ABSTRACT

SINGLE-PHOTON GENERATION THROUGH UNCONVENTIONAL BLOCKADE IN A THREE-MODE OPTOMECHANICAL CAVITY WITH KERR NONLINEARITY

by Avtej Singh Sethi

Quantum states of light with a fixed number of photons are useful for several quantum optical technologies. In this thesis, we theoretically study single-photon generation in a three-mode optomechanical cavity with a Kerr type nonlinear medium. We begin by presenting a detailed derivation of the master equation under Born-Markov approximations describing the dissipative dynamics of our setup. Next, we concentrate on two system parameters, the three-mode coupling rate (6) and the strength of the Kerr nonlinearity (*) and analyze the detuning (Δ) dependence of the second-order correlation function 6(2) to determine the photon statistics. As a key finding, we analytically conclude that the system can exhibit UPB for a wide range of detunings even with weak nonlinearities by adjusting the three-mode coupling. For instance, we find that the condition 6(2) (0) < 1 can be met with fixing * = 0.69^ and 6 = 2^ while ^ here represents the optical mode decay rate. Moreover, we observe that the minimum of 6(2) (0) shifts to higher detunings with increasing three-mode coupling rate and decreasing the Kerr nonlinear strength. With the current advancements in hybrid optomechanics, this work is experimentally feasible and can provide an alternate method for single-photon generation without relying on stringent conditions of CPB. SINGLE-PHOTON GENERATION THROUGH UNCONVENTIONAL BLOCKADE IN A THREE-MODE OPTOMECHANICAL CAVITY WITH KERR NONLINEARITY

Thesis

Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science by Avtej Singh Sethi Miami University Oxford, Ohio 2020

Adviser: Dr. Imran Mirza Reader: Prof. Samir Bali Reader: Dr. E. Carlo Samson

©2020 Avtej Singh Sethi This thesis titled

SINGLE-PHOTON GENERATION THROUGH UNCONVENTIONAL BLOCKADE IN A THREE-MODE OPTOMECHANICAL CAVITY WITH KERR NONLINEARITY

by

Avtej Singh Sethi

has been approved for publication by

College of Arts and Science and Department of Physics

Dr. Imran Mirza

Prof. Samir Bali

Dr. E. Carlo Samson Table of Contents

1 Introduction1

2 Preliminaries6 2.1 System Description ...... 6 2.2 Cavity Quantum Optomechanics ...... 8 2.3 Optical Kerr Effect ...... 12 2.4 Second-Order Correlation Function ...... 15

3 Theoretical Framework 18 3.1 Master Equation for a Damped Harmonic Oscillator ...... 19 3.2 Master Equation for a Three-mode Optomechanical Kerr Cavity . . . . . 29

4 Results 32 4.1 Conventional Photon Blockade ...... 32 4.2 Unconventional Photon Blockade in a Three-mode Optomechanical Kerr Cavity ...... 36 4.2.1 Transformed Hamiltonian ...... 36 4.2.2 Two-photon Two-phonon Truncation ...... 38 4.2.3 Dissipative Dynamics ...... 40

i 4.2.4 Second-Order Correlation Function: Plots and Discussion . . . . 41

5 Summary and Outlook 46

Bibliography 53

ii List of Figures

2.1 The nonlinear three-mode optomechanical cavity setup studied in this work. For the details about different parameters see the text...... 6 2.2 The consequence of the radiation pressure in a typical Fabry Peót cavity. Red arrows represent a drive field strong enough to move the end mirror

more than its zero-point fluctuations. The parameter 60 quantifies the strength of nonlinear optomechanical interaction...... 10 2.3 The setup for the third-harmonic generation...... 13 2.4 An illustration of different types of light sources and corresponding photon emission. The red dots on red straight lines represent the photon emission pattern (distribution)...... 16

4.1 Energy-level diagram (drawn up to three photons) describing a standard Jaynes-Cummings ladder. The anharmonicity in the energy spectrum under the strong-coupling regime is utilized to achieve the CPB. ~ = 1 in the diagram...... 34 4.2 Energy-level diagram describing all two-photon two-phonon states acces- sible in our model...... 40

iii 6(2) ( ) Δ 4.3 Second-order correlation function 01 0 as a function of the detuning for different experimentally accessible values of 6 and *. In this and the next plot we have selected Ω = 0.5^ while mechanical damping and the temperature of mechanical bath have been omitted...... 43 4.4 Density plots presenting how 6(2) (0) varies as a function of normalized Kerr nonlinearity strength */^ and normalized detuning Δ/^ for four values of three-mode coupling rate. In (a) 6 = ^ (b) 6 = 1.5^ (c) 6 = 2^ (d) 6 = 2.5^...... 45

iv Acknowledgements

I would like to express my deepest gratitude to Dr. Imran Mirza, for being my thesis advisor, and more importantly a mentor and a guide. He has always been a vital source of information, and helped tremendously whenever needed. I would also like the thank the Physics Department at Miami University, including all the faculty, staff and my peers who have been supportive and helpful and made this program an amazing experience. Finally, I would like to appreciate all my friends and family that have encouraged and motivated me to pursue my goals and achieve them.

v Chapter 1

Introduction

Computation and computers as we know them today, rely on principles of physics we discovered several decades ago. We have been optimizing and advancing this technology as it has slowly entered every aspect of our lives. As hard drives become smaller, and data stored denser, we are moving deeper into the nano-scale regime where quantum mechanical effects manifest themselves [1]. Quantum computation and algorithms have been argued to solve certain problems, classical computers struggle with. One such example is Shor’s algorithm invented by Peter Shor in 1994 [2]. Shor’s algorithm showed that a quantum computer can perform prime factorization of large strings of integer numbers exponentially faster than any classical computer. Since such a factorization plays a crucial role in the so-called RSA (Rivest–Shamir–Adleman) encryption scheme which is the backbone of our modern-day secure communication protocols, therefore, Shor’s algorithm showed how impactful quantum computing can be at the level of our everyday life. These and other advancements in quantum computer science are possible due to unique quantum properties such as superposition, and entanglement which have not been used in information technologies previously [3]. Quantum mechanics and it’s properties are

1 nonintuitive and typically observed at small scales and cold temperatures by isolating the quantum system from its environment. A photon is the quantum of light and can prove to be a useful tool in the development of quantum technologies due to its fast speed and non- interacting nature. Knill et al. [4] highlighted the value of single photons as they discussed the development of efficient quantum computation systems with linear optics. Thus, linear optical devices such as beam splitters, phase shifters, photo-detectors which are widely available and scalable used with a single-photon source can enable universal quantum computing. Generation of single photons reliably, however, has proven to be a challenge we have faced in the search and development of quantum technologies. Furthermore, using numerous beam-splitters for such a computation system has its limitations as this itself becomes a non-deterministic polynomial-time (NP) hard problem leading to the Boson sampling (sampling in an optical linear interferometer a particular output distribution of indistinguishable photons) [5]. Currently, there are a few different methods available to generate single photons [6]. One such method is the celebrated Spontaneous Parametric Down Conversion (SPDC) [7]. Dated back to the 1970s, SPDC is a weak optical process in which a higher energy photon (pump photon) is converted into two low energy photons (probe and idler photons) in a nonlinear material, such as BBO (Beta-Barium-Borate) crystal. Both energy and momentum are conserved during this process and the two photons generated in the process can be used separately as single photons or collectively as an entangled photon pair. However, SPDC is a weak process which means that roughly one in every million photons passing through the crystal undergoes the desired splitting. This puts serious restrictions on the consistency and reliability of using SPDC as a single-photon source. Another method to generate single photons utilizes the conventional photon blockade (CPB) [8]. In the absence of an emitter, light trapped inside a single-mode cavity behaves as a quantum harmonic oscillator with equally spaced energy levels. It turns out when

2 an emitter is brought in contact with such a single-mode light field then the energy levels (dressed states) of the combined atom-cavity system become anharmonic. Consequently, when the emitter is strongly coupled with the cavity field this produces an effect where after the absorption of one photon, the second and higher photon absorption becomes inaccessible–hence the name photon blockade. An interesting consequence of this effect is the process in which a laser beam entering the cavity interacts with the atom and due to CPB the photons are transmitted from the cavity in a singular sequential manner, effectively acting as a “photon turnstile" [9]. Despite such unique consequences, it is worthwhile to note that CPB in cavity quantum electrodynamics crucially relies on the condition of strong light-matter interaction which is not always straightforward to attain. Keeping into account the stringent conditions under which CPB can work in cavity QED setups, in 2010 Liew et al. theoretically showed that the same goal can also be achieved while relaxing the strong light-matter interaction [10]. In their proposal, they used two coupled optical cavities where instead of relying on the strong light-matter interaction, they used quantum destructive interference among different pathways leading to two-photon emission. Due to the perfect destructive interference, they showed that the two-photon emission can be completely suppressed and if the drive of the system is weak (mean number of photons in the laser ≤ 2) then only single photons will be emitted and hence such a system can be used as a single-photon source. In 2018 first experimental observation of UPB was reported independently by two groups: (1) one group used quantum dot cavity QED setup [11] and (2) other group used superconducting qubits coupled with microwave photons as their platform [12]. Since its first proposal in 2010, UPB has been argued to exist in a variety of settings [13]. One such setting is the cavity quantum optomechanics [14]. The radiation pres- sure exerted by light on everyday objects is negligibly small and hence is unobservable. However, with the advancement in the manufacturing of ultra-small optical cavities with

