Dissipative dynamics of an impurity with spin-orbit coupling Areg Ghazaryan Alberto Cappellaro Mikhail Lemeshko Artem G. Volosniev Institute of Science and Technology Austria ISTA am Campus 1 3400 Klosterneuburg Austria

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Dissipative dynamics of an impurity with spin-orbit coupling

Areg Ghazaryan, Alberto Cappellaro, Mikhail Lemeshko, Artem G. Volosniev

Institute of Science and Technology Austria (ISTA), am Campus 1, 3400 Klosterneuburg, Austria

(Dated: October 18, 2022)

Brownian motion of a mobile impurity in a bath is aﬀected by spin-orbit coupling (SOC). Here, we

discuss a Caldeira-Leggett-type model that can be used to propose and interpret quantum simulators

of this problem in cold Bose gases. First, we derive a master equation that describes the model

and explore it in a one-dimensional (1D) setting. To validate the standard assumptions needed for

our derivation, we analyze available experimental data without SOC; as a byproduct, this analysis

suggests that the quench dynamics of the impurity is beyond the 1D Bose-polaron approach at

temperatures currently accessible in a cold-atom laboratory – motion of the impurity is mainly

driven by dissipation. For systems with SOC, we demonstrate that 1D spin-orbit coupling can be

‘gauged out’ even in the presence of dissipation – the information about SOC is incorporated in the

initial conditions. Observables sensitive to this information (such as spin densities) can be used to

study formation of steady spin polarization domains during quench dynamics.

Dissipation of energy occurs naturally when a particle

with ﬁnite momentum moves through a medium. This

phenomenon is typically studied assuming that the mo-

mentum of the particle is decoupled from its spin degree

of freedom. This is however not the case for many con-

densed matter systems with strong spin-orbit coupling

(SOC). In particular, for externally driven setups with

non-trivial topological character, such as bosonic Kitaev-

Majorana chains [1, 2], Majorana wires [3, 4], as well as

systems featuring optical spin-Hall eﬀect [5, 6]. SOC is

also key for explaining transport of electrons through a

layer of chiral molecules [7, 8]. To understand equilibra-

tion processes in these systems and promote their use in

technologies, studies of dissipative dynamics with SOC

are needed. Cold-atom-based quantum simulators pro-

vide a natural testbed for such studies [9, 10] – they

complement the existing research of out-of-equilibrium

time evolution, see, e.g., [11–13], and SOC engineering

using laser ﬁelds [14, 15].

To enjoy the potential of quantum simulators, one re-

quires theoretical models that can be used to propose new

experiments and analyze the existing data [16]. In this

work, we present one such model designed to study an

impurity with SOC (see also recent Ref. [17] for a discus-

sion of a relevant Langevin-type equation). The impurity

is in contact with the bath that we model as a collec-

tion of harmonic oscillators. Using the Born and Markov

approximations, we derive a master equation, which ex-

tends the result of Caldeira and Leggett [18, 19] to a

spin-orbit-coupled impurity. To illustrate this equation,

we focus on one-dimensional (1D) systems. First, we test

it using the experimental data of Ref. [20], whose full

theoretical understanding is lacking, see, e.g., Ref. [21].

We ﬁnd that the Caldeira-Leggett model contains all in-

gredients to describe the observed breathing dynamics

of the impurity assuming that the initial condition is

the (only) tunable parameter. The calculations are an-

alytical, which simpliﬁes the analysis and allows us to

gain insight into the system: relevant time scales, short-

and long-time dynamics. Finally, we explore the dynam-

ics of the system with SOC. Without magnetic ﬁelds,

the 1D SOC can be gauged out so that the system can

be described using the Caldeira-Leggett equation with

SOC-dependent initial conditions. We present observ-

ables that are sensitive to these initial conditions and

can be used to study the eﬀect of SOC on time dynam-

ics, for example, formation of regions with steady spin

polarization. Our ﬁndings provide a convenient theoret-

ical model that can be used to propose and benchmark

quantum simulators of dissipative dynamics with SOC.

The particle-bath Hamiltonian. The Hamiltonian of

the system is given by Htot =HS+HB+HC. The three

terms account for, correspondingly, the (quantum) impu-

rity, the harmonic bath and the bath-impurity coupling.

