Fermat's Library | q-2020-02-20-232 annotated/explained version.

Control of anomalous diﬀusion of a Bose polaron

Christos Charalambous

, Miguel Ángel García-March

1,2

, Gorka Muñoz-Gil

, Przemysław Ryszard

Grzybowski

, and Maciej Lewenstein

1,4

ICFO – Institut de Ciéncies Fotóniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 València, Spain

Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland

ICREA, Lluis Companys 23, E-08010 Barcelona, Spain

February 18, 2020

We study the diﬀusive behavior of a Bose

polaron immersed in a coherently coupled two-

component Bose-Einstein Condensate (BEC).

We assume a uniform, one-dimensional BEC.

Polaron superdiﬀuses if it couples in the same

manner to both components, i.e. either at-

tractively or repulsively to both of them. This

is the same behavior as that of an impurity

immersed in a single BEC. Conversely, the po-

laron exhibits a transient nontrivial subdiﬀu-

sive behavior if it couples attractively to one of

the components and repulsively to the other.

The anomalous diﬀusion exponent and the du-

ration of the subdiﬀusive interval can be con-

trolled with the Rabi frequency of the coher-

ent coupling between the two components, and

with the coupling strength of the impurity to

the BEC.

1 Introduction

The phenomenon of anomalous diﬀusion attracts a

growing interest in classical and quantum physics, ap-

pearing in a plethora of various systems [1, 2]. In

classical systems, there has been a considerable eﬀort

to elucidate the properties and conditions of anoma-

lous diﬀusive behavior, with a large emphasis given

to the question of how this anomalous diﬀusion could

potentially be controlled. In many models, the ap-

pearance of the anomalous diﬀusion is attributed to

some random component of the system-environment

setup, usually distributed with a power-law. Exam-

ples include continuous time random walks [3], dif-

fusion on a fractal lattice [4], diﬀusivity (i.e. diﬀu-

sion coeﬃcient) that is inhomogeneous in time [5, 6],

or space [7–11] in a regular or random manner, the

patch model [12, 13], hunters model [14], etc. In quan-

tum systems, a paradigmatic instance of a highly con-

trolled system is that of a Bose Einstein Condensate

(BEC). It was shown that BEC with tunable inter-

actions, are promising systems to study a number of

diﬀusion-related phenomena, such as Anderson Local-

ization (AL) in disordered media [15–17], the expan-

sion of 1D BEC in disordered speckle potentials [15–

23], the subdiﬀusive behavior of the expansion of a

wave packet of a 1D quantum, chaotic and nonlinear

system [15, 22, 24–33], the Brownian motion of soli-

tons in BEC [34], as well as the superdiﬀusive motion

of an impurity in a BEC studied in [35–37].

In this work, we study how an impurity in a coher-

ently coupled two-component BEC shows a transient

anomalous diﬀusing behavior. We study this phe-

nomenon under experimentally relevant conditions, as

long as the BEC can be approximated as uniform and

one dimensional. We show that this transient anoma-

lous diﬀusing behavior can be controlled through the

strength of the interactions and the coherent coupling.

To this end, we treat the Bose Polaron problem within

an open quantum system framework. The open quan-

tum system approach has been used recently in the

context of ultracold quantum gases to study the diﬀu-

sion of an impurity and two impurities in a BEC [35–

37], for the movement of a bright soliton in a super-

ﬂuid in one dimension [38], see also[39–41]). On the

other hand, the eﬀect of contact interactions, dipole-

dipole interactions and disorder on the diﬀusion prop-

erties of 1D dipolar two-component condensates were

studied in [42], identifying again the conditions for

subdiﬀusion. The study of the diﬀusive behavior of

a 2D two-component BEC in a disordered potential

was undertaken in [43]. Finally, an important study

on an impurity immersed in a two-component BEC

was reported in [44].

The most important novel result of this work, is

that we show that under certain assumptions, one

can observe a transient subdiﬀusive behavior of the

immersed impurity. We study this for experimentally

feasible parameters, as long as the BEC can be ap-

proximated as uniform and one dimensional, and we

examine how the strength of the coherent coupling

and interactions modify this subdiﬀusive behavior.

To be more speciﬁc about the particularities of the

system considered in this work, we assume that an

external ﬁeld drives the population transfer (spin-

ﬂipping) between the two atomic levels. The popu-

lation transfer between the two levels turns out to be

described by Josephson dynamics, leading to what is

Accepted in Quantum 2019-12-28, click title to verify 1

arXiv:1910.01571v4 [quant-ph] 17 Feb 2020

known as internal Josephson eﬀect (see e.g. [45]). This

internal Josephson interaction controls the many-

body physics of multicomponent phase coherent mat-

ter. Importantly, after diagonalizing the Hamiltonian

through a Bogoliubov transformation, one obtains a

spectrum that has two branches: the density mode,

ungapped and with a linear behavior at low momenta;

and the spin mode, gapped, with a parabolic behavior

even at low momenta.

From the technical point of view, we identify how

under suitable assumptions, starting from the Hamil-

tonian describing the aforementioned system of an im-

purity in a coherently coupled two component BEC,

one can equivalently describe the impurity as a Brow-

nian particle in a bath, where the role of the bath

is played by the Bogoliubov modes of the coherently

coupled two-component BEC. Furthermore, we show

that the two branches obtained after the Bogoliubov

transformation, the density mode and the spin mode,

mentioned above, result in two distinct spectral den-

sities, which we derive in Section 3. We consider two

scenarios: same coupling among the impurity and the

two bosonic components, and repulsive coupling to

one component and attractive to the other. We show

that these scenarios correspond to the impurity cou-

pling either to the density or to the spin mode of the

two-component BEC, respectively. For the coupling

to the density mode there is no qualitative diﬀerence

in comparison to the case where the impurity is em-

bedded in a single BEC [35]. For the coupling to the

spin mode, we ﬁnd a diﬀerent spectral density, namely

a gapped sub-ohmic spectral density. We derive and

solve the equations of motion of the impurity. These

are obtained through the corresponding Heisenberg

equations for the bath and impurity particles and they

have the form of Generalized Langevin equations with

memory eﬀects. By solving numerically these equa-

tions we ﬁnd the eﬀect of the gapped sub-Ohmic spec-

tral density on the Mean Square Displacement (MSD)

of the impurity.

The paper is organized as follows. In Section 2

we introduce the model Hamiltonian and transform

it into the form of a Caldeira-Leggett one. In Sec-

tion 3 we derive the spectral densities for the cases

of coupling to the density or spin modes. In Section 4

we ﬁnd and solve the Langevin equations and in Sec-

tion 5 we present the results. We end the paper with

the discussion and outlook presented in Section 6.

