Schwinger-Keldysh path integral formalism for a Quenched Quantum Inverted Oscillator Sayantan ChoudhuryID1Suman Dey2Rakshit Mandish

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Schwinger-Keldysh path integral formalism for

a Quenched Quantum Inverted Oscillator

Sayantan Choudhury ID 1,∗Suman Dey2,†Rakshit Mandish

Gharat3,‡Saptarshi Mandal4,§and Nilesh Pandey1¶

1Centre For Cosmology and Science Popularization (CCSP),

SGT University, Gurugram, Delhi- NCR, Haryana- 122505, India,

2Department of Physics, Visva-Bharati University, Santiniketan, Birbhum-731235, India,

3Department of Physics, National Institute of Technology Karnataka, Surathkal, Karnataka-575025, India, and

4Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India,

In this work, we study the time-dependent behaviour of quantum correlations of a system of an

inverted oscillator governed by out-of-equilibrium dynamics using the well-known Schwinger-Keldysh

formalism in presence of quantum mechanical quench. Considering a generalized structure of a time-

dependent Hamiltonian for an inverted oscillator system, we use the invariant operator method to

obtain its eigenstate and continuous energy eigenvalues. Using the expression for the eigenstate, we

further derive the most general expression for the generating function as well as the out-of-time-

ordered correlators (OTOC) for the given system using this formalism. Further, considering the

time-dependent coupling and frequency of the quantum inverted oscillator characterized by quench

parameters, we comment on the dynamical behaviour, speciﬁcally the early, intermediate and late

time-dependent features of the OTOC for the quenched quantum inverted oscillator. Next, we study

a speciﬁc case, where the system of inverted oscillator exhibits chaotic behaviour by computing the

quantum Lyapunov exponent from the time-dependent behaviour of OTOC in presence of the given

quench proﬁle.

I. Introduction

In theoretical high energy physics, the underlying concept of the Feynman path integral is applicable to quantum

systems for which the vacuum state in far future is exactly the same as in far past. Clearly, the Feynman path

integral is not applicable for quantum systems with a dynamical background. The Schwinger-Keldysh [1–5] formalism

appears within the framework of quantum mechanics, where the physical system is placed on a dynamical background,

i.e., quantum dynamics of the system is far from equilibrium. Schwinger-Keldysh formalism seems to be the most

formidable choice to evaluate generating functionals and correlation functions in out-of-equilibrium quantum ﬁeld

theory and statistical mechanics [6]. It unravels the prodigious toolbox of equilibrium quantum ﬁeld theory to out-

of-equilibrium problems. There are large numbers of applications of out-of-equilibrium dynamics in various ﬁelds, for

instance, condensed matter physics[7–11], cosmology [12–15] (in the context of the primordial universe), black hole

physics[16,17], high energy physics - theory and phenomenology [18–22], and more.

The time-dependent states for dynamical quantum systems can be computed by solving Time-Dependent Schrodinger

Equation (TDSE). One of the ways to solve TDSE is by constructing the Lewis-Resenﬁeld invariant operator and this

is often termed as invariant operator representation of the wavefunction [23]. Some works following this approach to

compute the time-dependent eigenstates are reported in refs. [24–28].

Highly complex behavior of quantum systems can often be studied using a simple model of Standard Harmonic

Oscillator (SHO). Sister to SHO is the Inverted Harmonic Oscillator (IHO) that has remarkable properties which are

applicable throughout many disciplines of physics [13,20,29–48]. The quantum treatment of IHO is exactly solvable

similar to that of SHO. Unlike SHO, IHO can be used to model complicated types of out-of-equilibrium systems.

A key idea for quantifying the propagation of detailed quantum correlation at out-of-equillbrium, information theo-

retic scrambling via exponential growth, and quantum chaos can be easily studied via out-of-time-ordered correlators

or simply "OTOC" [49]. OTOC has been widely used as a tool to probe quantum chaos in quantum systems [50–57].

In a more general sense OTOC is a mathematical tool using which one can easily quantify quantum mechanical

correlation at out-of-equilibrium. Motivated by the work of [58], Kitaev [59] emphasized arbitrarily ordered 4-point

∗Corresponding Author: sayantan_ccsp@sgtuniversity.org, sayanphysicsisi@gmail.com

†dey.suman.vbu@gmail.com

‡rakshitmandishgharat.196ph018@nitk.edu.in

§saptarshijhikra@gmail.com

¶nilesh911999@gmail.com

arXiv:2210.01134v2 [hep-th] 2 Oct 2024

correlation functions, as "out-of-time-ordered correlators", in constructing analogies between eﬀective ﬁeld theories,

making the connection between superconductors [60] and black holes [61,62].

