Final Thesis

i

Quantum Studies of Optomechanical Oscillators

SADIQ NAWAZ KHAN

Department of Physics & Applied Mathematics

Pakistan Institute of Engineering and Applied Sciences (PIEAS)

Nilore, Islamabad 45650, Pakistan

October, 2012

ii

Quantum Studies of Optomechanical Oscillators

SADIQ NAWAZ KHAN

M. Phil Physics

A Thesis Submitted to the faculty of Applied Sciences of

Pakistan Institute of Engineering and Applied Sciences (PIEAS)

in partial fulfillment of the Requirement for the degree of

M.Phil. Physics

Department of Physics & Applied Mathematics

Pakistan Institute of Engineering and Applied Sciences (PIEAS)

Nilore, Islamabad 45650, Pakistan

October, 2012

iii

iv

Department of Physics & Applied Mathematics

Pakistan Institute of Engineering and Applied Sciences (PIEAS)

Nilore. Islamabad 45650, Pakistan

Declaration of Originality

I hereby declare that the work contained in this thesis and the intellectual content of

this thesis are the product of my own work. This thesis has not been previously

published in any form nor does it contain any verbatim of the published resources

which could be treated as infringement of the international copyright law.

I also declare that I do understand the terms ‘copyright’ and ‘plagiarism,’ and that in

case of any copyright violation or plagiarism found in this work, I will be held fully

responsible of the consequences of any such violation.

Signature: _______________________________

Name: Sadiq Nawaz Khan

Date: ____________________

Place: ____________________

v

Department of Physics & Applied Mathematics

Pakistan Institute of Engineering and Applied Sciences (PIEAS)

Nilore. Islamabad 45650, Pakistan

Certificate

This is to certify that the work contained in this thesis entitled: Quantum Studies of

Optomechanical Oscillators was carried out by: SADIQ NAWAZ KHAN, and in

my opinion, it is fully adequate, in scope and quality, for the degree of M.Phil.

Physics.

Approved by:

Supervisor:________________

Dr Shahid Qamar

Co-Supervisor:__________________

Muhammad Irfan

Head DPAM: ____________________

Dr Masroor Ikram

(October, 2012)

vi

Dedicated to

The memories of my Mother,

To my Family

and

all Special People in my life.

vii

Acknowledgements

I humbly thank Almighty Allah, The Merciful, The Beneficent, Whose bountiful

blessings made this job easier for me. I am very thankful to Dr. Shahid Qamar for

his excellent supervision, fruitful suggestions, cooperation, proper guideline and

encouragement during the period of this project. I would like to thank my co-

supervisor Mr. Muhammad Irfan for his helpful discussions and cooperation.

Special thanks to Dr. Sikander Majid Mirza for his precious and fruitful

suggestions. And at the end I would thanks my class fellows and my family members

whose encouragement, prayers and love was always there to keep my moral high

during the research period.

SADIQ NAWAZ KHAN

Nilore, Islamabad

October 2012

viii

Table of Contents Dedicated to ................................................................................................................ vi

Acknowledgements ...................................................................................................... vii

List of Figures ................................................................................................................ x

Abstract ......................................................................................................................... xi

Chapter 1 ...................................................................................................................... 12

Introduction ................................................................................................................ 12

1.1 Introduction ........................................................................................................ 12

1.2 Recent Developments ......................................................................................... 15

1.3 Thesis Layout ..................................................................................................... 17

Chapter 2 ...................................................................................................................... 18

Harmonic Oscillators ................................................................................................. 18

2.1 Theory of Classical Harmonic Oscillator ........................................................... 18

2.2 Quantum Harmonic Oscillator ........................................................................... 20

2.3 Radiation Pressure .............................................................................................. 22

Chapter 3 ...................................................................................................................... 24

The Micromechanical Resonators ............................................................................ 24

3.1 The system.......................................................................................................... 24

3.2 Dynamics of the System ..................................................................................... 25

3.3 Physics of the Laser Cooling .............................................................................. 28

3.4 Steady-State Solution ......................................................................................... 28

3.5 Equations for the Quantum Fluctuations ............................................................ 30

3.6 Stability of the system (Routh-Hurwitz Criterion) ............................................. 32

Chapter 4 ...................................................................................................................... 38

Spectrum and Effective Temperature of the System .............................................. 38

4.1 Position and Momentum Spectrum of the Resonator......................................... 38

Chapter 5 ...................................................................................................................... 46

Results and Discussions ............................................................................................. 46

Appendix A .................................................................................................................. 47

A.1 Parametric Amplification .................................................................................. 47

Appendix B .................................................................................................................. 52

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B.1 The equations of motion .................................................................................... 52

Appendix C .................................................................................................................. 54

C.1 The Steady-State Solution ................................................................................. 54

Appendix D .................................................................................................................. 56

D.1 The Stability Criterion ....................................................................................... 56

Bibliography ................................................................................................................ 58

VITA............................................................................................................................ 62

Appendix E .................................................................................................................. 63

x

List of Figures

Figure 1. The above fig. shows a Fabry-Perrot Cavity whose one mirror is heavier and

thus is stationary while the other is light and can vibrate due to radiation pressure and

thermal Brownian motion……………………………………………………………...4

Figure-2. Sketch of the cavity used to cool a micromechanical mirror. The cavity

contains a nonlinear crystal (OPA) which is pumped by a laser to produce parametric

amplification……………………………………………………………….…………15

Figure-3. Plot of the position of the mirror against the detuning when no OPA is

there………………………………………………………………………………..…27

Figure-4. Plot of the position of the mirror against the detuning when OPA is

there……………………………………………………………………………......…28

Figure-5. Plot of the Position spectrum against the frequency of the resonator. The

peaks determine the frequency range in which the resonator shows more

response………………………………………………………………………...…….33

Figure-6. A “zoomed in” view of the frequency response of the system. This narrow

region is of practical importance in calculating the integral of the spectrum (Discussed

in the next chapter)…………..…………………………………………………….…34

Figure-7. A plot of the denominator function. Its minimum is the position on the

frequency axis where the position spectrum has its

maximum……………………………………………………………………………..34

Figure-8. The effective temperature of the system verses the cavity detuning for

G=0……………………………………………………………………………..……36

Figure-A.1. The signal and idler waves satisfying the Phase wave Mixing

Condition…………………………………………………………………………..…41

xi

Abstract

In this thesis, we have studied the quantum mechanical features in an optomechanical

resonator. In particular, we considered a nanograms size micromirror inside a Fabry-

Perrot cavity which is coupled with a thermal reservoir. The coupling makes the

mirror to oscillate, as a result system exhibits the behavior similar to the harmonic

oscillator. Next a strong classical laser field of constant phase is applied to drive the

cavity. The cavity contain an optical parametric amplifier (OPA) i.e. a non-linear

crystal, which by parametric down conversion gives two photons for a single input

photon. The field inside the cavity which is considered to be quantized is also coupled

with the harmonic oscillator by radiation pressure. Under the action of these two

forces we have discussed a set of controllable parameters that gives us a stable

solution for the resonator’s position and momentum spectrum. The stability of the

system is discussed explicitly for both with and without OPA. We solved the

equations of motion for the system and find the region of cavity detuning that

corresponds to the minimum temperature.

12

Chapter 1

Introduction

1.1 Introduction

The fact that light can exert a force (by radiation pressure) is an idea that typically

originated from Kepler in the 17th century as a natural consequence of the (incorrect)

corpuscular theory of light. This theory was supported by Newton as an explanation

for the tilt in the comet’s tails. Later, Euler showed the existence of a repulsive

radiation force in the context of the longitudinal wave theory of light. Attempts were

made to measure the strength of this radiation force in the 18th century, but they

proved inconclusive [1]. The mechanical effects of electromagnetic radiation

(radiation pressure) were theoretically derived by Maxwell in his fundamental work

on electromagnetics (1873) but were experimentally confirmed a century ago (1901)

by Lebedev using a carefully calibrated torsion balance. This was independently

verified by Nichols and Hull (1903) [2,3]. Saha published a paper that suggested the

quantization of the momentum of light in relation to radiation pressure. This was

verified in 1923 by the experiment of Compton's scattering [4].

This radiation pressure was minute enough to be utilized at that time. Then

with the discovery of powerful and coherent light sources like lasers (which are

highly focusable as well), the radiation pressure became useful. It was then used for

trapping of very light particles (molecules, atoms, ions), cooling down gasses and also

for the formation of Bose-Einstein condensate [5-11]. With the improvements in the

micro-fabrication techniques, we are now able to control the dynamics of

nanomechanical systems (oscillators) with this radiation pressure. These

nanomechanical oscillators range from nanometers to centimeters in sizes and weighs

in nanograms. The effect of radiation pressure on massive resonators is negligible.