2 Hamiltonian

The dilute Bose-Einstein condensates created in

atomic gases [46] consist of bosons with internal de-

grees of freedom: the atoms can be trapped in diﬀer-

ent atomic hyperﬁne states. Soon after the ﬁrst obser-

vation of atom trap BECs, experimentalists succeeded

in trapping partly overlapping BECs of atoms in dif-

ferent hyperﬁne states that are (i) hyperﬁne split [47]

or (ii) nearly degenerate and correspond to diﬀerent

orientations of the spin [48]. We consider a two-

component Bose gas with both one-body (ﬁeld-ﬁeld)

and two-body (density-density) couplings, composed

of such atoms in diﬀerent hyperﬁne states. Further-

more, we assume that the two components are cou-

pled through a Josephson (one-body) type of cou-

pling. The two-body interaction results from short-

range particle-particle interactions between atoms in

diﬀerent internal states, while the one-body interac-

tion can be implemented by two-photon Raman opti-

cal coupling, which transfers atoms from one internal

state to the other. In present-day BEC experiments,

the internal Josephson or Rabi interactions, intercon-

verting atoms of diﬀerent internal states, consist in

two-photon transitions, induced by a laser ﬁeld or a

combination of a laser ﬁeld and oscillating magnetic

ﬁeld. To gain a perspective on the experimental rele-

vance of our study, we refer the readers to the work of

Refs. [47, 49–51]. Finally, we assume an impurity that

is immersed in the two-component BEC. This impu-

rity interacts with both components through contact

interactions. In Fig. 1 a sketch of the setup is shown.

The Hamiltonian of an impurity interacting with a

two-species bosonic mixture in one dimension reads

H = H

+ H

(1)

+ H

(2)

+ H

(12)

, (1)

where the impurity of mass m

is described by H

+ U(x), with U(x) being the trapping potential.

The interactions with the bosons are described by

. We study here only the case of free impurities,

hence we assume U(x) = 0. The terms of the individ-

ual bosonic species, labeled with the index j = 1, 2,

are

(j)

†

(x)

−

+ V

(x)

(x) dx

†

(x) Ψ

†

(x) Ψ

(x) dx,

where the intra-species contact interactions have

a strength given by the coupling constant g

4π~

(j)

, with a

(j)

the scattering length for the

species atoms, m

the mass of the atoms of these

species and the external potential for the atoms of the

species is denoted by V

(x). For simplicity, we as-

sume V

(x) = V

(x) = V (x) and m

= m

Furthermore we focus on the idealized case of an un-

trapped bath i.e. V (x) = 0 which results in a homo-

geneous density for the BEC. This is not a physical

scenario, but in practice, for a BEC trapped in a large

box, the impurity in the middle of this box would in-

deed approximately interact with a bath of constant

density.

The coupling Hamiltonian between the two bosonic

species consists in inter-species contact interactions,

with coupling constant g

, and a Rabi coupling Ω,

Accepted in Quantum 2019-12-28, click title to verify 2

which exchanges atoms between components, i.e.,

(12)

= g

†

(x) Ψ

†

(x) Ψ

(x) dx

+ ~Ω

†

(x) Ψ

(x) dx + H.c. (2)

where g

= 4π~

(12)

, with a

(12)

the scatter-

ing length for the intraspecies interactions. Without

any loss of generality, we will only consider Ω real

and positive. This is because even if a complex Rabi

frequency is assumed, this can always be cancelled

by introducing a counteracting phase for one of the

BECs, which can be shown to have no eﬀect on the

energy spectrum of the bath. The latter part of the

Hamiltonian, referred to as an internal Josephson in-

teraction, is a two-photon transition that is induced

by a laser ﬁeld or a combination of a laser ﬁeld and

an oscillating (rf) magnetic ﬁeld. This also introduces

an eﬀective energy diﬀerence between the two inter-

nal states/species of the BEC, which, assuming a low

intensity driving ﬁeld, is simply equal to the detuning

δ of the two-photon transition. This detuning does

not aﬀect our studies however, so for sake of clarity

and simplicity, we will assume it to be zero. We also

consider here only repulsive two-body coupling, i.e.

> 0.

Figure 1: We consider a setup of a coherently coupled two-

component BEC in which an impurity is immersed. By g

and g

we denote the intraspecies contact interactions of the

atoms of the ﬁrst and second species respectively. g

refers

to the coupling strength of the interspecies contact interac-

tion among atoms of the ﬁrst and second species. By Ω we

denote the Rabi frequency of the Raman coherent coupling

between the two species. Finally g

(1)

and g

(2)

indicate the

coupling of the impurity to the atoms of the ﬁrst and second

species respectively.

In what concerns the impurity-bosons interaction

part of the Hamiltonian, we assume that the interac-

tion is between the impurity and the densities of the

bosons, i.e. it has the form of a contact interaction:





j=1,2

(j)

†

(x)Ψ

(x) + h.c.





. (3)

with g

(j)

= 2π~

(j)

, where a

(j)

represents the

scattering length of the impurity with the bosons of

the j

BEC, and m

= m

/(m

+ m

) is the

relevant reduced mass. From this point onwards, we

assume that the BEC is one dimensional, which sim-

pliﬁes the analytical part of our studies. Nevertheless,

the main result of our work, which is the control of the

dynamics of an impurity in a coherently coupled two-

component BEC by coupling it either to the density

mode of the BEC or to the spin mode of the BEC, will

remain irrespective of the dimension. The diﬀusing

behavior itself however might change, since the spec-

tral density depends on the dimension of the BEC.

It is worth noting here, that the 1D case in fact is

peculiar since, in principle, in the purely 1D scenario

(not in the conﬁned 3D elongated/cigar shaped case)

the condensation is destroyed by the phase ﬂuctua-

tions [52]. However, if the phase coherence length is

larger than the band-size, then one can speak about

"true" BEC. In other words, as long as the physics

of interest happens on the scales smaller than the

phase coherence length, it is legitimate to use BEC

and Bogolubov-de Gennes theory, as we will do in our

work.

Furthermore, we comment here that, had we con-

sidered the more realistic case of a harmonically

trapped BEC, assuming it to be described by a

Thomas-Fermi density proﬁle, this would lead to a

discretized Bogoliubov spectrum, which would then

have to be treated under a similar approximation

as in the single BEC case in [36]. In principle, we

expect that the maim result of this work, i.e. be-

ing able to couple the impurity to two distinct types

of baths, should hold, but the related diﬀusing be-

haviours might change. We expect that when the gap

of the spin models is much bigger than the trap fre-

quency, the sub-diﬀusion, that we observe in Sec. 5,

should persist in the same form. The reader should

also have in mind that, as discussed in the appendix

of [36] the limit of the trapping frequency of the BEC

going to 0 would not lead to the homogeneous BEC

case. This scenario requires a more careful treatment,

that goes beyond the scope of our paper.

A further assumption we make in our studies is that

of a dilute gas of low depletion, since in this case, we

will be able to apply the Bogoliubov diagonalization

technique and obtain the energy spectrum of the BEC

bath. In the low-density sub-milikelvin temperature

regime of the atom trap experiments, we may assume

that the trapped atoms interact only through the par-

tial s-wave channel, and that the many-body proper-

ties are well described by assuming the particles to

interact as hard spheres. The radius of those spheres

is given by the scattering length a, which we assume

to be positive. We say that the system of particle

density n is dilute if the packing fraction of space oc-

cupied by the spheres na

 1. The assumption of

low depletion means that almost all particles occupy,

on average, the single particle state associated with

the condensate (k = 0, where k is the momentum for

the particular case of homogeneous BEC that we will

be considering). This implies that the temperatures

to be considered should be smaller than the critical

temperature.

Accepted in Quantum 2019-12-28, click title to verify 3

For a single BEC, all the bosons condensate at the

same state. However, this will not be the case for the

two-component BEC and one has to determine the

fraction of particles in each component, which will

depend on the ground state of the system. This is

determined by the parameters of the system.