3 high-quality factors, twelve years ago Kerry Vahala’s group and others showed that the radiation pressure can lead to observable effects when light is trapped inside micrometer- sized optical cavities [15]. Since the first experimental demonstrations such “quantum mirrors" have witnessed a tremendous amount of research activity due to their application in quantum information science [16–19], testing foundations of quantum mechanics [20] and building ultra-sensitive motion sensors (for example the ones which are used in the Laser Interferometer Gravitational-Wave Observatory (LIGO)) [21]. Recently, Sarma et al have theoretically studied UPB in a hybrid setting consisting of a three-mode optomechanical cavity filled with a Kerr-type nonlinear medium [22]. Since our goal is to use this system as a single-photon source, therefore, we revisit their work and focus on altering different system parameters (for example, detuning between laser and optical mode frequency, strength of nonlinear process, and three-mode coupling strength) to predict optimal parameter regime for single-photon emission. We use the second-order correlation 6(2) function to quantify single-photon emission and analytically address the problem under the two-photon truncation of the Hilbert space. Broadly speaking, we show that the UPB can be realized in our system at a varied range of operating parameters, and thus can be used as a source to generate single photons reliably within those parameters. In particular, we report that the detuning Δ dependence of the second-order correlation function with zero time delays i.e. 6(2) (0) remains less than 1 between 0.6^ ≤ Δ ≤ 1.5^ with * = 0.74^ and 6 = 2.5^. ^ here is the optical mode decay rate which is also used as the unit for the plots. By trying other different combinations of * and 6 we conclude that the antibunched light with a single Fock state generation is possible for other weaker nonlinearity strengths. However, the three-mode coupling strength is needed to be adjusted accordingly. Additionally, we notice that as we increase the three-mode interaction coefficient we see the optimum working ranges shift towards higher detuning values.

4 This thesis is organized as follows. In the second chapter, we discuss the preliminaries that include quantum optomechanics, photon-photon correlation functions, and Kerr-type nonlinearities. In the third chapter, we present a detailed derivation of the Born-Markov master equation describing the dissipative dynamics of our setup. In Chapter 4 we briefly revisit CPB and mainly concentrate on the results of this study where we identify the parameter regimes under which UPB could work and hence single photons can be generated. In the fifth chapter, we close with a summary of the main conclusions of this work and mention possible future directions.

5 Chapter 2

Preliminaries

2.1 System Description

As shown in Fig. 2.1, we consider the following model of our system. The optomechanical

Figure 2.1: The nonlinear three-mode optomechanical cavity setup studied in this work. For the details about different parameters see the text. system consists of a Fabry-Pérot cavity where the left mirror is fixed and can partially transmit the light shined on it. The right mirror, on the other hand, is perfectly reflecting and is ultra-thin such that when a strong enough laser is reflected this mirror can oscillate

6 due to the radiation pressure by the photons. In the quantum domain, these oscillations can be modeled as a quantum harmonic oscillator, and thus is shown with a spring to help visualize the motion. The optical cavity is assumed to support two independent modes with frequencies l1 and l2. Ω1(Ω2) represents the strength of the drive for optical mode

1(2). For simplicity, we have taken Ω1 = Ω2 = Ω in Fig. 2.1. Similar to optical modes, we assumed that the mechanical motion happens in a single-mode form with frequency l<.

The coefficient of interaction between the three modes, l1, l2, and l< is represented by 6. The strength of the nonlinear Kerr medium is defined as *. We have also added the decay rates to both optical modes and the mechanical mode as shown by ^1, ^2, and W, respectively. We start with defining the Hamiltonian of our system, which can be decomposed into three parts: the free Hamiltonian of the harmonic oscillators ˆ0, the Hamiltonian of the interaction ˆ8=C and the Hamiltonian describing the drive provided by the laser ˆ 3A . Putting everything together we can express the net system Hamiltonian as

ˆ BHB = ˆ0 + ˆ8=C + ˆ 3A , (2.1a) ˆ = l 0†0 + l 0†0 + l 1ˆ †1,ˆ 0 ~ 1 ˆ1 ˆ1 ~ 2 ˆ2 ˆ2 ~ < (2.1b) ˆ = *0†0†0 0 + *0†0†0 0 + 6(0†0 + 0†0 )(1ˆ + 1ˆ †), 8=C ~ ˆ1 ˆ1 ˆ1 ˆ1 ~ ˆ2 ˆ2 ˆ2 ˆ2 ~ ˆ2 ˆ1 ˆ1 ˆ2 (2.1c) ˆ = Ω (0 48l;C + 0†4−8l;C) + Ω (0 48l;C + 0†4−8l;C), 3A ~ 1 ˆ1 ˆ1 ~ 2 ˆ2 ˆ2 (2.1d)

where 0ˆ1, 0ˆ2, and 1,ˆ are the annihilation operators for the first optical mode photons, second optical mode photons and the mechanical mode phonons (quanta of vibrations for the mechanical motion), respectively. Note that in each of these harmonic oscillator

Hamiltonians we have neglected the zero-point energy. In ˆ8=C, the first two terms describe the nonlinear effect of the Kerr medium on each optical mode, and the last two terms

7 characterize the three-mode interaction. Here we have taken l1 − l2 = l< to ensure the cross-coupling between the optical modes via mechanical mode. For further details about such an interaction, we direct the reader to the reference [23]. Finally, ˆ 3A8E4 depicts the effect of the drive onto the two optical modes. Here l; is the laser frequency. Non-vanishing commutation relations among various ladder operators are

[0 , 0†] = , [0 , 0†] = , [1,ˆ 1ˆ †] = . ˆ1 ˆ1 1 ˆ2 ˆ2 1 and 1 (2.2)

In the next sections, we briefly discuss the topics of cavity quantum optomechanics and Kerr effect (which are the main ingredients in our model). We also revisit second-order correlation functions which we’ll use as a measure of quantification for single-photon generation in our results.

2.2 Cavity Quantum Optomechanics

In quantum optomechanics, we study the interactions between light and mechanical motion, typically at small (atomic/nanometer) scales. The radiation pressure exerted by light on everyday objects is negligibly small and hence is unobservable. However, with the advancement in the manufacturing of ultra-small optical cavities with high-quality factors, twelve years ago Kerry Vahala’s group and others showed that the radiation pressure can lead to observable effects when light is trapped inside micrometer-sized optical cavities [15]. In the simplest possible theoretical model of cavity quantum optomechanics, one can imagine a Fabry-Perót cavity driven by a strong coherent field (laser) in which one of the mirrors oscillates more than its zero-point fluctuations due to the radiation pressure exerted by the laser. Since the first experimental demonstrations such “quantum mirrors" have witnessed a tremendous amount of research activity due to their application in quantum

8 information science [16], testing foundations of quantum mechanics [20] and building ultra- sensitive motion sensors (for example the ones which are used in the Laser Interferometer Gravitational-Wave Observatory (LIGO)) [21]. Considering light-matter interactions, the simplest cases being that of light scattering of material, this could happen as elastic light scattering or inelastic light scattering. Elastic light scattering is when the outgoing scattered light has the identical frequency as the incoming light i.e. l0 = l. On the other hand, inelastic scattering is characterized as the outgoing light as having a shifted frequency 0 along with excitation or de-excitation of the material object, l = l ± l<, where l< is the vibrational frequency. Closely following [24], we consider a single-mode Fabry-Perót cavity driven by a laser as described above. The end mirror undergoes mechanical motion and thus the optical resonance frequency of the cavity is dependant by the displacement of the movable mirror. As such the Hamiltonian for such a system could be described as,

† † ˆ = ~l(G)0ˆ 0ˆ + ~l< 1ˆ 1.ˆ (2.3)

Here 0,ˆ 0ˆ† are the ladder operators for the modes of the optical cavity. Similarly 1,ˆ 1ˆ † are the ladder operators for the mechanical vibration of the end mirror. Here the displacement operator Gˆ can be expressed as a linear combination of the annihilation and creation † operators for quantum vibrations (phonons) i.e. Gˆ = GI? 5 (1ˆ + 1ˆ ), where GI? 5 is the magnitude of the zero-point fluctuations for the mechanical mode. Due to the mechanical oscillations, the optical resonant frequency l becomes dependent on the position of the vibrating mirror. Under this condition one can expand l(G) in the following form