Radiation pressure is used to cool down these oscillators to their quantum

ground state. This field of physics has been named as Optomechanics or cavity-opto-

mechanics [12-16]. A large range of applications employees mechanical resonators of

micro- and nanometers size, more frequently as sensors used in integrated electronic,

optical, and optoelectronical systems [17-20]. Changes in the motion of the resonator

are detected to a high degree of sensitivity by observing the radiation (or electronic

13

current) that interacts with the resonator. For instance, very small masses can be

detected by quantifying the change in frequency induced on the resonator, while small

displacements (or weak forces causing such types of displacements) can be calculated

by measuring the resulting shift in the phase of the light interacting with it (as in

gravitational waves detection) [21]. These resonators are continuously under the

action of thermal noise, which is due to the coupling with internal and/or external

degrees of freedom and is an important factor that limits the accuracy and sensitivity

of these devices. To get rid of this noise, these resonators must be cooled down.

Cooling of micromechanical resonators is a very important topic in many

fields of physics because it can help us in the development of ultra-high precision

measurements (in atomic force microscopes) [22], in the detection of gravitational

waves 23,24] and in the study of the transition of classical and quantum mechanical

behavior of a mechanical system (also called de-coherence) [25-27]. Radiation

pressure cooling of micromechanical resonator inside a Fabry-Perrot cavity is used for

many interesting applications, for example, to detect the quantum noise in the position

of a micromechanical resonator, and the phase noise of the quantized field in a Fabry-

Perrot cavity, in the designing of electromagnetically induced transparency and in the

generation of the entanglement of the micromirror with the field [28].

It is well known that when a photon strikes a mechanical surface, it imparts a

momentum of to the mechanical surface and reflects back. This imparted

momentum (or force) is very small however it becomes important in strong fields as

observed during the life cycle of a star. Stars host a very large scale of nuclear

reactions and due to these reactions some of the energy is emitted in the form of

radiation. This radiation imparts pressure on the gas particles of the star and stops

them from collapsing due to gravity. This pressure gives the star a life in billions of

years (which mean very stable system under the action of radiation pressure). In the

same way the radiation pressure can be used to control the motion of nanoscale

mechanical oscillators. Controlling the motion means restricting the oscillations of the

resonator to the smallest possible level so that the temperature of the resonator is

reduced to sub-kelvin domain [29]. Braginsky et al [30] (while working on the

detection of gravitational waves) were among the pioneers to detect the damping of an

oscillator due to radiation pressure. In recent years the search for obtaining the

quantum states on a macro-scale has spread beyond the gravitational wave detection

14

physics regime to a much wider area [31]. Recent studies has confirmed not only the

detection and measurement of Brownian noise due to thermal motion (very weak

force to measure) but also the mechanical damping of this motion by radiation

pressure[32]. All such effects are studied in a Fabry-Perrot cavity.

A typical Fabry-Perrot cavity is shown in Fig. 1. One mirror is heavier and

thus stationary. The other mirror is lighter (nanograms range) and therefore oscillates

due to coupling with thermal bath. The oscillations are along the cavity axis as shown

in Fig. 1. Radiation field of wavelength oscillate back and forth along the cavity

axis. The photons impart radiation pressure on the mirrors each time they strike the

surface. The cavity is high finesse cavity, which means that the field decay rate from

the cavity is small. Due to this small decay rate, the number of photons inside the

cavity is high and thus the radiation becomes significant. When laser field is shin on

the cavity, light inside the cavity starts reflecting back-in-forth. As a result of these

continuous reflections, the motion of the light mirror (also called cantilever) is

affected. The radiation pressure also changes the length of the cavity which in turn

changes the fundamental cavity mode. The intensity inside the cavity is also affected.

As a result we get optomechanical entanglement [29]. The oscillating mirror is treated

as quantum harmonic oscillator. As discussed above that the changing position of the

mirror changes the intensity of the field, thus the mirror’s position also changes the

effective mechanical force on the mirror. Under these conditions, the spring constant

modifies which is known as optical spring effect.

Due to this, the effective potential (

, with k the spring constant)

governing the dynamics of the movable mirror is also modified resulting in a multi-

stable solutions of the mirror’s position.

15

Figure -1. The above fig. shows a Fabry-Perrot Cavity whose one mirror is heavier and thus is

stationary while the other is light and can vibrate due to radiation pressure and thermal Brownian

motion.

For example bi-stability and tri-stability is observed [33]. The radiation force

acts on the mirror with a time lag. The details of this process is very complex however

due to this lagging (phase difference in the acting forces), there is always a negative

radiation force acting on the mirror that slows down the mirror i.e. ∮F.dx < 0. This is

called back action or self-cooling [29]. Due to back action the mirror’s effective

temperature is reduced. The mirror’s effective temperature can be reduced to milli

Kelvin if the thermal bath is at 1 K.

In this thesis we have considered a harmonic oscillator, which is basically one

mirror of a Fabry-Perrot cavity. One mirror is light (i.e. 10-12

kg) while the other

mirror is heavy. Radiation pressure can only affect the motion of light mirror, which

is coupled with a thermal bath. We also couple the mirror with radiation field, coming

from a laser. Under the action of these two forces, we look for a stable solution of the

system which corresponds to the lowest temperature. An optical parametric amplifier

is kept inside the Fabry-Perrot cavity to enhance the cooling effect. Both cases

(system without an optical parametric amplifier and with parametric amplifier) are

discussed. Stability conditions are also discussed for both of the cases.

1.2 Recent Developments

In the recent years, many experiments have been performed in an attempt to reach the

quantum ground state of such oscillators using the concept of radiation pressure.

Table 1 summarizes some of these results. The first optomechanical oscillator was

cooled to milli Kelvin temperature by coupling its 6 GHz mechanical mode. And this

16

temperature corresponds to a phonon number of around 0.07. The second oscillator

was an electromechanical oscillator and it was cooled to a phonon number of 0.34.

But scientists accept the lowest achieved phonon number (so far) of 9.

In Table 1, the temperatures are represented in terms of the phonon occupation

number ñ in the system (as ñ ω ≈ kBT ).

Table 1. A brief summary of the recent work in the field of laser cooling of optomechanical

oscillators is given here. Tb is the thermal bath temperature in kelvins [34].

Research Group

Tb [K] ñ Ref.

Cohadon et al. (1999) 300 8.2x105 [34]

Arcizet et al. (2006) 300 2.6 x105 [35]

Gigan et al. (2006) 300 6 x105 [36]

Schliesser et al. (2006) 300 4 x103 [37]

Naik et al. (2006) 0.003 25 [38]

Kleckner and Bouwmeester (2006) 300 2.3 x105 [39]

Corbitt et al. (2007) 300 8 x106 [40]

Poggio et al. (2007) 2.2 2.3 x104 [41]

Brown et al. (2007) 300 1.3 x108 [42]

Groblacher et al. (2008) 35 1 x104 [43]

Schliesser et al. (2008) 300 5.2 x103 [44]

Thompson et al. (2008) 300 1.1 x103 [45]

Vinante et al. (2008) 4.2 4 x103 [46]

Teufel et al. (2008) 0.05 140 [47]

Groblacher et al. (2009) 5 32 ± 4 [48]

Schliesser et al. (2009) 1.65 63 ± 20 [49]

Park and Wang (2009) 1.4 37 [50]

O'Connell et al. (2010) 0.025 <0.07 [51]

Rocheleau et al. (2010) 0.146 3.8 ± 1.3 [52]

Riviere et al. (2011) 0.6 9 ± 1 [53]

Teufel et al. (2011) 0.015 0.34 ± 0.05 [54]

Chan et al. (2011) 20 0.85 ± 0.08 [55]

Verhagen et al. (2011) 0.65 1.7 ± 0.1 [56]

17

1.3 Thesis Layout

In this thesis Chapter 1 discussed the history and introduction of the micromechanical

resonators. Chapter 2 introduces the mathematical model of harmonic oscillators due

to their importance in the understanding of optomechanical resonators. Our system is

basically a quantum mechanical harmonic oscillator, thus it is necessary to briefly

discuss harmonic oscillators first. A brief discussion about classical harmonic

oscillators and quantum harmonic oscillators is given in chapter 2. Details about the

mathematical derivations are given in the Appendices B, C and D. A detailed

discussion on the mathematical model of the system is presented in Chapter 3. The

process of degenerate parametric down conversion (which is a special case of

parametric down conversion) is briefly discussed in Appendix A. Appendix E

comprises of a technical paper. Most of the mathematical modeling of the system is

done in Chapter 3. Chapter 4 presents the system’s spectrum and also it’s effective

temperature. In the end results and conclusions are summarized.