With the above considerations in mind, we assume

that the two bosonic gases condense. This means that

we can apply mean ﬁeld theory and further assuming

that the ground state is coherent, the wavefunctions

(x) , Ψ

†

(x) for a homogeneous BEC are given by

(x) = Ψ

j,0

(x) + δΨ

(x), (4)

where Ψ

j,0

(x) = φ

(x)

iθ

, with θ

being the

phase of the coherent j

component, N

the num-

ber of bosons of the j

species and δΨ

(x) =

k6=0

j,k

(x)a

j,k

with φ

(x) =

√

ikx/~

the plane

wave solutions, with V

the corresponding bath’s vol-

ume. From here onwards we assume for simplicity

that V

= V

= V , i.e. that the two baths have the

same volume, and that we are dealing with homo-

geneous BECs. Here a

j,k

and a

†

j,k

are bosonic anni-

hilation and creation operatos. To proceed further,

we write the Hamiltonian in terms of these operators.

The bosonic parts read

(j)



†

j,k

k,k

†

j,k+q

†

j,k

−q

j,k

(12)

= g

k,k

†

1,k+q

†

2,k

−q

2,k

1,k

+ ~Ω

†

1,k

2,k

+H.c.,

with 

= k

/2m

. The zeroth order expectation

value (or mean ﬁeld value) of the Hamiltonian reads

j,0

1,0

2,0

+ ~Ω (Ψ

1,0

(Ψ

2,0

)

∗

+ (Ψ

1,0

)

∗

2,0

) (5)

2.1 Generalized Bogoliubov transformation

We now perform a generalized Bogoliubov transfor-

mation to diagonalize the Bosonic part of the Hamil-

tonian and hence obtain the energy spectrum of the

bath. We follow closely the results of [53–55] in

the rest of this section. This generalized Bogoliubov

transformation is understood to be composed of a ro-

tation, a scaling and one more rotation as in [53].

The derivation is based on a simple geometrical pic-

ture which results in a convenient parametrization of

the transformation. Following the generalized Bogoli-

ubov transformation, the initial bath operators are

transformed as

j,k

= Q

j,+,k

+,k

+ Q

j,+,k

†

+,−k

+ (−1)

j,−



j,−,k

−,k

+ Q

j,−,k

†

−,−k



†

j,−k

= Q

j,+,k

+,k

+ Q

j,+,k

†

+,−k

(6)

+ (−1)

j,−



j,−,k

−,k

+ Q

j,−,k

†

−,−k



with b

†

+(−),k

and b

+(−),k

and the creation/annhilation

operators for the ﬁnal spin (+) and density or phonon

(−) mode. In the latter, the total density ﬂuctuates,

while in the spin mode (+) the unlike particle densi-

ties ﬂuctuate out of phase. This, in the presence of

an internal Josephson interaction as in our case, is a

Josephson plasmon [56]. In Eq. (6) the parameters

are as follows: δ

1(2),−(+)

= 1, δ

1(2),+(−)

= 0 and

j,s,k

= (7)

j,s,k

(1−δ

j,s

) cos (γ

)+δ

j,s

sin (γ

)

cos θ

,s,k

(1−δ

) cos (γ

)+δ

sin (γ

)

sin θ,

where j

6= j, φ ∈ {0, 1}, s ∈ {+, −},

j(j

),s,k

[Γ

j(j

),s,k

+ (−1)

]/2Γ

j(j

),s,k

and R

= R

2−

(1, −1)

, R

= −R

1−

= (1, 1)

. Also, in Eq. (7),

sin (γ

) = (8)





1 −

[ω

1,k

− ω

2,k

]

(ω

1,k

− ω

2,k

)

+ 16Λ

1,k

2,k





with cos (γ

) deﬁned accordingly, and Γ

j,s,k

j,k

s,k

where

±,k

:= ~Ω

±,k

= (9)





j,k

(ω

1,k

− ω

2,k

)

+16Λ

1,k

2,k





with

j,k

= 

− (−1)



1+ (−1)

cos θ



~Ωn

j,k

+ 2Λ

j,k

= g

cos

(θ)

+ g

sin

(θ)+g

sin (2θ) cos (θ

= g

sin

(θ)

+ g

cos

(θ)−g

sin (2θ) cos (θ

√

−g

sin(2θ)

+ g

√

cos (2θ)cos(θ

where n

is the particle density of the j

bath,

and θ is the free parameter (angle) to be determined

Accepted in Quantum 2019-12-28, click title to verify 4

by the minimisation of the total energy. In the above

expression, θ

= θ

−θ

is the relative phase between

the two BEC. The minimization of the zeroth energy

with respect to the angle θ gives

tan(θ) =







, if θ

= π,

, if θ

= 0.

(10)

0.0 0.5 1.0 1.5 2.0

0.0

1.0

2.0

3.0

, =0

, =500

0.0 0.2

0.00

0.02

[ ]

0.01

0.0

0.2

0.000

0.001

0.002

Figure 2: (a) Energy spectrum for a coherentely coupled

two component BEC. There are two branches in the spectrum

corresponding to the density (-) and spin modes (+). We plot

both for diﬀerent values of the coherent coupling, Ω. First,

this illustrates that the gap opens for the spin mode (+);

and second, it shows that, while for Ω = 0 both branches

behave similarly, i.e., linearly for low k and quadratically for

large k, for ﬁnite Ω the (+) mode behaves quadratically even

at low k. This has direct implications on the behavior of the

spectral density in case 2, plotted in (b). When the Ω = 0

(blue line) the spectral density behaves as for the density

mode (i.e. with a w

-behavior). The red and green lines

(for Ω = 50π, 100πHz, respectively) show instead a diﬀerent

behavior. The inset shows a zoom, where we checked that

it ﬁts the simpliﬁed behavior in Eq. (50), i.e. has a lower

gap and behaves as

√

ω initially. In these plots we used

g = g

= 2.15 × 10

−37

J · m , n = 7(µm)

−1

, g

0.5 × 10

−37

J · m, with BEC and impurities made of Rb and

K atoms, respectively.

By minimizing with respect to the population im-

balance f =

−N

, one can obtain the following con-

ditions on the parameters of the system in order to

have an extremum of the energy,

∆ + Af − cos θ

(1 −f

)

1/2

= 0, (11)

where A =

−2g

4~Ω

is the mutual interaction

parameter, ∆ =

2δ+(g

−g

4~Ω

is the eﬀective detuning

parameter, and

A −cos θ

(1 −f

)

3/2

> 0, (12)

is the condition to have a minimum of the energy. In

[53], it was shown that to obtain the minimum energy

of the system, without imposing any condition on the

detuning δ, as is our case, then the relative phase

should be chosen to be θ

= π, referred to as the

π-state conﬁguration. From here on we assume the

symmetric case, i.e, g

= g

= g as this will allow us

to obtain analytically the spectral density in section

3. In this case, the equilibrium condition Eq. (11)

reads as



g − g

~Ω

√



− n

) = 0, (13)

which has two solutions

− n

= 0 (GS1) ,

− n

= ±n

1 −



2~Ω

(g−g



(GS2) ,

(14)

corresponding to neutral GS1 and polarized ground

states GS2. Here, we make the strong Josephson junc-

tion assumption

|A| < 1, (15)

which is also referred to as the miscibility condi-

tion. This implies that the minimum energy equi-

librium ground state has to be GS1 as is shown in

[53, 55]. Handable expressions for the spectral den-

sity obtained section 3 are possible over GS1. For the

regime in which ground state is GS2 we expect similar

qualitative behavior, but we did not obtained a form

for the spectral density which allows us to obtain the

diﬀusive behavior of the impurity. The study of the

impurity diﬀusion over GS2, and even at the phase

transition, falls out of the scope of this paper. Under

the miscibility condition (15), the energy spectrum

expressions simpliﬁes into

−,k

= (

(

+ (g + g

) n))

