A membrane-in-the-middle device for optomechanics

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A membrane-in-the-middle device

for optomechanics

THESIS

submitted in partial fulfillment of the requirements for the degree of

MASTER OFSCIENCE

in

EXPERIMENTAL PHYSICS

Author : Elger Vlieg

Student ID : 1266578

Group: Dirk Bouwmeester Lab

Supervisor : Wolfgang L ¨offler

2nd corrector : Michiel de Dood

Co-supervisor : Frank Buters

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A membrane-in-the-middle device

for optomechanics

Elger Vlieg

Faculty of Science, Leiden Institute of Physics Niels Bohrweg 2, 2333 CA Leiden, The Netherlands

August 31, 2017

Abstract

It has been a long term goal of physicists to control macroscopic quantum superposition states - cat states - since these connect to a number of open fundamental questions in physics: the transition from

the quantum to the classical world, the quantum measurement problem, and the area between quantum physics and theory of general

relativity. Optomechanics has been identified as a method for generating cat states, however, this is yet to be achieved. The scientific

community has developed increasingly improved optomechanical systems. About a decade ago, a promising optomechanical system has

been demonstrated that consists of a high-stress silicon nitride membrane in the middle of a Fabry-P´erot cavity. This project concerns the development of a membrane-in-the-middle device for our lab. Our main focus lies on developing an understanding about the connection between system design and optomechanical performance. In addition,

we demonstrate optomechanics for our device, and show that the initial optomechanical parameters are good. The availability of clearly

defined methods for improving upon the current system parameters implies that we are moving in the right direction towards quantum

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Chapter

1

Introduction

1.1 The limits of quantum theory

The discovery of quantum physics has changed our perception of nature funda-mentally: a quantum world, where systems may express multiple states clashes with our intuitive perception of the classical world that we live in. The apparent paradox of quantum superposition states is famously captured by Sch ¨odinger’s cat, who truly may be dead or alive (Figure 1.1).

The quantum and classical world thus exhibit radically different behavior, however, the fact that we separate between these worlds is purely artificial: there is only one world, but we simply do not understand the transition be-tween them (Figure 1.1).

Figure 1.1:Controversy exists about the way the quantum world connects to the

classi-cal world. Quantum behavior has been shown for system up to 104atomic mass units

(AMU). Systems with more than 108 _{AMU are assumed to be fully classical.}

Macro-scopic superpositions are called cat states.

The lack of understanding of the connection between quantum and classical physics is closely connected to the measurement problem. It concerns with the fact that there is no proven theory that describes the evolution of a wavefunc-tion into a operator eigenfuncwavefunc-tion state due to measurement (interacwavefunc-tion with macroscopic object), besides of the notion of the wavefunction collapse in

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ac-cordance to the Copenhagen convention [1]. Clearly, macroscopic (measurent) systems must exhibit some specific behavior towards quanta. Equivalenty, the measurement problem concerns the seemingly absence of macroscopic super-positions [1].

Various theories have been suggested beyond quantum theory that predict the irreversible breaking of superposition states into classical states (decoher-ence). These theories each have a specific scaling with particle number or mass. For sufficiently distinct superposition states, these theories deviate significantly from the decoherence rates predicted by standard quantum theory [1]. Thus, the experimental study of the decoherence of macroscopic superposition states would allow us to discriminate between these theories.

Besides these open questions, quantum theory is a very well established within its range of validity. For instance, the quantum superposition principle has been confirmed with high accuracy for photons.

Similarly, general relativity is a theory that is very well confirmed on macro-scopic scales, from millimeters to astronomical length scales [7, 15, 19, 26]. How-ever, the connection between quantum theory and gravity is lacking.

It is unclear how spatial superposition states can be connected with space-time curvature (Figure 1.2). In addition, the presence of vacuum quantum fluc-tuations would imply that spacetime is not a smooth geometry, contrary to macroscopic observations. The experimental exploration of spatial superpos-tion of heavy objects would shed light on the scarsely studied region between quantum physics and gravity [1].

Figure 1.2: Connecting quantum theory to theory of general relativity is problematic. Curvature of spacetime is a well defined phenomenon as long as the position of objects is defined. Then, the existence of spatial quantum superpositions prohibits the direct coupling of quantum theory with theory of general relativity.

The creation of spatial superstates of as-large-as-possible massive macro-scopic objects will enhance our understanding of: the connection between the quantum and classical world, the measurement problem, and the relationship between quantum theory and gravity. There is a world wide effort to push the limits of the largest ’Sch ¨odinger’s cat’ we can make, so that we may have a look at fundamental physics in region that is largely hidden away from us.

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1.2 Optomechanics 9

1.2 Optomechanics

Cavity optomechanics has been identified as an approach to study cat states. Optomechanics concerns the study of the interaction between light and me-chanics. A cavity optomechanical system consists of an optical cavity and at least one mechanical oscillator (Figure 1.3).

Figure 1.3: (from [4]) Our lab currently implements a system where a reflective mirror comprises the end of the cavity. Optomechanical action is generated by the radiation pressure (Figure 1.4), which presses the harmonic oscillator outwards.

If the laser that feeds the cavity is stationary in frequency, the harmonic os-cillator is driven due to optomechanical interaction (Figure 1.4). The radiation pressure pushes the harmonic oscillator outwards, thereby, the inter cavity field - radiation pressure - is reduced, and the oscillator moves back, and so forth. This is a typical classical example of optomechanics.

Figure 1.4: Blue denotes the resonant response of the optical cavity. Optomechanical action is reciprocal: not only does the harmonic oscillator experience a radiation pres-sure, the motion of the harmonic oscillator (Figure 1.3) couples back to the cavity field

as well. Namely, the motion of the oscillator (x(t)) shifts the cavity resonance frequency

(ωcav), thereby the intensity of the cavity field changes.

In case of optomechanical cat state generation, optics are implemented in or-der to manipulate and readout macroscopic mechanical quantum states. There-for, optics may be considered to take the role of a quantum accuracy toolbox.

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Phononic states of mechanical oscillators can be realized with an atomic mass unit ranging between 105-1014, thereby, mechanical oscillators provide a plat-form with the potential to study the complete unexplored regime between quan-tum and and classical physics (Figure 1.1) [1].

The success and feasibility of quantum optomechanical experiments is largely determined by the mechanical parameter Q (mechanical quality factor) and the optical parameterF (optical finesse). For instance, attaining the quantum groundstate is essential for quantum optomechanics, and to reach the quantum ground state by optical sideband cooling is largy a matter of attaining high val-ues for Q andF.

In the current experimental setup, the resonator functions as one of the cav-ity side mirrors (Figure 1.3). Thereby, the optical and mechanical properties of the optomechanical system are inherently linked. Therefore, in practice either the optical or mechanical parameters have to be compromised in order to en-hance the characteristics of the other.

Trampoline resonators are implemented as the mechanical device in the sin-gle ended cavity system (Figure 1.5). These samples possess cutting-edge opti-cal characteristics (reflectivity). However, around the stiff DBR mirror mechan-ical frictional losses are induced, therefore, the Q of these devices is in practice limited toO(105)[4]. The quantum ground state has been unattainable for these systems.

Figure 1.5: (from [4]) Nested trampoline resonators are comprised of a center trampo-line resonator, which is incorporated in another trampotrampo-line resonator that acts as a low pass filter for vibrations from outside. In the center of the inner trampoline resonator, a distributed Bragg reflector (DBR) mirror is fabricated to achieve high optical reflection.