We assume that HShas the form

HS=p2

2m+VSO(p,σ

σ) + Vext(q) + q2

j=1

2mjω2

,(1)

where mand qare the mass and the position of the impu-

rity, respectively. Vext is an external potential. VSO(p, σ

σ)

is the potential that describes SOC; it depends on the

Pauli vector, σ

σ, and the momentum of the impurity, p.

A particular form of VSO is speciﬁed below, see also

the Supplementary Material. The last term in Eq. (1)

is a standard harmonic counterterm, which makes Htot

translationally invariant for Vext = 0 [22]. The param-

eters ωjand mjare taken from the bath Hamiltonian,

HB=Pj[p2

j/(2mj) + 1

2mjω2

jx2

j]; cjdeﬁnes the strength

of the bath-impurity interaction HC=−q·Pjcjxj. [For

microscopic derivations that validate the form of HBand

HCfor weakly interacting Bose gases and Luttinger liq-

uids, see correspondingly Refs. [23] and [24].] To sum-

marize, we consider a single particle (impurity) linearly

coupled to an environment made of non-interacting har-

monic oscillators using the standard procedure [18, 25],

brieﬂy outlined below; this well-studied problem is ex-

tended here by subjecting the impurity to SOC.

Before analyzing Htot, we remark that there are a num-

ber of theoretical methods [26–30] that can be used for

interpreting experiments with impurities in Fermi [31–

34] and Bose gases [20, 35–39]. Time evolution of an

arXiv:2210.01829v2 [cond-mat.quant-gas] 17 Oct 2022

impurity in a Bose gas – the focus of this work – has

been studied using variational wave functions, T-matrix

approximations and exact solutions in 3D at zero [40–42]

and ﬁnite temperatures [43]. Many more methods exist

to address the 1D world. For example, experimentally

relevant trapped systems can be studied using numeri-

cally exact approaches [44, 45], for review see Ref. [46]. In

cases when those methods do not work (e.g., large energy

exchange or high temperature), it has been suggested to

connect a cold-atom impurity to quantum Brownian mo-

tion [23, 47–49]. Our work provides an example when

this idea leads to an accurate description of experimen-

tal data, setting the stage for testing assumptions behind

theoretical models of relaxation [19, 22] in a cold-atom

laboratory.

Born-Markov master equation with SOC. Time evo-

lution of the impurity-bath ensemble, deﬁned by Htot,

obeys the Von-Neumann equation: i~˙ρtot = [Htot, ρtot].

To extract dynamics of the impurity from ρtot, we rely

on the Born-Markov approximation [19, 50, 51], which

leads to the equation for the (reduced) density matrix

that describes the impurity, ρS:

dρS

dt =−i

~HS, ρS−1

~2Z+∞

ds C(s)q,Q(−s), ρS

~2Z+∞

ds χ(s)q,{Q(−s), ρS}.

(2)

We write Eq. (2) in the form standard for a Brownian

particle; the contribution of SOC is conveniently hidden

in Q(t), which is deﬁned as

Q(t) = i

~[HS,q] = q−p

m+vSO(σ

σ)t , (3)

where vSO =∂pVSO is the contribution to the ‘velocity’

of the particle due to SOC. Equation (2) contains the

bath autocorrelation functions C(t) and χ(t) in Eq. (2)

C(t) = ~Z+∞

dω J(ω) coth β~ω

2cos(ωt),

χ(t) = ~Z+∞

dω J(ω) sin(ωt),

(4)

where β= 1/kBT(Tfor temperature, and kBis

the Boltzmann constant). These functions assume

that all relevant microscopic information is encoded

in the spectral function J(ω), formally deﬁned as

J(ω) = Pjc2

jδ(ω−ωj)/(2mjωj). We choose J(ω) =

(2mγ/π)ωΩ2

c/(Ω2

c+ω2), recovering Ohmic dissipation

at ω→0; the phenomenological parameter Ωcdeﬁnes

the high-frequency behavior. The Ohmic spectral den-

sity is a standard choice in mesoscopic [19, 52, 53] and

in cold-atom physics [24, 54, 55]. We employ it here be-

cause it leads to a local-in-time damping that agrees with

the experimental data used below to validate the model

(see also Refs. [24, 56] for additional details about Ohmic

dissipation in 1D based upon long-wavelength approxi-

mations for superﬂuids). Super-Ohmic dissipation whose

relevance for Bose polarons is highlighted in Refs. [23, 48]

leads to strong memory eﬀects (non-local-in-time damp-

ing), thus, we do not consider it here.

