Anomalous energy exchanges and Wigner function negativities in a single qubit gate

2025-04-22 0 0 1.39MB 8 页 10玖币

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Maria Maﬀei,1Cyril Elouard,2Bruno O. Goes,1Benjamin Huard,3Andrew N. Jordan,4, 5 and Alexia Auﬀèves1

1Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France

2Inria, ENS Lyon, LIP, F-69342, Lyon Cedex 07, France

3Ecole normale supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

4Institute for Quantum Studies, Chapman University, 1 University Drive, Orange, CA 92866, USA

5Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA

Anomalous weak values and Wigner function’s negativity are well known witnesses of quantum contextuality.

We show that these eﬀects occur when analyzing the energetics of a single qubit gate generated by a resonant

coherent ﬁeld traveling in a waveguide. The buildup of correlations between the qubit and the ﬁeld is responsible

for bounds on the gate ﬁdelity, but also for a nontrivial energy balance recently observed in a superconducting

setup. In the experimental scheme, the ﬁeld is continuously monitored through heterodyne detection and then

post-selected over the outcomes of a ﬁnal qubit’s measurement. The post-selected data can be interpreted as

ﬁeld’s weak values and can show anomalous values in the variation of the ﬁeld’s energy. We model the joint

system dynamics with a collision model, gaining access to the qubit-ﬁeld entangled state at any time. We ﬁnd

an analytical expression of the quasi-probability distribution of the post-selected heterodyne signal, i.e. the

conditional Husimi-Q function. The latter grants access to all the ﬁeld’s weak values: we use it to obtain that of

the ﬁeld’s energy change and display its anomalous behaviour. Finally, we derive the ﬁeld’s conditional Wigner

function and show that anomalous weak values and Wigner function’s negativities arise for the same values of

the gate’s angle.

I. INTRODUCTION

Weak values have been originally deﬁned as the average

values for the results of weak measurements post-selected on

particular outcomes of a ﬁnal strong (projective) measure-

ment [1], where weak measurements are deﬁned as measure-

ments that minimally disturb the system [2]. Later on, the

concept of weak values has been generalized to any POVM

(positive operator valued measurement) and any choice of the

observable and the conditioning [3]. When the outcome used

for the post-selection is unlikely, weak values can exceed the

range of eigenvalues of the corresponding operators [4]. In

correspondence of such anomalous values, the Wigner func-

tion dictating the statistical distribution of the post-selected

measurements takes negative values [5]. Furthermore it can

be proven that both anomalous weak values [5–7], and Wigner

function’s negativity [8–10] are witnesses of contextuality.

Here we show that those eﬀects occur in a paradigmatic set-

ting of waveguide quantum electrodynamics where energy ex-

changes feature anomalous weak values. The setting is the so

called one-dimensional (1D) atom, a two-level emitter (qubit)

interacting with an electromagnetic ﬁeld propagating in 1D.

When the ﬁeld is prepared in a coherent state resonant with

the qubit’s transition this system implements a single qubit

gate with a ﬁdelity limited by the buildup of qubit-ﬁeld cor-

relations [11]. Weak values arise in the 1D atom when the

electromagnetic ﬁeld is monitored via heterodyne detection

and the data are post-selected over the outcomes of a qubit’s

projective measurement. This may be understood by regard-

ing the propagating ﬁeld as a weak measurement apparatus

for the qubit [12,13], see Fig. 1(a). Superconducting cir-

cuits represent the ideal setup to implement such a detection

scheme as they grant independent access to the states of qubit

and ﬁeld. A recent experiment on a superconducting single

qubit gate [14] showed anomalous weak values of the ﬁeld’s

energy change, i.e. values exceeding by far the single quan-

Figure 1. Schematics of the detection of the ﬁeld’s weak values in the

single-qubit gate. The gate is implemented by coherently driving a

1D atom with a pulse of area θ= Ωτ; the output ﬁeld is continuously

detected in the time interval [0, τ] with a heterodyne measurement

(weak measurement); at time τa projective (strong) measurement is

performed on the qubit and the heterodyne data acquired are post-

selected according on the outcome. The plot shows the values of the

change of the number of ﬁeld’s excitations as a function of time for

θ=0.93πand γτ =3/40. The gray line represents the unconditional

change of the number of excitations.

tum of energy that qubit and ﬁeld can physically exchange,

see Fig. 1(b).

