MSF phase, the

(1) particle

number symmetry breaks to

and the spectrum exhibits a gapless Goldstone mode in addition to

a gapped

-protected atom-like mode. In the ASF phase, the

(1) symmetry breaks completely,

leaving behind a Goldstone mode and an unprotected gapped mode. In both phases, the Goldstone

mode decays with a rate given by the celebrated Belyaev result, as in a single component condensate.

In the MSF phase, the gapped mode is sharp up to a critical Cherenkov momentum beyond which

it emits phonons. In the ASF phase, the gapped mode decays with a constant rate even at small

momenta. These decay rates govern the spectral response in microtrap tunneling experiments and

lead to sharp features in the transmission spectrum of atoms ﬁred through molecular clouds.

I. INTRODUCTION

The study of the Molecular Superﬂuid (MSF) phase of

weakly interacting ultracold Bose atoms, ﬁrst analyzed

theoretically some two decades ago [

], has recently

heated up again [

–

] due to groundbreaking experimental

progress [

] in coherently trapping ultracold cesium atoms

and controlling their Feshbach resonances to produce ce-

sium molecules (Cs

) [

–

]. The reﬁnement of these

trapping techniques enables a new generation of experi-

ments probing both equilibrium and dynamical properties

of both the MSF and proximate Atomic Superﬂuid (ASF)

phase. While the thermodynamics of the MSF-ASF sys-

tem are well-known theoretically [

] (see Fig. 1for a

phase diagram), its dynamical responses are more com-

plicated. A full dynamical theory of the system requires

an understanding of both the quasiparticle content and

their dissipative scattering properties.

In this work, we compute the near equilibrium decay

rates of the quasiparticles in the ASF and MSF phases

to one-loop order at zero temperature. The quasiparticle

decay rate determines the width of the spectral function,

schematically illustrated in the insets of Fig. 1. In princi-

ple, this is directly measurable by tunneling experiments

in which a micro-trap is placed in tunnel contact with the

MSF [13].

Single atom transmission spectroscopy provides an alter-

native, and perhaps more striking, experimental signature

of the quasiparticle dynamics in the MSF phase. An in-

cident atom evolves into the gapped quasiparticle mode,

which is sharp up to a critical momentum

. Beyond this

threshold, the quasiparticle Cherenkov radiates phonons

(the gapless mode). Thus, a slow atom ﬁred through a

MSF cloud propagates without dissipation, while faster

atoms slow down until they drop below the Cherenkov

threshold. This leads to a sharp feature in the energy

spectrum of the transmitted atoms as the incident energy

crosses the threshold – assuming the cloud is “optically

dense” enough to slow the atom before it passes through.

Using our computed scattering rates, we estimate that the

stopping power of a typical molecular cloud is suﬃcient

to observe these features (see Fig. 3). The existence of

follows from the symmetry structure of the MSF phase,

as we discuss below. Deep in the MSF phase

kc'mc

where

is the atomic mass and

the speed of sound (see

Eq.(42)).

We summarize here the equilibrium properties of the

system to contextualize our work and keep the presen-

tation self–contained. The system has a global

U(1)

symmetry associated with total atom number conserva-

tion

+ 2

, where

is the number of atoms

and

is the number of molecules. The simplest model

Hamiltonian for the system [

] includes kinetic contri-

butions, density–density interactions, and a Feshbach

interaction that coherently converts two atoms into a

molecule and vice versa - see equations

(1)

through

(6)

This interconversion occurs thanks to the hyperﬁne inter-

actions between the closed and open scattering channels

for atomic collisions [7,15,16].

With the formation of the condensate the global

U(1)

spontaneously broken. At low temperature there are two

distinct scenarios for this

U(1)

breaking: 1. when just

the molecules condense the phase is known as molecular

superﬂuid (MSF), and 2. when both the atoms and the

molecules condense as atomic superﬂuid (ASF). Due to

the coherent interconversion process, the condensation of

the atoms forces a condensation of the molecules, and thus

there is no phase where the atoms are condensed but the

molecules are not. In the MSF phase the global

U(1)

reduced to a global

, where the

charge is the parity

of the atom number, while in the ASF phase there is no

remaining symmetry. Prior work has argued [

] that

the zero temperature quantum phase transition between

the MSF and ASF phases is continuous and lies in the

quantum Ising class.