18

Chapter 2

Harmonic Oscillators

2.1 Theory of Classical Harmonic Oscillator

Harmonic oscillator is fundamental concept in physics. Its understanding is necessary

as its model can be employed to explain various physical phenomena found in nature

e.g. molecular vibrational modes on the micro scale to binary star systems on macro

scale. Any system that involves a change in any of its physical parameters repeatedly

about a mean value can be classified as a harmonic oscillator.

Consider as an example the following system. A mass m is moving under a

restoring force F about a mean position. Let its initial position is qo. If we do not

consider any external interruption in our system (i.e. an isolated system), then from

newton’s second law of motion says

(2.1)

where F is the sole restoring force acting on the system, m is the mass of the system, q

is the position of the system and is the spring constant of the system. The solution

of this equation gives the equation of motion (here the trajectory) of the system,

which is given by

( ) ( ) (2.2)

Here is the phase and A is the amplitude of the system, which can be

determined from the initial conditions that we impose upon the system. is the

natural frequency of the system given by

√

, where is the time

period of the oscillation. The system has maximum amplitude if the applied force’s

frequency is equal to (resonance condition). The total energy of the system is

always constant although the kinetic and potential energies changes with time i.e.

( )

( )

(2.3a)

and

19

( )

( )

(2.3b)

which gives the total energy (independent of time)

( ) ( )

(2.3c)

The total energy is independent of time, as must be for any non-decaying

(isolated) system. This total energy is also called the Hamiltonian of the system.

For any real physical system, we must include all the decaying channels into

the Hamiltonian. For example, if the system is coupled to the environment (sink) by a

coupling constant “ᵧ” then its equation of motion becomes

(2.4)

The solution of Eq. (2.4) can easily be obtained and is given by

( )

[√

(

) ]

(2.5)

The quality factor of the oscillating system is represented by Q and is a

measure of how much vibrations the system do before its amplitude decays to 1/e of

its initial value. Mathematically it is equal to

. If the system is coupled to a

bath that is driving the system (a bath) with a force F(t) then the equation of motion of

the system becomes

( )

(2.6)

if we take F(t)=F0 sin( ) then we obtain the same solution as mentioned by Eqs.

(2.2) and (2.5), however the amplitude now takes the following values:

⁄

√( )

(2.7)

with its resonance near the fundamental frequency. Its value is

20

√

(2.8)

It is interesting to see the Fourier transform of the solution to the equation of motion

of the system which is given by [57]

( )

( )

(2.9)

with ( ) being the Fourier transform of the driving force of the bath. If our bath is a

thermal bath at a certain temperature T, then there will be a Brownian force (or

thermal force) Fth (t). The position spectrum of any system is given by

( ) ⟨ ( ) ( )⟩ (2.10a)

Using Eq. (2.9) in Eq. (2.10a) (by taking Fth (t) constant), we obtain

( )

( )

(2.10b)

This position spectrum (Eq. (2.10b)) gives us ⟨ ⟩ by solving the following equation

⟨ ⟩ ∫ ( ) ( )

(2.11)

This is an important result that helps us to calculate the temperature T of the system.

From Equi-partition theorem, we have

⟨ ⟩ (2.12)

which is the temperature of the classical harmonic oscillator. Next we discuss the

quantum mechanical model of the harmonic oscillator.

2.2 Quantum Harmonic Oscillator

Quantum harmonic oscillator is a very fundamental concept and is important to be

discussed here as our main topic involves its study. The mathematical derivations start

from the Hamiltonian of the harmonic oscillator given in Eq. (2.3c). Introducing the

operators p, q for the physical quantities like position and momentum. The symbol

“q” is used for position in both classical and quantum mechanics but for momentum

(which is in classical mechanics), we use

, where is the

reduced Planck’s constant. Then Eq. (2.3c) takes the form

21

(2.13)

If we write the operators p and q in creation ( ) and annihilation ( ) operator

form then we have

√

( )

(2.14a)

and

√

( )

(2.14b)

It can easily check that [ ] and [ ] .

In terms of these operators our Hamiltonian takes the form

(

)

(2.15)

where we have used with n being the energy quantum state of the oscillator.

The wave functions corresponding to this energy are called Eigen functions of

the harmonic oscillator.

It is clear from Eq. (2.15) that the harmonic oscillator has energy even if it is

in the ground state of energy, which is known as the zero-point energy of the

oscillator. The wave function of the system in this energy state is the ground state

wave function given by [58]

( ) (

)

⁄

(

)

(2.16)

while the general solution for any energy is

( ) √

( ) ( )

(2.17)

22

The quantized electromagnetic radiation field can be represented by the

Hamiltonian of Eq. (2.15) such that and are the lowering and raising operators

of the field and n is called the number state of the electric field [59].

2.3 Radiation Pressure

Light is electromagnetic radiation that can exert pressure on material objects. Chapter

1 describes the developmental history of the mathematical background of radiation

pressure theory but for a systematic study it is discussed again briefly. Radiation

pressure of light is in fact very weak for small intensities but when the intensity is

large, its effects are phenomenal. The whole life cycle of a star is determined by the

radiation pressure of light emitting from that star. Sometimes radiation pressure can

overcome gravitational force, as can be seen in supernovas and in the white tails of

comets.

Fabry-Perrot cavity is a small cavity (with dimensions in mm) with two

mirrors. One mirror is partially reflecting while the other is completely reflective. The

reflective mirror is light and is taken as a damped harmonic oscillator of resonance

frequency .

Light is shined from the outside by laser. Laser light is mostly monochromatic

(say wavelength λ) and therefore the cavity length L must be adjusted so that it

satisfies the standing wave condition

where n is any number. The standing

waves formed in the cavity reflect back and forth thus imparting radiation pressure to

the mirrors. This pressure affects the motion of the small mirrors. In this thesis we

study it quantum mechanically in chapter 3. Here is a brief classical description of

radiation pressure driven damped harmonic oscillator.

The equation of motion for the damped harmonic oscillator is given by Eq.

(2.6), however the source term is replaced by the sum of radiation pressure force

( ( )) and thermal force i.e.

( ) ( ( ))

(2.18)

23

Thus Eq. (2.6) takes the form

( ( ))

(2.19)

The susceptibility χ of this system is defined as the response of the system to

the acting force. Thus using the definition of force given by Eq. (2.18) we have

( ) ( ) ( ( )) (2.20)

This ( ) is called effective susceptibility in the presence of radiation

pressure which is different from the natural susceptibility of the system. The effect is

similar to the spring constant modified by the radiation field, called optical spring

effect. Due to this modified spring constant (or susceptibility), the temperature of the

system is affected which is known as effective temperature.

24

Chapter 3

The Micromechanical Resonators

3.1 The system

In this section we develop the mathematical model of a quantum oscillator interacting

with a quantized electromagnetic field inside a Fabry-Perrot cavity. The system is

depicted in Fig. 2. Our system comprises of a Fabry-Perrot cavity of length L. One

mirror is heavier and is therefore fixed while the other one is lighter one having mass

m in nanograms. The radiation pressure and thermal coupling do not have any effect

on the heavier mirror. Only the lighter mirror experiences their effect. This light

mirror is subject to two forces. This mirror can only oscillate along the cavity axis.

One force is due to the coupling with the thermal bath at temperature T (called the

thermal Langevin force) and the other force is due to radiation pressure of the laser

light (arising from the bouncing back of the longitudinal modes of radiation from the

light mirror).

Coupling of light and therefore radiation pressure imparted by the radiation is

directly dependent on the power of the laser. The small is the input power of the

pumping laser the smaller is the radiation pressure on the mirror (due to weak

coupling). If the input laser is removed (switched off), then the mirror is be coupled to

the thermal bath only (because the bath is at a certain temperature T), thus the

resonator will be executing pure Brownian motion [60].

The oscillating mirror is coupled to the thermal bath by a constant which is

the energy decay rate of the mirror. This is defined by

⁄ where is

the (angular) oscillation frequency of the mirror and Q is the quality of the cavity. The

cavity frequency in this case is taken to be the fundamental mode ωc , which is related

to the cavity length by

, where c is the speed of light. The power of the

driving laser is P, related to the electric field amplitude of the laser by √

,

where is the frequency of the driving laser. The power P is usually taken in milli

watts. For example our system uses a 4 mW Nd:YAG laser.