, (16)

+,k

= [

(

+ (g − g

) n + 4~Ω)

+2~Ω [(g − g

) n + 2~Ω]]

. (17)

In Fig. 2 we plot the energy spectra as a function

of k for speciﬁc parameters, to illustrate the spin and

density branches. Furthermore, note that the bogoli-

ubov transformation elements satisfy the well-known

relation

)

− Q

)

= 1, (18)

where Q

1,+,k

1,−,k

2,+,k

2,−,k

with φ ∈ {0, 1},

that implies normalization. However, as is shown in

Accepted in Quantum 2019-12-28, click title to verify 5

[54], these Bogoliubov operators, do not fulﬁll the

bosonic commutations relations, which is understood

as a consequence of the fact that they are not orthogo-

nalized with respect to the quasicondensate functions

j,0

iθ

. In [54] it is shown that to overcome

this problem one needs to deﬁne some new transfor-

mation with components

j,s,k

, that are re-

lated to the previous ones as

j,s,k

= Q

j,s,k

−

j,0

j,s,k

∗

j,0

. (19)

The elements of this transformation, these new Bo-

goliubov operators, are expressed in terms of the Bo-

goliubov wave functions of our system f

j,s,k

+ (−1)

j,s,k

, (20)

where

1,−,k

= f

2,−,k





−,k



1/2

1,−,k

2,−,k



−,k

2



1/2

1,+,k

= f

2,+,k





+ ~Ω

+,k



1/2

1,+,k

2,+,k



+,k

2 (

+ ~Ω)



1/2

The spin mode branch is gapped while the density

mode branch is gapless. For the latter, at low values of

the momentum k the dispersion is linear, with a speed

of sound c

n (g + g

) /(2m

). On the contrary

for the gapped branch, the dispersion relation goes as

for low k, and at k = 0, it has a gap

gap

2~Ω [(g − g

) n + 2~Ω]. (21)

This corresponds to the Josephson frequency for small

amplitude oscillations. As we will see the fact that

there are two branches in the spectrum will give rise

to two diﬀerent noise sources.

Furthermore, one should note that, had we not in-

troduced the Rabi coupling term in the Hamiltonian,

the latter would commute with both n

and n

such

that we would have two broken continuous symme-

tries and both branches would be gapless (notice that

gap

→ 0 when ~Ω → 0). In this case, the low mo-

mentum excitations would be both phase-like, as it

has to be for Goldstone modes of the U (1)xU (1) bro-

ken symmetries [57]. Hence the introduction of the

Rabi coupling term, results in the system having only

one continuous broken symmetry, namely only n has

to be conserved now and not both n

and n

. The long

wavelength limit of the Goldstone mode corresponds

to a low-amplitude phonon ﬂuctuation in which the

total density oscillates and the unlike atoms move in

unison (i.e., with the same superﬂuid velocity). In

contrast, in the long wavelength gap mode the un-

like atoms move in opposite directions, while their

center of mass remains at rest. This ﬂuctuation is

then reminiscent of the motion of ions in an optical

phonon mode, which also exhibits a gapped disper-

sion. At zero momentum, the gap mode corresponds

to an inﬁnitesimal Josephson-like oscillation of the

populations in the distinguishable internal states. In

the strong Josephson coupling regime we have closed

orbits around a ﬁxed point for the Josephson Hamilto-

nian, with vanishing mean polarisation (or population

imbalance) and a phase diﬀerence around π if ~Ω ≥ 0,

giving rise to plasma-like oscillations.

2.2 Transfor med Impurity-Bath interaction

In terms of the original annihilation and creation op-

erators, the impurity-bath term reads as

k,q

(j)

(k) ρ

(q) a

†

j,k−q

j,k

(22)

k6=0

(k) V

(j)



j,k

+ a

†

j,−k



with ρ

(q) =

∞

−∞

−iqx

δ (x

− x) dx

, V

(j)

(k) =

(j)

δ (x − x

)] where F is the Fourier trasnform.

Furthermore, n

, with j = 1, 2, is the averaged den-

sity of the j

bath. The second line of Eq. (22) is a

consequence of the assumption that the bosons con-

dense. We will consider two cases for the coupling of

the impurity to the baths.

1. In the ﬁrst scenario the impurity couples to the

two baths in the same way

(1)

= g

(2)

= g

, (23)

2. while in the second scenario, the interactions are

attractive with one of the baths and repulsive

with the other the other,

(1)

= −g

(2)

= g

. (24)

After the Bogoliubov transformation the impurity-

bath term reads in both cases as

1. H

(−)

j,k6=0

(k) g

j,−,k

−,k

2. H

(+)

j,k6=0

(k) g

j,+,k

+,k

where x

±,k



±,k

+ b

†

±,k



. These equations show

that in case 1 the impurity only couples to the den-

sity (−) mode of the bosonic baths, while in case 2 it

Accepted in Quantum 2019-12-28, click title to verify 6

couples only to the spin (+) mode. For both cases,

we rewrite the impurity-bath terms as

j,k6=0

s∈{+,−}

j,s,k

ikx



s,k

+ b

†

s,−k



, (25)

where s = − for case 1 and s = + for case 2, and

j,s,k



j,s,k



. (26)

We note here that

1,s,k

2,s,k

such that V

1,s,k

= V

2,s,k

s,k

. We linearize the

interaction (see [35] for validity of this assumption)

to get

k6=0

s∈{+,−}

s,k

(I + ikx)



s,k

+ b

†

s,−k



. (27)

where V

s,k

= 2

s,k

. Thus, after a redeﬁnition b

s,k

→

s,k

−

s,k

I, the ﬁnal total Hamiltonian reads as

H = H

k6=0

s∈{+,−}

s,k

†

s,k

k6=0

s∈{+,−}

s,k

(28)

with g

j,s,k

= kV

s,k

/~ and π

s,k

= i



k,s

− b

†

k,s



the

momentum of the bath particles. We see that as in

[35], the coupling occurs between the position of the

impurity and the momentum of the bath particles.

However, in our work, the coupling can take place to

one of the two diﬀerent quasiparticle branches, de-

pending on the form of the impurity-baths interac-

tions.