Thompson et al. demonstrated that optomechanics can be performed in a membrane-in-the-middle (MIM) system (Figure 1.6) [22]. The immediate ben-efit of this system is that the optical and mechanical properties are no longer strongly connected, and there is much more flexibility in improving optical and mechanical properties individually.

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1.3 Relevance of this project 11

Figure 1.6: (left)(from [22]) Optomechanical coupling can be achieved by a reflective silicon nitride membrane inside a Fabry-P´erot cavity. (right) (from [23]) A phononic bandgap structure patterned in a silicon nitride membrane can isolate the mechanical resonator (center defect) to high degree from the environment.

For the mechanical device, typically, silicon nitride (SiN) films of tens of nanometers thickness are under-cut on a silicon frame to create a high tensile stress membrane. Such membranes can be obtained commercially, while dis-playing a Q factor ofO(106)[28]. Implementation of a MIM system thus poses significant benefits in terms of: optomechanical parameters, simplicity, costs, and time.

Membrane oscillators can be isolated to very high degree from the environ-ment by patterning phononic bandgap structures in the suspended silicon ni-tride film [23]. For silicon nini-tride thin film devices Q factors ofO(108)are within reach [23, 29]. This implies that the quantum ground state is attainable for these devices, and thus, a clear path towards quantum optomechanics has presented itself.

1.3 Relevance of this project

MIM is promising development in the pursuit of quantum optomechanics, there-fore, our lab has made the commitment to establish this technique. This project involved the development of a first membrane-in-the-middle experiment in Lei-den (page 139 of the appendix). We focus on establishing an as-complete-as-possible understanding of the relationship between system parameters and op-tomechanical performance, and perform optomechanics with our device strictly for demonstrational purposes. In chapter 3, we study the optics inside the Fabry-P´erot cavity in order to understand the connection between membrane position and cavity design and optomechanical performance. We made design choices for the MIM system based on our insights, and specified an optical sys-tem to couple a laser into the syssys-tem. Based on the reflection signal from the cavity, we were able to extract the finesse of the optical cavity.

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Incorporating a membrane oscillator into a Fabry-P´erot cavity is likely to involve securing the membrane. In chapter 4, we investigate the Q factor as function of different clamping forces. In addition, we explain how polarisation rotation due to birefringence can be amplified by the optical cavity, and how the tensile stress inside the SiN membrane can be mapped due to this effect. Also, we discuss what temperature profile is generated inside the membrane due to optical absorption.

In chapter 2, we demonstrate optomechanics with the MIM system by mea-suring the thermal motion of the resonator, and cooling down its fundamental mode to∼30 K. We numerically integrate the classical equations of motion for MIM to get familiar therewith, and elaborate on the importance of the optome-chanical parameters Q andF. Also, we outline an optomechanical experiment in which cat states are generated, to provide a clear notion of the type of experi-ment that we strive towards in our lab. In every section, we discuss the methods in detail, and we try to be as complete as possible in discussing the relevant as-pects for MIM optomechanics. In this regard, this report may be considered as a guide to classical MIM optomechanics.

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Chapter

2

Optomechanics

In this chapter, we investigate classical optomechanics for membrane-in-the-middle. It is highlighted by a demonstration of optical cooling of the mem-brane mode. Additionally, we elaborate on the meaning of the optomechanical parameters Q & F in relation to quantum experiments, in order to improve our understanding of the relevance of this project. This is made concrete by the outline of an experiment designed by our lab for optomechanical cat state generation.

2.1 Classical equations of motion

In this section, we study the classical equations of motions (CEOM) for the membrane-in-the-middle system (MIM). From these, we obtain the term for the optical damping. We follow Jayich et al. [11], but include additional back-ground information from the review of Aspelmeyer et al. [1].

2.1.1 Mirror-at-the-end

In the case where the membrane is perfectly reflective, the CEOM for the MIM system can be found as a slight adaptation from the CEOM of the mirror-at-the-end system (MAE) (Figure 1.3). Since the MAE system is slightly less compli-cated, we discuss the CEOM first for a single cavity MAE system (Figure 2.1).

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Figure 2.1:Here denoted are some of the relevant parameters for the CEOM of the MAE system that describes the time-evolution of the cavity field (α) and the membrane posi-tion (x). α gives the expectaposi-tion value of the bosonic lowering operator of the photons

in the cavity. The source feeds the cavity at a constant rate√κexαin. The dissipation of

optical energy occurs at a rate κ. The dissipation of mechanical energy goes at a rate

Γm. Since the energy (∼field squared) dissipation is an exponential process (Equation

(2.4)), the dissipation of the (linear) field goes with exactly half this rate. Here it is drawn as if the dissipation occurs to some external system/bath. However, these terms include any form of dissipation, for instance, transmission through the cavity mirror.

The CEOM for the MAE is derived from the quantum operator time-evolution by Aspelmeyer et al. [1]. The time-evolution of the boson lowering operators follows from the second quantized Hamiltonian to which the in-output formal-ism can be applied. The CEOM is then obtained by investigating the expectation values of the quantum operators, for which they find (Figure 2.1)

˙α = −κ

2α+i(∆+Gx)α+

√

κexαin (2.1)

m¨x = −mΩ2_mx−mΓm˙x+¯hG|α|2 (2.2)

with x the position of the membrane (with resting postion x = 0), and α the complex amplitude of the cavity field. αin∗ is the amplitude of the

incom-ing (laser) field, κex is the (external) coupling rate of the source into the cavity

mode,∆ is the laser detuning from the cavity resonance, κ is the optical energy dissipation rate, m is the effective mass of the membrane mode, Ωm is the

me-chanical frequency, andΓm is the mechanical energy dissipation rate. In Section

4.1.1 we elaborate onΓm andΩm.

In Equations (2.1) and (2.2), G is the linear frequency pull parameter, which may be considered as a radiation pressure coefficient

∗

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2.1 Classical equations of motion 15 Frp = −∂E ∂x = −¯h ∂|α|2ωcav ∂x ≈ −¯h|α| 2∂ωcav ∂x Frp ≡ ¯hG|α|2 thus G= −∂ωcav ∂x . (2.3)

The approximation ∂|α|2ωcav

∂x ≈ |α|

2 ∂ωcav

∂x is valid for most - if not all -

op-tomechanical devices. To enhance opop-tomechanical coupling, then, is a matter of maximizing ∂ωcav

∂x .

In Equation (2.1), the rotating wave approximation is used [1]. This means that the reference frame rotates with the resonance frequency of the cavity. In-deed, if this was not the case, we would expect a phase rotation of eiωcavt _{for α}

(Figure 3.13). Instead, the phase only rotates due to∆ and G·x. We notice this by solving Equation (2.1) (Figure 2.2),

α =Ce[i(∆+Gx)− κ 2]t− √ κextαin i(∆+Gx) −κ 2 ≡Ce[i(∆+Gx)−κ2]t− D _(2.4)

with C some constant. From this expression it also becomes clear that κ indeed is the energy (∼ |α|2) dissipation rate for the optical field.

In the classical picture, we often think about a complex field amplitude that is defined in space (Equation (3.5), Figure 2.3). α however is directly related to the bosonic lowering operator (hˆai). Then

|α|2 = hˆα†ˆαi = hˆni.

|α|2thus scales with the integral of the field squared (energy) inside the

cav-ity, and α is thus best viewed in its relation to the total optical energy inside the system.

