Surpassing the Standard Quantum Limit using an Optical Spring Torrey Cullen1 Ron Pagano1 Scott Aronson1 Jonathan Cripe1 Sarah Safura Sharif2 Michelle Lollie1 Henry Cain1 Paula Heu3 David Follman35 Garrett D. Cole345 Nancy

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Surpassing the Standard Quantum Limit using an Optical Spring

Torrey Cullen1,+,*, Ron Pagano1, Scott Aronson1, Jonathan Cripe1,**, Sarah Safura Sharif2,

Michelle Lollie1, Henry Cain1, Paula Heu3, David Follman3,5, Garrett D. Cole3,4,5, Nancy

Aggarwal6, and Thomas Corbitt1,*

1Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA, 70803

2School of Electrical and Computer Engineering, University of Oklahoma, Norman, OK

3Crystalline Mirror Solutions LLC and GmbH, Santa Barbara, CA, and Vienna, Austria

4Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics,

University of Vienna, A-1090 Vienna, Austria

*email: tcullen@caltech.edu; tcorbitt@phys.lsu.edu

5Present address: Thorlabs Crystalline Solutions, Santa Barbara, CA, USA

6Northwestern University Department of Physics and Astronomy, Evanston, IL

**Current Aﬃliation: Laboratory for Physical Sciences, College Park, MD 20740, USA

+Current Aﬃliation: California Institute of Technology - The Division of Physics,

Mathematics and Astronomy 1200 E California Blvd, Pasadena CA 91125

July 29, 2024

Quantum mechanics places noise limits and sensitivity restrictions on physical measure-

ments. The balance between unwanted backaction and the precision of optical measurements

imposes a standard quantum limit (SQL) on interferometric systems. In order to realize a

sensitivity below the SQL, it is necessary to leverage a back-action evading measurement tech-

nique, reduce thermal noise to below the level of back-action, and exploit cancellations of any

excess noise contributions at the detector. Many proof of principle experiments have been

performed, but only recently has an experiment achieved sensitivity below the SQL. In this

work, we extend that initial demonstration and realize sub-SQL sensitivity nearly two times

better than previous measurements, and with an architecture applicable to interferometric

gravitational wave detectors. In fact, this technique is directly applicable to Advanced LIGO,

which could observe similar eﬀects with a detuned signal recycling cavity. We measure a to-

tal sensitivity below the SQL by 2.8 dB, corresponding to a reduction in the noise power by

72±5.1 % below the quantum limit. Through the use of a detuned cavity and the optical spring

eﬀect, this noise reduction is tunable, allowing us to choose the desired range of frequencies

that fall below the SQL. This result demonstrates access to sensitivities well below the SQL at

frequencies applicable to LIGO, with the potential to extend the reach of gravitational wave

detectors further into the universe.

1 Introduction

The standard quantum limit (SQL) is a theoretical limit imposed on precision measurements by the Heisen-

berg uncertainty principle[1, 2]. While its name implies an ultimate limit on the precision measurements,

this limit can in fact be beaten through clever manipulations of the system under test. Ground based gravi-

tational wave detectors such as LIGO have achieved sensitivities that approach the SQL at frequencies near

100 Hz [3, 4]. As of the conclusion of the third observing run, the aLIGO detectors are within a factor of 2-3

arXiv:2210.12222v2 [quant-ph] 26 Jul 2024

of the SQL at 70 Hz [5, 6], with this quantity expected to improve with the implementation of frequency de-

pendent squeezing [7]. In this letter, we experimentally demonstrate a technique that allows interferometric

gravitational wave detectors to reach sensitivities below the SQL.

In an interferometric measurement such as LIGO, quantum noise exists in two parts: shot noise (impre-

cision noise) and quantum backaction (radiation pressure noise). Shot noise scales inversely with laser power

whereas radiation pressure noise (RPN) scales proportionally to laser power. The uncertainty principle for

these two quantities is given by

SimpSrpn ≥ℏ2/4 (1)

The SQL may be derived by analyzing this relationship. A spectral density for the imprecision and radiation

pressure noise can be given by [8, 9]

Simp =x2

zpf

4Γmeas

(2)

Srpn =ℏ2Γmeas

zpf

(3)

where Γmeas is a measurement rate and xzpf is the RMS of the oscillator’s zero-point ﬂuctuations [8, 9].

By analyzing the equation of motion for a damped harmonic oscillator the mechanical susceptibility of the

system is ˜

F=χ(Ω) = 1

m(Ω2

0−Ω2−iΓmΩ),(4)

where Γmis a mechanical damping rate and Ω0is the fundamental resonant frequency of the oscillator.

Equations 2, 3, and 4 lead to the total quantum noise (qn) in the system, given by

Sqn(Ω) = Simp +|χ(Ω)|2Srpn(Ω).(5)

This quantity can be minimized with respect to the measurement strength to determine the corresponding

level of minimum noise,

Γopt(Ω) = x2

zpf

2ℏ|χ(Ω)|.(6)

Plugging this into Eq 5 yields what is known as the SQL,

Sqn =SSQL =ℏ|χ(Ω)|.(7)

The calculations of the SQL used in this letter makes use of Equation 7 through code developed in Ref. [10].

This calculation shows that a lower limit exists on the total quantum noise in a system whose imprecision

noise and backaction remain uncorrelated. However, sub-SQL sensitivities can be achieved by correlating

and manipulating these two noise sources in a process referred to as quantum nondemolition (QND) [3, 11].

A variety of proof-of-principle sub-SQL QND techniques have been explored [12, 13, 14, 15, 16]. One

method involves injected squeezed light, something that was accomplished by the detector group at aLIGO.

In that work, a measurement of up to 3 dB below the SQL was realized by making use of injected squeezed

vacuum states and by subtracting unwanted classical noise from the measurement [6]. A second technique

also takes advantage of a variational readout which makes use of a second ﬁeld, often originating from the

same laser source, to modify the measurement quadrature [4, 6]. This letter makes use of a third method,

by amplifying the mirror’s motion via the optical spring eﬀect. This ampliﬁes the back action and any

signal resulting from the mirror’s motion at the optical spring frequency while simultaneously keeping shot

noise the same. A detailed calculation of why the mirror’s motion is ampliﬁed can be found in the following

section. Detailed calculations of the optical spring eﬀect can be found in Refs [17, 18, 19].

Not until recently has has it been possible to experimentally demonstrate any interferometric measure-

ment with sensitivity below the SQL [8, 20]. The Mason et al. experiment utilized a cryogenically cooled

high quality factor Si3N4membrane resonator dispersively coupled to a Fabry Perot cavity, which allowed

a sensitivity measurement up to 1.5 dB below the SQL, at megahertz frequencies. We go beyond this previ-

ously ground breaking measurement by performing the experiment in the audio band, and forgoing dispersive

coupling by instead employing a linear two-mirror cavity with a movable mirror. This operating range and

optical setup is more akin to the current design of ground based gravitational wave detectors.

Previous experiments performed by our group [21, 22] were carried out at room temperature. In order to

surpass the SQL, two main upgrades to our system must be made. First, thermal noise must be combatted.

This is realized with the addition of a cryostat cooler, bringing the cavity down to approximately 30 K.

Secondly, the frequency noise of the laser begins to limit noise levels at cryogenic temperatures. Frequency

noise is addressed with the addition of a delay line interferometer.

2 Explanation of Optical Spring Suppression

Here we dive deeper into the physics behind why the optical spring allows for a sub-SQL operation. To

understand the optical spring, consider the circulating power in a cavity given by,

PC=P0

1 + δ2,(8)

where P0is the maximum circulating power with the cavity on resonance and δis the detuning in terms

of linewidths of the cavity. In the case of a cavity in which the only loss is the transmission through one

mirror, in our case the input mirror in Figure 1, the max circulating power is given by P0=4

TPin, where

Tmirror transmission. The force due to radiation pressure on one mirror is then 2Pc

c. The optical spring

constant can be found by taking the derivative of this with respect to x,

KOS =2

dPc

dx =2

dPc

dδ

dx =−32πδPc

λcT (1 + δ2).(9)

Note that from Eq 9, a cavity detuning of δ < 0 corresponds to a positive optical spring constant and

therefore a restoring force.

All motion of the movable mirror in the cavity will be ampliﬁed at the optical spring resonance frequency,

given by ΩOS =qKOS +Km

m, where Kmis the mechanical spring constant and mis the mass of the movable

mirror. For this experiment, KOS ≫Km, and so we will approximate ΩOS ≈qKOS

m. This calculation

does not take into account the time delay in the cavity’s response that gives rise to an optical damping

force ΓOS . Additionally, this calculation assumes the mechanical damping rate is very small compared to

the optical damping force, Γm≪ΓOS . For our parameters, ΓOS ≪ΩOS , and therefore the magnitude of

the quality factor of the optical spring is large, QOS ≫1. This implies that the mirror motion, including

back action, is ampliﬁed at the optical spring resonance compared to the absence of the optical spring in a

similar system. Eﬀectively, at the resonance, the back action and any signal resulting from mirror motion

are ampliﬁed by a factor of QOS , while the shot noise remains the same. In the context of gravitational

wave detection, sensitivity is ordinarily referred to free mass displacement, or the equivalent displacement

sensitivity if the system were a free mass. This is done because the strain sensitivity may then be obtained

simply by dividing by the length of the cavity. At the resonance of the optical spring, this is accomplished

by dividing by QOS to obtain the original, unampliﬁed motion. In doing so, the shot noise is reduced by

the same factor QOS . Essentially, this results in a linear rescaling, such that in the optical spring picture,

all motion, including back action, is ampliﬁed by QOS and shot noise remains unchanged. In the free mass

picture, all motion, including back action, remains the same, but shot noise is reduced by QOS . Thus, in

evaluating the performance at ΩOS in the free mass picture, it is suﬃcient to consider only back action noise

and neglect shot noise.

To calculate the back action, we consider the radiation pressure imposed by the ﬂuctuations in the

circulating power of the cavity imposed by the shot noise of the incident light to the cavity,

xrp =1

mΩ2

2Pc

cr2hf

mΩ2

2Pc

cs8hf

PcT(1 + δ2).

(10)

To evaluate the performance, we can divide this by the free mass SQL,

xSQL =r2ℏ

mΩ2(11)

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SurpassingtheStandardQuantumLimitusinganOpticalSpringTorreyCullen1,+,*,RonPagano1,ScottAronson1,JonathanCripe1,**,SarahSafuraSharif2,MichelleLollie1,HenryCain1,PaulaHeu3,DavidFollman3,5,GarrettD.Cole3,4,5,NancyAggarwal6,andThomasCorbitt1,*1DepartmentofPhysics&Astronomy,LouisianaStateUniversity,Baton...

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Surpassing the Standard Quantum Limit using an Optical Spring Torrey Cullen1 Ron Pagano1 Scott Aronson1 Jonathan Cripe1 Sarah Safura Sharif2 Michelle Lollie1 Henry Cain1 Paula Heu3 David Follman35 Garrett D. Cole345 Nancy

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