 G  l(G) = l − + ... , 2 1 3 (2.4)

9 Figure 2.2: The consequence of the radiation pressure in a typical Fabry Peót cavity. Red arrows represent a drive field strong enough to move the end mirror more than its zero- point fluctuations. The parameter 60 quantifies the strength of nonlinear optomechanical interaction.

where 3 is the initial (non-expanded) length of the cavity which corresponds to the resonant frequency l2 and l(G) is the resonant frequency when the cavity has expanded to the length 3 + G. See Fig. 2.2 for better visualization of the situation. Under small- displacement limit (G << 3) the Hamiltonian given in Eq. (2.3) takes the form,

 l 0†0 l 1†1 6 0†0 1 1† . ˆ = ~ 2 ˆ ˆ + ~ < ˆ ˆ + ~ 0 ˆ ˆ( ˆ + ˆ ) (2.5)

The parameter 60, used to quantify the strength of the optomechanical interaction, is defined 6 l GI ? 5 through 0 = − 2 3 . Thus the interaction part of the Hamiltonian due to the single-mode cavity optomechanics turns out to be

 6 0†0 1 1† . ˆ$" = ~ 0 ˆ ˆ( ˆ + ˆ ) (2.6)

10 It is worthwhile to mention that the interaction is qualitatively representing a process in which average number of photons =ˆ = 0ˆ†0ˆ and mechanical oscillations (represented by displacement operator Gˆ ∝ 1ˆ + 1ˆ † appear in an interlinked fashion. Moreover, the interaction represents a nonlinear process with the photon (phonon) number remaining conserved (non-conserved) as the system evolves from one quantum state to another. The form of the above Hamiltonian can be easily extended to the situations in which the optical cavity supports multiple resonant modes. One such example is a bi-modal optical cavity where both optical modes independently interact with a single mechanical mode through the optomechanical interaction giving rise to the following Hamiltonian operator

ˆ = 6 (0†0 + 0†0 )(1ˆ + 1ˆ †). $"2 ~ 0 ˆ1 ˆ1 ˆ2 ˆ2 (2.7)

In the last equation, 0ˆ1, 0ˆ2 represent annihilation operators for optical mode 1 and 2, respectively. Another situation, which is particularly useful from the perspective of the model considered in this thesis (see Eq. (2.1c)), is the one in which both optical modes interact with each other through the single mechanical mode. In such a “three-mode coupling" when a photon is annihilated in mode 1, a photon is created in mode 2 along with the destruction or creation of a phonon. As such the photon number remain conserved but the phonon number may not be conserved. The Hamiltonian operator for this type of interaction takes the form shown below,

ˆ = 6(0†0 + 0†0 )(1ˆ + 1ˆ †). $"3 ~ ˆ2 ˆ1 ˆ1 ˆ2 (2.8)

6 parameter in the last equation characterizes the rate at which the three-mode coupling occurs which is needed to be compared with decay rates in a problem to determine strong or weak coupling regime.

11 2.3 Optical Kerr Effect

The Kerr effect describes the change in the refractive index of a nonlinear medium due to the presence of an electric field. The effect is observable in a variety of nonlinear media, for instance, crystals or glasses or even gasses. Kerr effect is one of the oldest discoveries in the field of nonlinear optics that dates back to 1875 when John Kerr for the first time reported the phenomena [25]. The key signature of the Kerr effect is that the altered refractive index is proportional to the square of the electric field of the electromagnetic radiation propagating through the medium. Since the Kerr effect is characterized as a third- order (j(3)) nonlinear interaction, therefore, in the following we begin from the third-order nonlinear processes. In a typical linear medium, the index of refraction = is a function of frequency l of the propagating field, which is also referred to as the dispersion relation. This means in linear media different frequencies associated with the light field experience different refractive indices. However, in a nonlinear medium with a j(3) type nonlinearity, Kerr effect shows that in addition to =(l) dependence, the refractive index is modified by a term proportional to  (intensity of the input field), as expressed below

= = =0(l) + =2. (2.9)

Here the first term describes the typical linear refractive index whereas the second term describes the modification factor due to j(3) effect and is intensity-dependent. Eq. (2.9) can be drived by considering the third harmonic generation (THG) in a j(3) nonlinear crystal. The setup for the process is shown in Fig. 2.3. An intense laser source emitting an electromagnetic field with carrier frequency l passes through a nonlinear crystal with j(3) nonlinearity. As a result, at the output, we obtain two fields, one with the same frequency

12 Figure 2.3: The setup for the third-harmonic generation. l but there also emerges another field with frequency 3l due to nonlinear effects. To see this mathematically, we consider a single-frequency (monochromatic) laser shined at a nonlinear crystal with j(3) susceptibility. The electric field associated with the input laser can be decomposed into positive and negative frequency components as

E E∗ E(C) = E cos(lC) = 0 48lC + 0 4−8lC 0 2 2

The polarization vector P(3) corresponding to this third order interaction is given by

n j(3) P(3) (C) = n j(3)E(C)E(C)E(C) = 0 (E3438lC + 3E2E∗48lC + 2.2.). (2.10) 0 8 0 0 0

In the above expression, the first term appearing with frequency 3l represents THG, while the second term refers to a process generating a light field with thee same input frequency l. We now look at the total response including the linear term, and consider terms at frequency l ignoring the THG term. We find

n j(1) 3n j(3) n E 48lC P(C) = 0 E 48lC + 0 E2E∗48lC + 2.2. = 0 0 j + 2.2. 2 0 8 0 0 2

j = j(1) + 3 j(3) ∗ = = p + j Δ= Where 4 E0E0. Further as 1 , we define as the change in the

13 refractive index due to the nonlinearity. As such we can write it as:

p q 1 ( ) 3 ( ) ∗ 1 ( ) 3 ( ) ∗ Δ= = 1 + j − 1 + j(1) ≈ 1 + j 1 + j 3 ì ì − 1 − j 1 ≈ j 3 ì ì 2 8 0 0 2 8 0 0

 = 2n = ∗/ We identify the intensity term as, 0 E0E0 2 which gives

3j(3) Δ= ≈  (2.11) 42n0=

Thus we have shown that Δ= ∝  which is the essence of the Kerr effect. Such a classical description of the Kerr effect, when applied to the quantum domain gives rise to an interaction Hamiltonian of the following form

† 2 2 ˆ 4AA = ~*(0ˆ ) (0ˆ) . (2.12)

In the above equation, the variable * quantifies the strength of the Kerr nonlinearity. It is worthwhile to mention that the full quantum theory of the Kerr effect (including the derivation of the Kerr Hamiltonian operator) can be developed using first-principle calculations based on density matrix formalism. However, the development of such a theory will consume a lot of space here. Therefore, we direct the interested reader to the Ref. [26]. Referring back to the system studied in this work, we point out that for bi-modal cavities subjected to a Kerr nonlinearity, one can easily extend the above Kerr Hamiltonian to

ˆ = * 0†0†0 0 + * 0†0†0 0 , 4AA2 ~ 1 ˆ1 ˆ1 ˆ1 ˆ1 ~ 2 ˆ2 ˆ2 ˆ2 ˆ2 (2.13)

which is the Kerr Hamiltonian used in (2.1c). *1(*2) is the Kerr interaction parameter for the 1(2) optical mode. The underlying assumption here is that both modes experience the

14 Kerr interaction individually which is a valid assumption to make if optical modes are not crossed coupled through the nonlinear process.

2.4 Second-Order Correlation Function

The second-order correlation function 6(2) (g) defines the joint-probability of detecting a single photon at time C + g at some detector position given another photon had already been detected at the same position at a previous time C [27]. g then represents the delay between first and second photon detection. For a single-mode problem with 0ˆ being the photon annihilation operator in that mode, 6(2) (g) function is defined as

0ˆ†(C)0ˆ†(C + g)0ˆ(C + g)0ˆ(C) 6(2) (g) = . 2 (2.14) 0ˆ†(C)0ˆ(C)

As seen in Eq. (2.14), 6(2) (g) is only dependant on g whereas the right hand side also has a dependence on C. This can be explained based on the fact that C here serves merely as some initial time when the first photon was detected which without loss of generality can be set to zero. The denominator in Eq. (2.14) ensures normalization. In the long time limit, typically the photon-photon correlations are washed out and the numerator factorizes such that 62(∞) = 1. In this work, we also utilize a special case of the second-order correlation function, where the detection delay time is ignored i.e. we set g = 0 and hence express 6(2) (0) as

0ˆ†0ˆ†0ˆ0ˆ 6(2) ( ) = . 0 2 (2.15) 0ˆ†0ˆ

The second-order correlation functions are useful to distinguish different types of statistical

15 Figure 2.4: An illustration of different types of light sources and corresponding pho- ton emission. The red dots on red straight lines represent the photon emission pattern (distribution).