Using Eqs. (3) and (4), we derive the master equation

dρS

dt =−i

~[HS, ρS]−iγ

~q,p, ρS−2mγ

β~2q,q, ρS

−imγq,vSO, ρS,(5)

where γdeﬁnes the strength of dissipation. The fre-

quency integrals leading to Eq. (5) are discussed in

the Supplementary Material. As expected, dissipa-

tive dynamics is aﬀected by SOC, see the last term in

Eq. (5). Finally, a proper Lindblad form for Eq. (5)

can be achieved by adding a minimally invasive term:

−γβ [p[p, ρS]] /(8m) [19, 59]; we employ this term in our

calculations.

To illustrate the master equation, we choose to con-

sider a 1D setting parameterized by the coordinate y.

Without loss of generality, we write the SOC term as

VSO =ασxpy. In this case, the master equation reads as

dρ

dt =dρ

dt α=0 −αF[ρ],(6)

where F[ρ] = σx∂yρ+∂y0ρσx+imγ

~(y−y0)σxρ+ρσx

with ρ≡ hy|ρS|y0i, and dρ

dt |α=0 describes time evolution

of the system without SOC (see the Supplementary Ma-

terial). The eﬀect of SOC is encoded in αF[ρ].

While the technical details leading to Eq. (2) are pre-

sented in the Supplementary Material, we recall here the

standard assumptions behind the Born-Markov approxi-

mation. First, the impurity-bath density matrix is sep-

arable throughout time evolution, such that ρtot(t)'

ρS(t)⊗ρB(t). Second, the bath is not aﬀected by the

impurity motion, namely, ρB(t)'ρeq

B. This assumption

is natural if the decay of bath correlations has the fastest

timescale τB; it implies that dynamical features ∼τBare

not resolved by our approach [19, 50]. To validate these

approximations, we shall demonstrate that the master

equation is capable of describing experimental data of

Ref. [20] that provide a benchmark point for us at α= 0.

Dynamics without SOC. First, we brieﬂy outline the

main features and ﬁndings of the experiment of Ref. [20].

In that work, a potassium atom was used to model an im-

purity in a gas of rubidium atoms. At t= 0, the impurity

was trapped in a tight trap created by a species-selective

dipole potential (SSDP) with ωSSDP/(2π) = 1kHz. At

t > 0, the dynamics was initiated by an abrupt removal

of the SSDP; the impurity was still conﬁned by a shal-

low parabolic potential, i.e., Vext (y) = ~2y2/2ml4, where

l=p~/mω and ω= (87 ×2π)Hz is the frequency of the

oscillator. The experiment recorded the size of the impu-

rity cloud ¯y=phy2i, and found that it can be ﬁt using

the expression

¯y= ¯y0+A1t− A2e−ΓΩtcos[p1−Γ2Ω(t−t0)],(7)

FIG. 1. (a)-(d) The width of the impurity (potassium) cloud ¯yas a function of time for diﬀerent values of the parameter η.

The dots with error bars show the experimental data of Ref. [20]. The curves are the ﬁts to Eq. (6). Panel (e) shows the values

of l0used in the ﬁt as a function of η. The panel also shows a linear ﬁt to these values (red line). The green curve shows the

eﬀective mass of the polaron calculated using the analytical methods outlined in Refs. [57, 58] (no ﬁtting parameters).

where A1,A2,Ω,Γ,¯y0, t0are ﬁtting parameters. The

key experimental ﬁndings of Ref. [20] were: (a) Ω (al-

most) does not depend on the impurity-boson interaction

parametrized by η; (b) by increasing ηone decreases the

amplitude of the ﬁrst oscillation; (c) at long times ¯yequi-

librates to about the same value, which is independent

of η. Point (b) was attributed to renormalization of the

mass of the impurity, i.e., to a polaron formation [60].

However, this posed several theoretical problems. In

particular, the breathing frequency of the polaron cloud

should depend on η, which contradicts observation (a),

see also discussions in Ref. [20, 21, 57, 61]. Our results be-

low suggest that one can understand the data of Ref. [20]

from the perspective of dissipative dynamics.

