Information Scrambling of the Dilute Bose Gas at Low Temperature Chao Yin1and Yu Chen2y 1Department of Physics and Center for Theory of Quantum Matter

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Information Scrambling of the Dilute Bose Gas at Low Temperature

Chao Yin1, ∗and Yu Chen2, †

1Department of Physics and Center for Theory of Quantum Matter,

University of Colorado, Boulder, CO 80309, USA

2Graduate School of China Academy of Engineering Physics, Beijing, 100193, China

(Dated: February 9, 2023)

We calculate the quantum Lyapunov exponent λLand butterﬂy velocity vBin the dilute Bose

gas at temperature Tdeep in the Bose-Einstein condensation phase. The generalized Boltzmann

equation approach is used for calculating out-of-time ordered correlators, from which λLand vB

are extracted. At very low temperature where elementary excitations are phonon-like, we ﬁnd

λL∝T5and vB∼c, the sound velocity. At relatively high temperature, we have λL∝Tand

vB∼c(T/T∗)0.23. We ﬁnd λLis always comparable to the damping rate of a quasiparticle, whose

energy depends suitably on T. The chaos diﬀusion constant DL=v2

B/λL, on the other hand, diﬀers

from the energy diﬀusion constant DE. We ﬁnd DEDLat very low temperature and DEDL

otherwise.

1. INTRODUCTION

Butterﬂy eﬀect, a deﬁning feature for classical chaotic

dynamics, also emerges in quantum settings and is cru-

cial for understanding strongly correlated systems. To

diagnose quantum chaos, out-of-time-ordered correlator

(OTOC) is ﬁrst introduced by Larkin and Ovchinikov to

study disordered superconductors [1]. This idea is rarely

visited until Kitaev recently revived it to understand the

shock wave back action in the black hole scattering prob-

lem [2,3]. To be speciﬁc, we deﬁne OTOC by two oper-

ators O,˜

Oas

C(t) = tr √ρ[O(t),˜

O(0)]†√ρ[O(t),˜

O(0)].(1)

Here ρ=Z−1

βe−βH with β= 1/T as the inverse tem-

perature, where we have set the Boltzmann constant

kB= 1, and Zβ= Tr e−βH as the partition func-

tion. His the system Hamiltonian that evolves oper-

ators by O(t)=eitH/~Oe−itH/~. For typical chaotic sys-

tems, OTOC grows exponentially as C(t)∼c0exp(λLt),

with c0being a non-universal constant. λLis the quan-

tum Lyapunov exponent that measures the growth rate

of quantum chaos, which shares similarities and diﬀer-

ences with its classical counterpart [4–6]. It was found

that λLis upper bounded by 2π/β [7], and the maximal

value is saturated by models with gravity duals [3,8–

10], including the Sachdev-Ye-Kitaev model [11,12] dual

to Jackiw-Teitelboim gravity [13–15]. Therefore, calcu-

lating λLis crucial for identifying holographic models

[16–18].

More generally, an information interpretation has been

discovered for OTOC [19]. Namely, λLmeasures how fast

local information scrambles to global ones, which reveals

the thermalization process in a closed quantum system.

∗chao.yin@colorado.edu

†ychen@gscaep.ac.cn

Moreover, for systems with a spatial structure, if we de-

ﬁne Oand ˜

Oas local operators whose locations are of

distance r, then the OTOC is vanishingly small unless

t&r/vB, for some constant vBcalled the butterﬂy ve-

locity [20–22]. vBcan be viewed as a ρ-dependent ex-

tension [23,24] of the Lieb-Robinson velocity [25], the

maximal speed information can propagate through the

system. Combining λLwith vB, one can deﬁne the chaos

diﬀusion constant DL=v2

B/λL. In the most chaotic sys-

tems, DLis argued to be universally comparable with

charge [21,26] and energy [27] diﬀusion constants.