25

The mirror’s oscillations about its equilibrium position that changes the cavity

length which produces shift in the phase of the cavity field. We also let ,

this condition is called adiabatic limit which means that the frequency of the resonator

is very small compared to the cavity frequency range so that scattering of photons into

modes other than the fundamental mode are easily neglected.

The OPA (optical parametric amplifier) is a nonlinear crystal which converts a

photon of frequency ω into two photons of frequency

, a process called degenerate

down conversion. One of the generated photons is called signal photon and the other

the idler photon. G is the non-linear parametric gain of the OPA. is the vacuum

noise (operator) entering the cavity, while is the vacuum noise leaving the

cavity. This vacuum noise is coming from vacuum fluctuations and it adds to the

system’s disorder.

Figure -2. Depiction of the Fabry-Perrot cavity that is used for cooling an optomechanical oscillator.

The Optical Parametric Amplifier (OPA) is pumped via a laser to produce parametric amplification.

3.2 Dynamics of the System

To derive the equations of motion for the system, we must write the Hamiltonian of

the system first. The Hamiltonian of the system (in the frame rotating at frequency

, which is the driving laser frequency here) is given by [60]

( )

(

)

( )

( )

(3.1)

Here ωC is the cavity frequency, is the driving field frequency, is

Planck’s constant, is the photon number in the cavity given by , while is

26

the annihilation operator of the field and is the creation operator,

is the

coupling constant (different from given in Eq.(2.20)) of the radiation field and the

cavity mirror, L is the cavity length (in milli meters). √ is the classical

normalized field outside the cavity, is the photon decay rate due to leakage from

the cavity, m is the mass of the mirror, is the angular frequency of the mirror, G is

the non-linear parametric gain, θ is the constant phase of the driving laser. The

momentum and position operators of the harmonic oscillator (mirror) are p and q

respectively. The finesse F of the cavity is related with by .

The first term in Eq. (3.1) corresponds to the Hamiltonian due to the cavity

field, this is defined by Eq. (2.15) but we have dropped the constant zero point energy

term i.e.

( ) . Also we are now measuring the energy of the field in the

reference frame of the driving laser frequency . The second term is coming due to

the coupling of the movable mirror with the cavity field (via radiation pressure), the

3rd term is the Hamiltonian of the movable mirror (with p and q momentum and

position operators of the mirror), the 4th term is the contribution of the coupling of the

driving laser and the quantized field of the cavity, while the last term is coming from

the interaction of the OPA with the cavity field. The nonlinearity of the parametric

down conversion is clear from the last term of the Hamiltonian i.e. it contains and

.

The equations of motion can be derived using Heisenberg Picture. In

Heisenberg picture, the equation of motion of an operator O is given by

[ ]

(3.2a)

In our case the system is open system because there is a leakage from the

cavity’s partially reflecting mirror. Also thermal bath is coupled to the resonator

which affects the systems dynamics. The fluctuation associated with the thermal bath

and dissipation (due to the leakage of the field through the cavity) affects the

dynamics of the system. Thus we have to use a more proper set of equations that can

also include the leakage and the thermal coupling of the system in the equations of

motion otherwise our study about the system will be incomplete. This is done by

introducing the non-linear quantum Langevin equations given below [61]

27

[ ] (3.2b)

This is specific for each operator. It can be zero for some

operators. Using Eq. (3.2b), we get the following equations of motion for the position

and momentum operators of the resonator and raising and lowering operators of the

cavity field (see Appendix B for details)

,

,

( ) √

( )

√

(3.3a)

(3.3b)

(3.3c)

(3.3d)

Eqns. (3.3) are the quantum Langevin equations for the system. Remember

that is there because of the viscous damping force that damps the mechanical

mode of the cantilever. In the above equations, which is the input vacuum noise

operator, satisfies the following correlation [62]

⟨ ( )

( )⟩ ( ) ,

⟨ ( ) (

)⟩ ⟨ ( ) (

)⟩ .

(3.4a)

(3.4b)

The mean value of this operator is zero, i.e. ⟨ ⟩ ⟨ ⟩ .

In Eqns. (3.3), is the Brownian noise operator (thermal force operator), it

represents a Gaussian quantum stochastic process and it has a non-Markovian

behavior (which means that neither its correlation function nor its commutator gives a

Dirac delta function), it is coming from the interaction of the thermal bath and the

mirror. This is a stochastic force and the mean value of this operator is also equal to

zero ⟨ ⟩ and it satisfies the following correlation for the bath temperature T [63]

28

⟨ ( ) ( )⟩

∫ ( ) [ (

) ] , (3.5)

where is the Boltzmann’s constant. We have taken the cut-off frequency of the

bath power spectrum to be infinite.

3.3 Physics of the Laser Cooling

The cooling of the mirror by the cavity field can be explained in the following

thermo-dynamical terms. The cavity field pressure is coupled with the cantilever by

the coupling constant . This cavity field serves as an effective reservoir for the

resonator when we have some detuning. This extra reservoir helps the resonator attain

an effective temperature that is less than the temperature of the thermal bath. In this

way by changing the cavity field parameters one can in principle achieve the

temperature which corresponds to the ground state of the resonator.

Adjusting cavity parameters include making (coupling of radiation pressure

to mirror) very larger than the decay rate (coupling to thermal reservoir). When the

coupling is strong, the cooling effect is also very strong. One can also increase the

coupling by increasing the strength of the cavity field. And this is possible if the

cavity is of high finesse and also the power of the input laser is large. With strong

cavity field , we get the steady-state solution for the system in equilibrium.

The radiation pressure force can be found by taking the spatial derivative of the

optomechanical coupling part of Eq. (3.1) i.e.

⟨ ⟩ ⟨

⟩, (3.6)

where is the 2nd

term of Eq. (3.1) and the space dependence is there in

due to optomechanical coupling. This force (Eq. (3.6)) is responsible for the multi-

stability, optically induced rigidity and the backaction cooling of the mirror.

3.4 Steady-State Solution

Now we find the steady-state solutions to the above equations. To do so, we take the

Taylor series of about , of about , of about , of

about and keep only the first two terms (0th

and 1st power of )

29

,

,

,

(3.7a)

(3.7b)

(3.7c)

(3.7d)

and now substitue these in Eqns. (3.3) and then equate the time derivatives in Eqns.

(3.3) to zero. Then we compare the same powers of on both sides to obtain (see

Appendix C)

,

,

,

(3.8a)

(3.8b)

(3.8c)

where

,

(3.9)

and

( )

.

(3.10)

Here is called the effective detuning of the cavity. This is the cavity

detuning modified by the radiation pressure. It is obvious from the above equation

that the motion of the resonator is affected by the field inside the cavity. And also the

interaction of the mirror with the field is changing the field and we get a new

stationary value for the field. The system attains these stationary values after a

specific transient time which is determined by the response of the system and the

strength of the coupling constants.

In Eqn. (3.8), is the new equilibrium (mean) position of the oscillator with

respect to the position in the absence of the radiation coupling. Also is the steady-

state value of the field inside the cavity. If the input field is zero, then the equilibrium

position of the oscillating mirror will be located at x=L, where x-axis is the cavity

axis. Both equations for and are not linear, this is because of the coupling of the

30

field with the cantilever. Both equations are 5th

order which means they have five

solutions. But here only three of the solutions are stable and the two are not stable.

If we choose and G=0 in the above equation for , then we get single

solution for . The system attains this steady-state situation after the transient time

of the system. During this transient time, the system experiences fluctuations in all the

above operators and the stability of the system can be derived from the equations of

the fluctuation operators.

3.5 Equations for the Quantum Fluctuations

To obtain the equations for the fluctuations in the operators, we substitue the

following values for the respective operators in Eqns. (3.3)

,

,

,

.

(3.11a)

(3.11b)

(3.11c)

(3.11d)

With , , and taking (thus neglecting terms containing

) we get the linearized equations

,

(3.12a)

as , so we get

.

(3.12b)

This is the linearized equation of motion for the fluctuations in the position of

the nanomechanical harmonic oscillator. Similarly

( ) (

)( )

( )

(3.13a)

again neglecting the non-linear terms (involving

and ), we

are left with

31

(

) , (3.13b)

and by following the above rules, we get for the field operators

√

(3.14a)

and

√

(3.14b)

Equations (3.12), (3.13) and (3.14) are the linearized quantum Langvin

equations for the fluctuations in these operators. These give us a crude picture of the

system’s dynamics during the transient time.

Next we introduce the cavity field quadrature

,

( ),

(3.15a)

(3.15b)

and

( )

(3.16a)

(3.16b)

Eqns. (3.16) can be written in terms of the following system of linear equations

( ) ( ) ( ), (3.17)

where ( ) is the fluctuations vector and ( )is the noise vector whose transposes are

given below.