Equation (6) leads naturally to the dynamics observed

in the experiment. To show this, we assume that the

initial density matrix of the impurity corresponds to a

Gaussian wave packet

ρ(y, y0, t = 0) = e−y2+y02

2l2

0/(√πl0),(8)

where l0is the parameter that determines the initial dis-

tribution of the impurity momenta; Eq. (8) is standard

for particles whose initial state is not precisely known.

We calculate the time dynamics for this initial condition

analytically using the method of characteristics (see the

Supplementary Material and Ref. [62]), which discovers

characteristic curves where the master equation can be

written as a family of ordinary diﬀerential equations [63].

The computed functional dependence resembles Eq. (7)

with ΓΩ = 2γ(see the Supplementary Material). Note

that our calculations have only two phenomenological pa-

rameters γand l0. All other parameters that appear in

Eq. (7), i.e., A1,A2,¯y0, t0and Ω, can be extracted from

our results. For example, Ω '2ωas in the experiment.

We present analytical results of the master equation

together with the experimental data in Fig. 1. The value

of γis restricted to be within the errorbars of the exper-

imentally measured value of ΓΩ (so that γ∼40Hz) [64].

The temperature is set to the value reported in the ex-

periment, i.e., T= 350nK [65]. The quality of the ﬁts

in Fig. 1 is comparable to what can be obtained with

Eq. (7), allowing us to conclude that the master equa-

tion provides a valuable tool for analyzing these data,

and cold-atom systems in general.

Let us brieﬂy discuss implications of our results for in-

terpretation of the experiment of Ref. [20]. First, the

weak dependence of Ω on ηis natural in our model:

the renormalization of the frequency is given by ωeﬀ '

ω(1 −γ2/(2ω2)), where γ/ω is a small parameter as in

the experiment. Second, the parameter ¯yfor t→ ∞ is

independent of γassuming that the thermal de Broglie

wavelength is small. Indeed, in this case, we derive

¯y'pkBT /(~ω)l≈15.42 µm in agreement with the

measurement.

The amplitude of the ﬁrst oscillation is determined

in our analysis by the initial condition, i.e., l0. To ex-

plain values of l0obtained in our ﬁt, one can specu-

late that the impurity forms a polaron state at t < 0

and that at t > 0 the dynamics is dominated by ﬁ-

nite temperature eﬀects. In this picture, the mass of

the impurity is renormalized (i.e., m→mp) only be-

fore the quench dynamics [66]; this might explain why

theoretical calculations can produce features of the am-

plitude but not of the frequency [20, 21, 61]. Renormal-

ization of the mass implies that the energy scale at t= 0

is given by ~ωSSDPpm/mp, which is incorporated into

Eq. (8) if l2

0∼~/(mωSSDP)pmp/m [67]. This expres-

sion agrees qualitatively with the outcome of our ﬁt, see

Fig. 1 (e). The linear increase of (mωSSDPl2

0/~)2how-

ever quantitatively disagrees with calculations of the ef-

fective mass [21, 57, 68, 69]. The agreement improves if

we disregard the point with η= 30, which (as suggested

in Ref. [20]) is already beyond a simple one-dimensional

treatment [70]. In any case, a further analysis of the

experimental data (beyond the scope of this paper) is

needed in light of our results.

Finally, we note that the inhomogeneity of the bath

as well as non-Markovian physics do not appear to be

important to describe dynamics discussed here. This

stands in contrast to what is known about properties of

the corresponding ground state [71] and low-energy dy-

namics [57, 61, 72], and results from a high temperature

and a large energy (∼1/l2

0) associated with the initial

impurity state.

Dynamics with SOC. We use the experimental pro-

FIG. 2. The spin polarization along the yaxis as a function

of position and time in the presence of SOC. The SOC am-

plitude is α= 40 Hz ·µm. l0for (a,b) case corresponds to

ω0/2π= 30 kHz, while lis given by ω/2π= 87 Hz. All other

parameters are as in Ref. [20], in particular T=350nK.

tocol of Ref. [20] also to illustrate the master equation

with SOC. The peculiarity of 1D is that the α-dependent

term can be gauged out from Eq. (6) via the transforma-

tion (in the position space) ρ=e−imασxy

~feimασxy0

~. The

function f(y, y0, t) then satisﬁes the standard Caldeira-

Leggett equation and can be solved exactly as without

SOC (see the Supplementary Material). Note that the

equation for fis spin-independent. The initial condi-

tion of the problem, ρ(y, y0, t = 0), deﬁnes the full spin

structure of the problem and time dependence of spin ob-

servables, as we illustrate below for ¯σy(y, t)≡Trspin(σy).

