Adiabatic-impulse approximation in non-Hermitian Landau-Zener Model Xianqi Tong1Gao Xianlong2and Su-peng Kou1y 1Department of Physics Beijing Normal University Beijing 100000 Peoples Republic of China

2025-05-06 0 0 546.72KB 11 页 10玖币

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Adiabatic-impulse approximation in non-Hermitian Landau-Zener Model∗

Xianqi Tong,1Gao Xianlong,2and Su-peng Kou1, †

1Department of Physics, Beijing Normal University, Beijing 100000, People’s Republic of China

2Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China

(Dated: October 25, 2022)

We investigate the transition from PT-symmetry to PT-symmetry breaking and vice versa in the

non-Hermitian Landau-Zener (LZ) models. The energy is generally complex, so the relaxation rate of

the system is set by the absolute value of the gap. To illustrate the dynamics of phase transitions, the

relative population is introduced to calculate the defect density in nonequilibrium phase transitions

instead of the excitations in the Hermitian systems. The result shows that the adiabatic-impulse

(AI) approximation, which is the key concept of the Kibble-Zurek (KZ) mechanism in the Hermitian

systems, can be generalized to the PT-symmetric non-Hermitian LZ models to study the dynamics

in the vicinity of a critical point. Therefore, the KZ mechanism in the simplest non-Hermitian

two-level models is presented. Finally, an exact solution to the non-Hermitian LZ-like problem is

also shown.

PACS numbers: Valid PACS appear here

I. INTRODUCTION

The quantum two-level system exhibiting an avoided

level crossing or level crossing plays an essential role in

quantum adiabatic dynamics. If the control parameter is

varied in time, the transition probability is usually cap-

tured by the Landau-Zener (LZ) theory [1,2]. Usually,

the quantum two-level system provides not only qualita-

tive but also quantitative descriptions of system proper-

ties. It has become the standard theory for investigating

many physical systems, e.g., the smallest quantum mag-

nets, and Fe8clusters cooled below 0.36 K, are success-

fully described by the LZ model [2,3].

In many cases, the relevant parameters (i.e., the energy

gap between the two levels on time) have the potential to

be more general than the original LZ process. This moti-

vates us to extend the level-crossing dynamics to level co-

alesce and various power-law dependencies in this paper.

Appropriate changes in the external parameters driving

the LZ transition can enable these LZ models to be ex-

perimentally realized in polarization optics [4], adiabatic

quantum computing [5,6], and non-Hermitian photonic

Lieb lattices [7–9].

Fundamental axioms of quantum mechanics impose

the Hermitian structure on the Hamiltonian. However,

recent developments have shown the emergence of rich

features for non-Hermitian Hamiltonians describing in-

trinsically non-unitary dynamics [10–14], which have also

been recently realized experimentally [15–17]. Although

the eigenvalues of the non-Hermitian Hamiltonians can

still be interpreted in terms of energy bands [18,19], the

signiﬁcance of their eigenvectors can no longer be handled

by conventional methods because they are not orthogo-

nal and thus already possess limited overlap without any

∗A footnote to the article title

†Corresponding author. Email: spkou@bnu.edu.cn

additional perturbations [20–25]. In this case, the excep-

tional points [26–31] (EPs) are particularly important,

where the complex spectra become gapless. These can

be seen as non-Hermitian counterparts of the conven-

tional quantum critical points [32–34]. In EPs, two (or

more) complex eigenvalues and eigenstates coalesce and

then no longer form a complete basis [35–37]. Our main

purpose is to study the linear quenching dynamics near

the critical point, which is captured by the Kibble-Zurek

(KZ) mechanism [27,28,38–43].

In this context, we present a successful combination

of the KZ [44–46] theory of topological defect produc-

tion and the quantum theory of the PT-symmetric non-

Hermitian LZ model [47–50]. Both theories play a promi-

nent role in contemporary physics. The KZ theory pre-

dicts the production of topological defects (vortexes,

strings) in the course of non-equilibrium phase transi-

tions [51–58]. This prediction applies to phase transitions

in liquid 4He and 3He, liquid crystals, superconductors,

ultra-cold atoms in optical lattices [59,60], and even

to cosmological phase transitions in the early universe

[61,62]. To the best of our knowledge, the KZ mecha-

nism in the simplest non-Hermitian two-level model has

not been discussed before.