3 Spectral densities

The spectral densities can be obtained from the self-

correlation functions [35] for each environment (cor-

responding to cases 1 and 2). These read

C (t) =

k 6= 0

s ∈ {+, −}

s,k

hπ

s,k

(t) π

s,k

(0)i/~. (29)

Using that the bath is composed of bosons for which

†

k,s

~ω

− 1

, (30)

we obtain

C (t)=

k6=0

s∈{+,−}

s,k



coth



~ω



cos (ω

t)−i sin (ω



= ν (t) −iλ (t) , (31)

where

ν (t) =

∞

s∈{+,−}

(ω) coth



~ω



cos (ωt) dω,

λ (t) =

∞

s∈{+,−}

(ω) sin (ωt) dω. (32)

In these deﬁnitions we used the spectral density,

(ω) = ~

k6=0

s,k

)

δ (ω − ω

) . (33)

The spectral density is then evaluated in the contin-

uous frequency limit as

(ω) = (34)

4ng

~ (2π)

dkk

d+1

s,k

)

δ (k−k

(ω))

∂

(k)|

k=k

(ω)

where D

is the surface of the hypersphere in the mo-

mentum space with radius k in d-dimensions. In the

particular case of 1D becomes D

= 2.

To obtain the expression for the continuous fre-

quency case, the inverse of the dispersion relation

from Eq. (9) is needed. For this general energy spec-

trum, obtaining such inverse function is not easy.

However this is indeed possible for the simpliﬁed case,

Eqs. (16)–(17). The inverse of the density (-) branch

which is the one to which the impurity couples for the

case 1 type of coupling, reads

−

(ω) = (35)

√





−1 +

1−







2ω







With this, for the density (−) branch (case 1 type of

coupling), the spectral density is

−

(ω) = ˜τ

−

(ω)

3/2

−

(ω)

, (36)

with

−

(ω) = 1 +



−



, (37)

−

(ω) = −1 +

−

(ω), (38)

˜τ

−

(2g

)

3/2

1/2

−

, (39)

and where Λ

−

= n (g + g

) /2~ is the cutoﬀ fre-

quency, which resembles the one in [35] when g is

replaced by

g+g

. In the limit of ω  Λ

−

, the spec-

tral density can be simpliﬁed to

−

(ω) = m

−

, (40)

where

−

(2η)

2πm

n (g + g

)

1/3

3/2

, (41)

Accepted in Quantum 2019-12-28, click title to verify 7

with η

−

g+g

. Thus, for the the π-state equilib-

rium conﬁguration one obtains a cubic spectral den-

sity.

3.1 Spin branch coupling

For the spin (+) branch (case 2 type of coupling), the

inverse of the spectrum reads as

(ω) =

√

ng× (42)





−1−

4~Ω

1−







2ω







In this case, the spectral density is

(ω) = m

eτ

(ω)

, (43)

where

(ω) = 1 +





(ω)=W (ω)





W (ω)+

−



gap







W (ω) = −

(ω) −

1 +



gap



eτ

(2g

)

3/2

1/2

πm

, (44)

with Λ

= n (g − g

) /2~. We note that to interpret

eτ

as a relaxation time, as is custom to do (see [35]),

one has to impose g ≥ g

to assure it remains a real

quantity. In other case, the spectral density will be

imaginary (note

(ω)

√

(ω)

is independent of the sign of

g − g

Let us ﬁnd how the spectral density in Eq. (43) sim-

pliﬁes in two limiting cases. First, in the absence of

coherent coupling, Ω = 0, the gap vanishes, E

gap

= 0.

In this case, Eq. (43) is equal to that of the density

mode, upon the interchange Λ

−

→ Λ

. Therefore,

on the long time limit ω  Λ

we obtain the same

cubic behavior of the spectral density. We illustrate

this case in Fig. 2. In panel (a) we show that the

two branches of the energy spectra have the same be-

havior, that is, linear at low k and parabolic for large

Second, we consider the case of ﬁnite Ω which im-

plies E

gap

> 0. A requirement which we impose on

the spectral density is that one cannot consider fre-

quencies lower than the gap energy E

gap

. Physically,

one can interpret this as follows: Since the energy

spectrum of the bath is gapped, with a gap given by

Eq. (21), the spectral density cannot assign a weight

at frequencies lower than this, because the bath can-

not excite the impurity with such frequencies since it

is not part of its spectrum. Then, we simplify the

spectral density as

(ω) = Θ (ω − E

gap

) J

(ω) , (45)

with Θ(.) the step delta function.

Let us now comment on the frequency region right

above the energy gap of our system. To this end, we

replace ω = E

gap

+ , where  > 0. We use  as the

small value expansion parameter in our case, i.e. we

consider the limit   E

gap

, such that ω ≈ E

gap

. We

furthermore introduce a cutoﬀ Λ, for which it holds

that   Λ − E

gap

. The expressions in the spectral

density will now read as

() = 1 +



gap



, (46)

W () =

−



gap



gap



gap

+ Λ



−

,

such that

() = (47)





−



gap









gap





gap

+Λ



−



Hence the spectral density is

() = Θ () τ



1/2

, (48)

with

= eτ

−

1 +



gap



gap

)



1 +



gap





1/4

. (49)

The ﬁnal form of the spectral density, after introduc-

ing a cutoﬀ Λ, to avoid the related ultraviolet diver-

gencies mentioned above, is

(ω) = (50)

Θ (ω − E

gap

) τ

[ω − E

gap

]

Θ (Λ + E

gap

− ω) .

We introduced a hard cutoﬀ to our spectral density

as this better describes the physical system we study.

Such a spectral density, i.e. with an exponent on the

frequencies less than 1, is often associated to sub-

diﬀusive impurity dynamics. Note that in the re-

sults that we present below, we assume g ≈ g

in [58], which signiﬁcantly simpliﬁes the expression

for the coeﬃcient of the spectral density τ

with-

out changing its behavior. In particular in this case

= (2g

)

3/2

1/2

πm

. In Fig. 2(b), we show

how the approximate spectral density in Eq. (50),

under the assumption   E

gap

, compares to the

Accepted in Quantum 2019-12-28, click title to verify 8

original spectral density. For vanishingly small val-

ues of E

gap

the spectral density approaches the form

of that of the density mode, i.e. it goes as ∝ ω

, as

expected. In the limit we are interested, that is, for ﬁ-

nite E

gap

, the spectral density behaves approximately

as in Eq. (50) (see inset in Fig. 2(b)).

Such gapped spectral densities as in Eq. (50), have

been studied extensively in the literature. In general,

they are usually related to semiconductors [59–61] or

photonic crystals (PC) [62]. In particular, the simpli-

ﬁed form of the spectral density, Eq. (50), is related in

particular with 3D PCs. The latter, are artiﬁcial ma-

terials engineered with periodic dielectric structures

[63]. If one considers an atom embedded in such a

material, it is known that if the resonant frequency of

the excited atom approaches the band gap edge of the

PC, strong localization of light, atom-photon bound

states, inhibition of spontaneous emission and frac-

tionalized steady-state inversion appear [64–67]. The

rapidly varying distribution of ﬁeld modes near the

band gap [62, 68] requires a non-Markovian descrip-

tion [69] of the reduced dynamics of quantum systems

coupled to the radiation ﬁeld of a PC [66, 67, 70–72].

This enhanced appearance of non-Markovian eﬀects

is also conﬁrmed by a recent study based on exact

diagonalization [73]. In this study it was observed

that for frequencies of the bath much larger than the

band gap, energy transfer between the system and

the bath is such that information and energy ﬂow ir-

reversibly from system to bath leading to Markovian

dynamics. Conversely, at the edges of the gaps, one

observes the largest backﬂow of information where the

energy bounces between the system and bath lead-

ing to non-Markovian evolution of the system. Fur-

thermore, deep within the band gap, less excitations

and energy are exchanged between the system and

the bath, which is shown to lead to localized modes

[74], expressed as dissipationless oscillatory behavior,

plus non-exponential decays (such as for example frac-

tional relaxations [75]).