Most of the recent works in many-body physics focus on studying dynamics of quantum systems where some time-

dependent parameter is varied suddenly or very slowly. This is commonly refereed to as a Quantum Quench. The

quench protocol is said to drive any out of equilibrium system and in turn can trigger the thermalisation process of

these systems, [63–65]. The eﬀects of quenches in quantum systems have even been explored experimentally for cold

atoms in [66–75]. A few of the works focusing on OTOCs in the context of quenched systems are [76–80].

Motivated by all these ideas, in this work, we compute the OTOC for inverted oscillators having a quenched coupling

and frequency, using Schwinger-Keldysh path integral formalism. In section II the continuous energy eigenvalues and

normalized wavefunction for a generic Hamiltonian of a time-dependent inverted oscillator are computed [81] using

the Lewis–Riesenfeld invariant method. In section (III), we construct the action for the Lagrangian of generic time-

dependent inverted oscillator. Using this action we derive the generating function for the inverted oscillator in section

IV. We modify this generating function using Shwinger-Keldysh path integral formalism in section V, following the

prescription of [6]. Using this generating function and the Green’s functions computed in Appendix Bwe ﬁnally

compute the OTOC for inverted oscillator in terms of Green’s functions in section VI. In the section (VII), we have

numerically studied dynamical behaviour of the OTOC by quenching coupling and frequency of the inverted oscillator.

We have also computed the Lyapunov exponent and commented on the chaotic behavior of the inverted oscillator in

this section. Section (VIII) serves as the ﬁnal section of our study before providing pertinent future possibilities and

directions.

II. Formulation of Time-dependent Generalized Hamiltonian Dynamics

In this section, we explicitly discuss the crucial role of a system described by a time-dependent quantum mechanical

inverted oscillator in a very generic way to deal with path integrals and diﬀerent types of quantum correlation

functions (anti time ordered, time ordered, out-of-time-ordered). A generalised Hamiltonian for a time-dependent

inverted oscillator can be written as:

H(t) = 1

m(t)ˆp2−1

2m(t)Ω2(t)ˆq2+1

2f(t) (ˆpˆq+ ˆqˆp).(1)

Here ˆq(t)denotes the time-dependent generalized coordinate, m(t)is the time-dependent mass, Ω(t)is the time-

dependent frequency, and f(t)is a time-dependent coupling parameter. In general these time dependent parameters

can be anything. It is signiﬁcant to note that, in contrast to the SHO, the second term, which represents the potential

energy of the inverted harmonic oscillator, has a negative sign. The main reason for the negative sign in the potential

energy is that for inverted oscillator, the harmonic oscillator frequency, ω(t), is replaced by iω(t). We can write the

Euler-Lagrange equation for the coordinate ˆq(t)of the inverted oscillator as:

Dt−Ω2

eff ˆq(t)=0,(2)

where the diﬀerential operator Dtis deﬁned as:

Dt=d2

dt2+dln m(t)

dt.(3)

The squared eﬀective time-dependent frequency in Eq.(2) is deﬁned by:

Ω2

eff (t)=Ω2(t) + f2(t) + ˙m(t)

m(t)f(t) + ˙

f(t).(4)

where the dot corresponds to the diﬀerentiation with respect to time "t". Then, the Euler-Lagrange equation in (3)

gives the equation of motion in the simpliﬁed form as:

ˆq+˙m

m˙

ˆq−Ω2

eff (t)ˆq= 0.(5)

Next, the TDSE for the above mentioned time dependent system can then be written as:

H(t)Ψ(q, t) = i∂Ψ(q, t)

∂t .(6)

Now, to derive the eigenstates and energy eigenvalues of the above TDSE, we will use Lewis-Resenﬁeld invariant

operator method, in the next section, which will be an extremely useful tool to deal with the rest of the problem.

A. Lewis–Riesenfeld Invariant Method

In this subsection we apply the technique of Lewis-Riesenﬁeld invariant operator method to calculate the time

dependent eigenstates and energy eigenvalues for the generalised Hamiltonian of inverted oscillator as given in Eq.(1).