Jayich et al. [11] write down the EOM for the MAE system in a slightly different form

˙α =i(∆−ω0x)α+κ

2(1−α) (2.5)

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Figure 2.2: The CEOM’s (Equation (2.1) and (2.2)) for the single cavity MAE system (Figure 1.3) numerically integrated. (upper left) The time evolution of α follows

Equa-tion (2.4) with C =

√

κextαin

i(∆+Gx)−κ

2. Then, the equilibrium value for the cavity field

|α|2 is

given by |C|2_{. (upper right) Integrating the CEOM shows that the membrane starts}

vibrating at its resonant frequency (Ωm = 400 kHz) when the cavity is pumped

on-resonance. (bottom left and right) the same as the upper figures, but with a blue de-tuned laser. here ω0 = −G = ∂ωcav ∂x , and P = G ∆−ωcav Eres

m † is the radiation pressure

con-stant.

The most significant difference between these notations is that in form of Jayich the field is normalized to the resonant equilibrium value, so that α=1 at resonance.

We will adopt the notation of Jayich et al. from now on. It is convenient not have to make statements about κext and αin in our analysis. Also, ω0 is more

intuitive to use than G in this case, for ω0is a positive number for a Fabry-P´erot

†_{Jayich et al. mention that the}_P_{is in units of Hz}2_{. However, x is not explicitly normalized.}

We rather keepPin units of m·Hz2so that ∂ωcav

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cavity with a membrane on the right (Equation (3.38)). Then, from Equation (2.5) it is directly clear that an postive offset in x can be countered by a postive offset in laser frequency∆ to counter the phase rotation of the optical field.

2.1.2 Membrane-in-the-middle

From Equations (2.5) and (2.2), it is trivial to write down the CEOM for the MIM system for case that the membrane is perfectly reflective

˙ ~αs = Ms~α+ κL 2 0 where ~α = αL αR ; Ms =i(∆−ω 0_x_{) −}κL 2 0 0 i(∆+ω0x) − κ₂R . and ¨x = −ωm2x−Γm ˙x+ P (|αL|2− |αR|2). (2.7) α_L/R are the complex field amplitudes in the left (L) and right (R) cavity.

Note that the cavity is fed by a laser only from the left side.

Coupling is introduced between the cavity fields due to photons tunneling though the membrane. Here, we thus no longer assume that the membrane is perfectly reflective. In Section, 3.2 we show that the frequency of the full cavity resonance shifts due to this coupling. It then makes sense to choose a complex coupling amplitude −ig, with g the real coupling strength, so that a detuning of the frequency is needed for the CEOM to be on resonance again. We have (Figure 2.3) ˙ ~α = M~α+ _κ_L 2 0 (2.8) where M = i(∆−ω 0_x_{) −}κL 2 −ig −ig i(∆+ω0x) −κ₂R . The time evolution of x is still given by Equation (2.7).

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Figure 2.3: Same as Figure 2.1 but for the MIM device. The CEOM of the MIM system

describes the time-evolution of αL/Rand x (membrane position). αLand αRare coupled

with strength g due to photons tunneling through the membrane. In the description

where the complex field amplitudes αL/Rare normalized to their resonance values in

the absence of optical coupling, the source feeds the left cavity at a constant rateκL

2. The

dissipation of optical energy occurs at a rate κL(κR) for the left (right) cavity.

The full cavity resonant frequencies can be found by setting detM(∆) = 0 whilst setting κL =κR =0 [11]. The latter procedure is equivalent to making the

cavity mirrors perfectly reflective, whereas the former ensures that the system of homogeneous linear equations represented byMhas a non trivial solution. I.e. there exists a non-zero~αfor which ˙~α =0. From detM(∆) =0 we retrieve

∆ωcav(x) = ±

q

g2_{+ (}_ω0_x₎2_. _(2.9)

We can now attune this to analytical expression for the frequency shift, which is shown in Equation (3.40), we find

g=c

L q

2(1−r) and ω0 = −ωL/(L/2)

with L the full cavity length, r the membrane reflectivity, and ωL the laser

frequency. Without knowing r, we can find a lower limit for it from κ_L/R by setting κL = κR, and assuming that the all cavity losses are caused by photons

tunneling through the membrane. In this case we have (half cavity length≈ L₂).

α ∼ (r)

c

Lt =eln(r)·2Lc t (2.10)

i.e. a slab of the beam hits the membrane every t = L_c, thereby only retaining r of the field. We also have (Equation (2.4))

α ∼e

−κt 2

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2.1 Classical equations of motion 19 then ln(r)c L = −κ 2 rmin =e− κ L 2c.

We then can also obtain an expression for the maximum coupling constant g=c

L q

2(1−rmin).

Figure 2.4:We used r= rminin order to integrate the CEOM of the MIM system

(Equa-tions (2.7) and (2.8)). We have not normalized |α|2, like in the CEOM presented by

Jayich et al. [11] We did not do this in order to have better comparability with the

CEOM for MAE (Figure 2.2). The normalization factor is given by ¯h−1Eres (|D|2,

Equa-tion (2.4)) and is shown in the figure instead. In the top figure, the correct value for

¯h−1Eresis found by substituting κ/2 inDfor κtot =κL/2+κR/2=κL. Likely, this only

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Figure 2.5: If the membrane is shifted from the center of the cavity, the degeneracy of the resonance frequency of the right and left cavity breaks. The resonance is restored for

the left (right) cavity for a detuning of∆= −∆_cav(∆_cav) (Equation (2.9)). The resonance

field amplitude is not shown in the bottom left picture, because the build-up of field is insignificant compared to it.

From Figures 2.4 and 2.5 it becomes clear how the MIM mechanics are in-corporated in the CEOM for r → 1. Namely, for r → 1, the MIM system acts as a double sided MAE, and will have a positive or negative radiation pressure based on which cavity is fed (to which the source is resonant). This is to be ex-pected because G mimics a single cavity MAE system almost completely for the MIM system when r →1 (Figure 3.21).

2.1.3 Optical damping

Here, we outline briefly how optical damping of the mechanical oscillator fol-lows from the COEM’s, we follow Jayich et al. [11].

There are no coupled field terms in the time evolution of x (Equation (2.7)), therefore, the derivation of the optical damping term is separable for the left

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and right cavity. We may derivative the term for one field, for which we simply write α. This means that it is sufficient to derive the optical damping for the MAE system.

The optical damping term is retrieved from the CEOM by expanding the parameters linearly around their steady-state solution [11]

α → ¯α+δα

x → ¯x+δx.

The optical CEOM is separable in δx and δα parts, which leads to

δα± =χα(±ω)δx

where α± follow from the decomposition into positive and negative

fre-quency parts

δα =δα+eiω +δα−e−iω

likewise

δx =δx+eiω +δx−e−iω

where δx+ =δx− =δx.

χα is the susceptibility that relates the response of the optical field to the

mechanical motion in the frequency domain

χα(ω) =

¯α

(∆−ω+i(κ/2))/ω0− ¯x.

It makes sense to Fourier transform δx and δα, because the CEOM’s are lin-ear, so that the the eigenfuctions of the differential equation are given by e±iω.

In this form, the motion of x is described by

δx±(ω) = χ(ω)f±(ω)

where f±is an external force on the membrane in the frequency domain, and χ(ω)is the susceptibility function.

We derive χ for the CEOM of x without optomechanics in Section 4.1.1. In the case of optomechanics, there is a modification of this function, in which the form of δα± = χα(±ω)δx is easy to recognize without doing the complete

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χ−1(ω) =χ−_mech1 (ω) − P (¯α∗χα(ω) +¯αχ

∗

α(−ω)) =χ

−1

mech+

∑

(ω)

where∑(ω)contains the optomechanical contribution.

In order to find the damping of the mechanical motion, we must solve for x in the absence of external force, so that we may find its natural motion ωnat. ωnat is then given by (Section 4.1.1)

χ−1(ωnat) = 0

and

x(t) = Aeiωnat_.