We study the 1D atom using a collision model [15] where

individual temporal modes of the electromagnetic ﬁeld locally

interact with the qubit in a sequential fashion [16,17]. When

the ﬁeld is prepared in a resonant coherent state, this method

provides the exact analytical expression of the qubit-ﬁeld en-

tangled state at any time [18]. From the collision model of

the driven 1D atom, we derive the analytical expression of

the ﬁeld’s Husimi-Q function conditioned on the outcomes of

the ﬁnal qubit’s measurement. This function gives access to

all the moments of the post-selected heterodyne distribution,

namely the ﬁeld’s weak values. Then we use the conditional

Husimi-Q function to derive the weak value of the ﬁeld’s en-

ergy change and we show that, in the typical working regime

of single qubit gates, it takes anomalous values as observed

arXiv:2210.05323v1 [quant-ph] 11 Oct 2022

in [14]. Finally we explore the relation between anomalous

weak values and negativity of the corresponding Wigner func-

tion. Exploiting the analytical expression of the ﬁeld’s state

obtained with the collision model, we compute the ﬁeld’s con-

ditional Wigner function and we show that anomalous weak

values and Wigner function’s negativities arise for the same

values of the gate’s angle.

The paper is organized as follows: In Sec. II we describe the

coherently driven 1D atom and present the collision model of

its dynamics. In Sec. III we present our main results: the ana-

lytical derivations of the ﬁeld’s conditional Husimi-Q function

and the weak value of the ﬁeld’s energy change. In Sec. IV we

derive the conditional Wigner function. Finally, in Sec. Vwe

draw the conclusions of our work.

II. SYSTEM AND MODEL

A. The coherently driven 1D atom

The 1D atom comprises a qubit coupled to a single-mode,

semi-inﬁnite waveguide. The waveguide ﬁeld constitutes a

reservoir of electromagnetic modes of frequencies ωkand lin-

ear momentum k=ωkv−1with vbeing the ﬁeld’s group ve-

locity taken as positive. These modes are destroyed (created)

by the operators ak(a†

k). The dynamics of the joint system is

ruled by the Hamiltonian:

H=

~ω0σ†σ+~∞

k=0

ωka†

kak

+i~g∞

k=0σ†ak−a†

kσ,(1)

where σ≡|gihe|, with |g(e)ibeing the ground (excited) state

of the emitter. Writing the above Hamiltonian we implic-

itly assumed that the light-matter interaction is weak enough

that only frequency modes close to ω0play a role (quasi-

monochromatic approximation) [17]. In this regime, the ro-

tating wave approximation is allowed [19], and the coupling g

can be considered uniform in frequency [20].

The ﬁeld’s lowering operator at the position xin the inter-

action picture [16–18] is given by

b(x,t)=s1

e−iωk(t−x/v)ak=b(0,t−x/v),(2)

where %is the modes’ density verifying the relation

Pke−iωk(t−t0)/% =δ(t−t0). The operators b(0,t) satisfy

the bosonic commutation relation, i.e. [b(0,t),b†(0,t0)] =

δ(t−t0) [17]. The qubit is located at the position x=0 of

the waveguide, such that the interaction picture Hamiltonian

reads:

HI(t)=i~√γσ†(t)b(0,t)−b†(0,t)σ(t),(3)

where we deﬁned the interaction picture lowering operator

σ(t)=e−iω0tσ, and the emitter’s decay rate γ=g2%. In

the regions where x<0 or x>0, the ﬁeld travels with-

out deformation, it is then natural to deﬁne ﬁeld’s input and

output operators respectively as bin(t)≡lim→0−b(, t), and

bout(t)≡lim→0+b(, t). These operators satisfy the mean

input-output relation hbout(t)i=hbin(t)i − √γhσ(t)i, in agree-

ment with the textbook input-output relation written in the

Heisenberg representation [20].