The Lewis-Riesenﬁeld invariant method is advantageous for obtaining the complete solution of an inverted harmonic

oscillator with a time-dependent frequency having PT symmetry. In this subsection, we will construct a dynamical

invariant operator to transit the eigenstates of the Hamiltonian from prescribed initial to ﬁnal conﬁgurations, in

arbitrary time. Let us assume an Invariant operator ˆ

I(t)which is a hermitian operator and explicitly satisﬁes the

relation:

dˆ

I(t)

dt =1

i[ˆ

I(t),ˆ

H(t)] + ∂ˆ

I(t)

∂t ,(7)

so that its expectation values remain constant in time. Here ˆ

H(t)is the time-dependent Hamiltonian of our system

deﬁned earlier in Eq.(1). If the exact form of invariant operator, ˆ

I(t), does not contain any time derivative operators,

it allows one to write the solutions of the TDSE as:

ψn(q, t) = eiµn(t)φn(q, t).(8)

Here, φn(q, t)is an eigenfunction of ˆ

I(t)with eigenvalue nand µn(t)is the time-dependent phase function. Note that

for the sake of simplicity, we are writing ˆqas qand the same for other operators. This means we further remove all

hats. Next, we consider that time-dependent linear invariant operator for the system in the form:

I(t) = α(t)q+K(t)p+γ(t),(9)

where α(t),K(t), and γ(t)are the time-dependent real functions. Using Eq.(7) we get:

˙α(t) + α(t)f(t) = −K(t)m(t)Ω2(t)

K(t)−K(t)f(t) = −α(t)

m(t)

˙γ(t)=0.

(10)

From Eq. (10) one can ﬁnd the following second order diﬀerential equation:

K+˙m

m˙

K−Ω2

eff (t)K= 0,(11)

where the squared eﬀective time-dependent frequency is deﬁned in Eq.(4) before. Therefore, we can write the Eq.(9)

as:

I(t) = K(t)p−m(t)h˙

K(t)−K(t)f(t)iq. (12)

Now we intend to ﬁnd the eigenstate φn(q, t)of I(t)using the following eigenvalue equation:

Iφn(q, t) = nφn(q, t)(13)

Here, the eigenstates satisfy the orthonormalization condition,which is, ⟨φn|φn′⟩=δ(n−n′). Subsituting the expres-

sion of I(t)from Eq. (12) in the above Eq.(13), one can ﬁnd the continuous eigenstates of the inverted oscillator:

φn=Nexp (im(t)

2K(t)[˙

K(t)−f(t)K(t)]q2+2n

m(t)q).(14)

It is very straightforward to calculate the normalizing constant, N, which is given by the following expression:

N=r1

2πK .(15)

One can ﬁnd the phase factor by inserting the eigenfunction of invariant operator in TDSE and solving the following

equation:

˙µnφn=i∂

∂t −Hφn.(16)

Using Eq.(8), we can write the wavefunction in the normalized form:

ψn=r1

2πK eiµn(t)exp (im(t)

2K(t)[˙

K(t)−f(t)K(t)]q2+2n

m(t)q).(17)

Now, to prove that the wave functions are always ﬁnite, we have to ﬁnd the integration over all possible continuous

eigenvalues, which is given by:

Ψ(q, t) = Z∞

−∞

g(n)ψn(q, t)dn. (18)

Here g(n)is the weight function. It helps one to ﬁnd which state the system is in. Let’s consider the weight function

as a gaussian function, which is of the following exact form:

g(n) = √a

(2π)1

exp −a2n2

4.(19)

Here ‘a’ is a real positive constant. Substituting the above form of weight function in Eq.(18) and integrating over all

possible continuous eigenvalues, one can write the wavefunction for inverted oscillator as:

Ψ(q, t) = 2A(t)

π1/4

exp iB2(t)+[iB1(t)−A(t)]q2.(20)

Here,we deﬁne the time dependent functions as:

A(t) =K2a21−4y2

a4−1

B1(t) = m(˙

K−fK)

2K+2y

K2a41 + 4y2

a4,

B2(t) = 1

2tan−12y

a2,

y(t) = Zt

dτ

m(τ)K2(τ).(21)

Now having obtained the normalization factor, we can write the expression for the energy of our desired quantum

system as:

E=r2A(t)

πZ∞

−∞

Ψ∗ˆ

HΨdq. (22)

Inserting the Eq.(20), ˆ

Hand conjugate of Eq.(20) into the Eq.(22) one can show that the time-dependent energy for

inverted oscillator is given by the following expression:,

E=1

2m(t)B2

1(t)

A(t)+A(t)+1

m(t)Ω2(t)

A(t)+f(t)B1(t)

2A(t).(23)

It is interesting to observe that the energy eigenvalues are independent of nbut energy is a continuous function of

time. Furthermore, there is no zero-point energy associated with this system of inverted oscillator.

