Structured light analogy of squeezed state Zhaoyang Wang1 2Ziyu Zhan1 2Anton N. Vetlugin3Qiang Liu1 2Yijie Shen4and Xing Fu1 2y 1Key Laboratory of Photonic Control Technology Tsinghua University Ministry of Education Beijing 100084 China

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Structured light analogy of squeezed state

Zhaoyang Wang,1, 2 Ziyu Zhan,1, 2 Anton N. Vetlugin,3Qiang Liu,1, 2 Yijie Shen,4, ∗and Xing Fu1, 2, †

1Key Laboratory of Photonic Control Technology (Tsinghua University), Ministry of Education, Beijing 100084, China

2State Key Laboratory of Precision Measurement Technology and Instruments,

Department of Precision Instrument, Tsinghua University, Beijing 100084, China

3Centre for Disruptive Photonic Technologies, SPMS,

TPI, Nanyang Technological University, 637371 Singapore

4Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom

(Dated: October 6, 2022)

Control of structured light is of great importance to explore fundamental physical

eﬀects and extend practical scientiﬁc applications, which has been advanced by ac-

cepting methods of quantum optics - many classical analogies of exotic quantum states

were designed using structured modes. However, the prevailing quantum-like struc-

tured modes are limited by discrete states where the mode index is analog to the

photon number state. Yet, beyond discrete states, there is a broad range of quan-

tum states to be explored in the ﬁeld of structured light – continuous-variable (CV)

states. As a typical example of CV states, squeezed state plays a prominent role

in high-sensitivity interferometry and gravitational wave detection. In this work, we

bring together two seemingly disparate branches of physics, namely, classical struc-

tured light and quantum squeezed state. We propose the structured light analogy of

squeezed state (SLASS), which can break the spatial limit following the process of

surpassing the standard quantum limit (SQL) with quantum squeezed states. This

work paves the way for adopting methods from CV quantum states into structured

light, opening new research directions of CV entanglement, teleportation, classical and

quantum informatics of structured light in the future.

Structured light has a great importance for develop-

ing both quantum and classical optical technologies,

bridging the gap between quantum and classical op-

tics and opening a platform for exchange of concepts

and methodologies [1]. Many quantum eﬀects could

be reproduced by designing spatial modes of structured

light [2–5]. For instance, the vortex beams carrying

orbital angular momentum were exploited to simulate

Landau levels and Laughlin states that unveil fermionic

topological order in bosonic matters [6–8]. By manip-

ulating a complex multi-vortex light ﬁeld, the Berezin-

skii–Kosterlitz–Thouless phase transition was observed,

opening the research of analogous thermodynamics in

photonic light ﬂuid [9]. The vector beam with spa-

tially unseparable polarization can resemble the quantum

entangled Bell states, paving the way for exploitation

of quantum measures for characterization of structured

light [10, 11] and enabling quantum-inspired applica-

tions such as local teleportation [12], turbulence-resilient

communication and encryption [13, 14]. Recently, vec-

tor vortex beams with multiple controlled unseparable

degrees of freedom were designed for higher-dimensional

entangled states [5, 15, 16], promising advanced applica-

tions from ultraprecise metrology to secure communica-

tions [17, 18].

However, the prevailing methodology mostly focused

on the classical-quantum parallelism of structured light

and discrete-variable (DV) quantum states, i.e. discrete

eigenmodes of structured light (spatial mode indices, po-

larization) and discrete quantum number (energy level,

photon spin). Further exploration is required to develop

classical-qauntum analogy between structured light and

continuous-variable (CV) quantum states, which are the

states of variables with continuous spectrum such as posi-

tion, momentum and amplitude of electromagnetic ﬁeld.

The best-known CV state is the squeezed state, where

the uncertainty of one of the variables is suppressed com-

pared to the coherent state [19]. This allows to surpass

the standard quantum limit (SQL) which is critical for

high-sensitive metrology.