From the work in [74], a relationship is suggested

of such long-lived oscillations that appear in the dy-

namics of a system coupled to a bath with a gapped

spectral density, with the fact that the Hamiltonian of

the system might have thermodynamic and dynamic

instabilities. This is the case when the Hamiltonian

is unbounded from below, i.e. non-positive. Physi-

cally this happens when one deals with Hamiltonians

that do not conserve the particles number, and this is

indeed the case for the Hamiltonian of the free quan-

tum Brownian motion we study here. Hence, as a

result, this unbounded Hamiltonian induces dynami-

cal instabilities in the long-time regime, correspond-

ing to the limit ω  Λ. In practice, as in the spirit

of [35], one can show that the eﬀect of the bath on

the impurity is not only to dissipate its energy, but

as well to introduce an inverse parabolic potential in

which the impurity is diﬀusing (which would work as

a renormalization of the trapping frequency had we

considered a harmonic trapping potential). This in-

verse parabolic potential is understood to be a con-

sequence of the unboundedness of the Hamiltonian,

and is what is resulting in the dynamical instability

of the long time solution of the impurity dynamics.

Unfortunately, contrary to the case in [35], we will not

consider a harmonic trap for the impurity, and hence

the positivity of the Hamiltonian is violated irrespec-

tive of the strength of the coupling of the impurity to

the bath. In practice one would study the impurity

constrained in a box of a certain size, which if in-

cluded in the modelling of the system, would result in

a positively deﬁned Hamiltonian, at the price of com-

plicating signiﬁcantly the analytical solution for the

impurity’s dynamics. Hence, we assume here that we

are looking at timescales where the eﬀect of the ﬁnite

sized box are not manifested. Theoretically, there are

also a number of other ways to circumvent this prob-

lem, even without referring to the presence of a box,

such as taking into account bilinear terms in the im-

purity’s or bath’s operators in the Hamiltonian as in

[76–79]. In any case, we will show below that for the

regime of the transient eﬀect that we are interested

in, this will not change our results.

Following the approach sketched above, we take ad-

vantage of the simplicity of the Fröhlich like Hamil-

tonian we are considering above. Furthermore, we

remind that we will look at the long time dynamics

of the impurity, i.e. ω  Λ, as was implied by the

above study on the spectral density’s form. In addi-

tion, one should take into account the dissipationless

oscillatory behavior, which can also be expressed as

an incomplete decay of the Green function impurity

propagator which we will study below. In fact by

identifying the equivalence of the appearance of these

oscillations with the incomplete decay of the Green

function, this provides us with a very simple condition

upon which the long-lived oscillations appear, that is

that the Green function has at least one purely imagi-

nary pole, which can be shown to only be possible for

frequencies within the band gap [62]. We will study

this in the next section.

4 Heisenberg equations and their solu-

tion

In this section, we derive the equation of motion for

the impurity, which will allow us to study its diﬀusive

behavior under various scenarios. To do so, we be-

gin with the Heisenberg equations of motion for both

the impurity and the bath particles. The latter set

of equations can be solved, and we use this solution

to obtain a Langevin like equation of motion for the

impurity. The Heisenberg equations for the bath par-

Accepted in Quantum 2019-12-28, click title to verify 9

ticles are

s,k

(t)

= i [H, b

s,k

(t)]

= −iΩ

s,k

(t) −~

j=1

(j)

s,k

x (t) ,

†

s,k

(t)

H, b

†

s,k

(t)

Ω

s,k

†

s,k

(t) −~

j=1

(j)

s,k

x (t) , (51)

and for the central particle

dx (t)

[H, x (t)] =

p (t)

, (52)

dp (t)

[H, p (t)]

[U(x),x (t)]−

k6=0

j={1,2}

s={+,−}

(j)

s,k

(t) .

Substituting the solutions of the equations of motion

for the bath into that of the central particle, one gets

¨x (t) +

∂

∂t

Γ (t − s) x (s) ds =

B (t)

, (53)

where

Γ (τ ) =

∞

j={1,2}

s={+,−}

(j)

(ω)

cos (ωτ ) dω,

B (t) = (54)

k6=0

j={1,2}

s={+,−}

(j)

s,k



†

s,k

(t) e

iω

− b

s,k

(t) e

−iω



are the damping and noise terms, respectively. Note

that in Eq. (53) we neglected a term −Γ (0) x (t).

This term may introduce dynamic instabilities in our

system in the long time regime. As in [35], we neglect

it as these instabilities are unphysical, that is, will

not occur in a physical realization of the system and

will only occur in the long time behavior. To be more

speciﬁc, for the coupling to the density mode this term

reads as,

−

(0) = τ

−

. (55)

For the coupling to the spin mode this term reads as

(0) = τ

− πE

1/2

Gap

+ 2 (Λ + E

Gap

)

0.5

(56)

× F

2,1



−

, −

;

Gap

Λ + E

Gap



As in [35], the solution of Eq. (53) takes the form

x(t) = (57)

(t)x(0) + G

(t) ˙x(0) +

(t −s)B(s)ds,

with the corresponding Green functions given by

(t)] =

+ zL

[Γ(t)]

z + L

[Γ(t)]

, (58)

(t)] =

+ zL

[Γ(t)]

, (59)

where L

[·] represents the Laplace transform. The

expressions for G

(t) and G

(t) depend on the spe-

ciﬁc type of bath we consider. For the ﬁrst scenario

(coupling to the density mode), in [35] it was found

that Γ(t) is

−

(t) = (60)

−



2Λ

−

t cos (Λt) −2



2 −Λ

−



sin (Λ

−



and under the assumption of z  Λ

−

[Γ

−

(t)] = τ

−

z + O





, (61)

which results in

(t)] =

(1 + Λ

−

) z

, (62)

and

(t)] =

(1 + Λ

−

) z

. (63)

Then, one obtains

(t) =

(1 + Λ

−

)

, (64)

and

(t) =

(1 + Λ

−

)

, (65)

where we see that the Green functions has an iden-

tical form to that of [35]. More importantly, G

(t)

diverges at t → ∞, a consequence of the fact that an

equilibrium state is not reached at this limit, as we

see in next section.