( ) ( ),

and

( ) ( √ √ )

(3.18a)

(3.18b)

And the coefficient matrix is given by

[

( )

( )

( ) ( )

( ) ( )]

.

(3.19)

32

This matrix gives us information about the stability of the system and is thus very

important to our study.

3.6 Stability of the system (Routh-Hurwitz Criterion)

The stability of any system is most important feature to be taken care of. Any

physically unstable solution for a system is identified and excluded in laboratory

experiments. The stability of our system can be checked from the Eigen values of the

matrix A. The Eigen values can be calculated from the characteristic equation of

matrix A by using

( ) , (3.20)

where I is a 4×4 identity matrix and is the Eigen value of the matrix A.

By substituting Eq. (3.19) in Eq. (3.20) and expanding, we get the characteristic

equation for the matrix A. This is a polynomial of of order 4. This equation is

complicated enough to be solved for .

[ ]

[ ]

( )

[ ]

[ ] (

[ ] [ ] ) (

[ ] [ ] )

( )

[ ](( ) )

[ ](( ) )

(3.21)

The roots of (Eigen values of A) comes out to be rather complicated. In

order to find the stability of the system, here we are following Routh-Hurwitz

criterion. From linear algebra we know that the system will be stable if and only if the

real parts of the Eigen values of A are negative. We have a useful criterion for the

stability of the system and that criterion does not involve direct calculation of the

Eigen values of A. This criterion is called Routh-Hurwitz criterion. According to this

criterion the coefficients of the various powers of can give us information about the

stability of a system. First all the coefficients of various powers of must be positive,

and then these coefficients must satisfy some ratios [64].

33

To find the stability criteria for the system, we first find the coefficients of

various powers of from the characteristic Eq. (3.21). The coefficients of various

powers of are given below

The coefficient of is

, (3.22a)

The coefficient of is

. (3.22b)

That of is

(3.22c)

That of is (3.22d)

The constant is a4 (coefficient of 0th

power of ), given by

[ ]

[ ]

( )

[ ](( ) )

[ ](( ) )

.

(3.22e)

These coefficients must satisfy the following conditions (inequalities)

, (3.23a)

This condition is satisfied by the coefficient .

The next condition is

, (3.23b)

which is also satisfied.

For the remaining conditions, we have to consider the following ratios

, (3.24a)

and

( ) , (3.24b)

and

34

(

)

(3.24c)

These ratios must obey

,

and

, which on solution gives the

following conditions imposed on the various parameters of the system for stability

(see Appendix D).

( ) ( ) , (3.25a)

( ) (

(

)

(

)

)

{( )

( )( )

[ ( )

]}

(3.25b)

and

( )

(

)

(

)

(3.25c)

Eqns. (3.25) are the conditions that must be satisfied by the system parameters

make the system stable. Eqns. (3.25) provide us with a broad range of parameter

values (of the system) we can use. If we operate our system in this window of

numerical values, we can achieve a stable cooled system. If there is no coupling (in

the absence of field inside the cavity), then and also thus the

stability conditions reduce to

, (3.26)

or the threshold condition (when the parameters just satisfy the criterion) for the

parametric oscillations is

, (3.27)

our simulations will be based on the fact that Eq. (3.26) is always obeyed.

35

If we remove the OPA from the system, then G=0, and Eqn. (3.25) reduces to

the stability conditions derived by Peternostro et. al. in [61], i.e.

( ) ( ) ,

(3.28a)

( ) (

)

{( ) ( )( )

[ ( )

]}

(3.28b)

and

( )

(3.28c)

As an example of the conditions imposed on the stability of our system by

these equations, we chooses some parameters by our choice (which are practically

realizable in the laboratory), then these stability conditions will tell us about the

region of frequency detuning in which this system will be stable. Let the cavity decay

, there is no parametric amplifier, then , mass of the mirror is

, the driving laser wavelength is

, the power of the

driving laser is , the natural frequency of the mirror is

,

and length of the cavity is , then according to conditions of equations

(3.25), the frequency detuning between the cavity and driving laser must be

.

For these parameters, we go back to Eqns. (3.8) with the approximation

for which we have a single solution for q. By plotting against in the range

defined by Eqns. (3.25), we obtain Fig. 3, which shows the behavior of in the

stability region. Due to the increase the detuning, the position of the resonator is

confined more and more.

36

Figure -3. Plot of the position of the mirror against the detuning when no OPA is there.

The above Fig. 3 shows the detuning dependence of the position of the

micromechanical resonator with G=0. When we introduce the optical parametric

amplification process (by keeping a nonlinear crystal inside the Fabry-Perrot cavity

and thus G non zero), we can change the oscillations amplitude very much. When we

take , ,

, ,

,

, K and G = 3.5× 107 then Eqns. (3.25) restrict the detuning to

.

The plot for verses in this case is given in Fig. 4. The confinement

scale of the position of the mirror is different in this case from Fig. 3. However the

temperature of the system is now much decreased (we will discuss it later) due to the

introduction of the OPA. This reduction in temperature shows that the fluctuations in

the position are now very small. Thus a large value of the position coordinate

(stationary equilibrium position) does not corresponds to large fluctuations in the

position.

37

Figure -4. Plot of the position of the mirror against the detuning when OPA is there.

Our next step is to find the spectrum of the system. Both the position and the

momentum spectrum tell us about the system’s behavior towards various cavity

detuning frequencies. These spectra are calculated in Chapter 4. After calculating

these spectra, we can calculate to temperature of the resonator.

38

Chapter 4

Spectrum and Effective Temperature of the

System

4.1 Position and Momentum Spectrum of the Resonator

In order to understand the dynamics of the system, we need to calculate the position

and momentum spectrum of the system. We have supposed that the frequency range

to which our system responds is extended to infinity. However there is a region in

which the response is maximum and then decreases very sharply to a value that is not

of much practical significance. Thus to know the region of frequency domain in

which the response is maximum, we must find the spectrum of the resonator. Both

position spectrum and momentum spectrum is of great importance. For this purpose

the Fourier transform of Equations (3.12), (3.13) and (3.14) is first calculated.

We know that Fourier transform of a continuous function ( ) is given by

( ) ∫ ( )

(4.1a)

and the Fourier transform of its derivative is

( ) ( ) (4.1b)

Applying these formulas to Equations (3.12), (3.13) and (3.14) we get

( ) ( )

(4.2a)

Similarly,

( ) ( ) ( ) ( )

( ( )

( ))

(4.2b)

and ( ) ( ) ( ) ( )

( ) √ ( )

(4.2c)

and

39

( ) ( ) ( ) ( )

( ) √ ( )

(4.2d)

These equations are coupled and we have to solve them simultaneously. On

solving Equations (4.2c) and (4.2d) simultaneously, we get the following

( )

, (4.3a)

where the constants a and b are defined below

(√ ⁄ ( ) √ √ ( )

√ √ ( ) √ √ ( )

( ) ( ) ( ) ( )

)

(4.3b)

and

( )

(4.3c)

Similarly

( )

(4.4a)

where

( √ √ ( ) ( ) ) (

)( √ √ ( ) ( )

)

(4.4b)

and (4.4c)

Substituting these values of ( ) and ( ) in Eq. (4.2b) and then solving it with

Eq. (4.2a) for ( ) gives the position fluctuations in the frequency domain

( )

( )([ ( ) )] ( ) √ {[(

) ]

( ) [( )

] ( )}),

(4.5a)

This is the fluctuation function of the position in frequency domain, with

( ) (

) (

)[ ( ) ]

(4.5b)

In Eq. (4.5a), the first term is coming from the contribution of the thermal reservoir

(because is present in that term), and the second one is coming from the coupling of

40

the mirror with the cavity field (because is present in that term). These oscillations

are under the influence of thermal and radiation pressure.

If radiation pressure is removed then the mirror’s fluctuations will be pure

Brownian fluctuations. The fluctuation function of the position operator will then look

like the following

( ) ( )

[ ( ) ]

(4.5c)

It should be noted that the mirror follows the Brownian motion function because it is

taken as single compact object (motion of its constituent particles is not considered).