Note that ωnatis a complex parameter in general.

The mechanical susceptibility has the following form

χ_mech−1 (ω)

m =Γ

2

m−iΓmω−ω2.

So in the mechanical case we can write (Section 4.1.1)

ωnat = s Ω2 m− Γ m 2 2 +iΓM 2 ωnat ≈Ωm+iΓM 2 . (2.11)

Clearly, Ωm gives the mechanical frequency andΓM the energy (∼ x2)

dis-sipation rate - which now we will call the damping - in the absence of optome-chanical coupling.

In order to assess the effects of optomechanics, we write χ−1 in the form of

χ−_mech1 [1] χ−1 m =Ω 2 m−iΓmω−ω2+Im(∑(ω)) mω ·ω+ Re(∑(ω)) 2mω ·2ω =Ω2_m+2ωRe(∑(ω)) 2mω −i Γm+Im (∑(ω)) mω ω−ω2 =Ω2_m+2ωδΩm(ω) −i Γm+Γopt(ω)ω−ω2 (2.12) where

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δΩm(ω) = Re(∑(ω))

2mω

is the optically induced frequency shift of the resonator, and Γopt(ω) = Im(∑(ω))

mω is the optical dampening.

The factor 2 is included for δΩm for

−ω2→ −(ω−δΩm)2 = −ω2+2δΩmω+ O(δΩ2m).

It is thus assumed that the frequency shift is small compared to the original resonance frequencyΩm.

We write down the frequency shift and optical damping as obtained by Aspelmeyer et al., by extracting Im(∑) and Re(∑), and assuming that Ωm+ δΩm ≈Ωm so that we may substitute ω →Ωm [1]. We substitute|¯α|2for ¯ncav.

δΩm = ¯h ¯ncav G 2 2mΩm ∆ +Ωm (∆+Ωm)2+ (κ/2)2 + ∆−Ωm (∆−Ωm)2+ (κ/2)2 (2.13) Γopt= ¯h ¯ncav G 2 2mΩm κ (∆+Ωm)2+ (κ/2)2 − κ (∆−Ωm)2+ (κ/2)2 . (2.14) We notice that cooling is most efficient if the laser is red shifted ∆ = −Ωm

from the cavity resonance. For most efficient heating, the laser should be blue shifted by this amount.

For the MIM system as described by (2.5) and (2.6) (and within their valid-ity), we simply get two times the terms in Equation (2.13) and (2.14), one for

|¯αL|2and|¯αR|2.

2.2 Optomechanical parameters Q and

F

The aim of this section is to get convinced that reaching the goal of performing quantum optomechanics experiments is to large extend governed by the im-provement of the optomechanical parameters Q andF. Since convincing is our goal here exclusively, a complete overview of the relationship between these parameters and optomechanics is not provided.

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2.2.1 The significance of Q

The degree to which a mechanical oscillator is able to retain its energy and is isolated from driving forces of the thermal environment is given by the Q fac-tor. In fact, these things are directly linked: the better the decoupling from the environment the better the oscillator is able to store its energy and vise versa. The Q factor is defined as [1]

Q =Ωm/Γm. (2.15)

Notice that the Q factor gives the phase traveled by the resonator in the time that is required to dissipate 1_ethof its energy (t =1/Γm).

Achieving good decoupling from environment - attaining a high Q - is im-portant for two (main) reasons: the coupling to the environment limits the low-est temperature the oscillator can be cooled down to, and it limits the coherence time of the quantum (ground) state. We will discuss both aspects separately. Quantum decoherence due to phonon absorption and dissipation

Here we provide two manners of conceptualizing the effect of Q on quantum decoherence, either way can be convenient for a specific scenario.

Firstly, one could view the environment as a heat bath with a constant num-ber of phonons. Then, Q determines the effective coupling to this bath, as it provides information on the amount of phonons in the bath (by Ωm) and the

rate at which they couple to the mechanical system (throughΓm). An increase

in Q then implies a decrease in the phonon coupling rate, thereby, a longer coherence time is achieved. We will refer to this scheme as the fixed phonon number picture.

Alternatively, one could picture the bath as having a fixed frequency density of phonons around the resonance frequency of the resonator, and the resonator as acting as a bandpass filter for this bath. Then, the Q gives information on the phonon density (Ωm) and the frequency window (Γm) wherein the resonator is

susceptible to phonons. An increase in the Q factor can then be viewed as a sharpening of the bandpass filter. This scheme will be referred to as the band-pass picture.

To derive the expression for the thermal coherence time, first, we need to obtain the thermal phonon number of the thermal bath around the mechanical resonance frequency ( ¯nth). The obtained expression can be viewed as a phonon

density by exchangingΩm for a variable frequency ω. We consider the grand

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2.2 Optomechanical parameters Q andF 25 ¯nth = hni = 1 Z ∞

∑

n=0 n exp −n·¯hΩm kBTbath ≡ 1 ZA

with Ωm the frequency of the resonator, ¯nth the average number of thermal

phonons, and Z= ∞

∑

n=0 exp −n·¯hΩm kBTbath = 1 1−exph− ¯hΩm kBTbath i

the partition function.

We find the reduced expression of the summation A exp ¯hΩ m kBTbath A−A= ∞

∑

n=0 exp −n·¯hΩm kBTbath exp ¯hΩ m kBTbath −1 A = 1 1−exph− ¯hΩm kBTbath i A = exp h _¯hΩ m kBTbath i (exph ¯hΩm kBTbath i −1)2. Then ¯nth = 1−exp − ¯hΩm kBTbath exp h ¯hΩm kBTbath i (exph ¯hΩm kBTbath i −1)2 ¯nth = 1 exph ¯hΩm kBTbath i −1. In the case that

Tbath ¯hΩm kB (2.16) we may approximate ¯nth ≈ 1 1+ ¯hΩm kBTbath −1+ O ( ¯hΩm kBTbath) 2 ¯nth = kBTbath ¯hΩm . (2.17)

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The high temperature approximation (Equation (2.17)) holds in general. For instance, for our system Ωm ≈ 400·103·2π rad/s, so that the requirement in

expression (2.16) translates to T2·10−5K. With our cryostat we are not able to cool lower than a few millikelvin, thus this approximation is valid for our setup.

An alternative way of deriving ¯nth is through the equipartition theorem,

which states that a system in thermal equilibrium attains an average energy of 1₂kBTbath per quadratic degree of freedom (x2 and v2 in this case). We can

then fill the system with phonons of energy ¯hΩm in order to match the average

energy ¯ E =2· 1 2kBTbath = ¯hΩm¯nth ¯nth = kBTbath ¯hΩm .

Thermal decoherence from the fixed phonon number picture In the fixed phonon number picture, the coupling to the bath is mediated by the mechani-cal energy dissipation rateΓm (Equation (4.9)). Expectation values of quantum

operators act classically. Then, we may write down the evolation of the expec-tation value of the phonon number (operator) directly [1]

d

dt¯n= −Γm(¯n− ¯nth) (2.18) then

¯n=e−Γmt₍_¯n

0− ¯nth) + ¯nth.

We find the thermal decoherence rate, by assessing the equation above for a system in the ground state [1]

1 τth = d dt¯ngs(t =0) = ¯nthΓm 1 τth = kBTbath ¯hΩm Γm (2.19)

with τth the thermal coherence time, which denotes to the average time for

a thermal phonon to enter the system.