For the second case (coupling to the spin mode),

as discussed in previous section the spectral density

is gapped and given by Eq. (50). To proceed, we

ﬁrst need an expression for the Laplace transform

[Γ

(t)] which can be shown to read as

[Γ

(t)] =

∞

dω

(ω)

cos (ωt)

−zt

= z

∞

dω

(ω)

ω (ω

+ z

)



gap



1.5



−

gap

+iz)F

2,1



;

;−

gap

−iz



−

gap

− iz) F

2,1



;

; −

gap

+ iz





gap

+ z



2,1



;

; −

gap



, (66)

Accepted in Quantum 2019-12-28, click title to verify 10

where F

2,1

(α, β; γ; z) is the hypergeometric function

2,1

(α, β; γ; z) =

∞

n=0

(α)

(β)

(γ)

, (67)

with (·)

being the Pochhammer symbol. Unfortu-

nately, to invert the Laplace transform in Eq. (59),

given Eq. (66), is rather complicated. For this rea-

son we restrain ourselves to study only the long-time

limit, determined by z  Λ. In this case the inverse

Laplace transform of the Green’s function in Eq. (59),

reads as

(t) = At, (68)

A=E

gap



gap

1.5

2,1



;

−Λ

gap



−

gap

2.5

2,1



;

; −

gap



+0.285714Λ

3.5

2,1



;

; −

gap



−1

which has the same time dependence as in case 1 (cou-

pling to the density mode). We also ﬁnd G

(t) = A

which is again a constant and hence does not play a

role in the time dependence of the MSD which we are

interested in. Unfortunately this is as far as we can get

analytically, as contrary to the coupling to the den-

sity mode, even though we have the Green function

at hand, using it to obtain an analytic expression for

the MSD of the impurity which is our ultimate goal

is not possible.

Equation (65) as well as Eq. (66) have been both

obtained at the long time limit, which implied ex-

panding the Laplace transform of the damping kernel

[Γ

−

(t)] , L

[Γ

(t)] at the ﬁrst order in z/Λ

−

, z/Λ.

In general, one could have considered higher orders

of the aforementioned expansion, but should then be

careful in inverting the Laplace transform to obtain

the Green function in deﬁning the relevant Bromwich

integral in the complex plane in such a way as to not

include the roots which correspond to divergent run-

away solutions [35]. Even if one would do so, the

result for the Green function would not change much,

and this can be proven by considering a numerical in-

version of the Laplace transform for the Green func-

tion, where the long time limit assumption is not

made. For the coupling to the density mode this

was shown using the Zakian method in [35]. For the

spin mode, we checked this using the same method.

Moreover, we contrasted its results to two other meth-

ods for numerically inverting a Laplace transform, in

particular, the Fourier and the Stehfest methods [80].

The Zakian method, gives the inverse of the Laplace

transform of a function F (z) in the following form

f (t) =

j=1







, (69)

where k

and β

are real and complex constants given

in [80]. With all of these methods, the Green function

behaves linearly with time for the range of parame-

ters we considered. In fact in the numerical results

presented in the next section, the Zakian method was

used to obtain the Green function, such that our re-

sults are not restricted just to the long time limit,

z  Λ.

In addition, we are also now in a position to check

the presence of the long-lived oscillations in our sys-

tem. As was mentioned before, this can only be the

case if the Green function exhibits a purely imaginary

pole, which if it exists, should correspond to a fre-

quency within the bandgap. As is shown in [81], this

will be the case, for the frequency that is a solution

+ Γ (0) − ∆ (ω) = 0, (70)

where

∆ (ω) := P

∞

(ω

)

ω − ω

dω

= −

2τ

1.5

2,1



;

; −

gap

−ω



3 (E

gap

− ω)

, (71)

is the bath self energy correction, where P denotes the

principal value. One can show that the expression of

Eq. (71) is always non-positive for ω < E

gap

and hence

the condition in Eq. (70) is never satisﬁed, such that

we do not have to worry about these oscillations in

the transient dynamics that we will study in the next

section.

Finally, one can evaluate the validity of the linear-

ity assumption which allowed us to consider a lin-

ear coupling between the BEC and the impurity (see

Hamiltonian in Eq. (27)) in terms of the physical pa-

rameters of the system. This assumption reads as

kx  1. In [35] it was shown that, as a function

of the temperature, there exist a maximum time for

which the linear assumption holds. In the system dis-

cussed here, since an expression for the MSD cannot

be found, for each set of parameters one has to eval-

uate numerically the long-time behavior of the MDS

and determine the maximum time for which the as-

sumption holds. To this end, one has to note that, dif-

ferently to [35], for the coupling to the spin mode the

momenta grows parabolically with ω even for small k

and there is an energy gap. Then, to evaluate the cri-

teria (kx  1), one has to use the expression for the

energy, Eq. (17) together with the numerically evalu-

ated MSD. We checked this condition in the numerical

examples presented in next section. A ﬁnal comment

is that, we also made sure that the assumption of a

purely 1D BEC was valid, by checking that the spa-

tial extend of the motion did not exceed the phase

coherence length [52].

Accepted in Quantum 2019-12-28, click title to verify 11

Log

[<x

(t)>/a

]

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

- 0.5

0.0

0.5

1.0

=1.5

(b)

Figure 3: Mean square displacement vs time for the case

of coupling to the spin mode. A cutoﬀ of Λ = 10

Ω was

used, where

Ω = 1000πHz. In (a) we plot it for diﬀerent

coherent couplings Ω and in (b) for diﬀerent couplings to the

bath. The MSD shows three regimes, where it behaves ap-

proximately as MSD(t) ∝ t

, and therefore linearly in log-log

pots, with a diﬀerent slope given by the anomalous exponent

α: (i) an initial short time behavior, where α ≈ 2; (ii) a

nontrivial transient subdiﬀusive behavior, where α < 1. We

plot a dashed orange line as a guide to the eye, to illustrate

the diﬀerent slopes in this regime; (iii) a long time ballistic

regime, with α = 2. In (a) we show that, as Ω is reduced,

the subdiﬀusive platteau enlarges and α gets smaller. In

(b) we show that increasing the couplings to the bath τ

also enlarges the plateau and reduces α. We consider Rb

and K atoms for BEC and impurities, respectively. We use

g = g

= 2.15 × 10

−37

J · m, density n = 7(µm)

−1

, and

impurity-BEC g

= 0.5 × 10

−37

J · m; We take τ

= 1 in

(a) and Ω = 100πHz in (b). The BEC was assumed to be

in the low temperature regime, i.e. when coth



~ω



→ 1

holds.

5 Results: Mean square displacement

With the Green propagator and the spectral density

at hand, we are now in a position to evaluate the

MSD. This, as shown in [35], is evaluated in the long

time limit ω  Λ

−

, Λ as

h[x (t) − x (0)]

i = MSD(t) = G

(t)h˙x

(0)i

dσG

(s)G

(σ)h{B(s), B(σ)}i

, (72)

where we assumed that the impurity-bath are initially

in a product state ρ (0) = ρ

⊗ ρ

(0), where ρ

the thermal Gibbs state for the bath at temperature

T . The initial conditions of the impurity and bath

oscillators are then uncorrelated. Then, averages of

the form h˙x (0) B(s)i vanish. To treat the second term

in Eq. (72), we note that

h{B(s), B(σ)}i

= 2~ν (s −σ) , (73)

where ν (t) is deﬁned as in Eq. (32).

In case 1 (coupling to density mode) the spectral

density reads as in Eq. (40). Then, the MSD behaves

the same way as in [35], with the only diﬀerence of

replacing g → g + g

. Hence the impurity will again

superdiﬀuse as

h[x (t) − x (0)]

i =



h˙x

(0)i +

−





, (74)

where ζ = 1 + τ

−

. Note that the superdiﬀusive

behavior hx

(t)i ∝ t

appears for both low temper-

ature (coth (~ω/2k

T ) ≈ 1) and high temperature

(coth (~ω/2k

T ) ≈ 2k

T/~ω) limits. Hence from

Eq. (74) we see that, eﬀectively, the contribution of

the Bogoliubov modes to the MSD behavior in this

case is just to modify the mass of the free particle.