The gate’s input ﬁeld is a square coherent pulse of ampli-

tude hbin(t)i=αt=αe−iω0t/√%, with αreal. Hence, the

ﬁeld’s state at the initial time (t=0−) reads |αi≡ D(α)|0i,

where D(α)=eαa†

0−α∗a0is the displacement operator of the

mode with frequency ω0which can be equivalently written as

D(α)=eRdt(αtb†(t)−α∗

tb(t))using the transformation (2). A uni-

tary driving on the qubit, HD(t), arises naturally when displac-

ing the interaction Hamiltonian in Eq. (3), D(−α)HI(t)D(α)=

HI(t)−Ωσy/2=HI(t)+HD(t), with Ω/2=gαand σy=

iσ†−iσ. In the classical limit of the ﬁeld [11], the qubit

reduced dynamics is solely dictated by HD(t) and hence it re-

duces to a pure rotation around the yaxis of an angle θ= Ωτ,

where τis the duration of the qubit-ﬁeld interaction. In this

limit, the light-matter interaction (int) is equivalent to the map:

|gi⊗|αiint(τ)

−−−−→ cos (θ/2) |gi+sin (θ/2) |ei⊗|αi.

Beyond the classical limit, the interaction entangles qubit

and ﬁeld resulting in a loss of purity of the reduced qubit’s

state and a degradation of the coherence of the input ﬁeld.

The joint qubit-ﬁeld state at time τcan be written as the pure

state:

|Ψ(τ)i=qPg(τ)g, ψg(τ)E+pPe(τ)|e, ψe(τ)i

with

|ψ(τ)i=D(α)





f(0)

(τ)−Rτ

0dt f (1)

(τ, t)e−iω0tb†(t)+. . .

√P(τ)



|0i,

(4)

where =g,e, and we adopted the short notation b(t)≡

b(0,t). The state in the parenthesis is the ﬁeld state in the

displaced reference frame, where the interaction Hamiltonian

is D(−α)HI(t)D(α), and the input ﬁeld is the vacuum. In

such a frame, the only mechanism responsible for the pho-

tons’ creation is the spontaneous emission, hence the ﬁeld

coincides with the emitter’s ﬂuorescence having amplitude

hbout(t)i − hbin(t)i=−√γhσ(t)i. The ellipsis represent the

components with j>1 spontaneously emitted photons, hav-

ing the form (−)jRdtjf(j)

(τ, tj)e−iω0t1b†(t1)..e−iω0tjb†(tj)|0i,

with tj={t1,t2, ...tj}. The functions f(j)

(τ, tj) are real and

their explicit expression has been derived in [18] and is re-

ported in the App. A. Since their amplitude is proportional

to γj/2, the components with j>2 can be neglected in

the regime usually considered in single-qubit gates where

Ωγ. We can deﬁne the probability that the qubit spon-

taneously emits jphotons during the evolution from |gito |i

as p(j)

(τ)≡Rdtj|f(j)

(τ, tj)|2/P(τ).

B. Collision model of the coherently driven 1D atom

We now deﬁne adimensional discrete-temporal modes of

the electromagnetic ﬁeld [16–18], i.e. bn≡√∆tb(tn), where

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AnomalousenergyexchangesandWignerfunctionnegativitiesinasinglequbitgateMariaMaei,1CyrilElouard,2BrunoO.Goes,1BenjaminHuard,3AndrewN.Jordan,4,5andAlexiaAuèves11UniversitéGrenobleAlpes,CNRS,GrenobleINP,InstitutNéel,38000Grenoble,France2Inria,ENSLyon,LIP,F-69342,LyonCedex07,France3Ecolenormalesupérie...

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Anomalous energy exchanges and Wigner function negativities in a single qubit gate

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