Now the spectrum is simply the Fourier transform of the correlations of the

position fluctuation function. Mathematically

( )

∫ ( ) ⟨ ( ) ( ) ( ) ( )⟩

(4.6)

Here we can use the following identity

⟨ ( ) ( ) ( ) ( )⟩ ⟨ ( ) ( )⟩ ⟨ ( ) ( )⟩

(4.7)

Then Eq. (4.6) takes the form

( )

∫ ( ) ⟨ ( ) ( )⟩

∫ ( ) ⟨ ( ) ( )⟩

(4.8)

We use the following correlations between the noise operators as given in [65]

⟨ ( ) ( )⟩

⟨ ( ) ( )⟩ ( ) ,

(4.9a)

(4.9b)

and

⟨ ( ) ( )⟩

⟨ ( )

( )⟩ ,

(4.9c)

(4.9d)

and also

41

⟨ ( ) ( )⟩ [ (

)] ( ) . (4.10)

On substituting these equations in Eq. (4.8) we get,

( )

( ) [ [( )

( )

( )]

[( )

] (

)]

(4.11)

This is the spectrum of the position fluctuations of the micro-mirror. One can

see the presence of in the first term referring to the contribution of radiation

pressure and the presence of temperature T in the second term showing contribution

from thermal reservoir.

Following the same procedure as we did for deriving Eq. (4.5a) and Eq. (4.11),

we can get the spectrum for the momentum fluctuations of the micro-mirror given by

( ) ( ) (4.12)

This spectrum is sometimes called density noise spectrum (DNS). This DNS

contain useful information about the system dynamics. The maximum of ( ) does

not occur at but is shifted. The reason of this shift is clear from Eq. (4.5b),

where the coupling to the thermal reservoir has brought the extra constant “ ” in the

denominator. One thing is clear, and that is the maximum of the spectrum will be at

that point at which we have a minimum of the denominator function in Eq. (4.5b) (see

Fig. 6. and Fig. 7. for example). The spectrum is plotted by using the approximation

(

)

, because at large temperatures .

If we take the values of Fig. 3 and Fig.4 except , G = 3.5×

107, then the spectrum looks like the following Fig. 5.

42

Figure -5. Plot of the Position spectrum against the frequency of the resonator. The peaks

determine the frequency range in which the resonator shows more response.

It is clear from Fig. 5 that the system shows maximum response in a very

narrow range of frequencies and is not responsive at all outside that range. A

magnified plot is shown in Fig. 6. We can see the narrow window in which the

response of the system is of practical importance.

The peaks are symmetric about zero. This symmetric behavior of the spectrum

is due to the fact that the spectrum function and the denominator function both are

even. The minimum of the denominator occurs exactly where the maximum of the

spectrum occur as shown in Fig. 7. It is also clear from Fig. 7 that the peaks are not

occurring at because of the coupling to the thermal reservoir.

The coupling has shifted these peaks away from zero.

43

Figure -6. A “zoomed in” view of the frequency response of the system. This narrow region is of

practical importance in calculating the integral of the spectrum (Discussed in the next chapter).

Figure -7. A plot of the denominator function. Its minimum is the position on the frequency axis where

the position spectrum has its maximum.

44

4.2 Effective Temperature of the Resonator

We have derived the position and momentum spectrum of the system in section 4.1.

Our next task is to find the effective temperature of the micromechanical mirror.

From thermodynamics and kinetic theory, we know that the temperature of a

body is related to the body’s degrees of freedoms by the Equi-Partition theorem.

According to Equi-Partition theorem each degree of freedom of a system that is

contributing a quadratic term of a coordinate or momentum to the total energy has an

average energy

. Mathematically

, (4.13)

with “ ” being the nth coefficient and x, y, …,t the position and/or momentum

coordinates and T is the temperature. The system that we are dealing with has two

degrees that contribute quadratic terms to the total energy (Hamiltonian of Eq. (3.1)).

Thus by the Equi-partition theorem the temperature of the micromirror is therefore

⟨ ⟩ ⟨ ⟩

(4.14)

This equation is for force-free, isolated system and our system is driven by

two forces and this formula is not applicable to our system. This is why our system’s

temperature is called effective temperature. In a driven system, we have

⟨ ⟩ ⟨ ⟩

, hence we cannot calculate the correct temperature of the system

using this relation. Thus we have to use the modified form of the above equation

given below to calculate the effective temperature [66]

⟨ ⟩ ⟨ ⟩

(4.15)

where

⟨ ⟩

∫ ( )

(4.16a)

and ⟨ ⟩

∫ ( )

(4.16b)

These Eqns. (4.16a) & (4.16b) are very difficult to solve analytically. Our

main task will be to find solutions to these equations (numerical). By definition, the

integration in Eqns. (4.16a) & (4.16b) runs from to but the effective

frequency range can be easily guessed from Fig. 5. and Fig. 7. We will check the

45

minimum effective temperature of the cavity (system) that can be attained by varying

the different parameters within the stability criteria.

When there is no OPA inside the cavity, we solve Eq. (4.15) for the following

parameters. We take , ,

, ,

, , K and G = 0, then the graph of the

temperature verses detuning curve looks like Fig. 8.

Figure -8. The effective temperature of the system verses the cavity detuning for G=0.

Fig. 8 shows the effective temperature verses cavity detuning graph of our

system. But there is a problem in the graph, and that is the vertical axis scale changes

very minute quantity. Our graph follows the pattern of the graph of reference [66] but

the minimum temperature in our case is very large. We get 298.42K while reference

[66] result is 15.23 K.

46

Chapter 5

Results and Discussions

In this thesis we have studied the optomechanical oscillators. We first considered a

simple case, i.e. a micromechanical mirror that is coupled with a thermal bath and a

radiation field (of constant phase). We derived the equations of motion for the various

parameters of the system, and studied the steady-state case for the cavity field and the

position (momentum) coordinates of the system. Then we calculated the equations of

motion for the quantum fluctuations in these parameters.

We introduced a nonlinear crystal (an Optical Parametric Amplifier) and

studied its effects on the dynamics of the system. We worked out the stability

conditions in the presence of the nonlinear crystal using the Routh-Hurwitz criterion.

We have also calculated the position and momentum spectra for the harmonic

oscillator. Our results show two peak spectrum for both the situations i.e. with OPA

and without OPA.

Finally we have calculated the effective temperature (or the cooling produced

by the radiation pressure) of the system by using numerical integration. The

qualitative behavior of our results is the same as discussed in earlier studies [66].

However, there appears quantitative difference that is due to the numerical errors.

Throughout our calculations, we have assumed that the phase of the driving

field is constant. However in practical situation, it is fluctuating about a mean value

and thus these calculations can be reproduced for a varying phase of laser light.

47

Appendix A

A.1 Parametric Amplification

We discussed parametric amplifier in Chapter 3. Parametric amplifier is a nonlinear

crystal which by the nonlinear process of parametric down conversion converts a

single photon into two photons such that the frequencies of the two photons when

added, gives the frequency of the first photon. Following is a brief mathematical

detail of the nonlinear optical process.

The propagation of electromagnetic waves through a medium is characterized

by the electric displacement vector

(A.1)

where P is the polarization of the medium. When the medium is non-linear, then P is

defined as

(A.2)

then

(A.3)

with

(A.4)

From Maxwell’s equations we know that

(A.5)

and

( )

(A.6)

which gives

48

(A.7)

or in 1-D

(A.8)

This the equation that governs the propagation of EM waves in a non-linear medium.

Putting

( ) (A.9)

in Eq. (A.8) gives

(

) ( )

(A.10)

By assuming that the wavelength of the EM waves is very small as compared to the

distance through which the amplitude of the wave varies significantly (called slowly

varying envelope approximation), we have

implying

or hence

then Eq. (A.10) gives

(

) ( ) (A.11)

By putting this in the non-linear equation of motion of EM waves Eq. (A.8), with

we get

(

) ( )

(A.12)

Now if we have two frequencies propagating inside the medium, i.e. then

the non-linear polarization is defined as

( )

(A.13)

49

Due to the non-linear nature of the susceptibility, there will be difference frequency

mixing giving rise to a field at . The amplitude of this field obeys

(from Eq. (A.12)),

√

( )

(A.14)

where . Eq. (A.14) shows that the amplitude (slowly varying) of

the generated field depends upon the amplitudes of the waves .

Now we discuss degenerate optical parametric amplification. Degenerate

parametric amplification is a special case of difference frequency mixing in which the

generated frequencies are the same. One frequency is called signal while the other is

called idler. The frequency that supplies energy for the non-linear process is .

Here , and . Consider a cavity that can support a mode

. is generated by the following frequency difference process

(A.15)

This wave now enters in another frequency difference process with and gives

the following wave

(A.16)

Thus we have two consecutive frequency mixing processes. Our system involves

degenerate frequency mixing, thus

. But this will only happen if the

non-linear crystal is oriented in front of the waves such that the phase wave matching

conditions is satisfied i.e.

thus in Eq. (A.14) as is shown in the following Fig.A.1.