Due to the above implications, general properties of

OTOC have arisen a lot of interest (see [28] for a re-

cent review). For example, OTOC has been theoreti-

cally calculated in many-body-localized systems [29–33],

integrable systems [34], and diﬀusive metals [35–37], and

experimentally measured in NMR systems [38–40], ion

traps [41,42] and superconducting circuits [43–45]. How-

ever, OTOC remains to be studied for the dilute Bose gas

in Bose-Einstein condensation (BEC), realizable in cold

atom experiments [46]. Moreover, unlike models studied

before, BEC hosts two temperature regimes with qual-

itatively diﬀerent elementary excitations. How does in-

formation scramble in the crossover temperature regime?

In this paper, we ﬁll this gap using the generalized Boltz-

mann equations (GBE) approach [47–50].

The rest of the paper is structured as follows. In Sec-

tion 2, our model is introduced, where we focus on the

BEC regime TTBEC. We identify a crossover temper-

ature T∗TBEC, where quasiparticle excitations change

from phonon-like at TT∗to particle-like at TT∗.

In Section 3, we apply the augmented Keldysh formalism

to derive GBE that govern the evolution of OTOC, to the

leading nontrivial order of the interaction strength g. In

Section 4, we extract λLfrom GBE for the whole tem-

perature regime TTBEC, and get λL∝T5for TT∗

and λL∝Tfor TT∗. We further show that λLis

comparable to the damping rate of a quasiparticle at a

suitably deﬁned energy, which can be extracted from tra-

ditional Boltzmann equations. In Section 5, we present

our results on vB. It is of the order of the sound veloc-

arXiv:2210.03025v2 [cond-mat.quant-gas] 7 Feb 2023

ity cat TT∗, and grows as a power law vB∼T0.23

for TT∗. We further show that for both temperature

regimes, the chaos diﬀusion constant DLand the energy

diﬀusion constant DEare not related to each other. We

ﬁnally conclude in Section 6.

2. MODEL

Here we introduce our model. Consider Nbosons con-

tained in a 3-dimensional box of volume V=L3. Using

ψ(x) to be the complex ﬁeld operator that annihilates

a boson at space position x, we study the homogeneous

Bose gas with Hamiltonian

HBG =HK+HV,(2)

where the kinetic energy is

HK=Zdxψ†(x)−~2

2m∇2ψ(x),(3)

with mbeing the boson mass. The interaction HVis

given by

HV=g

2Zdxψ†(x)ψ†(x)ψ(x)ψ(x),(4)

where we have assumed the temperature is suﬃciently

low, so that pairs of bosons feel a delta function pseu-

dopotential [51]

v(x−x0) = 4πas~2

mδ(x−x0)≡gδ(x−x0),(5)

determined by the s-wave scattering length as(or equiv-

alently, the interaction strength g). In the momentum

space, we deﬁne the boson annihilation operator at wave

vector kby ak=V−1Rdxψ(x)e−ik·x. Then HBG can

be rewritten as

HBG =X

ka†

kak+g

2VX

k1,k2,k3

a†

k1a†

k2ak3ak1+k2−k3,

(6)

where k=~2k2/2m,k=|k|, and ktakes values in

{2πn/L :n∈Z3}.

There are three independent length scales in this

model: the scattering length as, the inter-particle spac-

ing n−1/3where n=N/V , and the thermal wavelength

λT=r2π~2

mT .(7)

We focus on the dilute and low-temperature limit

na3

s1, nλ3

T1,(8)

where perturbation theory applies. In this regime close

to equilibrium, nearly all of the Nbosons condense in the

zero-momentum state, forming a BEC [51]. As in stan-

dard Bogoliubov theory for a homogeneous BEC, we ap-

proximate the zero-momentum creation/annihilation op-

erators in (6) by a large c-number √N0≈√N:

a0=a†

0=√N, (9)

where we have ignored higher order corrections N−N0∝

pna3

s[51]. Moreover, we use the standard Bogoliubov

transformation to obtain the eﬀective Hamiltonian from

(6)

H=H0+H1,where (10a)

