Improving Kerr QND Measurement Sensitivity via Squeezed Light Stepan Balybin1 2Dariya Salykina1 2and Farid Ya. Khalili2 1Faculty of Physics M.V.Lomonosov Moscow State University Leninskie Gory 1 Moscow 119991 Russia

2025-05-08 0 0 418.58KB 8 页 10玖币

侵权投诉

Improving Kerr QND Measurement Sensitivity via Squeezed Light

Stepan Balybin,1, 2, ∗Dariya Salykina,1, 2 and Farid Ya. Khalili2, †

1Faculty of Physics, M.V.Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia

2Russian Quantum Center, Skolkovo IC, Bolshoy Bulvar 30, bld. 1, Moscow, 121205, Russia

In Ref. [1], the scheme of quantum non-demolition measurement of optical quanta that uses a resonantly

enhanced Kerr nonlinearity in optical microresonators was analyzed theoretically. It was shown that using the

modern high-𝑄microresonators, it is possible to achieve the sensitivity several times better than the standard

quantum limit. Here we propose and analyze in detail a signiﬁcantly improved version of that scheme. We show,

that by using a squeezed quantum state of the probe beam and the anti-squeezing (parametric ampliﬁcation) of

this beam at the output of the microresonator, it is possible to reduce the measurement imprecision by about one

order of magnitude. The resulting sensitivity allows to generate and verify multi-photon non-Gaussian quantum

states of light, making the scheme considered here interesting for the quantum information processing tasks.

I. INTRODUCTION

An ideal quantum measurement described by von Neu-

mann’s reduction postulate [2] does not perturb the measured

observable 𝑁. The suﬃcient condition for implementation of

such a measurement is the commutativity of the operator ˆ

𝑁

with the Hamiltonian ˆ

𝐻of the combined system “measured

object+meter” [3,4]:

[ˆ

𝑁, ˆ

𝐻]=0,(1)

where

𝐻=ˆ

𝐻𝑆+ˆ

𝐻𝐴+ˆ

𝐻𝐼,(2)

𝐻𝑆,ˆ

𝐻𝐴are, respectively, the Hamiltonians of the object and

the meter, and ˆ

𝐻𝐼is the interaction Hamiltonian. In the article

[5], the term “quantum non-demolition (QND) measurement”

was coined for this type of measurement.

In many cases, a sequence of measurement of a variable

𝑁(𝑡)is required, instead of a single measurement. The typical

example is detection of external force acting on the object.

In this case the value of ˆ

𝑁have to be preserved between the

measurements, which leads to another (also suﬃcient) com-

mutativity condition:

[ˆ

𝑁, ˆ

𝐻𝑆]=0,(3)

The observables which satisfy both conditions (1) and (3) are

known as QND observables.

It follows from Eqs. (1), (3) that the interaction Hamiltonian

have to commute with the measured observable:

[ˆ

𝑁, ˆ

𝐻𝐼]=0,(4)

In the particular case of the electromagnetic energy or number

of quanta measurement, this means that nonlinear interaction

of the electromagnetic ﬁeld with the meter has to be used. A

semi-gedanken example of such a measurement based on the

radiation pressure eﬀect was considered in Ref. [6].

∗sn.balybin@physics.msu.ru

†farit.khalili@gmail.com

Later a more practical scheme based on the cubic (Kerr) op-

tical non-linearity [7] was proposed. In this scheme, the signal

optical mode interacts with another (probe) one by means of

the optical Kerr nonlinearity. As a result, the phase of the

probe mode 𝜑𝑝is changed depending on the photon number

𝑁𝑠in the signal one (the cross phase modulation, XPM). The

subsequent interferometric measurement of this phase allows

to retrieve the value of 𝑁𝑠with the precision depending on

the initial uncertainty of 𝜑𝑝, see details in Sec. IV of Ref. [1].

In the ideal lossless case, the photon numbers in both modes

are preserved. However, due to the XPM eﬀect, the phase of

the signal mode is perturbed proportionally to the probe mode

energy uncertainty, fulﬁlling thus the Heisenberg uncertainty

relation.

The natural sensitivity threshold for the QND measurement

of the number of quanta is the standard quantum limit (SQL)

Δ𝑁SQL =√𝑁 , (5)

where 𝑁is the mean number of the measured quanta. It

corresponds to the best possible sensitivity of a linear non-

absorbing meter [4], for example a phase-preserving linear

ampliﬁer [8]. Starting from the initial work [9], many proof-

of-principle experiments based on the XPM idea were done,

see the review articles [10–12]. The sensitivity exceeding the

SQL was demonstrated in these experiments, but the ultimate

goal of the single-photon accuracy was not reached due to

the insuﬃcient values of the optical nonlinearity in highly

transparent optical media.

The recent advantages in fabrication of high-𝑄monolithic

and integrated microresonators [13], which combine very low

optical losses with high concentration of the optical energy

promises the way to overcome this problem. This possibil-

ity was analyzed in detail in Ref. [1]. It was shown that the

sensitivity of the Kerr nonlinearity based QND measurement

schemes is limited by the interplay of two undesirable eﬀects:

the optical losses and the self phase modulation (SPM) of the

probe mode, which perturbs the probe mode phase proportion-

ally to the energy uncertainty of this mode.