50

Figure -A.1. The signal and idler waves satisfying the Phase wave Mixing Condition

Phase matching condition take care of both the conservation of energy and

momentum in the down conversion process. The equation of motion for the signal

waves now becomes ( )

√

( )

(A.17)

But and , then

(A.18)

with

√

( )

( )

= non-linear coupling constant. The complex

equation for is

(A.19)

Let the pump field apmlitude , with = real no. and = phase of the

driving field, and , then if the medium is non-absorbing we have

and to make we must take

, then

(A.20)

and

(A.21)

51

Adding and subtracting Eqns. (A.20) and (A.21), we get

(

) ( ) (A.22)

and

(

) ( ) (A.23)

now put ( ) to get

(A.24)

and

(A.25)

Solving simultaneously, we get

( )

(A.26)

and ( )

(A.27)

Equations (A.26) and (A.27) shows that the field is growing exponentially called

amplification and the field is decaying exponentially called de-amplification. This

is how degenerate parametric amplification takes place [67].

52

Appendix B

B.1 The equations of motion

We write Eq. (3.2b) with O replaced by the position operator q of the harmonic

oscillator and H from Eq. (3.1)

[ ]

(B.1)

Knowing the fact that q commutes with itself, with and the second powers

of these operators, we get

[ ] (B.2)

Putting this in Eq. (B.1) we get

(B.3)

This is the first equation of Eq. (3.3). Remember there is no velocity noise term to add

to the RHS of Eq. (B.3).

Now again take Eq. (3.2a) with O replaced by the momentum operator p of the

harmonic oscillator and H from Eq. (3.1)

[ ]

(B.4)

p also commutes with itself, with and the second powers of these operators,

thus we get

[ ] (B.5)

and

53

[ ] (B.6)

Using these in Eq. (B.4) we get

(B.7)

And the noise term for the force is due to two factors. One is due to thermal force and

the other is due to mechanical decay. Thus adding these terms to Eq. (B.7) gives

(B.8)

This is the 2nd

equation of Eq. (3.3).

Now writing c in place of O in Eq. (3.2b) and again the same Hamiltonian gives

[ ]

( )

(B.9)

Now c commutes with itself, with p and q and with their second powers. But it do not

commute with its conjugate. Thus we have

[ ] [ ] (B.10)

Using these in Eq. (B.9) with the corresponding noise term we get

( ) √ (B.11)

The term shows the cavity leakage of photons.

And if we take the complex conjugate of Eq. (B.11), then we get

( )

√

(B.12)

Equations (B.10) and (B.11) are the last two equations of Eq. (3.3).

54

Appendix C

C.1 The Steady-State Solution

We write Eqns. (3.3) here

,

,

( ) √

( )

√

(C.1a)

(C.1b)

(C.1c)

(C.1d)

Now let’s write the Taylor series of about , of about , of

about and keep only the first two terms (0

th and 1

st power of )

.

(C.2a)

(C.2b)

(C.2c)

(C.2d)

Now let’s equate Eqns. (C.1) to zero and then put Eqns. (C.2) in them, thus we have

(C.3)

Comparing the similar powers of on both sides we get

(C.4)

which is Eq. (3.8a).

Now from Eq. (C.1b)

55

(C.5)

Put , and (

)(

) in the above equation to get

( ) (

)( ) (

)

(C.6)

As , we compare the similar powers of of on both sides to get (for the 0th

power of )

( ) (

)( ) ( )

(C.7)

Re-arranging gives

(C.8)

which is Eq. (3.8b) with

(C9)

Now simultaneously solving (C.1c) and (C.1d) with their time derivatives equal to

zero, we get

(C.10)

with ( )

(C.11)

56

Appendix D

D.1 The Stability Criterion

We start with the characteristic equation of matrix A

[ ]

[ ]

( )

[ ]

[ ] (

[ ] [ ] ) (

[ ] [ ] )

( )

[ ](( ) )

[ ](( ) )

(D.1)

After collecting various coefficients of we have

, (D.2a)

. (D.2b)

(D.2c)

(D.2d)

and

[ ]

[ ]

( )

[ ](( ) )

[ ](( ) )

.

(D.2e)

Now according to Eqns. (3.24)

, (D.3a)

and

( ) , (D.3b)

and

(

)

(D.3c)

57

and (D.3d)

We first find

After performing calculations we find that

( )( )

(D.4)

The numerator of Eq. (D.4) is nothing but Eq. (3.25a).

Now we calculate

. Calculating

gives nothing but the result of Eq. (D.4). i.e.

(D.5)

So we have to calculate the remaining two conditions from

Calculating the

ratio

gives us a very long expression. The numerator of this expression is as under

( ( [ ] [ ])

( )

( ) (( )

) )( ( [ ]

[ ]) ( ) (( )(

) ( )

) ( ) ( )

( [ ]

[ ])( ) (( ) ) )

(D.6a)

And the denominator is

(( )

)( ( )

( ( ) ))

(D.6b)

These Eqns. (D.6a) and (D.6b) are nothing but conditions (3.25b) and (3.25c).

58

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62

VITA

The author Sadiq Nawaz Khan was born on 15 February 1986 in village Mirzabad of

district Lower Dir. He did his matriculation in 2002 from Govt. High school

Timergara. He did his F.Sc in Pre-Medical in 2004 from Jamal English Education

Academy Chakdara, Lower Dir. Then he studied additional math from Board of

Malakand in 2005. Meanwhile he was selected in UET Peshawar but his love for

Physics compelled him to study Physics. He did his B.Sc from Govt, Degree college

Timergara in 2007. In 2010 he got his Master degree in Physics from University of

Peshawar.

He was awarded a fellowship for M.Phil. Physics at Pakistan Institute of

Engineering and Applied Sciences (PIEAS) in 2010. He is currently pursuing M.Phil.

Physics at PIEAS. This thesis is a part of project done at PIEAS. His research interest

fields include Laser physics, Optomechanics, Quantum Optics, Astro Physics,

Nuclear Physics and computational Physics.

63

Appendix E

Quantum studies of Optomechanical Oscillators

Sadiq Nawaz Khan, Shahid Qamar, Muhammad Irfan

DPAM, Pakistan Institute of Engineering and Applied Sciences (PIEAS), PO Box Nilore, Islamabad

45650, Pakistan.

ABSTRACT:

We develop the mathematical model of an optomechanical oscillator inside a Fabry-Perrot cavity in the

presence of an optical parametric amplifier and discuss its properties. We find a steady state solution

for the position and momentum operators of the oscillator and for the field inside the cavity. The

equations of motion for the fluctuation operators are also calculated and the stability of the system is

discussed. Finally the position and momentum spectra are derived.

I. INTRODUCTION

The mechanical effects of electromagnetic radiation

(radiation pressure) were theoretically derived by Maxwell

in his fundamental work on electromagnetics (1873) but

were experimentally confirmed a century ago (1901) by

Lebedev using a carefully calibrated torsion balance. This

was independently verified by Nichols and Hull (1903)

[1]. Saha published a paper that suggested the

quantization of the momentum of light in relation to

radiation pressure. This was verified in 1923 by the

experiment of Compton's scattering [2]. This radiation

pressure was minute enough to be utilized at that time.

Then with the discovery of powerful lasers, the radiation

pressure became useful [3-8]. With the improvements in

the micro-fabrication techniques, we are now able to

control the dynamics of nanomechanical systems

(oscillators) with this radiation pressure [9-14].

Braginsky et al were among the pioneers to detect the

damping of an oscillator due to radiation pressure [15,16].

Recent studies has confirmed not only the detection and

measurement of Brownian noise due to thermal motion but

also the mechanical damping of this motion by radiation

pressure. All such effects are studied inside a Fabry-Perrot

cavity [17]. A typical Fabry-Perrot cavity is shown in Fig.

1. One mirror is heavier and thus stationary. The other

mirror is lighter and therefore oscillates due to coupling

with thermal bath. The oscillations are along the cavity

axis as shown. Field of wavelength oscillates back and

forth along the cavity axis. The cavity is high finesse

cavity (the photon decay rate from the cavity is small).

Figure 2. A Fabry-Perrot Cavity whose one mirror is heavier and

stationary while the other is light and vibrate.

When laser light is shin on the cavity, light inside the

cavity starts reflecting back-in-forth. As a result of these

continuous reflections, the motion of the light mirror is

affected.

The oscillating mirror is treated as quantum harmonic

oscillator. The mirror is under a net thermal and radiation

force. This force modifies the spring constant of the

mirror, an effect called optical spring effect. Due to optical

spring effect, the effective potential governing the

dynamics of the movable mirror is also changed resulting

in multi-stable solutions of the mirror’s position [18].