In this paper, we propose a generalized mathemat-

ical model for structured light analogy of squeezed

state (SLASS), accompanied by paraxial wave equa-

tion (PWE). Typical forms of squeezed states, includ-

ing squeezed vacuum states and squeezed number states,

are mimicked by a new family of spatial structured

modes. Analogues to quantum tomography techniques,

we also assemble the experimental setup for generat-

ing and detecting the SLASS. Moreover, inspired by

the superiority of squeezed state in surpassing the SQL,

we demonstrate the SLASS overcoming spatial diﬀrac-

tion limits with deep subwavelength and superoscillation

structures. The classical-quantum analogy, presented in

this paper, allows bringing the concepts and methodolo-

gies of CV quantum optics into classical regime and de-

velop novel approaches for such applications as super-

resolution imaging and microscopy with classical light.

At the same time, a new family of structured light can

be utilized in quantum technologies such as structured

photon manipulation, ultra-sensitive measurement, and

arXiv:2210.02105v1 [physics.optics] 5 Oct 2022

quantum kernel computing [20].

Results

Concepts. In quantum optics, eigenstates of two

modes with indices j= 1,2 are the solution

of the stationary Schr¨odinger equation, ˆ

H|n, mi=

En,m |n, mi, with the two-mode Hamiltonian operator

H=1

j=1,2ˆp2

j+ ˆq2

j[21]. Here ˆqj=ˆaj+ ˆa†

j/√2

and ˆpj=ˆaj−ˆa†

j/√2iare canonical variables sat-

isfying commutation relation [ˆqj,ˆpj] = i, ˆa†and ˆaare

the creation and annihilation operators, En,m is the en-

ergy of the state |n, miwhich is the two-mode num-

ber state with nphotons in mode 1 and mphotons in

mode 2 (nand mare integers). The set of all states

|n, miform a full orthonormal basis, allowing to ex-

press any pure two-mode quantum state as a superpo-

sition of number states. For instance, the two-mode

squeezed vacuum state (SVS) can be represented as an

inﬁnite sum, P

cn|n, ni, where cnis the amplitude (Sup-

plementary Material). Alternatively, this states can be

represented as a squeezing operator acting on the two-

mode vacuum state: ˆ

S(τ)|0,0iwhere τis the squeez-

ing parameter deﬁning the magnitude and orientation

of squeezing (Supplementary Material). Similarly, the

displaced squeezed vacuum state (DSVS), the squeezed

number state (SNS) and displaced squeezed number state

(DSNS) can be expressed as ˆ

D(α)ˆ

S(τ)|0,0i,ˆ

S(τ)|0, Ni

and ˆ

D(α)ˆ

S(τ)|0, Ni, respectively, where Nis an inte-

ger, ˆ

D(α) is the displacement operator with parameter

αdeﬁning the magnitude and direction of displacement

(Methods).

In classical optics, the Maxwell’s equations under

paraxial approximation are reduced to the transverse

eigenmode equation ˜

HSLun,m(ξ, η) = Cn,mun,m(ξ, η),

where Cn,m is the eigenvalue and un,m(ξ, η) is the

transverse mode function [22, 23]. The analytic form

of un,m(ξ, η) in the Cartesian coordinate is Hermite-

Gaussian (HG) mode. Here, ˜

HSL =1

j=x,y ˜p2

j+ ˜q2

j,

j=x, y represents two orthogonal directions in two-

dimensional space, ˜pj=−i∂

∂j , ˜qj=jsatisfying the Pois-

son bracket relation, {˜qj,˜pj}= 1. By denoting un,m(ξ, η)

as |un,mi, we get ˜

HSL|un,mi=Cn,m|un,miwhich has the

form of the stationary Schr¨odinger equation above. The

“number state of structured light” |un,miconstitutes an

inﬁnite-dimensional space similar to the Hilbert space of

quantum number states. In addition, it is worth noting

that during structured light propagation, longitudinal co-

ordinate zcould also be seen as analogy of time variable

tof quantum states (details in Supplementary Material).

Based on the similar mathematical framework, we con-

struct the structured light analogy of CV quantum states.