This work mainly focuses on the dynamical evolution

of the PT-symmetric non-Hermitian LZ model, including

adiabatic and impulse regimes during the slow quench of

a system parameter. A real-to-complex spectral transi-

tion, which is usually called the PT transition, occurs

in the non-Hermitian LZ model. In the PT-symmetric

regime, the eigenvalues are real, ensuring the probabil-

ity is conserved. When the energy gap is large enough

away from the EP, the adiabatic theorem ensures that a

system prepared in an eigenstate remains in an instan-

taneous eigenstate. This is in contrast to diabatic evolu-

tions included by a very fast parameter change. This sit-

uation is more involved in the PT-broken regime where

the eigenvalues are complex conjugates. Furthermore,

the probability is no longer conserved because there is

arXiv:2210.12709v1 [quant-ph] 23 Oct 2022

Figure 1. (a) The energy level of the Hamiltonian (1); dot-

ted line: ν= 0 case. Note the EP at γ=±ν√δ−1 and

asymptotic form of eigenstates: |1i,|2i. (b) The inverse of

the energy gap in the non-Hermitian LZ model. The four

dashed lines correspond to the instants in the PT-symmetry

and PT-symmetry breaking regimes, which separate the adi-

abatic and impulse regimes.

exponential growth and an exponential decay level, i.e.,

only the exponential growth state is left under the adia-

batic evolution. Thus, the adiabatic conditions of the NH

system were modiﬁed [63]. Near the EP, however, due to

the reciprocal of the absolute value of the energy gap be-

ing greater than the change of parameters, the dynamics

cannot be adiabatic, and the system gets excited. Then,

since the modulus of the system is not conserved during

the evolution, the relative occupation is proposed to cal-

culate the excitation rather than the projection on the

excited state. This scenario is captured by the adiabatic-

impulse (AI) approximation. Finally, we also give non-

trivial exact solutions for the non-Hermitian LZ model,

successfully obtaining the theoretical free parameters in

AI approximation.

In this paper, we want to illustrate the AI approxima-

tion from the simplest non-Hermitian two-level LZ model

and discuss it in three diﬀerent parts. Sect. II presents

the adiabatic-impulse distinction in the PT-symmetric

region, and the exact solutions of two quenching pro-

cesses. In Sect. III, we also discuss the AI approximation

solution of the PT-broken regime under diﬀerent initial

conditions. In Appendix A, we explain the exact solution

of the non-Hermitian Landau-Zener-like problem. De-

tails of analytic calculations are in Appendix B(diabatic

solutions in the PT-symmetric regimes).

II. PT-SYMMETRY

The PT-symmetric non-Hermitian LZ model we con-

sider is

H(t) = 1

2(−1)nγ ν

ν(1 −δ) (−1)n+1γ,(1)

where the γ= ∆tis time-dependent, ∆ is a time-

independent constant, and n= 0 in this part. The sys-

tem experiences adiabatic time evolution when ∆ →0,

and ∆ → ∞ means diabatic evolution. In this model,

ν,δ > 0 are constant parameters. We set ν= 1 as an

energy unit that does not inﬂuence the results.

For a non-Hermitian Hamiltonian H, let |iRidenote

the ith left eigenstates with the (generally complex)

eigenenergy Ei, i.e., iLH=iLEi. Note that the mth

right eigenvector |jRisatisﬁes HjR=EjjRand the

biorthonormal relation hiL|jRi=δij .