Distribution Description Photon flow 6(2) (0) Super-Poissonian Chaotic/Thermal Bunched >1 Poissonian Coherent/Laser Random 1 Sub-Poissonian Quantized/single photons Antibunched <1

Table 2.1: Photon statistics: For each distribution, a description of the type of light source, the flow of photons and the corresponding second-order correlation function 6(2)(0) values are expressed. distributions followed by the photons emitted by a source. Possible distributions can be categorized into three groups, namely, Poissonian, Sub-Poissonian, and Super-Poissonian distributions. As seen in Figure 2.4, the Poissonian distribution of light corresponds to coherent light, where the mean and variance of the distribution are the same and 6(2) (0) = 1. Super-Poissonian distribution is a characteristic of thermal or chaotic light with 6(2) (0) >1. This indicates a stream of bunched photons stream meaning there is a higher probability to detect a photon right after already detecting one. Finally, a sub-Poissonian distribution

16 of light represents the case where photons are emitted at regular gaps as compared to a random source, and thus are antibunched with 6(2) (0) <1 [27]. Since we are interested in the single-photon generation therefore, we will be working out the optimal condition on system parameters to achieve 6(2) (0) <1. Table 2.1, summarizes the above information in a condensed tabular format.

17 Chapter 3

Theoretical Framework

No quantum system can be completely isolated from its environment. Even at very low temperatures with proper shielding from stray electric and magnetic fields, a quantum system can be impacted by the environmental vacuum fluctuations in non-trivial ways. Consequently, to predict the behavior of any realistic quantum system one needs to include the environmental degrees of freedom. Since single-photon sources are no exceptions to the rule, therefore, in this chapter we focus on developing a theoretical framework to describe the open dynamics of the setup studied for single-photon generation in this thesis. Referring to Fig. 2.1, we notice that our setup consists of two optical modes in the Fabry-Perót cavity and a mechanical oscillator based on a vibrating mirror. All of these subsystems can be modeled as quantum harmonic oscillators. Accordingly, we begin by considering a simplified situation of a single damped quantum harmonic oscillator in the next section and present a microscopic derivation of the dynamic equation (master equation) followed by the oscillator in the presence of environmental interactions.

18 3.1 Master Equation for a Damped Harmonic Oscillator

For this section, we closely follow Carmichael [28] and start with introducing a general model for open quantum systems through the following Hamiltonian

ˆ = ˆ ( + ˆ  + ˆ ( , (3.1)

The above Hamiltonian has been decomposed into three pieces: ˆ ( describes free system

Hamiltonian, ˆ  models the free environment Hamiltonian, and ˆ ( is the Hamiltonian describing the interaction between the system and environment. We describe the state of the global system (system and environment combined) by the density operator bˆ(C). The von-Neumann equation followed by bˆ(C) is given by

¤ 8 bˆ = − ,ˆ bˆ, (3.2) ~

Since we are interested in the consequences of the system-environment couplings, therefore, we transform into the interaction picture. Therein by noticing that the free Hamiltonians of the system and environment commute, one can easily show that Eq. (3.2) transforms to

¤ 8   b˜ = − ˜ ( , b˜ , (3.3) ~ where ˜ ( is defined as

8 8 (ˆ(+ˆ  )C − (ˆ(+ˆ  )C ˜ ( := 4 ~ ˆ ( 4 ~ . (3.4)

19 For the solution of Eq. (3.3), we formally integrate both sides of this equation from some initial time C0 = 0 to some present time C. We obtain

¹ C 8  0 0  b˜(C) = b˜(0) − ˜ ( (C ), b˜(C ) 3C.. (3.5) 0 ~

Substituting this solution back into Eq. (3.3) yields the following integro-differential equa- tion obeyed by b˜(C)

8 1 ¹ C b˜¤ = − ˜ (C), b˜( ) − ˜ (C), ˜ (C0), b˜(C0)3C0. ( 0 2 ( ( (3.6) ~ ~ 0

To focus on the system dynamics alone, we next perform the trace operation (represented by CA) over the environmental degrees of freedom on both sides of the last equation. The equation of motion followed by the system reduced density operator d˜(C) is then given by

8 1 ¹ C  d¤ = − ˜ (C), b˜( ) − ˜ (C), ˜ (C0), b˜(C0)3C0 , ˜ tr ( 0 2 tr ( ( (3.7) ~ ~ 0 where the system density operator d˜(C) is related to the global density operator b˜(C) through the partial trace operation: d˜(C) = tr {b˜(C)}. Up to this point, our analysis was exact. However, to proceed further with the analytic treatment we’ll now apply a few approximations.

• As the first approximation, we assume that the interaction between system and

environment is turned on just after time C0 = 0. This implies that initially system and environment exist in an uncorrelated state i.e. we can express b˜(0) as a tensor product of the following form

b˜(0) = d˜(0) ⊗ ˜ (0). (3.8)

20 Here d˜(0) and ˜ (0) are the initial density operators for the system and environment, respectively. With this assumption, the first term appearing on the right-hand side of Eq. (3.7) can be removed. Note that this step is equivalent to extracting a term

tr (ˆ ( ˜0) from the interaction Hamiltonian and absorbing it into ˆ (. Also, notic- ing that the trace operation and temporal integration are commuting mathematical operations here, therefore, we can rearrange these to arrive at

1 ¹ C d¤(C) = − ˜ (C), ˜ (C0), b˜(C0) 3C0. ˜ 2 tr ( ( (3.9) ~ 0

In the above equation, we observe that the density operator evolution depends on the all-time history of the system due to the integration on the time interval [0,C]. Such a dependence still makes the above equation complicated to solve.

• To simplify the matters further, we next apply the so-called Born approximation. The Born approximation assumes that the coupling between system and environment is weak and the environment is extremely large compared to the system size such that the density operator of the environment remains unaffected throughout the evolution of the system. Mathematically, wee can express Born approximation as ˜ (C) ≈ ˜ (0) which simplifies Eq. (3.9) to

1 ¹ C d¤(C) = − ˜ (C), ˜ (C0), d(C0)˜ ( ) 3C0. ˜ 2 tr ( ( ˜ 0 (3.10) ~ 0

• With the Born approximation, the equation of motion for the system density operator has been considerably simplified. However, still, the structure of the equation is non-local in time i.e. the future evolution of the system on all past times. At this point, we make another approximation, called the Markov approximation, to relax this restriction. Under the Markov approximation, we assume that the future

21 time evolution of the system only depends on its present state i.e. d˜(C0) ≈ d˜(C). The Markov approximation relies on the separation of time-scales. We notice that there are two time scales in the present problem. One time scale is set by how quickly the state of the system is evolving and the other time scale is fixed by the time it takes to develop correlations between system and environment due to interaction (environment memory time). The Markov approximation implies that the environment memory time is extremely short compared to the system dynamics time scale. It is worthwhile to notice that the Markov approximation, in a sense, is consistent with the Born approximation which treats the environment as a larger system that remains unaltered. Hence, we arrive at

1 ¹ C d¤(C) = − ˜ (C), ˜ (C0), d(C)˜ ( ) 3C0. ˜ 2 tr ( ( ˜ 0 (3.11) ~ 0

Eq. (3.11) is still not fully Markovian because it depends on an initial time C = 0 as seen in the lower integration limit. Therefore, we make the substitution for C0 = C − g and further we extend the upper limit of integration to infinity, using the argument that the time scales comparison mentioned above. We thus write

1 ¹ ∞ d¤(C) = − ˜ (C), ˜ (C − g), d(C)˜ ( ) 3g. ˜ 2 tr ( ( ˜ 0 (3.12) ~ 0

Eq. (3.12) is commonly known as the Born-Markov master equation in the literature [29]. Notice that at times (not valid for our optomechanical system case) the Markov approximation is not followed by an open quantum system. Examples of such non- Markovian dynamics can be found in the situations when either system-environment coupling is strong or the size of the environment is compared to the system size

22 [30, 31].