H0=X

kEkα†

kαk,and (10b)

H1=g

√VX

k1,k2

Mk1,k2α†

k1α†

k2αk1+k2+ h.c.,(10c)

where αkand α†

kare the annihilation and creation oper-

ators for the Bogoliubov quasiparticle, with boson com-

mutation relation

[αk, α†

k0] = δk,k0.(11)

In (10), the quasiparticle has spectrum

Ek=pk(k+ 2gn),(12)

and collision matrix [52]

Mk1,k2=√nE1+E2−E3+ 3E1E2E3

4√E1E2E3

,(13)

with Ei≡ki/Ekiand k3=k1+k2. In deriving

(10), we have discarded a c-number term, and higher

order terms in 1/N. We have also discarded the term

∝αk1αk2α−k1−k2+ h.c., which describes the process

that creates or annihilates three quasiparticles simulta-

neously. At leading order, such oﬀ-shell processes do not

contribute to the kinetic equations that we will derive.

(10) is then our starting point for a ﬁeld-theoretic cal-

culation for information scrambling, and in the end of

Section 3we will justify the Bogoliubov approximation

(9) in this nonequilibrium context. Note that, although

strictly speaking, the sums over kin (10) should avoid

the k=0point, this makes no diﬀerence for latter cal-

culations, since Ekand Mk1,k2both become zero when

one of the karguments (including k3) is set to 0.

(12) suggests a crossover behavior for the quasipar-

ticles. Deﬁning the characteristic momentum k0≡

√mgn/~=√4πasn, the quasiparticles change from

phonon-like Ek≈~ck at kk0, where the sound veloc-

ity c=pgn/m, to particle-like Ek≈kat kk0. The

corresponding crossover temperature is T∗≡~2k2

0/m =

~ck0=gn. Thus we expect OTOC also behaves diﬀer-

ently at the two temperature regimes: the very low tem-

perature TT∗, and the relatively high temperature

T∗TTBEC.

𝜌𝜌

𝒪

#(0)

𝒪

#(0)

𝒪(𝑡)

𝑢 +

𝑢 −

𝑑 +

𝑑 −

𝜌!

−

Φ = 𝜙", 𝜙#$

𝜙(𝒔)

=𝒪"⊗ 𝒪#2(𝑡)

FIG. 1. The augmented Keldysh contour C(left) for OTOC

in (1), is equivalent to the conventional Keldysh contour Cc

(right), where the ﬁelds are doubled, and the initial state ρc

includes the perturbation from ˜

3. THE AUGMENTED KELDYSH FORMALISM

In this section we set ~= 1. We ﬁrst remark on our

regularization in (1), namely inserting two √ρs between

the commutators. The advantage is threefold: It avoids

potential ultraviolet divergences, and is the one for which

the chaos bound [7] is proved. Moreover, in kinetic theory

it has a clear physical meaning related to classical chaos

[53].

(1) contains four terms that can be arranged as

C(t) = 2 Re ˜

C(t) + TOC,where

C(t) = tr √ρO(t)˜

O(0)√ρO(t)˜

O(0).(14)

Here TOC stands for time-ordered correlations, and we

have assumed the operators to be Hermitian for simplic-

ity. We focus on ˜

C(t) because TOC does not host expo-

nential growth.

3.1. relation between OTOC and TOC in a

doubled system

To calculate OTOC in (14), we ﬁrst introduce the time

contour Cshown on the left of Fig. 1, which contains

two parts: up(u) and down(d), with each part contain-

ing two branches: for example ucontains u+ and u−.