II. MODEL

Our system is a quantum oscillator interacting with a

quantized electromagnetic field inside a Fabry-Perrot

cavity of length L. The system is depicted in Fig. 2. One

mirror is heavier and is therefore fixed while the other one

is lighter. The radiation pressure and thermal coupling do

not have any effect on the heavier mirror. The light mirror

is subject to two forces. One force is due to the coupling

64

with the thermal bath at temperature T and the other force

is due to radiation pressure of the laser light.

Figure 3 Sketch of the cavity used to cool a micromechanical

mirror.

The oscillating mirror is coupled to the thermal bath by a

constant which is the energy decay rate of the mirror.

This is defined by

⁄ where is the

(angular) oscillation frequency of the mirror and Q is the

quality of the cavity. The cavity frequency in this case is

taken to be the fundamental mode ωc , which is related to

the cavity length by

, where c is the speed of light.

The power of the driving laser is P, related to the electric

field amplitude of the laser by √

, where is the

frequency of the driving laser. The power P is usually

taken in milli watts. For example our system uses a 4 mW

laser.

We also let , this condition is called

adiabatic limit which means that the frequency of the

resonator is very small compared to the cavity frequency

range so that scattering of photons into modes other than

the fundamental mode are easily neglected.

The OPA is a nonlinear crystal which converts a photon

of frequency ω into two photons of frequency

, a

process called degenerate down conversion. G is the non-

linear parametric gain of the OPA. is the vacuum noise

entering the cavity, while is the vacuum noise leaving

the cavity [19]. The Hamiltonian of the system is given by

[19]

( )

(

) ( )

( )

(1)

Here ωC is the cavity frequency, is the driving field

frequency, is Planck’s constant, is the photon number

in the cavity given by , while is the annihilation

operator of the field and is the creation operator,

is the coupling constant of the radiation field and

the cavity mirror. √ is the classical normalized

field outside the cavity, is the photon decay rate due to

leakage from the cavity, m is the mass of the mirror, is

the angular frequency of the mirror, θ is the constant phase

of the driving laser. The momentum and position operators

of the harmonic oscillator are p and q respectively.

The first term in Eq. (1) is the Hamiltonian due to the

cavity field. The second term is due to the coupling of the

movable mirror with the cavity field, the third term is the

Hamiltonian of the movable mirror, the fourth term is due

to the coupling of the driving laser to the field of the

cavity, and the last term is due to the interaction of the

OPA with the cavity field.

In Heisenberg picture, the equation of motion of an

operator O is given by

[ ]

(2)

Or for a system exposed to some noise we have

[ ] (3)

Using Eq. (3), we get the following equations of motion

,

,

( )

√ ( )

√

(4a)

(4b)

(4c)

(4d)

Eqns. (4) are the quantum Langevin equations for the

system. In the above equations, which is the input

vacuum noise operator, satisfies the following correlation

⟨ ( )

( )⟩ ( ) , ⟨ ( ) (

)⟩ ⟨ ( ) (

)⟩ .

(5a)

(5b)

In Eqns. (4), is the Brownian noise operator and the

mean value of this operator is also equal to zero ⟨ ⟩

and it satisfies the following correlation for the bath

temperature T

⟨ ( ) ( )⟩

∫ ( ) [

(

)

] ,

(6)

III. STEADY-STATE SOLUTION

Now we find the steady-state solutions to the above

equations. To do so, we take the Taylor series of

about , of about , of about ,

of about and keep only the first two terms (0th

and 1st power of )

65

,

,

,

(7a)

(7b)

(7c)

(7d)

and now put these in Eqns. (4) and compare the same

powers of on both sides to get

,

,

,

(8a)

(8b)

(8c)

where

, (9)

and

( )

.

(10)

is called the effective detuning of the cavity.

IV. QUANTUM FLUCTUATIONS

To obtain the equations for the fluctuations in the

operators, we substitue the following values for the

respective operators in Eqns. (4)

,

.

(11a)

(11b)

(11c)

(11d)

With , , and taking (thus

neglecting terms containing ) we get

, (12a)

as , so we get

. (12b)

Similarly

( )

( )( )

( )

(13a)

again neglecting the non-linear terms (involving

and ), we are left with

(

) ,

(13b)

and by following the above rules, we get for the field

operators

√

(14a)

and

√

(14b)

Equations (3.12), (3.13) and (3.14) are the linearized

quantum Langvin equations . Next we introduce the cavity

field quadrature

,

( ),

(15a)

(15b)

and

( )

(16a)

(16b)

Eqns. (3.16) can be written in terms of the following

system of linear equations

( ) ( ) ( ), (17)

where ( ) is the fluctuations vector and ( )is the noise

vector whose transposes are given below.

( ) ( ),

and

( ) ( √ √ )

(18a)

(18b)

And the coefficient matrix is given by

[

( )

( )

( ) ( )

( ) ( )]

.

(19)

This matrix gives us information about the stability of

the system and is thus very important to our study.

66

The stability conditions for the system can be

calculated by using Routh-Hurwitz criterion [20]. We

get the following conditions

( ) ( ) , (20a)

( ) (

(

)

(

)

)

{( ) ( )( )

[ ( )

]}

(20b)

and

( )

(

)

(

)

(20c)

These are the stability conditions under which the

system will be stable. For example if ,

, mass ,

,

,

, and , then

according to conditions of Eqns. (20), the frequency

detuning between the cavity and driving laser must be

. The plot for under these

conditions is given in Fig. 3. For G = 3.5× 107 , Eqns.

(20) restrict the detuning to . The

plot for for this G is shown in Fig. 4.

Figure. 3. Plot of the position of the mirror against the

detuning when no OPA is there.

Figure. 4. Plot of the position of the mirror against the

detuning when OPA is there.

We next take the Fourier transform of Eqns.

(14) to get the fluctuations in the position coordinate

of the oscillator. On solving the Fourier transform of

Eqns. (14) simultaneously for ( ) we obtain

( )

( )([ ( )

)] ( ) √

{[( ) ]

( ) [( )

] ( )}),

(21)

with

( ) (

) (

)

[ ( ) ]

(22)

67

Now we take the Fourier transform of the

correlation of Eq. (21) to get the position spectrum.

By solving

( )

∫ ( ) ⟨ ( ) ( )⟩

∫ ( ) ⟨ ( ) ( )⟩

(23)

We obtain the position spectrum of the position

fluctuations of the oscillator. We use the

correlations ⟨ ( ) ( )⟩ and

⟨ ( ) ( )⟩ ( )in solving Eq.

(23).

( )

( )

[ [( )

( )

( )]

[(

)

] (

)]

(24)

This is the position spectrum of the oscillator. The

momentum spectrum can be obtained by the same

procedure. It is given by ( ) ( )

We plot this spectrum for the same parameters that

we used for Fig. 3 and get Fig. 5

Figure. 5. Plot of the Position spectrum against the

frequency of the resonator.

These peaks occur at the position at which the

minimum of the denominator of Eq. (24) occur.

Fig. 6 shows the plot of the denominator function

of Eq. (24).

Figure. 6. A plot of the denominator function. Its

minimum is the position on the frequency axis where the

position spectrum has its maximum.

Next we find the effective temperature of the

oscillator by employing equi-partition theorem.

The temperature of any system is given by [19]

⟨ ⟩ ⟨ ⟩

(25)

Where

⟨ ⟩

∫ ( )

(26a)

and ⟨ ⟩

∫ ( )

(26b)

However we replace the infinite limits by the limits

calculated from Fig. 5 and Fig. 6 and finally get the

effective temperature of the system given in Fig. 7

Figure. 8. The effective temperature of the system

verses the cavity detuning for G=0.

This plot is showing the trend of the temperature

calculated by [19]. Our plot is different

68

quantitatively form the plot of [19] due to the

inaccuracies in the numerical integration.

V. CONCLUSIONS

We developed the mathematical model for the

optomechanical oscillator with an OPA inside a

Fabry-Perrot cavity. We found the steady state

solutions for the field and the momentum and

position of the system. We discussed the equations

for the fluctuation operators and calculated the

spectrum. We saw that the spectrum is effective in

a narrow region of frequency only and it drops very

quickly to zero outside this window. We found the

limits from this spectrum. Finally we found the

effective temperature of the oscillator.

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A, 13, 307, (1901).

[2] M. Saha, Astrophysical Journal , 50, 220,

(1919).

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[4] A., Dziedzic, J. E. & Chu, S. Ashkin, Optics

Letters, 11, 288. (1986).

[5] T. W. & Schawlow, A.L. Hansch, Optics

Communications, 13, 68, (1975).

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Wineland, Physical Review Letters, 40,1639,

(1978).

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69