From expression (2.19) is becomes clear why Q is a convenient parameter to use in optomechanics. Based on the expression (2.19), two parameters of the

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resonator are required in order to assess the coherence time from the ground state: Ωm defining the amount of thermal phonons, andΓm defining the rate at

which they couple to the mechanical resonator. We capture both aspects in the Q factor

1

τth =

kBTbath

¯hQ . (2.20)

Thermal decoherence from the bandpass picure It is not trivial how a high-Q oscillator may act as a bandpass filter for phonons, especially when we consider that the mechanical oscillator only gets more susceptible for external forces with an increase of Q (=decrease inΓm, Figure 4.1).

The reason why an increase in susceptibility at the mechanical resonance frequency leads to effective bandpass filtering for thermal phonons lies in the thermal nature of the process. By the equipartition theorem ¯nth is set by Tbath

(if the energies of the phonons are equivalent within the resonance peak in the susceptibility function). Then, an increase in Q implies that in thermal equi-librium the probability of finding phonons closely matching Ωm goes up, as

the resonance peak in the susceptibility goes up. Since ¯nth is a constant of Q,

this implies that there are less off-resonance phonons inside the resonator, and effectively the thermal phonon bath has been (increasingly) bandpass filtered.

The phenomenon described above does not necessary imply an increased coherence time for the ground state resonator, as it does not take into consid-eration the coupling rates of the phonons. Increased time coherence in thermal equilibrium is however directly observed in the classical picture. We will con-sider this effect as justification that in the quantum regime coherence times also must go up. Also, keep in mind that the goal of this paragraph is to extend the conceptualization of Equation (2.20), and not to provide rigorous mathematical proof of the justification thereof.

The Fourier transform of the classical motion of the oscillator over some measurement of lenght τ is given by [1]

˜x(ω) = √1 τ Z τ 0 x(t)e iωt dt.

The Wiener-Khinchin theorem connects this to the noise power spectral den-sity introduced by the thermal environment

Sxx(ω) =

Z ∞

−∞hx(t)x(0)ie iωt_dt

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lim

τ→∞

h|˜x(ω)|2i = Sxx(ω).

From the expressions above we obtain [1]

Z ∞

−∞Sxx(ω)

dω 2π = hx

2_i_. _(2.21)

This results implies that the variance of the motion hx2i in a staionary sys-tem can be obtained from the surface integral of the noise power spectral den-sity Sxx. This becomes important in thermal equilibrium, where the fluctuation

dissipation theorem relates the noise to the dissipative (imaginary) component of the susceptibility (linear response) [1]

Sxx(ω) =2kBTbath

ω Imχm(ω).

In section 4.1.1 we derive χ for the harmonic oscillator, its imaginary part can be approximated as Imχm(ω) ≈ 1 2meffΩ2m Γm 2 ω (|ω| −Ωm)2+ (Γ₂m)2 then Sxx(ω) = kBTbath meffΩ2m Γm 2 (|ω| −Ωm)2+ (Γ₂m)2 . (2.22)

This is a doubly peaked Lorentzian (figure 2.6) for which the surface integral has a standard solution

Z ∞ −∞Sxx(ω) dω 2π = kBTbath meffΩ2m 1 2π Z ∞ −∞ Γm 2 (|ω| −Ωm)2+ (Γ₂m)2 dω ≈ kBTbath meffΩ2m 1 π Z ∞ −∞ Γm 2 (ω−Ωm)2+ (Γ₂m)2 dω = kBTbath meffΩ2m .

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Figure 2.6: (from [1]) the noise power spectral density (Sxx, Equation (2.22)) is a Lorentzian of which the surface area is independent of the mechanical energy

dissi-pation rateΓm. Sxxis a symmetric function of ω, so that there is another resonance peak

at ω= −Ω_m. This is important for the surface integral in (2.21).

It should not come as a surprise that the surface integral of Sxx is

indepen-dent of Γm: its surface integral gives hx2i, which is coupled to the average

en-ergy by the equipartition theorem. Indeed, the same expression can be obtained from the equipartition theorem. For the classical energy of the harmonic oscil-lator as function of x we have

E(x) = 1

2kx

2

with k the spring constant, which is related to the resonance frequency Ωm = s k meff . We obtain 1 2kBTbath = 1 2khx 2_{i =} 1 2Ω 2 mmeffhx2i hx2i = kBTbath meffΩ2m .

Now that we have obtained the noise power spectrum density of the ther-mal motion of the resonator, we may link this to the motion of the mechanical oscillator, and the coherence thereof (figure 2.7).

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Figure 2.7:(from [1]) The thermal movement of the oscillator is goverened by the reso-nance frequency, however, the thermal motion deviates from a perfect sinusoid due to off-resonance frequency components in the noise power spectral density (Figure 2.6). In particular, frequency differences in the wave packet (2.6) create an amplitude and phase modulation of which the shortest time scale (largest frequency difference) is given by the full-width-at-half-maximum (FWHM) of the noise power spectral density. Since the

FWHM is given byΓm (Equation (2.22)), the temporal response of the thermal noise is

limited byΓ−_m1. We then observe that the motion of the mechanical decoheres with time

constantΓ−_m1.

We thus see that in the classical pictureΓm defines a frequency window for

the thermal phonons to which the resonator is susceptible (Figure 2.6). Based on this, we portray the leaking in of thermal phonons as

1

τth

∼(Frequency density of phonons in bath)·(frequency window) 1 τth ∼ ¯nthΓm = kB_¯hΩTbath m Γm 1 τth ∼ kBTbath ¯hQ .

The most important thing to realize here is that the decoherence time from the quantum ground state connects in the same way to Γm as in the classical

picture (Figure 2.7). This must be, for we derived the thermal coherence time from the expectation value of the phonon number (which behaves classically) (Equation (2.18)).

Q improves the optical cooling of the mechanical mode

Here, we show how Q increases the effectiveness of optical cooling. Then, the fabrication of high-Q devices is also important to achieve ground-state cooling for quantum mechanical experiments.

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In Section 2.1.3, we discussed how optomechanical coupling may lead to optical damping of the mechanical mode. The total energy dissipation rate then is

Γtotal=Γm+Γopt (2.23)

where Γm includes all mechanical dissipation channels to the thermal

envi-ronment, andΓoptincludes the effects of the laser.

In the previous section, we obtained that the thermal occupation of a me-chanical mode is independent of the thermal phonon coupling rate Γm

(Equa-tion (2.18) and (2.17)). Γ−_m1only gives the coherence time of the thermal motion (Figure 2.7). Thus, an increase inΓmcan never result in cooling of the

mechani-cal mode.

The reason why Γopt leads to effective cooling (or heating) is that it does

not describe a coupling to the thermal bath, but to a laser bath instead (Figure 2.8). Note that, even though Γopt links to a different thermal environment, the

mechanical response (susceptibility) of the resonator is fully defined by Γtotal

(Section 2.1.3), thus also the shape of the thermal noise spectral density (Figure 2.6, Equation (2.21)) is governed byΓtotal(Figure 2.14).

Figure 2.8: The optomechanical damping rateΓoptcouples the mechanical mode to a

laser bath - with temperature T_laser- rather than the thermal environment - with T_bath.

Therefore, the effective temperature of the mechanical oscillator Teff may shift from

Tbath.

Another important difference betweenΓoptandΓmis that the former is does

not necessarily reciprocal. For Γm it is implied that an increase therein both

results in faster energy dissipation from the resonator, as well as faster coupling from the thermal bath back to the resonator. However,Γmconsists of a negative

term that strictly heats, and a positive term that cools (Equation (2.14)). Thus, an increase in out-coupling rate, does not necessarily increase the in-coupling rate and vise versa. This is equivalent to viewing Tlaseras a dynamic parameter

based on the laser frequency. We explore this for laser cooling in the classical limit, we have (Equation (2.14))

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Γopt= ¯h ¯ncav G 2 2mΩm κ (∆+Ωm)2+ (κ/2)2 − κ (∆−Ωm)2+ (κ/2)2

Γopt ≡Γopt,+−Γopt,-.