arXiv:2210.06237v1 [cond-mat.quant-gas] 12 Oct 2022

Molecular Superﬂuid Atomic Superﬂuid

Residual Symmetry: Residual Symmetry: None

Quasiparticle Spectrum Quasiparticle Spectrum

FIG. 1. The mean ﬁeld phase diagram as a function of the binding energy

and total condensate density

+ 2

, with

two insets showing the quasiparticle spectral weight. The white lines are the mean ﬁeld results for the quasiparticle dispersion

discussed in Sec.IV. The solid white line in the MSF side inset indicates that the excitation is inﬁnitely long lived: in the MSF

phase the

symmetry keeps the spectral line for the gapped excitation sharp up to a threshold momentum

, above which

decay by phonon emission is allowed. In the ASF phase there is no such symmetry protection, the gapped mode is damped at

arbitrarily low momenta and is in fact very diﬀuse. In both phases the gapless mode is always damped. To give the reader a

reference for the momentum scales we marked

on the

axis: it is the scale where the gapless excitation changes nature from

predominantly phonon–like to particle–like. The phase boundary depicted assumes 2gam > gm, see Eq.(16).

There are two experimentally tunable parameters which

control the phase diagram: the total number of atoms n

and the molecular binding energy

. When

is large and

positive, the molecular state is anti–bound, so we expect

ASF to be the equilibrium phase, while for large and

negative

forming a molecule is energetically favorable

and we expect MSF. No such simple argument can be

made for the eﬀect of the particle number

on the phase.

Indeed, the transition is re-entrant as a function of particle

number at the mean–ﬁeld level (see Fig. 1).

The quasiparticle content of the ASF and MSF phases is

as follows. Since the

U(1)

symmetry is broken in both

the ASF and MSF phases, there is one Goldstone boson

on each side of the transition. In the MSF phase, the

molecules are at lower energy than the atoms and the

Goldstone mode is molecular in nature, carrying a

even

charge. A second, gapped branch of the spectrum is atom-

like, carrying odd

charge. By

conservation, the

upper mode can lose energy and momentum into the Gold-

stone mode, but cannot disappear. At zero temperature,

such emission can only happen above a Cherenkov momen-

tum

, where the group velocity matches the Goldstone

velocity. The decay width above the Cherenkov transition

is proportional to

∼(k−kc)3

and is given by Eq.

(52)

The gapless mode in the MSF phase has a similar lifetime

to that of a single component BEC [

] and is given by

Eq.(55).

In the ASF phase the Goldstone mode has mostly atomic

character while the high energy mode has mostly molec-

ular character. However, there is no

conservation so

no decays are forbidden. The decay rate of the high en-

ergy mode scales with the gap, ∆, as ∆

and is given

by Eq.

(70)

. The decay rate of the Goldstone mode in

the ASF phase is of similar form to Eq.

(55)

however with

2m→m.

To obtain these results, we derive a low energy eﬀective

theory on each side of the transition from the microscopic

theory of Eq.

(1)

. This approach signiﬁcantly simpliﬁes

the calculation compared to using the microscopic theory

directly, and is applicable as long as we restrict our at-

tention to excitations with wavelengths longer than the

healing length of the condensate and energies below the

multiparticle threshold. Using the eﬀective theory, we

compute for each excitation the one–loop imaginary part

of the self energy in the on–shell approximation, and thus

estimate the decay rates of the gapped modes.

The paper is organized as follows: in Sec.II we set up the

problem, then in Sec.III we brieﬂy review the mean–ﬁeld

phase diagram. In Sec.IV we review the Bogoliubov mean

ﬁeld theory of the spectrum and in Sec.Vwe summarize

our results for the decay rates of the gapped mode in

each phase. The interested reader will ﬁnd details of the

calculation of certain integrals in Appendix A.

II. EUCLIDEAN ACTION AND SYMMETRIES

The number of atoms

and the number of molecules

are not separately conserved, only the total number

N=Na+ 2Nm. The Euclidean action of the system is

S=Zdrdτ T+Ha+Hm+Ham +HF(1)

T=¯

Ψm∂τΨm+¯

Ψa∂τΨa(2)

Hm=¯

Ψm−∇2

4m−2µ+νΨm+gm

2|Ψm|4(3)

Ha=¯

Ψa−∇2

2m−µΨa+ga

2|Ψa|4(4)

Ham =gam |Ψa|2|Ψm|2(5)

HF=−α¯

Ψ2

aΨm+¯

ΨmΨ2

a(6)

The subscript adenotes the atoms and mthe molecules.