III. Evaluation of Action

In this section we start with a Lagrangian of the inverted oscillator and derive the representative action for the

generalised inverted oscillator. We express the action in terms of Green’s functions which are derived in Appendix B.

The action thus obtained will be useful for deriving the generating function of inverted oscillator. Here, it is important

to note that the generating function in the present context physically represents the partition function in Euclidean

signature.

As the Green’s function for inverted oscillator, given in Eq.(B12) is hyperbolic in nature, one can use the hyperbolic

identities. Inferring from [6] one can then write the classical ﬁeld solution for the inverted oscillator as,

¯q(t) = 1

G(T)qfG(t−t0) + q0G(tf−t) + Ztf

dt′[G(T)Θ(t−t′)G(t−t′)−G(t−t0)G(tf−t′)] J(t′).(24)

The time-derivative of this ﬁeld can be given by,

¯q=1

G(T)qf˙

G(t−t0)−q0˙

G(t−tf) + Ztf

dt′[G(T)Θ(t−t′)˙

G(t−t′)−˙

G(t−t0)G(tf−t′)]J(t′).(25)

Here T≡tf−t0, such that t0and tfdenote the initial time and ﬁnal time respectively. Also, ¯q(t0) = q0and ¯q(tf) = qf

are the intial and ﬁnal ﬁeld conﬁgurations respectively. Applying the boundary condition ˙

G(0) = 1 one can ﬁnd,

¯q(t0) = 1

G(T)qf−q0˙

G(T)−Ztf

dt′G(tf−t′)J(t′).(26)

Furthermore, using the hyperbolic identity,

G(T)G(tf−t)−˙

G(T)G(tf−t) = G(t−t0),(27)

we obtain:

¯q(tf) = 1

G(T)qf˙

G(T)−q0+Ztf

dt′G(t′−t0)J(t′)(28)

In general, the action for any ﬁeld is given by:

S=Ztf

[L+J(t)q(t)]dt. (29)

Here the term J(t)is an auxiliary time dependent ﬁeld and Lis the Lagrangian of the system. For an inverted

oscillator the the representative Lagrangian can be written as:

L=1

2m(t) ˙q2+m(t)ω(t)2q2−f(t)m(t) ˙qq (30)

Substituting, Eq.(30) in Euler Lagrange equation the equation of motion for the inverted oscillator becomes,

m(t)¨q+˙

m(t) ˙q−"m(t)ω(t)2−f(t)˙

m(t)−m(t)˙

f(t)#q= 0 (31)

Substituting Eq.(30) in (29) the action for the inverted oscillator can be expressed as:

S=Ztf

t01

2m(t) ˙q2+1

2m(t)ω2(t)q2−f(t)m(t) ˙qq+J(t)q(t)dt. (32)

Using the equation of motion i.e. Eq.(31) and the deﬁnition of classical ﬁeld solution as stated in Eq.(24), we ﬁnd

the classical limit for the action of inverted oscillator as,

Scl(q0, qf|J)≡S(¯q) = 1

2m(t)¯q˙

¯q

t0−1

2f(t)m(t)¯q2

2Ztf

dtJ(t)¯q(t).(33)

The ﬁnal expression for the action can be obtained using Eq.(26) and Eq.(28), as given below:

S(q0, qf|J) = 1

2m(tf)1

G(T)q2

f˙

G(T)−q0qf+Ztf

dt′qfG(t′−t0)J(t′)−m(t0)1

G(T)qfq0−q2

0˙

G(T)+

Ztf

dt′q0G(tf−t′)J(t′)−1

2f(tf)m(tf)q2

f−f(t0)m(t0)q2

0+1

21

G(T)Ztf

dt[G(t−t0)qf+G(tf−t)q0]J(t)

−1

G(T)Ztf

dt Ztf

dt′[Θ(t−t′)G(tf−t)G(t′−t0) + Θ(t′−t)G(tf−t′)G(t−t0)]J(t)J(t′).

(34)

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Schwinger-KeldyshpathintegralformalismforaQuenchedQuantumInvertedOscillatorSayantanChoudhuryID1,∗SumanDey2,†RakshitMandishGharat3,‡SaptarshiMandal4,§andNileshPandey1¶1CentreForCosmologyandSciencePopularization(CCSP),SGTUniversity,Gurugram,Delhi-NCR,Haryana-122505,India,2DepartmentofPhysics,Visva-Bha...

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Schwinger-Keldysh path integral formalism for a Quenched Quantum Inverted Oscillator Sayantan ChoudhuryID1Suman Dey2Rakshit Mandish

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