The time evolution of the expectation value for the phonon number due to optical dampening then is

d¯n

dt = −Γopt¯n= −Γopt,+¯n+Γopt,-¯n. We write this into the form of a thermal process

d¯n dt = −Γopt,+(¯n− Γ opt,-Γopt,+¯n ) = −Γopt,+(¯n− ¯nth,laser) with ¯nth,laser = ¯n Γ opt,-Γopt,+.

We find the temperature of the laser bath by comparing this to a thermal occupation ¯nΓ opt,-Γopt,+ = kBTlaser ¯hΩm Tlaser = ¯n Γ opt,-Γopt,+ ¯hΩm kB . (2.24)

From Equation (2.24) and (2.14) we obtain that Tlaser changes a function of

laser detuning.

For laser cooling we may approximate (Equation (2.14) Γopt,+ ≈Γopt

so we write d¯n

dt = −Γopt(¯n−¯nth,laser) −Γm(¯n− ¯nth,bath) dTeff

dt = −Γopt(Teff−Tlaser) −Γm(Teff−Tbath). (2.25) This formula then describes the coupling as depicted in Figure 2.8.

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We obtain the expression the equilibrium temperature, from Equation (2.25) we have

(Tbath−Teff)Γm+ (Tlaser−Teff)Γopt =0.

We solve for Teff

Teff =

ΓmTbath+ΓoptTlaser

Γm+Γopt . (2.26)

For laser cooling (Equation (2.14)), the addition of phonons from the laser bath is only significant at very low phonon numbers (quantum limit), thus, in the classical limit we may approximate Tlaser =0 (Figure 2.8)

Teff = Tbath 1+Γopt/Γm . (2.27) We have Γm = Ωm Q so that Teff = Tbath 1+_ΩQ mΓopt . (2.28)

Γopt is limited by a maximum laser power - at which point the optical

ab-sorption in the resonator causes effective heating again - so Teff is limited the

rate at which the mechanical system heats (∼ Γm ∼ Q−1). Thus, an increase in

Q allows a mechanical system to be cooled to lower temperatures optically. Based on Equation (2.23) and (2.15), we notice that we can boost the Q factor with optical heating, since Γtotal → 0. This seems ideal, for then the thermal

coherence time τth → ∞ (Equation (2.20)). Unfortunately, this will not help us

to stay coherent in the mechanical ground state for quatum experiments. The mechanism that is implemented here to boost τthis a high phonon occupation of

the oscillator. So, the mechanical ground state will decohere faster, rather than slower, in this scheme, and Equation (2.20) is not valid when optical damping is included.

2.2.2 The significance of

F

The optical finesse is defined as the full-width-at-half-maximum of the resonant (transmission) peaks in the optical response of the cavity, relative to its free

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spectral range (Figure 2.9). The finesse is thus given by

F = ωFSR

κ (2.29)

with ωFSR (Equation (3.39)) the free spectral range of the system, and κ

(Equation (2.4)) the optical energy dissipation rate. We will discuss both of these aspects separately in order to outline the significance ofF.

Potential quantum decoherence experiments are performed on the mechan-ical system. Therefore, optmechan-ical parameters do not - contrary to Q - directly im-pose restrictions on the coherence time of the mechanical cat state once it is generated. However, the optical field may raise the environmental bath tem-perature due to optical absorption. An increase in bath temtem-perature will in fact lower the coherence time of the mechanical ground state, even some time after the optical field has drained (Equation (2.20)). Additionally, in the experimen-tal procedure, the optical field can decohere the system by carrying quantum information out of the system [24], this we will not discuss here.

The significance of κ lies in its influence on the way the mechanical sys-tem can be manipulated with optics, both in terms of the extend and strength thereof.

The importance of amplifying optomechanical interaction strength with κ may not be obvious at first sight, considering that this amplification can always be achieved by increasing the number of photons inside the system ¯ncav

(Equa-tions (2.13) and (2.14) provide an example). However, increasing the number of photons will also increase the power absorption inside the membrane, thereby, increasing the bath temperature.

A reduced κ allows for stronger optomechanical interaction with less pho-tons, therefore, the mechanical system can be manipulated with less induced heating. Without going into detail, Figure 1.4 may improve understanding of why a decreased linewidth of the cavity enhances optomechanical coupling. The increase in optomechanical coupling then scales with κ−1.

In addition to κ, the linear optomechanical coupling strength is largly gov-erned by the frequency pull parameter G‡(Equation (2.3)). For G in a mebrane-at-the-end system (Figure 1.3) we have [11]

G = −ωL

L ∼ L

−1

‡_{Equivalently, g}

0or g can be used here. g0is the vacuum optomechanical coupling strength,

and links the classical G to the displacement induced by a phonon (g0=Gxzpf), thereby, linking

it to the quantum discription. g = √¯ncavg0 gives the total optomechanical coupling rate in

the linearized description around the expectation value [1]. The key realization here is that G∼g0∼g.

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with ωL the laser frequency, and L the cavity length. Note that G takes a

slightly different form for the membrane-in-the-middle system (Section 3.2.3), however, still the linearized frequency pull parameter exhibits G ∼ L−1in gen-eral (Figure 3.18).

The loss rate through the cavity mirrors scales with L−1 (Equation (2.10)). However, we thus obtain that increasing κ in this manner does not necessarily benefit optomechanical interaction. Indeed, often the coupling governed by the ratio G_κ. For instance, the single mechanical excitation probability by a laser pulse scales with [24]

p ∼np

G2

κ2

with npthe photons in the pulse.

Because of the ratio G_κ in relation to optomechanical coupling, it can be con-venient to describe the optical system by its finesse

F = ωFSR κ . Since (Equation (3.39)) ωFSR ∼L−1 we have F ∼ G κ.

The finesse relates the amount of times a photon traverses the system, and thus the amount of times is may interact with the mechanical oscillator. In the case that the optical dissipation is governed by transmission through the cavity mirrors, κ−1 ∼L andF is a constant as function of cavity length.

Not all optomechanical action is governed by the ratio G_κ, and F does not necessarily provide a complete picture of the optical performance. However, any process that scales with

¯ncavG n+1 κn

can typically be directly related toF as well.

In some experimental scenarios, the optomechanical interaction may not be limited by heating. In this case, κ improves optomechanical action additionally by increasing ¯ncav. We have

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¯ncav ∼κ−1 and ¯ncavG n+1 κn ∼ Gn+1 κn+1 ∼ F n+1_. _(2.30)

The absorption inside the the membrane scales with L−1, because (Section 3.2.1)

E2(~r)|ncav ∼ L

−1

.

Then, if the optomechanical interaction is limited by heating, we may in-crease the number of photons linearly with L and

¯ncavG n+1 κn ∼ Gn κn ∼ F n . (2.31)

Optomechanical cooling is a process that scales with ncavG

2

κ (Equation (2.14)).

We will discuss this in more detail in the remainder of this subsection. Optical cooling

From Equation (2.14), we notice that the maximum cooling is given for ∆= −Ωres

withΩres the mechanical resonance frequency.

This gives in the classical limit (Teff 0)

Γopt ≈Γopt,+ = ¯h ¯ncav G 2 2mΩm κ (∆+Ωm)2+ (κ/2)2 ∆=−Ωm = 4¯h 2mΩm ¯ncavG2 κ . (2.32)

Clearly, as κ goes down, the optical damping goes up, and the resonator will be able to cool down closer to the temperature of the laser bath (Figure 2.8, Equation (2.28)). From Equation (2.32), we notice that the optical cooling rate scales with ¯ncavG

2

κ .