It was shown in Ref. [14], that in the ideal lossless case,

the SPM eﬀect can be compensated using the measurement of

the optimal quadrature of the output probe ﬁeld instead of the

phase one. However, in presence of the optical losses, only

partial compensation is possible, limiting the sensitivity by the

arXiv:2210.00857v2 [quant-ph] 14 Aug 2023

value

(Δ𝑁𝑠)2=1

Γ2

𝑋1

4𝜂𝑁𝑝+ (1−𝜂)𝑁𝑝Γ2

𝑆,(6)

see Eq. (32) of [1]. Here Δ𝑁𝑠is the measurement error, 𝑁𝑝is

the photon number in the probe mode, Γ𝑋,Γ𝑆are the dimen-

sionless factors of, respectively, the cross and the self phase

modulation, see Eqs. (25) and (40), and 𝜂is the quantum ef-

ﬁciency of the measurement channel. The second term here

stems from the SPM. Due to this term, the optimal value of

𝑁𝑝exists, which provides the minimum of Δ𝑁𝑠:

(Δ𝑁𝑠)2=Γ𝑆

Γ2

𝑋

𝜖 , (7)

where

𝜖=1−𝜂

𝜂(8)

is the normalized loss factor. It was shown in Ref. [1], that

using the best microresonators available now, the sensitivity

Δ𝑁𝑠∼102-103could be achieved.

Eqs. (6), (7) imply that the probe mode is prepared in the

coherent quantum state. At the same time, in was shown by

C.Caves in the work [15], that the sensitivity of optical inter-

ferometric measurements can be improved without increasing

the optical power by using the non-classical squeezed states

of light. Currently, this method is routinely used in the laser

gravitational-wave detectors [16,17]; see also the review [18].

In the same work [15], C. Caves proposed also to use an addi-

tional degenerate optical parametric ampliﬁer (anti-squeezer)

at the output of the interferometer to reduce impact of the

losses in the optical elements located after this ampliﬁer, in-

cluding the photodetector(s) ineﬃciency. Note that usually, it

is the output losses constitute the major part of the total losses

budget in the optical interferometers. Recently, this method

was demonstrated experimentally [19].

In the current work we show that sensitivity of the QND

measurement scheme, considered in Ref. [1], can be radically

improved using these techniques. This paper is organized

as follows. In Sec. II, we derive the linearized input/output

relations for the microresonator. In Sec.III, we calculate the

measurement error, taking into account the SPM eﬀect and

the losses in the probe mode. In Sec. IV, we estimate the

sensitivity, which can be achieved using the best available

microresonators. In Sec. Vwe consider the eﬀect of optical

losses in the signal mode both in the measurement and in the

quantum state preparation scenarios. In Sec. VI, we discuss

possible applications of the considered scheme to quantum

information processing. We summarize the results of this

paper in Sec. VII.

FIG. 1. The scheme of the QND measurement of the photon number

in signal mode via XPM eﬀect. DOPA: degenerate optical parametric

ampliﬁer, HD: homodyne detector.

II. EVOLUTION OF THE OPTICAL FIELDS IN THE

MICRORESONATOR

The measurement scheme is shown in Fig. 1. Here the

strong coherent beam is split by the unbalanced beamsplitter

with the transmissivity 𝑇≪1 into the probe and the reference

beams. The probe beam is squeezed by the degenerate op-

tical parametric ampliﬁer DOPA 1 and then injected into the

microresonator. There it interacts with the signal beam and

then passes through the second optical parametric ampliﬁer

DOPA 2 and ﬁnally is recombined with the reference beam in

the homodyne detector.

Taking into account that the main goal of this paper is to

show the advantages provided by the squeezing, and in or-

der to simplify the calculations, we, similar to the paper [1],

do not take into account explicitly the optical losses in the

microresonator, but assume instead that

𝑄load

𝑄intr ≈𝜏

𝜏∗≪𝜖2,(9)

where 𝑄load is the loaded quality factor, 𝑄intr is the intinsic

one, 𝜏is the interaction time, and 𝜏∗is the relaxation time of

the microresonator.

In this case, the interaction of the signal and probe modes

in the non-linear microresonator can be described by the fol-

lowing Hamiltonian [1]:

𝐻=

𝑥=𝑝,𝑠 ℏ𝜔𝑥ˆ

𝑁𝑥−ℏ𝛾𝑆

2ˆ

𝑁𝑥(ˆ

𝑁𝑥−1)−ℏ𝛾𝑋ˆ

𝑁𝑝ˆ

𝑁𝑠,(10)

where

𝑁𝑝,𝑠 =ˆ𝑎†

𝑝,𝑠 ˆ𝑎𝑝,𝑠 (11)

are the photon number operators in the probe and signal modes,

respectively, ˆ𝑎𝑝,𝑠, ˆ𝑎†

𝑝,𝑠 are the corresponding annihilation and

creation operators for these modes, 𝜔𝑠, 𝑝 are their frequen-

cies, 𝛾𝑆,𝛾𝑋— coeﬃcients of SPM and XPM interactions,

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ImprovingKerrQNDMeasurementSensitivityviaSqueezedLightStepanBalybin,1,2,∗DariyaSalykina,1,2andFaridYa.Khalili2,†1FacultyofPhysics,M.V.LomonosovMoscowStateUniversity,LeninskieGory1,Moscow119991,Russia2RussianQuantumCenter,SkolkovoIC,BolshoyBulvar30,bld.1,Moscow,121205,RussiaInRef.[1],theschemeofquant...

展开>> 收起<<

Improving Kerr QND Measurement Sensitivity via Squeezed Light Stepan Balybin1 2Dariya Salykina1 2and Farid Ya. Khalili2 1Faculty of Physics M.V.Lomonosov Moscow State University Leninskie Gory 1 Moscow 119991 Russia.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Improving Kerr QND Measurement Sensitivity via Squeezed Light Stepan Balybin1 2Dariya Salykina1 2and Farid Ya. Khalili2 1Faculty of Physics M.V.Lomonosov Moscow State University Leninskie Gory 1 Moscow 119991 Russia

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: