Ponderomotive Squeezing of Light

AuthorJacob L. Beckey

SupervisorDr. Haixing Miao

August 1, 2017

Contents

1 Introduction 2

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Purpose and Structure of the Report . . . . . . . . . . . . . . . . . . . . . . . 3

2 Analysis of Ponderomotive Squeezing in MichelsonInterferometer 4

2.1 Derivation of Input-Output Relation . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Optical Spring and Mechanical Transfer Function . . . . . . . . . . . . . . . . 6

2.3 Matrix Form of Input-output Relation . . . . . . . . . . . . . . . . . . . . . . 8

3 Ponderomotive Squeezing with One Mirror 9

3.1 Relating Power Fluctuation to Mirror Motion . . . . . . . . . . . . . . . . . . 9

3.2 Calculating Squeezing Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Calculating Squeezing Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Calculating Cutoff Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Implementing Ponderomotive Squeezing in FINESSE 14

4.1 Converting Quadratures to Sidebands . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Current Status and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 15

A Detailed Derivations 16

A.1 Michelson Interferometer Input-output Relation . . . . . . . . . . . . . . . . . 16

A.2 Laser Power Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A.3 Squeezing Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A.4 Cutoff Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1

Chapter 1

Introduction

1.1 Background

Figure 1.1: Gaussianrandom fluctuations inamplitude and phase dueto vacuum noise. Semi-classically, the quantumnoise can be representedby the complex numberq(ω) = nA(ω) + inϕ(ω)

Generally, this project aimed to explore how quantum mechanicscan be leveraged to increase the sensitivity of gravitational wavedetectors. More specifically, we investigated a technique calledponderomotive squeezing. This is a method of generating squeezedstates of light using moving mirrors. At the fundamental level, weare talking about the quantum uncertainty that results from thefact that photons are discrete quantum particles. Similar to anelectron cloud, which represents the “quantum fuzziness” associ-ated with an electron’s position, we also have this fuzziness whenwe study light. In our case, it is uncertainty in amplitude andphase of an optical field.

It is a direct result of the Heisenberg Uncertainty Prin-ciple and we model this phenomenon by imagining quantum fluc-tuations superimposing themselves onto the carrier optical field,e.g. the laser in an interferometer. These fluctuations limit thesensitivity of gravitational wave detectors, so it is of interest tounderstand the phenomenon so we may increase the sensitivity ofour instruments.

The name ‘squeezed light’ comes from the above repre-sentation of the quantum noise that couples with the optical field. In the vacuum state,it is spherical and Gaussian. The goal is the squeeze that ball of quantum noise into anellipse. Physically, this represents decreasing the uncertainty in amplitude and increasing itin phase, or vice versa. Luckily, it is a phase measurement that we are interested in making,so increasing the uncertainty in amplitude is of no consequence.

2

1.2 Purpose and Structure of the Report

My goal in writing this report was to create a document that both summarizes my researchand also serves as a transition between reading the Living Review [3] that is cited throughout,and reading a quantum mechanical treatment of the same phenomena [1], [2]. Because I havenot had a rigorous quantum mechanics course, I absolutely did not understand the quantumapproach to modelling these phenomena at first. Thus, I spent the first half of my timeattempting to understand the classical and semi-classical approaches so that I could thentackle the less intuitive quantum description. When I was finally in the position to read thequantum papers, I was amazed by how elegant and concise the more rigorous approach is.Thus, I want this report to serve the following function: after having read at the very leastchapters 1, 2, and 6 of the Living Review [3], someone should look at Haixing’s book chapter[1] and have a general idea of what was being done, and then this report will facilitate I nicetransition from the sideband picture to the quadrature picture. In other words, this reportserves as a transition from describing things classically, to describing them more elegantlyand properly using quantum operators.

The report begins with the analysis of a Michelson interferometer. From there, weexamine a simpler one-mirror case; however, we find (spoiler alert) that they are identicalsystems when talking about ponderomotive squeezing. Once these systems are describedanalytically, we compare these expressions with the FINESSE software. Seeing agreementbetween thoroughly tested software and carefully derived expressions gives us confidencethat we understand ponderomotive squeezing theoretically. Then, the final goal of theproject, is to create a ponderomotive squeezing element in FINESSE which allows the userto specify relevant parameters and easily model the effect in their system of interest. Foranyone interested in understanding how some of the expressions were derived in Haixing’sbook chapter [1], I have done several very careful calculations in the appendix.

3

Chapter 2

Analysis of PonderomotiveSqueezing in MichelsonInterferometer

2.1 Derivation of Input-Output Relation

The goal of this section is to analytically describe what happens to an optical field after ithas interacted with the components of a Michelson interferometer. It is helpful to first readchapter two of the Bond et al [3]. There, a semi-classical approach is used to model lightwith upper and lower quantum sidebands. This description of light interacting with variousoptical systems was easier to start with, and provided the theoretical background needed tounderstand the less untuitive, but much more elegant quantum description.

Figure 2.1: Schematic plot of a simple Michelson interferometer locked on dark fringe (left) and itsmathematical model with propagating optical fields (right).

The system we are interested in studying, shown above, is a simple Michelsoninterferometer with equal arm lengths and mirrors of identical mass. The assumption that

4

the arm lengths are identical implies that the optical field entering each port returns onlyto that port and we say that the interferometer is locked on dark fringe. We are interestedin the quantum noise that enters the dark port of the interferometer. We still consider thecarrier entering the common port because it is the laser power, or the fluctuation in power,that supplies the varying radiation pressure force that leads to the ponderomotive squeezing.The goal of this derivation is to be able to describe the optical field outputted at the darkport of the interferometer. The naming convention used for the optical fields is shown onthe right side of figure 2.1.

The field entering the common port (or bright port) is a superposition of quantumfluctuations in amplitude, c1, and phase, c2, upon a carrier field produced by the laser. Thisoptical field takes the form

Einc (t) = [

√2I0hω0

+ c1(t)] cos(ω0t) + c2(t) sin(ω0t) (2.1)

The field entering the differential port (dark port) is purely comprised of vacuumfluctuations and thus takes the form

Eind (t) = a1(t) cos(ω0t) + a2(t) sin(ω0t) (2.2)

We restrict our discussion to the case of a 50:50 beam splitter, in which half ofthe field that hits the splitter is transmitted, and the other half is reflected. The physicaldescription of what happens to an optical field at a surface can be obtained by solvingMaxwell’s equations at a boundary. These solutions are called Fresnel’s Equations but theyare outside the necessary scope of this brief report, so only the pertinent result will be stated.Namely, that a field gains a phase shift of π radians upon transmission through an opticalsurface, in this case the beam splitter. Using the conventions outlined in the diagram, thisphase shift manifests itself as a negative sign throughout the derivation.

Because we are dealing with a half-half beam splitter, we have reflection and trans-mission coefficients given as R = T = 1

2 . Thus, the fields propagating towards the end testmasses gain a factor of 1√

2and are given as

EinA (t) = [Einc (t)− Eind (t)]/√

2 (2.3)

EinB (t) = [Einc (t) + Eind (t)]/√

2 (2.4)

The fields interact with the mirror and the fluctuations in amplitude cause a tinydisplacement of the mirror which modulates the phase of the field. The resultant fieldsheading back to towards the beam splitter are written very compactly as

EoutA,B(t) = EinA,B(t− 2τ − 2xA,B/c) (2.5)

This equation is essentially saying that the field that comes out of the cavity isjust like the field that entered except for a phase change due to free propagation throughthe macroscopic length of the cavity (indicated by the 2τ term, where τ ≡ L

c ) and the phasechange due to the microscopic mirror displacement, x. The factor of 2 in front of the mirror

5

displacement term arises from the fact that the field goes out and comes back through thattiny displacement.

Finally, the output at the dark port will simply be a superposition of the re-flected/transmitted fields returning from their journeys through the interferometer arms.The outputted field of interest is

Eoutd (t) = [EoutB (t)− EoutA (t)]/√

2 (2.6)

Then, to relate the input field amplitudes to the outputted ones, we make thefollowing definition.

Eoutd (t) = [EoutB (t)− EoutA (t)]/√

2 ≡ b1(t)cos(ω0t) + b2(t)sin(ω0t) (2.7)

This definition just says we are going to eventually equate the coefficients on thecosine and sine terms with output coefficients b1(t) and b2(t). This gives us the first concretedefinition of the so-called ‘input-output relation.’ Essentially, we must carry out a lengthysubstitution and simplification process that will result in some terms in front of a cosineterm and some in front of a sine term, and the equating of these terms gives us the desiredinput-output relation. The full derivation can be found in the appendix, but the result isgiven as

b1(t) = a1(t− 2τ) (2.8)

b2(t) = a2(t− 2τ) +

√2I0hω0

ω0

cxd(t− τ) (2.9)

This relation tells us that when quantum fluctuations enter the dark port of aninterferometer, they superimpose themselves onto the carrier field and cause a power fluc-tuation which causes the end test masses to move. This mirror motion causes a phasemodulation in the outputted quantum noise. So, fluctuations in amplitude of the opticalfield induce a modulation of the phase fluctuation. In this way, the noise sources becomecorrelated when they interact with the interferometer.

2.2 Optical Spring and Mechanical Transfer Function

We now consider the radiation-pressure force acting on the two test masses. The expressionfor this interaction is taken from Haixing’s book chapter, but should be derived in depth atsome point. To the first-order in fluctuations/modulations, we have

FA(t) =2I0c

(1 +

√hω0

2I0

(c1(t− τ)− a1(t− τ)

))(2.10)

FB(t) =2I0c

(1 +

√hω0

2I0

(c1(t− τ) + a1(t− τ)

))(2.11)

6

Taking the difference between these two equations gives the differential-moderadiation-pressure force FB(t) − FA(t). This manipulation is much more straightforward.You simply expand (1.10) and (1.11) and then carry out the subtraction, most terms cancel,and you are left with

FB(t)− FA(t) =

√8I0hω0

c2a1(t− τ) (2.12)

Using Newton’s second law, we can relate the differential-mode radiation-pressureforce to the acceleration of the end mirrors with equal masses of m. The general idea ofmodeling a cavity as an optical spring is to (insert general idea when you have actually readabout it). The new, more complex differential equation is a very familiar one. It is the ODEthat describes a damped, driven harmonic oscillator, and it is given by

m¨xd(t) =

√8I0hω0

c2a1(t− τ)− kxd(t)− b ˙xd(t) (2.13)

Rearranging to isolate the force, we will have the standard form for the damped, drivenharmonic oscillator. We will then proceed by taking the Laplace transform of both sides ofthe following equation. This will convert our equations of time into equations of complexfrequency, s, where s ≡ σ + iΩ.√

8I0hω0

c2a1(t− τ) = m¨xd(t) + b ˙xd(t) + kxd(t) (2.14)

√8I0hω0

c2La1(t− τ) = mL

¨xd(t)

+ bL

˙xd(t)

+ kLxd(t)

√8I0hω0

c2e−iΩτ a1(Ω) = m

(s2xd(s)− sxd(0)− ˙xd(0)

)+ b(sxd(s)− ˙xd(0)) + kxd(s)

Assuming the initial state of the optical spring to be one that has xd(0) = 0 and ˙xd(0) = 0,and noting that we are only considering solutions in which σ = 0 in our complex frequencys = σ + iΩ, we will get

√8I0hω0

c2e−iΩτ a1(Ω) = xd(s)(ms

2 + bs+ k)

= xd(Ω)(−mΩ2 + biΩ + k)

So, our solution is of the form

xd(Ω) = e−iΩτ√

8I0hω0

c21

−mΩ2 + biΩ + ka1(Ω) (2.15)

7

It is only a small step from here to easily see what form our transfer function takes. Thedefinition of a mechanical transfer function, H(Ω), applied to our specific case, is

H(Ω) =xd(Ω)

Fd(Ω)

Thus, from eq. (1.15), it is fairly easy to see that if we divide the differential force termsover, we will be left with only the fraction on the right-hand-side of the equation. This isour mechanical transfer function, and it is written as

H(Ω) =1

−mΩ2 + biΩ + k(2.16)

This mechanical transfer function tells us how much something, in this case thetest masses, will be displaced when a certain force is exerted upon them. Now that we havethis, we can synthesize our input-output relation into a nice, compact form.

2.3 Matrix Form of Input-output Relation

Our goal is now to synthesize equations (2.8), (2.9), and (2.15) into a compact input-outputmatrix. Equations (2.8) and (2.9) must be transformed into the frequency domain first.Doing this using a Laplace transform yields

b1(Ω) = e−2iΩτ a1(Ω) (2.17)

b2(Ω) = e−2iΩτ a2(Ω) + e−iΩτ√

2I0hω0

ω0

cxd(Ω) (2.18)

Each term picks up a phase factor due to the time delay involved. The general resultused can be found in any Laplace transform table and is given as Lf(t− τ) = e−sτF (s),where in our case s = iΩ Now, we are ready to substitute our expression for xd(Ω), equation(2.15), into equation (2.18). This yields

b2(Ω) = e−2iΩτ a2(Ω) + e−iΩτ√

2I0hω0

ω0

c

(e−iΩτ

√8I0hω0

c21

−mΩ2 + biΩ + ka1(Ω)

)

b2(Ω) = e−2iΩτ a2(Ω) + e−2iΩτ 4I0ω0

c21

−mΩ2 + biΩ + ka1(Ω)

Let’s define the opto-mechanical coupling constant, κ ≡ 4I0ω0

c21

−mΩ2+biΩ+k . Now we canexpress our input-output relation in compact matrix form

[b1(Ω)

b2(Ω)

]= e−2iΩτ

[1 0κ 1

] [a1(Ω)a2(Ω)

](2.19)

We have now fully described the quantum noise entering and returning to the darkport of a Michelson interferometer locked on dark fringe.

8

Chapter 3

Ponderomotive Squeezing withOne Mirror

We now consider a simpler case of ponderomotive squeezing in which we have a laser hitting asingle suspended test mass. This is of interest because it should give us a solid understandingof how FINESSE handles these situations and will inform us as to what parameters mustbe specified in order to implement a ponderomotive squeezing element in FINESSE. In casesomeone would like to check the derivation of the input-output relation, it will be placed inthe appendix. It is almost identical to the more complex Michelson case; however, there area few minor differences one can easily see by comparing the two results.

3.1 Relating Power Fluctuation to Mirror Motion

The input-output relation, before transforming into the frequency domain, for the one-mirrorcase is given as

b1(t) = a1(t− 2τ) +

√2I0hω0

(3.1)

b2(t) = a2(t− 2τ) + 2

√2I0hω0

ω0

cxd(t− τ) (3.2)

We now have described the output field in terms of the input field and an, as ofyet, unknown mirror modulation, xd. In the previous chapter, we assumed a result givenin Haixing’s book chapter for the radiation pressure force on a mirror. Here, we will derivethat result more carefully.

P = hω0

∣∣∣Ein(t− τ)∣∣∣2 (3.3)

9

When this expression is expanded, there are multiple terms in which fluctuationsare multiplied together. These values will be incredibly small and are therefore dropped. Inother words, the expressions that follow are kept to first order approximations. The detailsof this derivation can be found in appendix A.2. The important thing to note is that whenthe squared trigonometric functions are reduced, they become dependent on 2ω0 which isan extremely high frequency. Thus, these terms cannot be resolved by the photodetectorand are therefore dropped. So, our final expression is

P = I0 +√

2hω0I0a1(t− τ) (3.4)

It is very clear that the power is made up of a constant, I0, and a fluctuation,the second term. This fluctuation is what drives the mirror that modulates the quantumnoise we are interested in studying. We can now very easily determine the fluctuation inradiation-pressure force on the mirror due to the power fluctuation. This is given generallyas δF = 2δP

c , or in our case

δF =

√8hω0I0c2

a1(t− τ) (3.5)

The deltas used here are there to emphasize the fact that this force is different from the totalradiation-pressure force on the mirror. We are only interested in the fluctuation in forcedue to the fluctuation in power. Now, we will set up the differential equation describing oursystem and solve it using a Laplace transform.

√8I0hω0

c2a1(t− τ) = m ¨xd(t) + b ˙xd(t) + kxd(t)

The Laplace transform yields a complex-valued function of the real variable Ω given as

xd(Ω) = e−iΩτ√

8I0hω0

c21

−mΩ2 + biΩ + ka1(Ω) (3.6)

We could stop here, because this is a perfectly valid analytic solution; however, we havethe eventual goal of comparing our solutions with FINESSE, so we are going to make thefollowing substitutions k = ω2

0m and b = ω0mQ . This yields the final form of our solution

which is given as

xd(Ω) = e−iΩτ√

8I0hω0

c2a1(Ω)

1

m

1

(ω2 − Ω2) + iΩω0

Q

(3.7)

Following identical steps to those in section 2.3, we can synthesize our three frequencydomain equations into a compact matrix representation. This is given as

[b1(Ω)

b2(Ω)

]= e−2iΩτ

[1 0κ 1

] [a1(Ω)a2(Ω)

](3.8)

Where we have a new coupling constant, κ, given as κ = 8I0ω0

c21m

1

(ω2−Ω2)+iΩω0Q

.

10

3.2 Calculating Squeezing Amplitude

Figure 3.1: Homodynereadout scheme with in-jected local oscillator.

We now want to compare it with FINESSE to make sure the well-tested detectors agree with our derivations. To do this, however,we must calculate the power spectral density of the optical field ofinterest. The power spectral density tells us how much of a signalis present at a given frequency. Again, the detailed derivations willbe in the appendix and the crucial expressions will be given here.

In reality, we must use a homodyne detection scheme inorder to measure the quadratures of the optical field (figure 3.1).However, to start, we are just going to compare our analytic resultwith the nonphysical qd-detector in FINESSE, which magicallymeasures the amplitude of a quantum quadrature. We begin withthe equation of the electric field which now contains a local oscilla-tor with power IHD and homodyne readout angle θ. Note that theelectric field is a function of time; however, this information playsno role in the derivation so the notation is simplified.

E =

(√IHD cos θ + b1

)cosω0t+

(√IHDsinθ + b2

)sinω0t (3.9)

To calculate power rigorously, we need to compute

P =1

2(EE† + E†E) (3.10)

The details of this calculation are in the appendix. As throughout the report, the higherorder quantum fluctuation terms are dropped, as are quickly oscillating terms. This resultsin the fairly simply expression

P =

√IHD2

((b†1 + b1) cos θ + (b†2 + b2) sin θ

)(3.11)

Now we are ready to calculate the power spectral density of the modulated quantum noise.Assuming the power operator commutes, we just have to calculate

Sm = P P † (3.12)

Doing this calculation yields

Sm = IHDπ

2

(κκ∗sin2θ + (κ+ κ∗) cos θ sin θ

)(3.13)

To compare with FINESSE’s qd-detector, we have to determine the power spectral density ofthe vacuum fluctuations and then it is the square root of this ratio, given as Sqz =

√Sm/S0

that gives us the squeezing amplitude, or amount of squeezing. These expressions are givenas

S0 = IHDπ

2(3.14)

11

Sqz =√κκ∗sin2θ + (κ+ κ∗) cos θ sin θ (3.15)

Finally, we are ready to compare our analytic expression with FINESSE. First, wemust convert to decibels using the equation Sqz′ = 20 log10(Sqz)

Figure 3.2: Comparison between FINESSE software and analytic expressions for amount of squeez-ing. Squeezing amplitude is roughly 1.7 dB.

Luckily, we have perfect agreement with FINESSE. This gives us confidence thatwe are on the right track to understanding ponderomotive squeezing. With the eventualgoal of creating a ponderomotive squeezing element for FINESSE, we need to investigatethe parameters we will be specifying the model with in the end. So, we have amplitude nowand we know how to describe it. Now we turn our attention to squeezing angle.

3.3 Calculating Squeezing Angle

The squeezing angle we are attempting to calculate is the angle θ that is an maximum orminimum of the Sqz function. From calculus, we know we must differentiate our functionwith respect to theta and then set that derivative equal to zero and then solve for theta.The derivation can be found in appendix A.3, but the result is given analytically as

θ =1

2cot−1

(−κκ∗

κ+ κ∗

)(3.16)

This tells us at what homodyne angle θ we have our maximum squeezing and agreeswith Kimble, et al. in the case of a real coupling constant [2]. In the next section, we willfix our squeezing function at this value of theta and graph the function with respect to themechanical frequency of the mirror. This will generate a squeezing spectrum from which wecan determine the bandwidth of our squeezing.

12

3.4 Calculating Cutoff Frequency

Here, we calculate a cutoff frequency after which we say the optimal squeezing no longeroccurs. In other words, this cutoff frequency will allow us to define a bandwidth for oursqueezing element in FINESSE. Graphically, we can visualize this using a squeezing spec-trum. This spectrum is generated by fixing our homodyne angle at the squeezing angledefined in the previous section. In other words, we are fixing our angle to be the one atwhich maximal squeezing occurs, and then we are going to sweep over a frequency range.The result will be our squeezing spectrum.

Recall from above that the function which represents the amount of squeezing isgiven as

Sqz(θ) =√κκ∗sin2θ + (κ+ κ∗) cos θ sin θ

Before, we were interested in seeing what happens when we vary the homodyneangle, theta. Now, we are going to fix theta to be the angle at which optimal squeezingoccurs. We will call this θs, and it is given by our expression for squeezing angle fromSection 3.3

θs =1

2cot−1

(−κκ∗

κ+ κ∗

)(3.17)

We are going to vary the frequency of our mirror, Ω, which is contained withinthe optomechanical coupling constant κ. Additionally, we are going to assume an infiniteQ factor throughout the rest of this section. This simplifies our model and is not necessaryfor our purposes of creating a squeezer in FINESSE. Graphing this squeeze function withrespect to mirror frequency yields

Figure 3.3: Squeezing spectrum of the quantum fluctuations at the dark port of a Michelson inter-ferometer locked on dark fringe (blue) and cutoff frequency (red).

13

Chapter 4

Implementing PonderomotiveSqueezing in FINESSE

4.1 Converting Quadratures to Sidebands

The upper and lower sideband operators are defined in (Haixing’s book chapter) becausethey are useful for studying quantum noise in ground-based GW detectors. The computa-tions are far simpler in this picture; however, FINESSE uses sidebands to model everything,so our final expressions must be converted to the sideband picture before they can be im-plemented in FINESSE. The upper and lower sideband operators are given as b+ ≡ bω0+Ω

and b− ≡ bω0−Ω. From these, we can define the amplitude b1 and phase b2 quadratures as

b1 =b+ + b†−√

2b2 =

b+ − b†−i√

2(4.1)

These two equations can easily be expressed in matrix form as

[b1b2

]=

[1√2

1√2

1i√

2−1i√

2

] [b+b†−

](4.2)

This allows us to express our input-output relation in a way that FINESSE canhandle. Recalling our expression for the input-output relation of a simple Michelson inter-ferometer, we can now write

[1√2

1√2

1i√

2−1i√

2

] [b+b†−

]= e−2iΩτ

[1 0κ 1

] [a1(Ω)a2(Ω)

](4.3)

14

4.2 Current Status and Future Work

The ponderomotive squeezing element is currently being implemented in FINESSE by DanielBrown and will be tested this year by myself and others in the group. I will be continuingmy research project by examining several new systems. Of particular interest is finding outhow higher order modes are squeezed and what happens to the squeezing spectrum whenmore realistic mirror surfaces are included in the model. Furthermore, I am interested incoupling this squeezer with a cavity and seeing if more realistic parameters are then able tobe used for mirror mass and laser power. If these investigations are fruitful, there should bemore than enough content for a publication to be submitted by February or March of thiscoming year.

15

Appendix A

Detailed Derivations

A.1 Michelson Interferometer Input-output Relation

This derivation picks up from equation (1.6) in section one of this document. What followsis essentially a careful substitution and simplification process in which small quantities aredropped, trigonometric identities are used in order to apply small angle approximations, andfinally terms are collected in such a way that the result is an incredibly compact descriptionof the input-output relation of a Michelson interferometer.

So, we begin with a substitution into equation (1.6) from section one. As wesubstitute all of the relevant fields, the expression quickly becomes very long, as you can seehere

Eoutd (t) =

1√

2

(EoutB (t) − EoutA (t)

)

=1√

2

(EinB (t − 2τ − 2xB/c) − E

inA (t − 2τ − 2xA/c)

)

=1√

2

(1√

2[Einc (t − 2τ − 2xB/c) + E

ind (t − 2τ − 2xB/c)] −

1√

2[Einc (t − 2τ − 2xA/c) − E

ind (t − 2τ − 2xA/c)]

)

=1

2

(Einc (t − 2τ − 2xB/c) + E

ind (t − 2τ − 2xB/c) − E

inc (t − 2τ − 2xA/c) + E

ind (t − 2τ − 2xA/c)

)

=1

2

((√√√√ 2I0

hω0

+ c1(t − 2τ − 2xB/c)

)cos(ω0(t − 2τ − 2xB/c)) + c2(t − 2τ − 2xB/c)sin(ω0(t − 2τ − 2xB/c))

+ a1(t − 2τ − 2xB/c)cos(ω0(t − 2τ − 2xB/c)) + a2(t − 2τ − 2xB/c)sin(ω0(t − 2τ − 2xB/c))

−(√√√√ 2I0

hω0

+ c1(t − 2τ − 2xA/c)

)cos(ω0(t − 2τ − 2xA/c)) − c2(t − 2τ − 2xA/c)sin(ω0(t − 2τ − 2xA/c))

+ a1(t − 2τ − 2xA/c)cos(ω0(t − 2τ − 2xA/c)) + a2(t − 2τ − 2xA/c)sin(ω0(t − 2τ − 2xA/c))

)

We can drop the 2τ phase shifts from the cos(ω0(t − 2τ − xA,B/c)) and sin(ω0(t − 2τ −xA,B/c)) terms because τ = L

cand because we define L = nλ0 = n2πc

ω0. Thus, we see that ω02τ =

2ω0n2πcω0c

= 4nπ, where n is an integer. Because these integer multiple of 2π terms do not change the

absolute phase of the sine and cosine terms, they can be dropped. Additionally, the xA,B/c terms

can be dropped from the time delay in the amplitude and phase terms, a1,2(t− 2τ − 2xA,B/c) and

c1,2(t− 2τ − 2xA,B/c) because 2τ xA,B/c.

16

Eoutd (t) =

1

2

((√√√√ 2I0

hω0

+ c1(t − 2τ)

)cos(ω0(t − 2xB/c)) + c2(t − 2τ)sin(ω0(t − 2xB/c))

+ a1(t − 2τ)cos(ω0(t − 2xB/c)) + a2(t − 2τ)sin(ω0(t − 2xB/c))

−(√√√√ 2I0

hω0

+ c1(t − 2τ)

)cos(ω0(t − 2xA/c)) − c2(t − 2τ)sin(ω0(t − 2xA/c))

+ a1(t − 2τ)cos(ω0(t − 2xA/c)) + a2(t − 2τ)sin(ω0(t − 2xA/c))

)

We are going to simplify the cos(ω0(t−2xA,B/c)) and sin(ω0(t−2xA,B/c)) terms usingthe following identities

cos(α)cos(β) + sin(α)sin(β)

sin(α)cos(β)− cos(α)sin(β)

where α = ω0t and β = 2ω0cxA,B in our case. Once we expand each cosine and sine term using the

above identities, we will also apply small angle approximations to the terms involving 2ω0cxA,B .

The small angle approximations are essentially truncated Maclaurin expansions to a second-orderapproximation given as

sin(β) = β

cos(β) = 1 +β2

2

Also note that we are deriving a first-order approximation. Thus, the β2 terms will simply be

dropped.

Eoutd (t) =

1

2

((√√√√ 2I0

hω0

+ c1(t − 2τ)

)(cos(ω0t) + sin(ω0t)(2

ω0

cxB)

)+ c2(t − 2τ)

(sin(ω0t) − cos(ω0t)(2

ω0

cxB)

)

+ a1(t − 2τ)

(cos(ω0t) + sin(ω0t)(2

ω0

cxB)

)+ a2(t − 2τ)

(sin(ω0t) − cos(ω0t)(2

ω0

cxB)

)

−(√√√√ 2I0

hω0

+ c1(t − 2τ)

)(cos(ω0t) + sin(ω0t)(2

ω0

cxA)

)− c2(t − 2τ)

(sin(ω0t) − cos(ω0t)(2

ω0

cxA)

)

+ a1(t − 2τ)

(cos(ω0t) + sin(ω0t)(2

ω0

cxA)

)+ a2(t − 2τ)

(sin(ω0t) − cos(ω0t)(2

ω0

cxA)

))

This bulky equation is greatly shortened when we collect all like terms. Collecting like

terms yields the following equation

Eoutd (t) =

1

2

((√√√√ 2I0

hω0

+ c1(t − 2τ) + a1(t − 2τ)

)(cos(ω0t) + sin(ω0t)(2

ω0

cxB)

)

+

(c2(t − 2τ) + a2(t − 2τ)

)(sin(ω0t) − cos(ω0t)(2

ω0

cxB)

)

+

(−

√√√√ 2I0

hω0

− c1(t − 2τ) + a1(t − 2τ)

)(cos(ω0t) + sin(ω0t)(2

ω0

cxA)

)

+

(− c2(t − 2τ) + a2(t − 2τ)

)(sin(ω0t) − cos(ω0t)(2

ω0

cxA)

))

To complete the final simplification by hand would be an onerous exercise that would

surely result in a number of silly errors, so the symbolic manipulation package in Python, called

SymPy, was used. To program the expressions in SymPy more easily, the (t − 2τ) parts were left

off of the amplitude and phase quadrature terms, and the (t − τ) information was dropped off of

the displacement terms. The direct output, before any approximations are made, is given as

17

Eoutd (t) =

(a1 −

a2

cω0xA −

a2

cω0xB +

c2

cω0xA −

c2

cω0xB

)cos (ω0t)

+

a1

cω0xA +

a1

cω0xB + a2 −

c1

cω0xA +

c1

cω0xB −

ω0

cxA

√√√√ 2I0

hω0

+ω0

cxB

√√√√ 2I0

hω0

sin (ω0t)

All of the bold terms above are extremely small because both the quantum fluctuations

in amplitude and phase, and the microscopic mirror motion are tiny, thus these terms can be

dropped. Moreover, we can convert this SymPy output back into the proper notation we were

using previously and then make one final simplification using the definition of our differential-mode

motion xd(t− τ) ≡ xB(t− τ)− xA(t− τ) . Doing this, we have

Eoutd (t) = a1(t− 2τ) cos (ω0t) +

(a2(t− 2τ)− ω0

cxA(t− τ)

√2I0hω0

+ω0

cxB(t− τ)

√2I0hω0

)sin (ω0t)

Eoutd (t) = a1(t− 2τ) cos (ω0t) +

(a2(t− 2τ) +

√2I0hω0

ω0

cxd(t− τ)

)sin (ω0t)

Finally, after that lengthy calculation, we are ready to produce the desired result. Wesimply define the coefficient on cos(ω0t) to be b1(t) and the coefficient on sin(ω0t) to be b2(t). Withthis definition, we arrive at the input-output relation for a Michelson interferometer, which is givenas

b1(t) = a1(t− 2τ)

b2(t) = a2(t− 2τ) +

√2I0hω0

ω0

cxd(t− τ)

A.2 Laser Power Fluctuation

We know that power is proportional to the absolute value of the electric field squared. Haixingchooses to use a proportionality constant (for reasons he will hopefully be able to explain) of hω0.From this starting point, we will derive the expression for laser power.

P = hω0|Ein(t− τ)|2

Now, the equation for the electric field impinging on the mirror is given as

Ein(t− τ) = [

√2I0hω0

+ a1(t− τ)]cos(ω0(t− τ)) + a2sin(ω0(t− τ))

Expanding the square of the electric field is a lengthy expression given as

P = hω0

(2I0hω0

cos2ω0t+ a21(t− τ)cos2ω0t+ 2a1(t− τ)a2(t− τ) sin0 t cos0 t

+ 2a1(t− τ)

√2I0hω0

cos2ω0t+ a22(t− τ)sin2ω0t+ 2a2(t− τ)

√2I0hω0

sinω0t cosω0t

)

18

The bold terms in the previous expression will be dropped because they are second orderin quantum fluctuations and are thus very, very small. And we will then use trigonometric identitiesto simplify further.

P = hω0

(2I0hω0

cos2ω0t+ 2a1(t− τ)

√2I0hω0

cos2ω0t+ 2a2(t− τ)

√2I0hω0

sinω0t cosω0t

)= 2I0cos

2ω0t+ a1(t− τ)√

8I0hω0cos2ω0t+ a2(t− τ)

√8I0hω0 sinω0t cosω0t

= 2I0

(1

2+

cos 2ω0t

2

)+ a1(t− τ)

√8I0hω0

(1

2+

cos 2ω0t

2

)+ a2(t− τ)

√8I0hω0

(sin 2ω0t

2

)Again, the bold terms will be dropped. This time the justification is because they oscillate soquickly (Terahertz) that they are not resolved by the photodiode. We are left with our beautifullysimple expression of the form Ptotal = Pconstant + Pfluctuation

P = I0 +√

2hω0I0a1(t− τ)

A.3 Squeezing Angle

The squeezing angle is defined to be the homodyne angle at which maximum squeezingoccurs for a given system. So, we simply need a function that takes in a homodyne angleand outputs an amount of squeezing (we usually convert to dB in the end but that isunimportant for this derivation. We obtained such a function and it is given in equation(3.15). So, we will simply differentiate this expression with respect to theta, set that equalto zero, and solve for theta.

Sqz =

(κκ∗ sin2 θ + (κ+ κ∗) cos θ sin θ + 1

) 12

d

dθ[Sqz] =

2κκ∗ sin θ cos θ + (κκ∗)(cos2 θ − sin2 θ)

2√

2κκ∗ sin2 θ + (κ+ κ∗) cos θ sin θ + 1

0 =2κκ∗ sin θ cos θ + (κκ∗)(cos2 θ − sin2 θ)

2√

2κκ∗ sin2 θ + (κ+ κ∗) cos θ sin θ + 1

0 = 2κκ∗ sin θ cos θ + (κ+ κ∗)(cos2 θ − sin2 θ)

0 = κκ∗ sin 2θ + (κ+ κ∗)(1

2+

1

2cos 2θ − (

1

2− 1

2cos 2θ))

0 = κκ∗ sin 2θ + (κ+ κ∗) cos 2θ

−(κ+ κ∗) cos 2θ = κκ∗ sin 2θ

cot 2θ =−κκ∗

κ+ κ∗

θ =1

2cot−1

(−κκ∗

κ+ κ∗

)

19

A.4 Cutoff Frequency

The vertical red line shown in the previous plot is indicating the cutoff frequency. This iscalculated by considering the difference between the squeezing that occurs when the mirroris fixed and the squeezing that occurs when it oscillates with a given frequency. You thenset a cutoff criteria, and then solve for mirror frequency Ω. Mathematically, we have

κ− κ0

κ0= x

κ

κ0− 1 = x

κ

κ0= x+ 1

κ = (x+ 1)κ0

8I0ω0

mc21

Ω20 − Ω2

c

= (x+ 1)8I0ω0

mc21

Ω20

1

Ω20 − Ω2

c

=x+ 1

Ω20

Ω20 − Ω2

c =Ω2

0

x+ 1

Ω20 −

Ω20

x+ 1= Ω2

c

Ω20

(1− 1

x+ 1

)= Ω2

c

Ωc = Ω0

√x

x+ 1

20

Bibliography

[1] Haixing Miao and Y. Chen. ”Quantum Theory of Laser Interferometer GravitationalWave Detectors.” Advanced Gravitational Wave Detectors. Cambridge, UK: CambridgeUniversity Press, 2012. 277-96. Print

[2] H.J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, Sergey P. Vyatchanin.Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics. Physical Review D,Volume 65, 022002

[3] Charlotte Bond, Daniel Brown, Andreas Freise, Kenneth Strain. Interferometer Tech-niques for Gravitational-wave Detection. Living Reviews in Relativity 13, 2010.

21

Acknowledgements

Firstly, I would like to thank Daniel Toyra, Dr. Daniel Brown, and Dr. Haixing Miaofor their time and effort in assisting me. Before I arrived in Birmingham I had not had Idedicated course in quantum mechanics, so their assistance along the way was absolutelycrucial to my understanding of the project. Perhaps more importantly, they have mademe look very forward to pursuing a PhD in the field. Secondly, I would like to thankDr. Andreas Freise, Dr. Bernard Whiting, and Dr. Guido Mueller for their collaborativeorganization of this wonderful opportunity. Moreover, I extend my gratitude to the manypeople at NSF, Univ. of Florida, and elsewhere that were involved in this process. Theimportance of this program to my education and to my future simply cannot be overstated.I am incredibly grateful to have had the opportunity to learn so much and meet so manyinteresting people.

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