Up till this point, we have not made any specification about the nature of the system- environment interaction. To proceed further with our analytic analysis, we now assume a particular decomposed form of ˜ ( which is appropriate for a large class of quantum optical problems. This form is given as

Õ ˜ ( = ~ B˜8 (C) ⊗ 4˜8 (C). (3.13) 8

Note that the above equation is written in the interaction picture where we defined B˜8 (C) := 8ˆ C/ −8ˆ C/ 8ˆ C/ −8ˆ C/ 4 ( ~B8 (C)4 ( ~ and 4˜8 (C) := 4 ( ~48 (C)4 ( ~. Next, we insert Eq. (3.13) into Eq. (3.11), expand the commutators and apply the trace operation on the environmental operators. We thus obtain

Õ ¹ ∞ h d˜¤(C) = B˜8 (C)d˜(C)B˜9 (C − g) tr {4˜8 (C)˜ (0)4˜9 (C − g)} − B˜8 (C)B˜9 (C − g)d˜(C)× 8, 9 0

tr {4˜8 (C)4˜9 (C − g)˜ (0)} − B˜9 (C − g)d˜(C)B˜8 (C) tr {4˜9 (C − g)˜ (0)4˜8 (C)} i + d˜(C)B˜9 (C − g)B˜8 (C) tr {˜ (0)4˜9 (C − g)4˜8 (C)} 3g. (3.14)

Using the cyclic property of trace, we further simplify the above equation to

¹ ∞ Õ h  d˜¤(C) = − B˜8 (C)B˜9 (C − g)d˜(C) − B˜9 (C − g)d˜(C)B˜8 (C) 4˜8 (C)4˜9 (C − g) 8, 9 0   i + − B˜8 (C)d˜(C − g)B˜9 (C − g) + d˜(C − g)B˜9 (C − g)B˜8 (C) 4˜9 (C − g)4˜8 (C) 3g, (3.15)

23 where the expectation value terms (indicated by h...i) are given by

4˜8 (C)4˜9 (C − g) = tr {4˜8 (C)4˜9 (C − g)˜ (0)} = tr {4˜9 (C − g)˜ (0)4˜8 (C)}, (3.16)

4˜9 (C − g)4˜8 (C) = tr {˜ (0)4˜9 (C − g)4˜8 (C)} = tr {4˜8 (C)˜ (0)4˜9 (C − g)}. (3.17)

The application of Markov approximation restricts the environmental correlations to be extremely short lived as compared to the system evolution time. Under such an extreme condition, we assume the two-time correlations mentioned above 4˜8 (C)4˜9 (C − g) and

4˜9 (C − g)4˜8 (C) to be Dirac delta functions in time i.e. we set 4˜8 (C)4˜9 (C − g) ∝ X(g) and

4˜9 (C − g)4˜8 (C) ∝ X(g). Next, we further specialize in the above master equation to the case of a quantum har- monic oscillator weakly coupled with a Markov environment (a damped quantum harmonic oscillator). Since the quantization of a single-mode optical field trapped inside an optical cavity also turns out to be equivalent to the problem of a quantum harmonic oscillator, therefore, this case is also applicable to the open system dynamics of an optical cavity coupled with a thermal bath. For this particular problem, the Hamiltonian ˆ takes the form

ˆ l0†0 Õ l A†A Õ :∗0A† : 0†A . = ~ ˆ ˆ + ~ 9 ˆ9 ˆ9 + ~( 9 ˆ ˆ9 + 9 ˆ ˆ9 ) (3.18) | {z } 9 9 ˆ ( | {z } | {z } ˆ  ˆ(

In the system Hamiltonian ˆ (, l is the single-mode frequency of the cavity mode with annihilation operator 0ˆ. The environment is assumed to be a multimode quantum harmonic oscillator bath with 9th mode frequency given by l 9 with annihilation operator Aˆ9 . Non- [0, 0†] [A , A†] X vanishing commutation relations are ˆ ˆ = 1 and ˆ9 ˆ: = 9 : . Finally, the interaction between system and 9th mode of the environment is characterized by the complex coupling

24 parameter : 9 . We notice that the form of interaction Hamiltonian is the same as taken in Eq. (3.13). Therefore, by the following identifications

B 0, B 0†, 4 4† Õ ^∗A†, 4 4 Õ ^ A ˆ1 −→ ˆ ˆ2 −→ ˆ ˆ1 = ˆ −→ 9 ˆ9 ˆ2 = ˆ −→ 9 ˆ9 (3.19) 9 9 we can rewrite Eq. (3.15) by expanding the summations over the indices 8 = 1, 2 and 9 = 1, 2 as

¹ ∞  d˜¤(C) = − 0ˆ0ˆ†d˜(C − g) − 0ˆ†d˜(C − g)0ˆ 4−8lg 4˜†(C)4˜(C − g) + ℎ.2. 0    + 0ˆ†0ˆd˜(C − g) − 0ˆd˜(C − g)0ˆ† 48lg 4˜(C)4˜†(C − g) + ℎ.2. 3g, (3.20) where the conversion of operators into the interaction picture can be performed by applying the Baker–Hausdorff Lemma. We thus have used

8ˆ(C/~ −8ˆ(C/~ −8lC 8ˆ(C/~ † −8ˆ(C/~ † 8lC B˜1(C) = 4 04ˆ = 04ˆ , B˜2(C) = 4 0ˆ 4 = 0ˆ 4 ; (3.21) Õ Õ 4 (C) = 48ˆ  C/~4†4−8ˆ  C/~ = :∗A†48l 9 C, 4 (C) = : A 4−8l 9 C . ˜1 ˆ1 9 ˆ9 ˜2 9 ˆ9 (3.22) 9 9

The remaining piece in this calculation is to evaluate the two-time environmental correla- tions which turn out to be four in total. In the following, we mention the final values of these correlations that can easily be found by choosing a multimode Fock state basis set

25 for the trace operation.

Õ −8l

4−~l

Here : is the Boltzmann constant. To further evaluate the non-vanishing environmental correlations, we replace the summation over the discrete mode number < to an integral assuming a continuum of allowed frequencies o. To fix the dimensions we introduce the density of states function 6(o). The physical interpretation of the factor 6(o)3o is the number of oscillators lying in the frequency range o to o + 3o. These considerations yield

¹ ∞ † 8og 2 4˜ (C)4˜(C − g) = 3o4 6(o) :(o) =¯(o, )), (3.28) 0 ¹ ∞ † −8og 2 h i 4˜(C)4˜ (C − g) = 3o4 6(o) :(o) =¯(o, )) + 1 . (3.29) 0

26 Inserting Eq. (3.28) and Eq. (3.29) into Eq. (3.20) and applying the Markov approximation on d˜(C − g) which allows us to assume d˜(C − g) = d˜(C). After some rearrangements of terms we arrive at the following form of the master equation

¹ C ¹ ∞  † †  −8(o−l)g 2  † d˜¤(C) = 0ˆd˜(C)0ˆ − 0ˆ 0ˆd˜(C) 3g 3o4 6(o) :(o) + 0ˆd˜(C)0ˆ 0 0 ¹ C ¹ ∞ † † †  −8(o−l)g 2 + 0ˆ d˜(C)0ˆ − 0ˆ 0ˆd˜(C) − 0ˆ0ˆ d˜(C) 3g 3o4 6(o) :(o) =¯(o, )) + ℎ.2. 0 0 (3.30)

To evaluate the double integrals in the above equation, we notice that the time C sets the scale for the system evolution while the integration on g takes into account reservoir correlation times which are much shorter according to the Markov approximation. We, therefore, extend the integration limit on g from 0 to ∞ and perform the integration by introducing a convergence factor 4nC

! ! ¹ ∞ n o − l 3g4−8(o−l−8n)g = − 8 lim lim 2 2 lim 2 2 n→0 0 n→0 (o − l) + n n→0 (o − l) + n  1  = cX(o − l) + P . (3.31) o − l

The real part of the integrand is a Lorentzian function that after integration gives a Dirac delta function with area c. The imaginary part is evaluated through Cauchy’s theorem with P (...) representing the Cauchy principal value. Using this integral result, Eq. (3.30) simplifies to

 † †   2   † d˜¤(C) = 0ˆd˜(C)0ˆ − 0ˆ 0ˆd˜(C) c6(l) :(l) + 8Δ + 0ˆd˜(C)0ˆ

† † †   2  + 0ˆ d˜(C)0ˆ − 0ˆ 0ˆd˜(C) − 0ˆ0ˆ d˜(C) c6(l) : (l) =¯(l, )) + 8Δ + ℎ.2., (3.32)

27 where Δ and Δ are the principle value contributions which are given by

" 2 # " 2 # ¹ ∞ 6(o) :(o) ¹ ∞ 6(o) :(o) Δ = 3o , Δ = 3o =(o, )) . P l o P l o ¯ (3.33) 0 − 0 −

2 Finally, by fixing c6(l) : (l) := ^/2 and =¯(l, )) := =¯ and combining terms we arrive at the master equation

h i ^     d˜¤ = −8Δ 0ˆ†0,ˆ d˜ + 20 ˆd˜0ˆ† − 0ˆ†0ˆd˜ − d˜0ˆ†0ˆ + ^=¯ 0ˆd˜0ˆ† + 0ˆ†d˜0ˆ − 0ˆ†0ˆd˜ − d˜0ˆ0ˆ† . 2 (3.34)