When the quantum regime is approached, the finite laser temperature starts to become important (Equation (2.24)). In the resolved sideband regime (Figure 2.9) we have for the optical driving term during optical cooling

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2.2 Optomechanical parameters Q andF 37 Γopt,- =¯h ¯ncav G 2 2mΩm κ (2Ωm)2+ (κ/2)2 Γopt,-κ Ωm = ¯h ¯ncav G 2 2mω κ 4Ω2 m . (2.33)

Then, we have for the classical laser temperature (2.24) Tbath ∼ Γ opt,-Γopt,+ Tbath ∼ κ 4Ωm 2

In the resolved sideband regime, it can be shown that then the minimum cooling temperature in the quantum regime is given by [1]

ˆnmin = κ 4Ωm 2 . (2.34)

We thus observe that κ not only influences optomechanical coupling strength, but also the extend to which the resonator may be influenced. Specifically, lower

κallows for extended optical cooling.

Figure 2.9: (from [1]) The motion of the mechanical oscillator around its resonance

frequency Γm generates sidebands around the laser frequency at the ±Ωm. Since the

full-width-at-half-maximum around the resonance frequency is given by κ, we refer to

κΩm as the sideband resolved regime.

Figure 2.9 provides an interesting perspective on optical cooling (Equation (2.14)). Without going into details, the strength at which the cavity interacts with light of a certain frequency scales with its transmission profile (Figure 2.10). In the quantum picture, the transmission profile translates to a density of optical states inside the cavity, so that interaction with the cavity is strong if the frequency is close to cavity-resonance (Figure 2.11).

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Figure 2.10: Cavity transmission response for a finesse of 100 (left) and 1000 (right).

κ−1 is 10 times higher in the latter case (Equation (2.29)), so that the line width is 10

times smaller (Figure 2.9).

Note how the shape of the two detuned lorentzian sidebands can be recog-nized in the expression of the optical damping (Equation (2.14)). The blue side band corresponds to an increase in photon energy as opposed to the laser, so to the absorption of a phonon by a photon. Then, the red sideband corresponds to the reverse process.

If the laser is detuned −Γm from the cavity resonance, the blue sideband

overlaps with the cavity resonance whereas the red sideband is off-resonance (Figure 2.11). Since here the density of states is sharply peaked around the opti-cal resonance, the optomechaniopti-cal process where a phonon is extracted from the resonator will be heavily favored in this configuration. Then, efficient cooling achieved for this laser detuning.

Figure 2.11: (adapted from [1]) Optical cooling is initiated by overlapping the blue sideband with the position where the cavity density of states is highest. In this

configu-ration, the phonon absorption rate is stimulated linearly with κ−1 (Equation (2.32)),

whereas the phonon generation rate is suppressed by κ−1_{. An decrease in cavity}

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2.3 Cat state generation by mechanical mode swapping 39

2.3 Cat state generation by mechanical mode

swap-ping

Our lab has designed a promising new experiment to achieve macroscopic su-perposition states by swapping non-degenerate mechanical modes with op-tomechanics (Figure 2.12) [24]. We briefly outline this experiment in order to provide a clear notion of the type of experiment we strive towards with quan-tum optomechanics. In addition, the scope of the importance of Q and F is thereby further demonstrated.

The design of this experiment stems from the realization that its much easier to generate superpositions between two mechanical modes, as opposed to an optical and mechanical state. This is because mechanical modes can be coupled much stronger optomechanically [3].

Figure 2.12: (adapted from [24]) The experimental procedure of measuring decoher-ence of a cat states by mechanical mode swapping goes as follows: (i) the mechanical modes are cooled down to the ground state, (ii) mechanical mode 1 is projected into the first excited state, (iii) a macroscopic superposition state is generated by a π/2 pulse from the swapping lasers, (iv) the system evolves for time τ, during which the cat state

rotates with ei(Ω1−Ω2)t _{around the z-axis of the Bloch sphere, (v) a π/2 pulse rotates}

the state vector again 90◦ around the x-axis of the Bloch sphere, this will reveal if the

superposition decohered in step (iv) upon (vi) read-out of the state of mechanical mode 1.

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Hamiltonian for a system with two non-degenerate mechanical modes takes the following form [24] ˆ H = " ¯hωcavˆα†ˆα+ 2

∑

n=1 ¯hΩnˆbn†ˆbn # +¯hJ ˆb†₁ˆb2+ˆb1†ˆb2

with ˆα (ˆbn) the bosonic lowering operator for the cavity photons (phonons

in mode ’n’), and ωcav(Ωn) the frequency of the cavity mode (mechanical mode

’n’).

The outer right term in the Hamiltonian generates coherent swapping be-tween the mechanical modes at rate J, upon switching on the swapping lasers [3, 24, 25].

In the experimental procedure, firstly, the mechanical modes are optically cooled down to their ground states. Whether the ground state can be reached is determined by the optomechanical parameters Qn and F (Qn quality factor of

the mechanical oscillator ’n’,F finesse of the system, Section (2.2)).

For the next step, mechanical mode 1 is projected into its first excited state by a laser pulse - blue detuned off-resonance byΩ1 - and consecutive post

se-lection. The post selection is done by measuring a single red detuned photon transmitted from the cavity (the one that transfered its energy to the resonator). Post selection is inperfect due to the possibility of reaching a higher order exited state, or the detector giving a false positive due to dark counts. The probability of finding the mechanical resonator in its first exited state scales withF2_{. Here,}

we assume that the procedure is perfect, we have the following mechanical state [24]

|ψi(ii) = |10i.

The swapping lasers are turned on to apply a π

2 pulse, thereby, a cat state is

generated

|ψi(iii) =

1

√

2[|10i + |01i].

During the pulse, the system should not decohere due to light carrying in-formation of the mechanical systems out of the cavity, this requires a low optical decay rate κ [24].

After this, the system is left to evolve for time τ. During this time, the de-coherence time is limited by the intrusion of thermal phonons. The thermal coherence time τth scales with Q (Equation 2.20), we should thus set τ < τth.

Other significant limiting conventional decoherence channels are the emission of black body radiation by the resonator and collision by gas molecules.

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2.4 Experimental membrane-in-the-middle optomechanics 41

The coherent state after time τ becomes

|ψi(iv) = 1 √ 2 h eiΩ1τ_|₁₀_{i +}_eiΩ2τ_|₀₁_ii_.

On the Bloch sphere this looks like a rotation around the z-axis with ei(Ω1−Ω2)τ_.

Alternatively, the system may have decohered in time τ, upon which we obtain the a 50/50 mixed state between|10iand|01i.

Another π/2 swapping pulse is applied to manipulate these states so that we can discriminate between them, we either retrieve

|ψi(v) =sin((Ω2−Ω1)τ/2) · |10i ±i cos((Ω2−Ω1)τ/2) · |01i

in the coherent case, or a 50/50 mixed state between |10i ±i|01iin the de-cohered case.

Finally, the state of mechanical mode 1 is read-out by a laser pulse - red de-tuned byΩ1- and a single photon detector. Thereby - after many repetitions of

the experiment - we retrieve the ampltiude of the states |10i and |01i, and ob-tain whether the system has decohered after time τ. Then, after performing the experiment whilst varying τ, we retrieve the decoherence time of the system.