At any instantaneous time, the right eigenstates of this

Hamiltonian can be written on the time-independent ba-

sis |1iand |2i. The ground state |↓ (t)Riand the excited

state |↑ (t)Riare given by the following equation,

| ↑ (t)Ri

| ↓ (t)Ri= −i

√δ−1cosh θ

2isinh θ

−1

√δ−1sinh θ

2cosh θ

2!|1i

|2i,(2)

where cosh θ=/√2−1, sinh θ= 1/√2−1, =

γ/ν√δ−1. We consider only θ∈[0, π]. If θis not a real

number, the system’s PT symmetry is broken, and this

part is not considered. The energy spectrum is depicted

in Fig. 1with the energy gap ∆g=pγ2+ν2(1 −δ). It

can be seen that the energy gap ∆g= 0 at the excep-

tional points γEP =|ν√δ−1|, which is accompanied by

the coalesce of eigenvalues and eigenstates.

The density of topological defects can be introduced

in the non-Hermitian LZ model in the following way.

Suppose the state |1iis the eigenstate of arbitrary non-

Hermitian operate ˆ

O:ˆ

O|1Ri=n|1Ri(n=±1,±2, ...),

while the state |2Rialways corresponds to the 0 eigen-

value, i.e. ˆ

O|2Ri= 0 |2Ri. So for any normalized state

can be written as |Ψi=a|1Ri+b|2Ri(|a|2+|b|2=

1,hiL|jRi=δij in the PT-symmetry regime). The unity

density defect is determined by the expectation value of

operator ˆ

O,hˆ

Oi=hΨ|ˆ

O|Ψi/n =|hΨ|1i|2. But for non-

unitary evolution in the PT-broken area, the probability

of time evolution state on the instantaneous states is not

conserved, i.e., |a|2+|b|26= 1. Then, hOiis replaced with

the relative occupation

Dr=h1L|Ψi2

|h1L|Ψi|2+|h2L|Ψi|2,(3)

where |nLiis the nth left eigenstate. When we dis-

cuss only the PT-symmetric regime, where h1L|Ψi2+

h2L|Ψi2= 1, Drreturns to the Hermitian case, i.e.,

Dr=hˆ

Oi.

Suppose the system evolves adiabatically from the

ground state of (1) at |t|→∞to the ground state across

the EP. Therefore, the state of the system will go from

a density-free phase to a density-defected one, that is,

undergo a phase transition from |1Rito |2Ri. If the time

evolution fails to be adiabatic, which is usually the case,

the state at the end is a superposition of states |1Riand

|2Ri, so the expected value of the operator ˆ

Ois nonzero.

Then we will show that the KZ theory can well predict

the topological density (3) in the non-Hermitian LZ sys-

tem.

The analogy of relaxation time scale, relative temper-

ature, and quench time scale are determined as follows.

First, the KZ theory neatly simpliﬁes the evolution of

system dynamics. The simpliﬁcation is the essence of

the KZ mechanism that suggests splitting the quench into

the regime near the EP and the quasiadiabatic regime far

from the EP, that is to say, the state becomes change-

less (impulse), or can adjust to changes in the parame-

ter. This is the key concept from Zurek [64–66], and the

switch from adiabatic to impulse is determined by the

relevant time scale. And the relevant time scale equals

the reciprocal of the energy gap, which is small when the

parameter is apart from the EP and relatively large in the

impulse regime. According to the adiabatic theorem, as

long as the reciprocal of the gap is small enough, the sys-

tem will evolve from the ground state and remain in the

ground state. This naturally shows that the reciprocal of

the gap must be small in the adiabatic evolution regime,

which can be regarded as the equivalent relaxation time

scale introduced above: τ= 1/pγ2+ν2(1 −δ). The

dimensionless distance = ∆t/ ν√δ−1of the system

from the exceptional point is the relative temperature.

The quench time is τQ=ν√δ−1/∆. Then, ν√δ−1 is

identical with 1/τ0. Finally, the relaxation time can be

rewritten as

τ=τ0

√2−1,  =t

τQ

.(4)

For ||  1, the relaxation time τ≈τ0/|ε|will be back

to topological defect density in liquid 4He [64–66], which

will be discussed in details below.