The above master equation is in the interaction picture. If one wants, it is straightforward to transform back into the Schrödinger picture using the identity

h i ¤ 1 −8ˆ(C/~ 8ˆ(C/~ dˆ = ˆ (, dˆ + 4 d4˜¤ . (3.35) 8~

† Using ˆ ( = ~l0ˆ 0ˆ and Eq. (3.34) in Eq. (3.35) we obtain the desired Lindblad master equation in the Schrödinger picture for a damped harmonic oscillator (for example a single-mode optical cavity):

h i ^   ^=¯   dˆ¤ = −8l0 0ˆ†0,ˆ dˆ + (=¯ + 1) 20 ˆdˆ0ˆ† − 0ˆ†0ˆdˆ − dˆ0ˆ†0ˆ + 20 ˆ†dˆ0ˆ − 0ˆ0ˆ†dˆ − dˆ0ˆ0ˆ† , 0 2 2

(3.36) l0 = l+Δ where 0 represents the frequency shift in the harmonic oscillator natural frequency l due to coupling with the environment. The physical interpretation of the first term on the right-hand side of Eq. (3.36) is of closed system dynamics (which is also known as the von- Neumann equation) while the remaining two terms are purely arising due to environmental

28 interactions with the system. As one can see these terms give rise to energy loss from the system in the form of damping of quantized oscillations (photon decay in the example of optical cavities). The parameter ^ defines the rate at which such a damping/decay happens. Due to this classification of terms sometimes above master equation is also expressed in the following short form ¤ h † i dˆ = −8l0 0ˆ 0,ˆ dˆ + L [dˆ]. (3.37)

Here the Liouvillian L [dˆ] is a superoperator that presents the decay terms in the master equation. Sometimes, based on the Liouvillian form, it becomes possible to obtain the time-evolution of the density operator by directly integrating Eq. (3.37).

3.2 Master Equation for a Three-mode Optomechanical Kerr Cavity

The hybrid setup studied in this thesis consists of three subsystems namely, two independent l l photonic modes trapped inside an optical cavity with resonant frequencies 1 and 2 ; and a phononic mode describing the quantized mechanical oscillations with resonant frequency l<. We direct the reader to Fig. 2.1 to recall the setup. Therein, the system Hamiltonian

ˆ BHB consists of several terms mainly categorized into a free Hamiltonian ˆ0, an interaction

Hamiltonian ˆ8=C and the drive Hamiltonian ˆ 3A . To describe the dissipative dynamics ˆ ˆ ˆ ˆ of our system we consider three environments 1 , 2 and 3 . We suppose 1 is ˆ coupled with the first optical mode, 2 is coupled with the second optical mode, and ˆ 3 is coupled with the mechanical mode. Since after quantization both optical and the mechanical mode behave like quantum harmonic oscillators, consequently, the problem essentially becomes an extension of a single damped quantum harmonic oscillator to three damped harmonic oscillators. Therefore, by following the last section we introduce the

29 total Hamiltonian ˆ of our open quantum system as

Õ † Õ † Õ † ˆ = ˆ BHB + l 9 Aˆ Aˆ9 + l 9 Aˆ Aˆ9 + l 9 Aˆ Aˆ9 ~ 1 91 1 ~ 2 92 2 ~ 3 93 3 91 92 93 | {z } | {z } | {z } ˆ ˆ ˆ 1 2 3 Õ ∗ † † Õ ∗ † † Õ ∗ † † + (: 0ˆ Aˆ + : 9 0ˆ Aˆ9 ) + (: 0ˆ Aˆ + : 9 0ˆ Aˆ9 ) + (: 1ˆAˆ + : 9 1ˆ Aˆ9 ) . ~ 91 1 9 1 1 1 1 ~ 9 2 2 92 2 2 2 ~ 93 93 3 3 91 92 93 | {z } | {z } | {z } ˆ(1 ˆ(2 ˆ(3 (3.38)

Here all three environments are assumed to be multimode quantum harmonic oscillators 9 9 9 l l l with 1th, 2th, and 3th mode frequency given by 91 , 91 , and 91 , respectively with respective annihilation operators Aˆ1, Aˆ2 and Aˆ3. We further assume that these environ- ments are independent that becomes more evident through specifying the non-vanishing † † † commutation relations [Aˆ9 , Aˆ ] = 1, [Aˆ9 , Aˆ ] = 1 and [Aˆ9 , Aˆ ] = 1. 1 91 2 92 3 93 Following the last section’s line of calculations (now for three damped harmonic oscil- lators) one can perform a rather lengthy but straight forward calculations to show that the final Lindblad master equation takes the form

¤ 1 h i dˆ = ˆ BHB, dˆ 8~ ^   ^   + 1 (=¯ + 1) 20 ˆ dˆ0ˆ† − 0ˆ†0ˆ dˆ − dˆ0ˆ†0ˆ + 1 =¯ 20 ˆ†dˆ0ˆ − 0ˆ 0ˆ†dˆ − dˆ0ˆ 0ˆ† 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 ^   ^   + 2 (=¯ + 1) 20 ˆ dˆ0ˆ† − 0ˆ†0ˆ dˆ − dˆ0ˆ†0ˆ + 2 =¯ 20 ˆ†dˆ0ˆ − 0ˆ 0ˆ†dˆ − dˆ0ˆ 0ˆ† 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ^   ^   + 3 (=¯ + 1) 21ˆ dˆ1ˆ † − 1ˆ †1ˆ dˆ − dˆ1ˆ †1ˆ + 3 =¯ 21ˆ †dˆ1ˆ − 1ˆ1ˆ †dˆ − dˆ1ˆ1ˆ † , (3.39) 2 3 2 3 where ^1, ^2, and ^3 represent the decay rates of optical mode 1, optical mode 2, and me- chanical mode, respectively. The absolute temperatures of these thermal baths (multimode

30 quantum harmonic oscillators) are given in terms of average photon number for bath 1 (=1), average photon number for bath 2 (=2), and average phonon number for bath 3 (=3). In the shorter form, the above master equation can be expressed as follows by defining three Liouvillian operators each describing the dissipation of a subsystem due to the coupling with the environment

¤ 1 h i dˆ = ˆ BHB, dˆ + L1 [dˆ] + L2 [dˆ] + L3 [dˆ]. (3.40) 8~

It is worthwhile to mention that the Lindblad structure of the optomechanical master equations becomes questionable under the ultra-strong optomechanical coupling regime in which the optomechanical coupling strength 6 becomes comparable to the mechanical vibration frequency l< i.e. 6 ∼ l<. Since such a regime is beyond the scope of this thesis, therefore, we direct the interested reader to the references [32, 33]. In the next chapter, we’ll use the master equation mentioned in the last two equations to quantify the single-photon generation by calculating the second-order correlation functions.

31 Chapter 4

Results

4.1 Conventional Photon Blockade

The photon blockade [34] as briefly discussed in the Introduction Chapter is a physical effect where a second photon is hindered or blocked from emission from an absorbing medium if it is already occupied by a photon. A similar phenomenon exists in the condensed matter physics known as the Coulomb blockade [35], wherein the flow of an electron is blocked through an electrical circuit due to the presence of another electron. However, the mechanisms are different. In the case of Coulomb blockade a strong Coulombic repulsion is responsible for blocking the flow of further electrons. As photons are chargeless particles, therefore, such an interaction cannot exist between them, rather an indirect coupling is needed to preclude further photon emission. It turns out that there are two ways to achieve such a coupling. One is referred to the conventional photon blockade (CPB) [36] and other the unconventional photon bloackde (UPB) [10]. In the following we first discuss the CPB. The CPB has been predicted and observed in the cavity QED setups in the strong coupling regime which requires the atom-cavity coupling rate 602 is greater than the loss

32 rates (cavity decay rate ^ and spontaneous emission rate W from the atom) [37]. Modeling a the coupling of a two-level emitter (with a lowering operator fˆ and transition frequency l46) a single-mode optical cavity (with the annihilation operator 0ˆ and resonant frequency l0), we can express the total Hamiltonian using the standard Jaynes-Cummings model as

 1 l f†f l 0†0 6 0f† f0†
. ˆ  = ~ 46 ˆ ˆ + ~ 0 ˆ ˆ + ~ 02 ˆ ˆ + ˆ ˆ (4.1) 2

To find out the eigenspectrum of the above Hamiltonian, we note that nature of interaction allows us to define the basis set either as {|6, = + 1i , |4, =i}. Here = is the eigenvalue of the number operator for the cavity mode and |6i(|4i) is the ground (excited) state of the atom/emitter. This basis set also represents the “bare/non-interacting" states of the atom-cavity Hamiltonian. Inclusion of the interaction part, the eigenvalues take the form

  q _ l = 1 ~ 2 62 = , ± = ~ 0 + ± Δ02 + 4 02 ( + 1) (4.2) 2 2 while Δ02 = l46 −l0. Corresponding eigenvectors which are also called the dressed states of the Jaynes-Cummings model turns out to be

|=, +i = cos(Φ=/2) |4, =i + sin(Φ=/2) |6, = + 1i , (4.3)

|=, −i = − sin(Φ=/2) |4, =i + cos(Φ=/2) |6, = + 1i . (4.4)

The angle Φ= depends on detuning, atom-cavity coupling rate and number of photons in the cavity mode. It is defined as √ 2602 = + 1 tan (Φ=) = . (4.5) Δ02

In Fig. 4.1 we draw the combined energy-level diagram before and after the Jaynes-

33 Figure 4.1: Energy-level diagram (drawn up to three photons) describing a standard Jaynes- Cummings ladder. The anharmonicity in the energy spectrum under the strong-coupling regime is utilized to achieve the CPB. ~ = 1 in the diagram.