2.4 Experimental membrane-in-the-middle

optome-chanics

We measured the thermal motion of the membrane and the effect of optical damping experimentally. The data of these experiments is presented here. This data was obtained in strong collaboration with Fernando Jose Luna, Sameer Sonar, Vitaly Fedoseev, Wolfgang Loeffler, and Frank Buters. For details on the experimental methods and setup, please refer to Eerkens at al. [8].

In these experiments, the full cavity length amounted to ∼ 99.5 mm. This was determined by measuring the free spectral range of the cavity (Equation (3.39)). In Section 3.1, we propose that the cavity length should be ∼98 mm in order to minimize scattering losses induced by the membrane. In addition, we suggest how the laser source is best coupled into this system. During this ex-periment coupling optics were used that were designed for a half-cavity length of ∼ 50 mm (∼ 100 mm full cavity length). Therefore, the full cavity length needed to be extended to provide sufficient coupling into the fundamental cav-ity eigenmode.

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Figure 2.13: (left)The thermal noise power spectral density (Sxx) (Figure 2.6) of the

mechanical resonator is given by a Lorentzian (Equation (2.21)). Sxx characterizes the

thermal motion of the membrane: its surface area scales with the average thermal en-ergy (phonon occupation) inside the mechanical mode (Figure 2.6), and its shape relates to the frequency components in thermal motion (Figure 2.7). Since the thermal occu-pation of the mechanical mode scales with the bath temperature (Equation (2.17)), the

surface integral of Sxx scales with Tbath as well (Figure 2.6). (right) The thermal

mo-tion power spectral density in the case that the mechanical resonator is optically cooled

with 195 µW laser power. Since the optical dampingΓopt (Equation (2.14)) relates to

a coupling to a laser bath, the mechanical mode has effectively cooled down so that

Teff < Tbath (Figure 2.8, Equation (2.28)).

Figure 2.14: Same as Figure 2.13 but displayed in comparative fashion. Here we also include the power spectral density for a blue detuned laser.

In Section 2.2.1 we showed that the surface integral of the thermal noise spectrum density (Sxx) scales with mechanical mode temperature (Figure 2.6),

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2.4 Experimental membrane-in-the-middle optomechanics 43 Teff(P) = R∞ −∞Sxx(ω, P)dω2π R∞ −∞Sxx(ω, 0)dω_2π ·T0

where T0is the ambient mode temperature, and P is the cooling laser power.

I.e., we find the surface integral relative to Sxx(ω, 0) for which we know the

temperture T0=293 K.

Figure 2.15: As the laser power goes up, the optical damping increases (Equation (2.14)). Therefore, the effective mode temperature decreases. The errorbar denotes 2

σin y-direction and an estimated error of 5 µW on the x-axis. The y-error was extracted

from the residuals of the least-squares-fit.

2.4.1 Concluding remarks

Data analysis was also performed by Frank Buters, the extracted values match the ones that we found here (Figure 2.16).

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Figure 2.16:(from Frank Buters) Analysis of the same data as in Figures 2.13, 2.14. In the

(left)figure, the laser power has been exchanged for the effective damping (Γeff = Γ_Γopt_m).

The data then fits Teff ∼Γeff−1(Equation (2.27)).

The previous - mirror-at-the-end - setup displayed superior damping char-acteristics versus laser power (Figure 1.3 and 1.5). For a comparableΓm

(Equa-tion (2.27)), Eerkens et al. were able to cool down the mechanical mode of the trampoline resonator sub 10 K with 3 µW laser power [8].

The reason for the superior optomechanical performance of the previous de-vice is twofold: 1) the finesse of the MIM system was one order lower (Equation (2.30), Figure 3.27), and 2) the linear frequency pull parameter G (Equation (2.3)) is much lower for our system (Equation 2.14, Figure 3.19). Note that we did not attempt to optimize G for this experiment.

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Chapter

3

Optics

In this chapter, we discuss the optics of the device in relation to the the speci-fications of the system. In particular, in regards of the length of the cavity and the position of the membrane.

We study the transverse and longitudinal aspects of the optics inside the cavity separately. The combination thereof is complicated, but by the investi-gating two simple models we may still attempt to create a complete overview. Specifically, the study of longitudinal modes provides information on the effect of membrane displacements on a scale defined by the wavelength of the cavity mode, whereas, by discussing transverse modes we may understand the effects of system design on a larger scale.

This section is concluded by a measurement of the optical finesse with and without a membrane inside the Fabry-P´erot cavity.

3.1 Transverse modes in a Fabry-P´erot cavity

In this section, we discuss the beam optics inside the Fabry-P´erot (FP) cavity. This concerns the optics besides some wavelength-frequency oscillatory part ei(ωt−kz)_{. The implications of this oscillation are discussed in chapter 3.2 instead.}

In particular, in this section we are interested in finding the eigenmodes of the cavity. These are fields that retain their shape up to a (complex) constant upon a round trip, which implies that they can be trapped effectively inside the cavity. In addition, we discuss how the cavity geometry affects the shape of these modes and how this influences scattering losses. Lastly, we consider how a laser is best coupled to the system, again in relation to the shape of the eigenmodes.

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its action on an electro-magnetic field. In general, this problem will have the following mathematical form for the complex field amplitude upon a round trip [20] ˜ Enm(1) =e−2ikL Z Z plane ˜ K(x, y, x0, y0)E˜ (0) nm(x0, y0)dx0dy0 =e−2ikLγnm˜ E˜nm0 (3.1)

where ˜γnmis the complex eigenvalue of mode ’nm’, and L the length of the FP

cavity, and ˜K the kernel of the system.

The kernel ˜K describes the evolution of the field from ˜E(0) to ˜E(1) upon a round trip, evaluated at some plane in the system. Typically, ˜K changes its form depending on the choice of the location of this plane [20].

Clearly, it is anything but trivial to find an expression for ˜K, nor to construct eigenmodes from it. Luckily, there exist a class of wave forms for which the action of the idealized FP cavity can be described by a simple 2 x 2 matrix, such that the eigenmodes are easily recognizable within this class. This class is characterized by the fact that its elements are solutions of the paraxial wave equation∗.

3.1.1 Paraxial wave equation

The derivation of the paraxial wave equation will give us insights in the way its solution space is restricted as opposed to the general wave equation, and in the extend of its validity.

Throughout this chapter, we will restrict our attention to the electric field components of waves, since the magnetic field can be recovered by using Maxwell’s equations in vacuum ~ ∇x~B = 1 c2 ∂~E ∂t.

The time evolution of electro-magnetic fields is described by the wave equa-tion [10] ∇2~E= 1 c2 ∂2 ∂t2 ~_E

with c the speed of light in vacuum.

A complete orthonormal basis of the linearly-polarized solution space of the wave equation is given by plane waves

∗_{Note that small opening angles (= paraxial) are implied for these beams by the fact that the}

A membrane-in-the-middle device for optomechanics

A membrane-in-the-middle device

for optomechanics

A membrane-in-the-middle device

for optomechanics

Elger Vlieg

Abstract

Contents

Chapter

1

Introduction

1.1

The limits of quantum theory

1.2

Optomechanics

1.3

Relevance of this project

Chapter

2

Optomechanics

2.1

Classical equations of motion

2.1.1

Mirror-at-the-end

2.1.2

Membrane-in-the-middle

2.1.3

Optical damping

∑

2.2

Optomechanical parameters Q and

F

2.2.1

The significance of Q

∑

∑

∑

2.2.2

The significance of

F

2.3

Cat state generation by mechanical mode

swap-ping

∑

2.4

Experimental membrane-in-the-middle

optome-chanics

2.4.1

Concluding remarks

Chapter

3

Optics

3.1

Transverse modes in a Fabry-P´erot cavity

3.1.1

Paraxial wave equation