Quantum resetting in continuous measurement induced dynamics of a qubit Varun Dubey1 Raphael Chetrite2 Abhishek Dhar1

2025-05-01 0 0 712.46KB 28 页 10玖币

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Quantum resetting in continuous measurement

induced dynamics of a qubit

Varun Dubey1, Raphael Chetrite2, Abhishek Dhar1

1International Centre for Theoretical Sciences, Tata Institute of Fundamental

Research, Bengaluru 560089, India,

2Universit´e Cˆote d’Azur, CNRS, LJAD, Parc Valrose, 06108 NICE Cedex 02, France

E-mail: varun.dubey@icts.res.in, raphael.chetrite@unice.fr and

abhishek.dhar@icts.res.in

Abstract. We study the evolution of a two-state system that is monitored

continuously but with interactions with the detector tuned so as to avoid the Zeno

aﬀect. The system is allowed to interact with a sequence of prepared probes. The

post-interaction probe states are measured and this leads to a stochastic evolution

of the system’s state vector, which can be described by a single angle variable. The

system’s eﬀective evolution consists of a deterministic drift and a stochastic resetting

to a ﬁxed state at a rate that depends on the instantaneous state vector. The detector

readout is a counting process. We obtain analytic results for the distribution of number

of detector events and the time-evolution of the probability distribution. Earlier work

on this model found transitions in the form of the steady state on increasing the

measurement rate. Here we study transitions seen in the dynamics. As a spin-oﬀ

we obtain, for a general stochastic resetting process with diﬀusion, drift and position

dependent jump rates, an exact and general solution for the evolution of the probability

distribution.

1. Introduction

The problem of repeated measurements on quantum systems is of great interest in the

context of monitoring and controlling its time evolution and in the context of answering

questions such as that of the time of arrival. Of particular interest is the situation

where a system is coupled to a probe and repeated measurements are performed on

the probe. An obviously interesting question is as to what these measurements on the

probe can tell us about the system. A number of recent experiments have looked at

the trajectories of quantum systems subjected to repeated measurements [1, 2, 3, 4].

General discussions of measurements and quantum trajectory theory can be found in

Refs. [5, 6, 7, 8, 9, 10, 11, 12].

A quantum system that is continuously monitored via direct measurements remains

frozen in its state. This is the well known quantum Zeno eﬀect [13, 14, 15, 16]. The Zeno

freezing can be avoided under the scheme of indirect measurements where the system is

allowed to interact for a period τwith a probe, with an interaction strength that scales as

arXiv:2210.15188v2 [quant-ph] 10 Apr 2023

Quantum resetting in continuous measurement induced dynamics of a qubit 2

τ−1/2, and then the state of the probe is measured projectively [17]. Depending upon the

result of this measurement, one obtains partial information about the state of the object

system. If this process is repeated with identically prepared probes, then the Zeno efect

is avoided and system evolves stochastically. It can be shown [5, 6, 7, 8, 9, 10, 11, 18]

that, for a two-state object system interacting with a sequence of identically prepared

probes (which are also two-state systems), the state of the object system evolves via a

stochastic Schr¨odinger equation with jumps. Furthermore, when the interaction strength

between the object system and the probes scales as τ−1/2, the reduced density matrix

of the system evolves via a Lindblad equation. Any given stochastic trajectory of the

wavefunction corresponds to the system’s evolution for a particular outcome of the

measurement sequence. Averaging over these outcomes corresponds to the case of blind

measurements and the entire information of the system’s evolution is contained in the

reduced density matrix.

The basis of the current work is the model described in Ref. [19]. In this work, the

authors have considered a measurement problem on a two-state system similar to the

one described in the last paragraph. The principal conclusion is that upon variation

of the relative strength λ(deﬁned below, see Eq.(18)) of measurement, the system

exhibits transitions which mark various stages in the onset of the quantum Zeno eﬀect.

Note that usual Zeno eﬀect refers to the phenomena whereby a system’s dynamics gets

frozen as a result of continuous measurements on it. This Zeno eﬀect is avoided with

the choice of interaction strength scaling as τ−1/2. But what Ref. [19] ﬁnds for their

model is that in the limit of inﬁnite measurement strength there is again a freezing of

the dynamics. Interestingly, signatures of this freezing appear even at large but ﬁnite

mesaurement strengths, with parts of the Hilbert space becoming inaccessible — this is

referred to as Zeno eﬀect appearing in stages.

Following [19], we model the detector readouts as a counting process and investigate

the onset of the Zeno regime in the counting statistics of the readout process. We note

that similar investigations have been carried out in [20]. However, in the model we

consider, the counting process has a stochastic intensity [21] given by the rate function αt

in Eq. (27). Our calculations reveal that for λ= 2, the mean count E[Nt] of the counting

process exhibits a topological transition as remarked in [19]. Measurement induced

entanglement transitions and topological phase transitions based on Zeno physics have

gained considerable attention, in particular we note the recent studies [22, 23, 24, 25, 26].