Cummings interaction. On the left side of diagram we see the energy structure of a non-interacting atom with two energy states separated by energy l4 (~ = 1 and assuming ground state energy to be zero). In the middle of the diagram, we draw the energy levels for a free single-mode optical cavity which being a quantum harmonic oscillator represents l an energy ladder configuration with equally spaced levels with energy difference of ~ 0. On the right hand side we have drawn the dressed states (eigenstate) produced due to atom-cavity coupling. For Δ02 = 0, we notice that the interaction Hamiltonian splits bare l (= + 1 ) atom-cavity states into two states that have an equal separation about ~ 0 2 , with √ displacement ± =~602. Now, as shown in the figure by orange colored vertical arrows, if

34 we have a drive with a single photon energy l? then after the absorption of that photon, the energy of the second photon remains off-resonant for higher energy levels and as such this system cannot be excited to the higher energy unless the first photon is emitted. Thus, there establishes an effective repulsion for the second photon as long as the first photon is in the system. This is the essence of the CPB [8]. However, as we can see that the splitting between the dressed states crucially relies on the atom-cavity coupling strength

602 or on the strong coupling regime of the cavity QED. We can use cooperativity factor C to distinguish between strong and weak coupling regimes of cavity QED. It is defined  62 ^W ^ W as = 02/2 where is the optical mode decay rate and is the spontaneous emission from the two-level emitter. In recent years C values as high as 163 has been achieved using Ce atoms cavity QED [38]. This is still not a straight forward process and has its own challenges, which poses limitations on the demonstration of the CPB. Interestingly, there exists another approach to accomplish the same task that is alternate to the conventional photon blockade, and thus aptly named the unconventional photon blockade (UPB). Rather than relying on the strong atom-cavity coupling, the UPB utilizes quantum destructive interference between different excitation pathways leading to states with two-photon emission [39]. The UPB was first theoretically proposed by Liew and Savona in 2010 [10] and was experimentally observed in 2018 [12] by Snijders et al. In the next section, we focus on our main setup (a three-mode optomechanical cavity with Kerr nonlinearity) and use it as an example to describes how UPB can be used to produce single-photon Fock states of light.

35 4.2 Unconventional Photon Blockade in a Three-mode Optomechanical Kerr Cavity

4.2.1 Transformed Hamiltonian

We begin with the total Hamiltonian for our system (Eq. (2.1b)-(2.1d)) consisting of a three-mode optomechanical cavity filled with a Kerr-type nonlinear medium

ˆ = l 0†0 + l 0†0 + l 1ˆ †1ˆ + *0†0†0 0 + *0†0†0 0 ~ 1 ˆ1 ˆ1 ~ 2 ˆ2 ˆ2 ~ < ~ ˆ1 ˆ1 ˆ1 ˆ1 ~ ˆ2 ˆ2 ˆ2 ˆ2 + 6(0†0 + 0†0 )(1ˆ + 1ˆ †) + Ω (0 48l;C + 0†4−8l;C) + Ω (0 48l;C + 0†4−8l;C). ~ ˆ2 ˆ1 ˆ1 ˆ2 ~ 1 ˆ1 ˆ1 ~ 2 ˆ2 ˆ2 (4.6)

Notice that here to ensure the cross-coupling between optical and mechanical modes (three- mode coupling), we have supposed that the difference between optical mode frequencies l l l is equal to the mechanical mode frequency i.e. 1 − 2 = <. We now apply a few transformations on ˆ for the sake of simplicity. The first transformation is introduced to remove the time-dependent phase factors in the drive terms. To this end, we transform l  into a frame rotating with laser frequency ;. As a result, the transformed Hamiltonian e1 turns out to be related to ˆ through the following relation

 = *ˆ †ˆ*ˆ − ˆ , e1 1 1 1 (4.7)

36 −8ˆ1C/~ where the unitary operator *ˆ1 is defined as *ˆ1 ≡ 4 and we have chosen ˆ1 = l (0†0 + 0†0 ) Δ ≡ l − l ~ ; ˆ1 ˆ1 ˆ2 ˆ2 . With detuning 1 ; one can find

 = Δ0†0 + (Δ − l )0†0 + l 1ˆ †1ˆ + *0†0†0 0 + *0†0†0 0 e1 ~ ˆ1 ˆ1 ~ < ˆ2 ˆ2 ~ < ~ ˆ1 ˆ1 ˆ1 ˆ1 ~ ˆ2 ˆ2 ˆ2 ˆ2 + 6(0†0 + 0†0 )(1ˆ + 1ˆ †) + Ω (0 + 0†) + Ω (0 + 0†). ~ ˆ2 ˆ1 ˆ1 ˆ2 ~ 1 ˆ1 ˆ1 ~ 2 ˆ2 ˆ2 (4.8)

Next, we simplify the above Hamiltonian further by introducing another unitary transfor- ˆ † *ˆ ≡ 4−82C/~ ˆ = l (1ˆ †1ˆ + 0 0 ) mation 2 where 2 ~ < ˆ2 ˆ2 . The purpose of this transformation is to remove the frequency shift in the second optical mode term and completely remov- ing the mechanical oscillator free Hamiltonian. Additionally, we apply the rotating wave approximation assuming the three-mode coupling to be much weaker than the mechanical vibration frequency i.e. 6 << l<. We thus arrive at a much simplified form of the system  = *ˆ †ˆ *ˆ − ˆ Hamiltonian e2 2 1 2 2

 = Δ(0†0 + 0†0 ) + *(0†0†0 0 + 0†0†0 0 ) + 6(0†0 1 + 0 0†1ˆ †) e2 ~ ˆ1 ˆ1 ˆ2 ˆ2 ~ ˆ1 ˆ1 ˆ1 ˆ1 ˆ2 ˆ2 ˆ2 ˆ2 ~ ˆ1 ˆ2 ˆ1 ˆ2 + Ω (0† + 0 ). ~ 1 ˆ1 ˆ1 (4.9)

Due to the application of the above mentioned transformations, we note that the Ω2 term has disappeared. We would like to point out that in an actual experiment the drive would populate both optical modes. However, in no way our transformed Hamiltonian is in contradiction with the experiments, rather it has been expressed in the above form for the sake of mathematical convenience only. We now incorporate dissipation in our model Hamiltonian by adding decay terms by hand. Note that since all decay channels are independent therefore, such a phenomenological inclusion of decay terms generates the same dynamics as obtained from the master equation analysis. To this end, we modify the optical mode detuning Δ by adding an imaginary optical leakage rate ^ term. Similarly,

37 we add a complex mechanical damping term with a rate of W. Eq. (4.9) then becomes

8 ^ 8 W  8^  8 W 0  ~ 0†0 0†0 ~ 1†1 0†0 0†0 ~ 1†1 ˆ = e2 − ( ˆ ˆ1 + ˆ ˆ2) − ˆ ˆ = ~ Δ − ( ˆ ˆ1 + ˆ ˆ2) − ˆ ˆ 2 1 2 2 2 1 2 2 + *(0†0†0 0 + 0†0†0 0 ) + 6(0†0 1 + 0 0†1ˆ †) + Ω (0† + 0 ). ~ ˆ1 ˆ1 ˆ1 ˆ1 ˆ2 ˆ2 ˆ2 ˆ2 ~ ˆ1 ˆ2 ˆ1 ˆ2 ~ 1 ˆ1 ˆ1

(4.10) This Hamiltonian will be our main working Hamiltonian for the rest of this chapter. We point out that in Eq. (4.10) while adding the decay terms we have assumed that the absolute temperatures of all three environments are zero i.e. =¯1 = =¯2 = =¯3 = 0. We also observe that the three-mode cross-coupling terms represent two sets of physical processes each occurring at a rate of 6. In the first process, a photon is created in the first optical mode at the expense of annihilating a photon from optical mode 2 and a phonon from the mechanical mode. In the other process (which is the reverse to the first) a photon is destroyed in the first optical mode and consequently, a photon is created in the second optical mode along with a phonon in the mechanical mode.