In the following, we will consider the dynamics of the

LZ model described by the time-dependent Schr¨odinger

equation id

dt |Ψi=ˆ

H(t)|Ψi, see Eq. (1). When the

whole evolution begins at time ti=−∞, the initial

state is chosen to be the ground state |φGi, and lasts

till tf→ −γEP . Since the eigenvalues coalesce at EP, no

matter how slowly the parameters are driven, it is impos-

sible for the system’s quantum state to evolve adiabati-

cally near the critical point. This paper aims to quantify

this unavoidable excitation level in non-Hermitian sys-

tems. At the beginning of the evolution, the energies

are real, and the energy gap is large enough so that the

states of the system evolve adiabatically. On the con-

trary, when the time-dependent parameter of the Hamil-

tonian is gradually approaching an exceptional point, the

time-evolved state can not follow the change of the pa-

rameter of the Hamiltonian. The evolved state becomes

an impulse near the EP. Furthermore, when the system

is in the PT-broken regime, only one state dominates the

population, i.e., the eigenstate with the largest imagi-

nary eigenvalue. So, the whole evolution stage can be

divided into two diﬀerent regimes in PT-symmetric or

(PT-broken) regime:

t∈(−∞,−ˆ

t) : |Ψ(t)i ≈ (phase factor) |φG(t)i,

t∈[−ˆ

t, −γEP ] : |Ψ(t)i ≈ (phase factor) |φG(−ˆ

t)i,(5)

which is same for the evolution t∈[γEP ,+∞), because

the energy spectrum is real and symmetric on both sides

when |γ|> γEP .

Figure 2. Transition probability as a function of τQin the

PT-symmetric regime for two diﬀerent: D↑(upper curves),

D↓(lower curves). Dots: numerical data by solving the

Schr¨odinger equation, solid lines: AI approximation (8), (11)

in which the α= 0.06 is determined from the exact solu-

tion of Appendix A. Upper (lower) curves all corresponding

to θ0= 0.50π(0.01π).

According to Eq. (5), the state will be impulsed in

the regime near the EP that only has a diﬀerent phase

factor. And if the state can adiabatically evolve during

the time, we can approximately consider the state as the

instantaneous eigenstate of the Hamiltonian with a dif-

ferent phase factor. Of course, the process will return to

the adiabatic evolution when the real energy gap is big

enough again. The assumption behind Eq. (5) above is

based on how well the KZM works in the non-Hermitian

LZ model.

However, the instants ±ˆ

tare still unknown. It is ﬁrstly

calculated from the equation proposed by Zurek in the

paper on classical phase transition [64–66]

τ(ˆ

t) = αˆ

t, (6)

and α=O(1) is a constant independent of τQ[44,45]. In

the PT-symmetry regime, the solution of Eq. (6) reads

ˆεs=εs(ˆ

t) = 1

√2v

ts1 + 4

+ 1, xα=ατQ

τ0

.(7)

For fast transition, i.e., xα→0, we get ˆ

t=pτ0τQ/α.

Then, the ﬁrst case we consider is completely in the

PT-symmetric region. The initial state |Ψ(ti)iis set to

ground state |↓ (ti)i, and the start evolution point away

from the EP, i.e., from −∞ to −γEP . Thus, we can cal-

culate the occupation of the ﬁnal state on the left eigen-

state, rather than the right eigenstate in the Hermitian

system:

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Adiabatic-impulseapproximationinnon-HermitianLandau-ZenerModelXianqiTong,1GaoXianlong,2andSu-pengKou1,y1DepartmentofPhysics,BeijingNormalUniversity,Beijing100000,People'sRepublicofChina2DepartmentofPhysics,ZhejiangNormalUniversity,Jinhua321004,People'sRepublicofChina(Dated:October25,2022)Weinvestig...

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Adiabatic-impulse approximation in non-Hermitian Landau-Zener Model Xianqi Tong1Gao Xianlong2and Su-peng Kou1y 1Department of Physics Beijing Normal University Beijing 100000 Peoples Republic of China

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