These transitions have been identiﬁed with presence of exceptional points [27] in

the spectrum of non-Hermitian Hamiltonian which evolves the quantum state under

continuous measurement and post-selection. In [28], the spectral approach is employed

to investigate the properties of Markov processes that are reset to a ﬁxed state at times

picked from an exponential distribution. [29, 30] consider exceptional points of non-

Hermitian Hamiltonians as well Liouvillians governing open system dynamics.

We point out that the stochastic dynamics of our qubit can be interpreted as the

overdamped motion of a particle in a tilted periodic potential with a resetting of the

position to a particular point, at a rate that depends on the particle position. Stochastic

Quantum resetting in continuous measurement induced dynamics of a qubit 3

resetting has been widely studied in the classical context [31, 32, 33, 34] but there are few

studies in the quantum context [35, 36, 37]. Our study provides a simple example where

stochastic resetting in a quantum system appears naturally as a result of measurements.

As our second main result, we use the renewal approach to compute the exact time-

dynamics of the probability distribution of the wavefunction.

The plan of this paper is as follows. In Sec. 2, we describe the basic setup and the

measurement protocol, discuss the emergence of the stochastic Schr¨odinger equation

and summarize known results from earlier work. We also discuss the Bloch sphere

representation of the qubit and the particular simpliﬁcation that occurs for the system

we study. In Sec. 3 we present the calculation to obtain explicit expressions for the

generating function for the number of clicks and from it the mean number of clicks.

Sec. 4 contains the calculation for the time-evolution of the system using a renewal

approach. We then discuss a second approach based on the non perturbative formula

for the resolvent (Green’s function). In Sec. 5 we present some results on spectral

properties of the probability evolution operator and use it to write another solution for

the time evolution. We conclude in Sec. 6.

2. Basic setup and summary of earlier work

2.1. Deﬁnition of the model and dynamics

Consider a 2−level system Swhose Hilbert space HSis spanned by the vectors

|ψ0i="1

0#,|ψ1i="0

1#.(1)

The system evolves with the Hamiltonian

HS="0γ0

γ00#=γ0σx(2)

where γ0is a positive frequency. σx, σy, σzrepresent the Pauli matrices. At any instance

t, the state of Sis given by the normalized vector

|ψ(t)i=a(t)|ψ0i+b(t)|ψ1i="a(t)

b(t)#.(3)

At this instance, Sis allowed to interact with another 2−level system Dfor a short time

interval τ. The Hilbert space HDis spanned by {χ0, χ1}deﬁned similarly as in Eq. (1).

At the start of the interaction, Dis assumed to be in the state χ0. The combined state

of the system Sand the detector Dis the uncorrelated vector

|Ψ(t)i=|ψ(t)i⊗|χ0i(4)

in the tensor product space H=HS⊗ HD. We adopt the convention that for states

or operators in H, the ﬁrst factor corresponds to Sand the second factor to Din all

summands. The state Ψ(t) evolves in the interval τby the Hamiltonian

H=HS⊗I+rγ

τπ1⊗σy,(5)

Quantum resetting in continuous measurement induced dynamics of a qubit 4

where we note that the interaction part of the Hamiltonian is scaled as 1/√τand γis

a non-negative coupling frequency. In Eq. (5), the projector π1=|ψ1ihψ1|and Iis the

identity operator. It follows that the combined state after the interval τis given by

|Ψ(t+τ)i= exp [−iτ H]|Ψ(t)i

=|ψ(t)i⊗|χ0i+ (−iτ)hHS−iγ

2π1|ψ(t)ii⊗ |χ0i − i√γτ[π1|ψ(t)]i ⊗ [σy|χ0i] + O(τ3

2).

Now a projective measurement in the basis |χ0i,|χ1i ‡ is performed to determine the

state of the detector. If the detector is found to be in the state |χ0i, then the un-

normalized state |˜

ψ(t+τ)iof the system, up to ﬁrst order in τis given by

ψ(t+τ)i=hI−iτHs−iγ

2π1i|ψ(t)i,(6)

The probability of this event, i.e, of the readout to be χ0, up to ﬁrst order in τis given

p0= 1 −γτ hψ|π1|ψi= 1 −γτ |b(t)|2.(7)

If the readout is χ1, then the un-normalized state and the probability of the readout are

ψ(t+τ)i=√γτπ1|ψ(t)i(8)

p1=γτhψ|π1|ψi=γτ|b(t)|2.(9)

This completes description of one measurement cycle. Subsequently the object system

is coupled to another detector initialized in χ0and the process is repeated sequentially.

Every time a detector is measured to be in the state χ1corresponds to a ’click’.