We use decomposition of CV quantum state in the basis

of the number states, |n, mi, and reproduce this superpo-

sition with the transverse eigenmode |un,mi, where the

amplitude of the state |n, miis used as the amplitude

of the mode |un,mi. Similar to CV quantum states dis-

cussed above, we reproduce analogies of the SVS, DSVS,

SNS and DSNS states - SLASS, using structured light.

Following the methodology of quantum optics, these

states can be represented via action of the correspond-

ing operators to the transverse eigenmode: ˜

S(τ)|u0,0i,

D(α)˜

S(τ)|u0,0i,˜

S(τ)|u0,N iand ˜

D(α)˜

S(τ)|u0,N i. Here

the operators ˜

S(τ) and ˜

D(α) have the same form as

their quantum counterparts, but applied to the states

|un,mi, where these analogous operators in structured

light regime satisfy the same relation as their quantum

counterparts (see Methods).

Structured light analogy of squeezed state. Quan-

tum two-mode vacuum state |0,0iposses an isotropic

uncertainty in the quadrature amplitudes, as shown in

Fig. 1 a1. Here we use the wave function to character-

ize the distribution of quadrature amplitudes of quan-

tum state for a certain mode [24]: the left subplot shows

the wave function in phase space, and the right subplot

shows the wave function in time evolution (varying θ)

on an arbitrary quadrature. The uncertainty of number

state |0, Niis shown in Fig. 1 d1. The wave function

of SVS and SNS are shown in Fig. 1 b1 and e1, where

the uncertainty is squeezed in the vertical direction but

enlarged in the horizontal direction (left subplots). The

uncertainty in the vertical direction also oscillates with

varying θat a period of π(right subplots). The wave

functions of DSVS and DSNS are overall displaced by

the displaced operator, shown in Fig. 1 c1 and f1.

To develop the analogy of the CV quantum states in

the structured light regime, we map the two-mode num-

ber state |n, mito the HGn,m mode (|un,mi). The two-

mode quantum vacuum state is, thus, mapped to the

fundamental HGn,m mode with (n, m) = (0,0), shown in

Fig. 1 a2. The transverse proﬁles of SLASS at various

planes are shown in Fig. 1 b2,c2,e2 and f2, respectively.

The beam waist of HG mode (marked with white dotted

circles) can be seen as the counterpart for the SQL, and

we name it as SSL. Similarly, the beam waist of SLASS

can be seen as the counterpart for the quantum limit of

quantum squeezed states, and we name it as the spatial

limit. The spatial limits of SLASS are ellipses (marked

with blue dotted circles), exhibiting anisotropic in diﬀer-

ent quadrature directions, analogous to the behavior of

squeezed state in surpassing SQL.

Moreover, the longitudinal evolution of SLASS is il-

lustrated in Fig. 1 a3-c3 and d3-f3, where the squeezed

quadrature direction of structured light varies in prop-

agation direction z. This behavior qualitatively repro-

duces that of the quantum squeezed state, where the di-

rection of squeezing varies (oscillates) in time. Due to

the diﬀraction of light, though, the distribution becomes

broader for structured light as it propagates, which is not

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StructuredlightanalogyofsqueezedstateZhaoyangWang,1,2ZiyuZhan,1,2AntonN.Vetlugin,3QiangLiu,1,2YijieShen,4,andXingFu1,2,y1KeyLaboratoryofPhotonicControlTechnology(TsinghuaUniversity),MinistryofEducation,Beijing100084,China2StateKeyLaboratoryofPrecisionMeasurementTechnologyandInstruments,Departmentof...

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Structured light analogy of squeezed state Zhaoyang Wang1 2Ziyu Zhan1 2Anton N. Vetlugin3Qiang Liu1 2Yijie Shen4and Xing Fu1 2y 1Key Laboratory of Photonic Control Technology Tsinghua University Ministry of Education Beijing 100084 China.pdf

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Structured light analogy of squeezed state Zhaoyang Wang1 2Ziyu Zhan1 2Anton N. Vetlugin3Qiang Liu1 2Yijie Shen4and Xing Fu1 2y 1Key Laboratory of Photonic Control Technology Tsinghua University Ministry of Education Beijing 100084 China

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