In case 2 (coupling to the spin mode), analytical

expressions for the Eq. (72) cannot be found. We

remind again that we are interested in the transient

eﬀects attributed to the bath frequencies right above

the band gap, after making the assumption for ω =

gap

+  that ω ≈ E

gap

i.e.   E

gap

. In this case the

Green function reads as in Eq. (50), while the noise

kernel at low temperatures, where coth



~ω



→ 1,

can be shown to be equal to

ν (t) = τ

(Λ−E

gap

)

1.5

(75)



cos (E

gap

t)F

1,2



;

; −

gap

−Λ)



t (E

gap

− Λ) cos (E

gap

×F

1,2



;

; −

gap

− Λ)



where

1,2

(α; β, γ; z) =

∞

n=0

(α)

(β)

(γ)

. (76)

In this case, we were not able to obtain an analytic

solution for the MSD. We evaluate it numerically, with

the results being valid for ﬁnite Ω, as we used the

simpliﬁed version of the spectral density, Eq. (50). In

all calculations we checked that all assumptions made

are fulﬁlled.

In Fig. 3 we show the numerically evaluated MSD

as a function of time according to Eq. (72) and dif-

ferent Rabi frequencies and interaction strengths. We

remind that initially, h˙x

(0)i = 0. In Fig. 3 (a) we

show how decreasing Ω both enlarges the duration of

Accepted in Quantum 2019-12-28, click title to verify 12

the subdiﬀusive plateau and reduces the anomalous

exponent α. We should note that the results are valid

only for ﬁnite Ω: since we use the simpliﬁes spectral

density, Eq. (50), we are never able to describe the

smooth transition to the cubic spectral density, which

will show an smooth change to ballistic behavior for

the whole range. Then, the eﬀect of reducing Ω is

merely to reduce the gap, not to change the form of

the spectral density. As a consequence, the plateau is

enlarged. In Fig. 3 (b) we show how the MSD varies

as a function of the coupling stength of the impurity

to the BECs. Here, we observe that increasing the

coupling strength results in more subdiﬀusive motion

and an increase in the duration of the subdiﬀusive

plateau. Furthermore, note that, in Fig. 3, time is

measured in units of the inverse of

Ω = 1000πHz and

hence the transient subdiﬀusive phenomenon appears

in time of the order of ms.

In general, from the results presented in Fig. 3,

we numerically ﬁnd three regimes of behavior for the

MSD, and in each regime it behaves as MSD(t) ∝ t

where α is diﬀerent at each regime. The exponent

α is known as the anomalous exponent. In regime

(i), there is an initial short time behavior where, as

expected, the MSD grows more or less ballistically

with time. So here, α ≈ 2; In regime (ii), there is a

plateau where α < 1. This is a transient subdiﬀusive

behavior; Finally, in regime (iii), which is the long

time behavior, the impurity superdiﬀuses with α = 2.

We interpret this behavior as follows: the impurity

performs free motion initially. Then after interact-

ing with the large frequencies of the bath, the im-

purity begins to perform a subdiﬀusive motion since

it screens the part of the spectral density that de-

pends on the square root of the bath modes frequen-

cies. At long times, and after undergoing dissipation

for some time, the impurity again eﬀectively only in-

teracts with the lower frequencies of the bath which

have zero eﬀect on the motion of the impurity and

hence the impurity performs a ballistic motion.

Finally, we comment here that the results of the nu-

merical integrations undertaken to obtain the MSD in

this section, indeed appear to agree with the analyti-

cal calculations we performed in the previous section.

This is to be understood in the following sense. The

analytical results of the previous section, implied that

the coupling of the impurity to the spin mode, results

in the impurity interacting with a bath giving rise

to a diﬀerent spectral density, which intuitively one

expects to give a diﬀerent MSD behaviour. This is

indeed what we observe in the results presented in

Fig. 3.

6 Conclusions

In this work, we studied the diﬀusive behavior of

an impurity immersed in a coherently coupled two-

component BEC, that interacts with both of them

through contact interactions. We showed how start-

ing from the standard Hamiltonian that would de-

scribe such a scenario, one can recast the problem

into that of a quantum Brownian particle diﬀusing

in a bath composed of the Bogoliubov modes of the

two-component BEC. We discussed the under certain

assumptions and conditions required to obtain this

description.

We found that the main diﬀerence of this scenario

compared to that of the impurity being coupled to

a single BEC studied in [35], is that for the scenario

of the impurity being coupled diﬀerently to the two

BECs, namely coupled attractively to one of them and

repulsively to the other but with the same magnitude,

results in the impurity being coupled to the spin mode

of the coherently coupled two-component BEC. This

implies that its dynamics is determined by a quali-

tatively diﬀerent spectral density. In particular this

new spectral density is gapped and subohmic close to

the gap. We demonstrate numerically, that such a

spectral density gives rise to a transient subdiﬀusive

behavior. Furthermore, we show that this transient

eﬀect can be controlled by the magnitude of the Rabi

frequency, as well as by the strength with which the

impurity couples to the two BECs. These can control

the time duration for which this subdiﬀusive behavior

appears. A mechanism for inducing a transient con-

trolled subdiﬀusion in Brownian motion has been also

proposed in [82], but with a completely diﬀerent way

for achieving it and most importantly not considering

the system from a microscopic perspective. Moreover,

we comment that the setup we studied, thanks to the

appearance of this gapped subohmic spectral density,

could also serve for simulating quantum-optical phe-

nomena, that could be seen for example in photonic

crystals, using instead cold atoms, as was proposed

also in [83] for the case of optical lattices. In addi-

tion, we note that our studies could be extended to

the scenario of having two impurities in the coherently

coupled two-component BEC, and study as in [37],

the eﬀects that the coupling to the spin mode could

have on the bath-induced entanglement between the

two impurities. Finally, we could also study the eﬀect

that this new gapped spectral density could have on

the functioning of the impurity as a probe to measure

the temperature of the two-component BEC, as in

[84]. Last but not least, it should be noted here that,

if one considers the scenario of attractive two-body

coupling, i.e. g

< 0, and includes the Lee-Huang-

Yang corrections to the Hamiltonian, one will obtain

the scenario of quantum droplets studied theoretically

in [85] and recently proven experimentally in [86] .

This we expect to lead in diﬀerent interesting dynam-

ics for the immersed impurity in the two-component

coherently coupled BEC.

Accepted in Quantum 2019-12-28, click title to verify 13

Acknowledgments

We (M.L. group) acknowledge the Spanish Min-

istry MINECO (National Plan 15 Grant: FISI-

CATEAMO No. FIS2016-79508-P, FPI), the Ministry

of Education of Spain (FPI Grant BES-2015-071803),

EU FEDER, European Social Fund, FundaciÃş

Cellex, Generalitat de Catalunya (AGAUR Grant No.

2017 SGR 1341 and CERCA/Program), ERC AdG

OSYRIS and NOQIA, EU FETPRO QUIC, and the

National Science Centre, Poland-Symfonia Grant No.

2016/20/W/ST4/00314. MAGM acknowledges fund-

ing from the Spanish Ministry of Education and Voca-

tional Training (MEFP) through the Beatriz Galindo

program 2018 (BEAGAL18/00203).

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