For the sake of discussion, as the initial condition we

consider the state that is spin-polarized along the z-axis:

ρ(y, y0,0) = 1

2√πl0

e−y2+y02

2l2

0| ↑ih↑ |; (9)

other parameters of the system are taken from Ref. [20].

We use γ= 40 Hz, which was typical in that experiment.

The strength of SOC, α, can be tuned in cold-atom set-

ups, see, e.g., [73–75]. We assume that α¯y/(ωl2)1 to

demonstrate that even weak SOC can lead to an observ-

able eﬀect in dynamics.

Time evolution of ¯σy(y, t) is shown in Fig. 2. Note

that Eq. (9) is not an eigenstate of the system with SOC

– dynamics occurs even without a change of the trap,

i.e., l0=l[see Figs. 2 (c) and (d)]. Without dissipation

(γ= 0), we observe oscillation of the spin density with

¯σy>0 for y > 0 and ¯σy<0 for y < 0 [see Figs. 2 (a)

and (c)]. This eﬀect is solely due to SOC, and can be

easily understood from a one-body Schr¨odinger equation.

Eﬀects of temperature and dissipation are most visible in

Fig. 2 (d): the impurity is heated up by the presence of

the bath, which creates regions with steady spin polar-

ization along the ydirection. Spatial extension of these

FIG. 3. The same as in Fig. 2 but with an additional magnetic

ﬁeld along the ydirection. The amplitude of the magnetic

ﬁeld is µBB/~= 100 Hz.

regions is determined by the temperature; the time scale

for their formation is given by 1/γ (similarly to the dy-

namics without SOC, see Fig. 1). This eﬀect can be

observed in cold-atom systems by analyzing populations

of the involved hyperﬁne states.

Finally, we remark that Eq. (5) allows us to include

Zeeman-type terms, which naturally appear in ultracold

atoms with synthetic SOC [14, 15]. To this end, we add

the term µBB·σto HS. Its presence strongly modiﬁes

the spin dynamics because SOC cannot be gauged out.

Theoretical analysis also becomes more involved, since

we cannot solve the system analytically for all values of

αand B. Still, we obtain closed-form expressions using

tools of perturbation theory for α→0 (see the Sup-

plementary Material). The eﬀect of the magnetic ﬁeld

is illustrated in Fig. 3. Initially, the dynamics with the

magnetic ﬁeld is similar to the dynamics presented in

Fig. 2. However, at later times we observe spin preces-

sion possible only in the presence of both SOC and the

magnetic ﬁeld. Spin precession leads to an exchange of

domains with positive and negative values of ¯σy, and can

be used for engineering the spin structure.

Conclusions. We analyzed Brownian-type motion of

a spin-orbit coupled impurity with the goal to develop

a simple theoretical tool that can be used to propose

and analyze cold-atom-based quantum simulators. We

introduced a master equation suitable for the problem.

We tested it and illustrated its usefulness by interpreting

available experimental data without SOC [20]. Our re-

sults suggested that the impurity does not experience any

mass renormalization during quench dynamics at exper-

imentally accessible temperatures. Finally, we demon-

strated that systems with SOC can be studied analyti-

cally, and calculated observables that measure changes in

population of the hyperﬁne states of the impurity atom.

A comparison between results of our theoretical study

and experimental data (when available) can be used to

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Dissipativedynamicsofanimpuritywithspin-orbitcouplingAregGhazaryan,AlbertoCappellaro,MikhailLemeshko,ArtemG.VolosnievInstituteofScienceandTechnologyAustria(ISTA),amCampus1,3400Klosterneuburg,Austria(Dated:October18,2022)Brownianmotionofamobileimpurityinabathisaectedbyspin-orbitcoupling(SOC).Here,we...

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Dissipative dynamics of an impurity with spin-orbit coupling Areg Ghazaryan Alberto Cappellaro Mikhail Lemeshko Artem G. Volosniev Institute of Science and Technology Austria ISTA am Campus 1 3400 Klosterneuburg Austria

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