In the limit τ=dt →0 the stochastic evolution of the normalized state is thus

given by

|ψ(t+dt)i=









|ψ(t)i+dt −iHS−γ

2π1+1

2αt|ψ(t)i,with prob. p0= 1 −αtdt,

|ψ1i,with prob p1=αtdt,

(10)

where αt≡γhψ(t)|π1|ψ(t)i=γ|b(t)|2.(11)

Equivalently, we can write also the complex non linear stochastic equation

d|ψ(t)i=−iHS−γ

2π1+αt

2|ψ(t)idt +√γπ1

√αt−I|ψ(t)idNt.(12)

Note that for the second outcome in Eq. (10) we should include a factor b(t)/|b(t)|.

However, rigorously, we should interpret the equation Eq. (12) (and all the others of

this articles) for the corresponding one-point projector |ψ(t)ihψ(t)|. In more explicit

vectorial way

d a(t)

b(t)!=" γ

2|b(t)|2−iγ0

−iγ0−γ

2+γ

2|b(t)|2!dt + −1 0

0−1 + 1

|b(t)|!dNt# a(t)

b(t)!.(13)

‡Note that the fact that the associated projectors π0, π1are rank 1, which is natural in this

bidimensional HD, is in fact for a multidimensional HDthe main hypothesis which permits to conserve

the purity of the system state and to write, as in the following, an equation for the pure state instead

of the (impure) density matrix.

Quantum resetting in continuous measurement induced dynamics of a qubit 5

In these equations, Ntis a Poisson counting process which counts the number of

clicks in any ﬁnite interval [0, t]. For almost all realizations one may take N0= 0. The

change dNtat the instance tbe deﬁned as the Ito-diﬀerential with the usual properties

dNt=Nt−Nt−, dNtdNt=dNt, dNtdt = 0,(14)

where Nt−= limt0→t−Nt0(here we assume the trajectories of Ntare right continuous

with left limits) and with the expected value of the Poisson increment, conditioned upon

the fact that the state of the ket of system takes the value |ψ(t)i, is equal to

E[dNt] = αtdt =γ|b(t)|2dt. (15)

Equations (12, 13) are sometimes called stochastic Schr¨odinger equations [9] or

quantum trajectory for pure state. The ﬁrst appearance of this type of equation with

Poisson noise in this set-up was in [38] and in [39]. Since then, diﬀerent justiﬁcations

have been given for the fact that they model quantum systems which are subject to

continuous indirect measurements. The general case of quantum trajectories is for mixed

states and includes also Gaussian white noise [8, 9, 10].

For blind measurements one considers an average over the outcomes and the density

matrix ρ(t) = h|ψ(t)ihψ(t)|i evolves via

∂tρ(t) = −i[HS, ρ(t)] + γ

2(2π1ρ(t)π1− {π1, ρ(t)}).(16)

which is the form of the Lindblad equation [40, 41] with only one Krauth operator π1

which is moreover self-adjoint.

Physically, two phenomena are in competition in equations (12, 13) :

(i) Collapsing in basis |ψ0i,|ψ1ithanks to continuous measurement. More precisely,

when γ0→0 (i.e. HS→0), (12, 13) models the continuous measurement of π1.

As π1is a diagonal matrix in the basis |ψ0i,|ψ1i, this basis is said to be of non-

demolition form with respect to the measurement [9]. This will lead at large time

to [42, 43] the collapse in the basis |ψ0i,|ψ1iwith the born law with respect to the

initial ket |ψ(0)i, i.e. :

lim

t→∞ |ψ(t)i=(|ψ0iwith probability |hψ0|ψ(0)i|2

|ψ1iwith probability |hψ1|ψ(0)i|2.(17)

(ii) Rabi (coherent) oscillation due to unitary evolution. More precisely, when γ= 0,

then (14,15) dNt= 0 and the equation (12, 13) is the free (unitary) evolution with

Rabi Hamiltonian HSwhich leads to classical Rabi oscillation [44].

Note that in the general case with ﬁnite γ0and γ, as the commutator [HS, π1]6= 0,

the unitary evolution comes to prevent the asymptotic collapse (17). The asymptotic

behavior will then be a smooth invariant density that we will exhibit below. Note

also that the competition between continuous non demolition measurement and

thermalization (instead of free unitary evolution here) has recently been extensively

studied (see e.g Refs: [45, 46, 47]).

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摘要：

QuantumresettingincontinuousmeasurementinduceddynamicsofaqubitVarunDubey1,RaphaelChetrite2,AbhishekDhar11InternationalCentreforTheoreticalSciences,TataInstituteofFundamentalResearch,Bengaluru560089,India,2UniversiteC^oted'Azur,CNRS,LJAD,ParcValrose,06108NICECedex02,FranceE-mail:varun.dubey@icts.res...

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Quantum resetting in continuous measurement induced dynamics of a qubit Varun Dubey1 Raphael Chetrite2 Abhishek Dhar1

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