Here

is the molecular binding energy; a negative

means it is energetically favorable to make bound states.

The minus sign in front of the Feshbach term is chosen so

that

α >

0 in equilibrium the phases of the condensates

are locked to the same value; Without loss of generality

we assume

α >

0, as it is always possible to absorb

its sign with the ﬁeld redeﬁnition Ψ

m→ −

. The

chemical potential

appears with a factor of 2 in the

molecular part because two atoms bind to form a molecule

and only the total number is conserved. Under the

(1)

symmetry associated to this conservation law the atomic

ﬁeld transforms with unit charge and the molecular one

with double charge

Ψa→eiθΨaΨm→ei2θΨm(7)

The system has two nontrivial low–temperature phases:

the atomic superﬂuid (ASF) phase, in which the

(1) sym-

metry is completely broken –both Ψ

and Ψ

condense–,

and the molecular superﬂuid (MSF) phase, in which the

U(1) breaks down to Z2

Ψa→ −ΨaΨm→Ψm(8)

As such we expect any continuous transition between the

two phases to be Ising class [2].

III. MEAN FIELD PHASE DIAGRAM

We brieﬂy review the mean ﬁeld phase diagram (see Fig.1)

of the system, to provide context for our calculations. We

map the phase diagram in terms of the total density

+ 2

and the binding energy

, which are exper-

imentally tunable. As anticipated, there are two stable

thermodynamic phases at zero temperature (ASF and

MSF), separated by an Ising class transition line [

If 2

gam > gm

the transition is reentrant as a function of

for small negative

(see Eq.

(16)

). Parametrizing the

ﬁelds in polar form

Ψj=√njeiθjj=a, m (9)

clariﬁes the role of the relative phase between the con-

densates. In the absence of an external potential the

equilibrium solution is uniform, so we drop the gradient

terms. The mean ﬁeld energy density is then

EMF =1

2gan2

a+1

2gmn2

m+gamnanm

−2αna√nmcos (2θa−θm)

−µ(na+ 2nm−n) + νnm

(10)

which is minimized by

0 = αna√nmsin(2θa−θm) (11)

µ=gana+gamnm−2α√nmcos(2θa−θm) (12)

2µ−ν=gmnm+gamna−αna

√nm

cos(2θa−θm) (13)

These equations have two nontrivial solutions:

MSF phase: na= 0, nm=n/2>0. At mean ﬁeld

2µ−ν=gmnm(14)

as usual for a weakly interacting Bose gas. The

combination 2

µ−ν

acts as an eﬀective chemical

potential for the molecules.

ASF phase: na>

, nm>

0. While there is a closed

form expression for

and

, it is complicated and

we omit it. In the large

limit at ﬁxed density

, we

ﬁnd to lowest order in

the relations

√nm

αn/ν

and

gan

, which means that the system behaves

as a simple atomic BEC.

There is no phase where the atoms are condensed but

the molecules are not. Since

α >

0, in the ASF phase

the phases of the atomic and molecular condensates lock

together:

2θa−θm= 0 (15)

Eliminating the chemical potential and imposing

= 0

we ﬁnd the phase boundary (see Fig.1)

gam −gm

2n−2α√2n=ν(16)

The phase boundary crosses the ν= 0 axis at

n×= 32α2/(2gam −gm)2(17)

and the leftmost point is at

|ν?|= 4α2/|2gam −gm|(18)

so that using a negative binding energy with

|ν|<|ν?|

the reentrant nature of the transition is visible. Finally,

we point out that the phase diagram and the order of

the transition are sensitive to the sign of

gagm−g2

(for

concreteness we assume

gagm> g2

) but the realized

phases are the same. For a detailed discussion see [2].

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LifetimeofExcitationsinAtomicandMolecularBose-EinsteinCondensatesMatteoBellitti,GarryGoldstein,andChrisR.LaumannDepartmentofPhysics,BostonUniversity,Boston,MA02215,USA(Dated:October13,2022)RecentexperimentalprogresshasproducedMolecularSuperuids(MSF)inthermalequilibrium;thisopensthedoortoanewclassofe...

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Lifetime of Excitations in Atomic and Molecular Bose-Einstein Condensates

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