Such Cis called the augmented Keldysh contour intro-

duced in [47]: if there is only one part (up or down)

instead, then it is the conventional Keldysh contour [54]

that is used for calculating TOC. We parametrize Cby

the contour time s, which goes from t= 0−(the time

slightly before 0) to t= +∞and back to t= 0−in

the up part of C, and then goes to +∞and back to 0−

again in the down part of C, completing one cycle of the

whole contour. Equivalently one can describe the contour

time by the doublet s= (κ, t), where the Keldysh label

κ=u+, u−, d+, d−denotes the branch that the conven-

tional time t∈(0−,+∞) lives in. Deﬁne the contour

Hamiltonian H(s)

H(s) = H−i

2βHδ(t+ 0) κ=u+, d+

−H κ =u−, d−,(15)

where the delta function at (u+,0−) and (d+,0−) ac-

counts for the thermal density matrix ρ. Then (14) can

be rewritten on this contour C:

C(t) = DTCOd−(t)˜

Od+(0)Ou−(t)˜

Ou+(0)e−iRCdsH(s)E

=Z[Dφ]Od−(t)˜

Od+(0)Ou−(t)˜

Ou+(0)eiS[φ],(16)

where in the ﬁrst line, TCtime orders the operators by

its position in the contour C, and h·i =Z−1

βTr (·). In the

second line we used the path integral representation by

replacing operators αkand α†

kwith classical ﬁelds φk(s)

and ¯

φk(s) that live on the contour C, and deﬁned the

contour action

S[φ] = S0[φ] + S1[φ],(17)

where S0and S1correspond to H0and H1in (10) re-

spectively, whose expressions are given later. We pro-

vide several remarks on (16). First, the Keldysh κlabels

are not unique because the operator insertions can move

along the contour: Ou−(t) can be replaced by Ou+(t) for

example. Second, for notational simplicity we omit the

functional dependence of Son ¯

φ, which is also integrated

in R[Dφ]. Lastly, we use Oκ(t) for both the quantum

operator Oat s= (κ, t), and its path integral represen-

tation that is a function of φ(s) and its time derivatives.

We have expressed (14) as a path integral along the

augmented Keldysh contour C, which gets rid of oper-

ators and their time ordering. As a result, there is an

equivalent perspective that turns out to be useful: The

path integral can be viewed as one along a conventional

Keldysh contour Ccas shown on the right of Fig. 1in-

stead, by merging the up and down parts of C, so that

there are two sets of ﬁelds φu(s) and φd(s) that live on

the contour Cc. Here sis the contour time for Cc, and we

combine the ﬁelds to a two-component one Φ= (φu, φd)t

with its conjugate ¯

Φ= ( ¯

φu,¯

φd). The operator inser-

tions are also combined, where the initial perturbations

Oare absorbed into the initial state ρc. Later we will

ﬁnd the speciﬁc form of O,˜

Oand ρcis irrelevant for us

to extract λLand vB. The action governing the con-

tour evolution in 0 < t < ∞factorizes to up and down

contributions, so that the OTOC is converted to a TOC

hOu−(t)Od−(t)i, for a doubled system: the original one,

u, together with its augmented ancilla system d. Here

we call hOu−(t)Od−(t)ia TOC because it can be calcu-

lated on a single Keldysh contour. To be more precise,

it can viewed as h(O ⊗ O)(t)I(0)i, where the two Os are

combined to one operator O⊗O, and an identity opera-

tor is inserted at time 0 to make the time order manifest.

The two subsystems have the same Hamiltonian (10) for

time evolution, and do not couple to each other. How-

ever, there is a price to pay: The initial state ρc, for

the average h·i appearing in the TOC, includes the per-

turbation ˜

Oand becomes a complicated entangled state

shared by the two subsystems, which is expressed picto-

rially in Fig. 1. (Without the perturbation, the density

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InformationScramblingoftheDiluteBoseGasatLowTemperatureChaoYin1,andYuChen2,y1DepartmentofPhysicsandCenterforTheoryofQuantumMatter,UniversityofColorado,Boulder,CO80309,USA2GraduateSchoolofChinaAcademyofEngineeringPhysics,Beijing,100193,China(Dated:February9,2023)WecalculatethequantumLyapunovexponent...

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Information Scrambling of the Dilute Bose Gas at Low Temperature Chao Yin1and Yu Chen2y 1Department of Physics and Center for Theory of Quantum Matter

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