4.2.2 Two-photon Two-phonon Truncation

For a laser-driven optomechanical cavity, if the strength of the drive is strong enough mean- ing Ω1 > {^, W} then, in general, a very large of photons populate the optical modes and hence many phonons can be generated. Such a situation becomes intractable analytically unless one enters the classical regime where certain approximations can be made. Since we are interested in the single-photon Fock state generation (which are purely quantum mechanical states of light) therefore to proceed with the analytic results we apply the two- photon two-phonon truncation. According to this truncation, we restrict our Hilbert space

38 to have at most two photons and two phonons basis elements in it, which produces the following quantum state of the system

E Õ  (C)  (C) |

We are following the notation that the first, second, and third slots in our state represent the number of photons in the first optical mode, the number of photons in the second optical mode, and the number of phonons in the mechanical mode, respectively. We assume that initially, the system is in the ground state which means there are no photons in either optical mode and no phonons in the mechanical mode. This state exists with the probability amplitude of 000. The drive from the laser and the interaction between the modes can then evolve the system to any one of these 18 different excited states. However, when we combine all such allowed excited states subjected to our truncation, it is found that the system can only be excited to 5 other states as shown in Figure 4.2 and expressed in the state form as

|Ψ(C)i = 000(C) |000i + 100(C) |100i + 011(C) |011i + 200(C) |200i

+ 022(C) |022i + 111(C) |111i . (4.12)

The coefficient 8 9 : is the probability amplitude which allows us to calculate the probability of finding the system in that state. Note that the other states as shown in Figure 4.2 are not accessible by any transition described by our Hamiltonian.

39 Figure 4.2: Energy-level diagram describing all two-photon two-phonon states accessible in our model.

4.2.3 Dissipative Dynamics

Since in our model we can have at most two photons in the system, therefore, the photon blockade will refer to the situation when the probability of two-photon emission will be zero.

For mode 01 this requires 200 = 0. To this end, we solve the time-dependent Schrödinger’s equation and take the steady-state limit due to the presence of a constant drive. Projecting the wavefunction onto this we can arrive at 6 coupled differential equations as can be seen

40 below

3 000 = −8Ω , 3C 100 (4.13a) 3  8^  √ 100 = −8 Δ −  − 86 − 8Ω( 2 +  ), (4.13b) 3C 2 100 011 200 000 3  8^  −8W  011 = −8 Δ −  − 8  − 86 − 8Ω , (4.13c) 3C 2 011 2 011 100 111 3  8^  √ √ 200 = −8 Δ −  − 28* − 286 − 28Ω , (4.13d) 3C 2 200 200 111 100 3  8^  −8W  022 = −8 Δ −  − 8  − 286 − 286 , (4.13e) 3C 2 022 2 022 111 111 3  8^  −8W  √ 111 = −8 Δ −  − 8  − 86(2 + 2 ) − 8Ω . (4.13f) 3C 2 111 2 111 022 200 011

¤ Solving these differential equations for the steady-state wherein 8 9 : = 0 where 8, 9, and : take values corresponding to the six possible eigenstates of the system. We can then solve the set of equations for the coefficients as functions of Δ, 6, and *.

4.2.4 Second-Order Correlation Function: Plots and Discussion

For the analytic results, we concentrate on the second-order correlation function with zero time-delays. Note that the inclusion of detection delays either requires numerical analysis [40] or the application of the quantum regression theorem [41] which we leave for future work. After solving Eq. set(4.13) under steady-state conditions, we can use the value of the amplitudes to solve for the second-order correlation function introduced in Eq. (2.15). Assuming that both optical modes are subjected to identical conditions, we define the

41 correlation function for the first optical mode as

D E 0ˆ†0ˆ†0ˆ 0ˆ ( ) 1 1 1 1 6 2 (0) = , (4.14) 01 D E2 0†0 ˆ1 ˆ1

6(2) ( ) the second optical mode correlation function 02 0 will follow an analogous equation. In the above equation, we solve for the numerator first. To this end, we make use of Eq. (4.12) and find that only 200 amplitude contributes in the numerator i.e.

† † † † † † 2 0ˆ 0ˆ 0ˆ0ˆ = CA{0ˆ 0ˆ 0ˆ0ˆdˆ} = hΨ| 0ˆ 0ˆ 0ˆ0ˆ |Ψi = 2|200| . (4.15)

The denominator which serves as a normalization factor just represents the square of the average photon number in the first optical mode. We observe that for denominator not only

200 but also other amplitudes (100 and 111) contribute in the following way

 2 † 2 †
2 †
2 2 2 2 0ˆ 0ˆ = CA{0ˆ 0ˆ} = hΨ| 0ˆ 0ˆ |Ψi = |100| + 2|200| + |111| . (4.16)

Inserting Eq. (4.15) and Eq. (4.16) into Eq. (4.14), the second-order correlation function can be expressed in terms of probability amplitudes as,

 2 (2) 2| 200| 60 (0) = . (4.17) 1 2 2 2
2 |100| + 2|200| + |111|

Before reporting our results, we notice that there are mainly three parameters in this problem which can be varied to obtain the optimal conditions for single-photon detection: l l (1) detuning Δ between the laser frequency ; and first optical mode frequency 1 , (2) three-mode coupling rate 6, and (3) strength of Kerr nonlinearity *. In Fig. 4.3 we plot

42 6(2) ( ) Δ Figure 4.3: Second-order correlation function 01 0 as a function of the detuning for different experimentally accessible values of 6 and *. In this and the next plot we have selected Ω = 0.5^ while mechanical damping and the temperature of mechanical bath have been omitted. the second-order correlation function vs Δ while taking different values of {6, *} in each curve. All parameters are defined in the unit of ^ which represents the rate at which photon can be lost from the cavity. We note that for certain detunings, the system does undergo 6(2) ( ) < UPB in all 4 cases plotted (i.e. 01 0 1). And in achieving the UPB we have chosen weak nonlinearity strengths which makes the experimental realization of these results less challenging. The physical reason of achieving UPB can be understood from the two-photon energy-level diagram that is Fig. 4.2. The values for detuning at which either zero or at most on photon is detected correspond to the perfect destructive interference which completely suppresses system to populate thee two-photon state |200i. Additionally from Fig. 4.3 we 6(2) ( ) −→ are notice that for very large detunings 01 0 1, showing that the input drive leaves 6(2) ( ) the system almost unchanged. Moreover, the minima of 01 0 shifts towards higher

43 detuning as we increase the three-mode interaction strength. This shows that the minimum 6(2) ( ) value of 01 0 (which defines the best working point for the single-photon generation) is tunable by adjusting the parameter 6 while fixing all other parameters to constant values. As discussed earlier, in the CPB strong light-matter interaction creates a nonlinear energy spectrum which turns out to be an essential ingredient for the single-photon genera- tion. However, in our setup, the nonlinearity is introduced by the Kerr medium. Therefore, it is worthwhile to discuss how Kerr nonlinearity impacts the antibunching of light. For this purpose, in Fig. 4.4 we plot ;>6(6(2) (0)) as a function of both the normalized Kerr strength */^, and normalized detuning Δ/^, for four values of the coupling coefficient 6. We notice that the threshold of the condition for the UPB (that is when 6(2) (0) < 1) is set by the light to dark blue colored density regions in all four plots. As can be seen, for all 4 values of 6, UPB is achievable and occurs for a quite wide working range, including low nonlinearity strengths, where CPB would not function. We see that with increasing 6, the density region showing 6(2) (0) < 1 shifts towards higher detuning as consistent with Fig. 4.3. We also observe that these UPB regions can be shifted towards lower Kerr medium strengths */^ at the cost of tuning 6/^ to higher values.

44 Figure 4.4: Density plots presenting how 6(2) (0) varies as a function of normalized Kerr nonlinearity strength */^ and normalized detuning Δ/^ for four values of three-mode coupling rate. In (a) 6 = ^ (b) 6 = 1.5^ (c) 6 = 2^ (d) 6 = 2.5^.

45 Chapter 5

Summary and Outlook

In this thesis, by revisiting [22], we discussed the statistics of photons emitted by a laser-driven three-mode optomechanical cavity with Kerr type nonlinearity. Our main focus was to indicate the parameter regime in which such a system can be utilized as an experimentally feasible source for generating single photons. After briefly reviewing cavity quantum optomechanics and Kerr effect in optics, we presented the detailed derivation of the master equation followed by our setup under the Born-Markov approximations of open quantum systems. With the master equation at hand, we theoretically calculated the second-order correlation 6(2) function under the two-photon and two-phonon truncation of the Hilbert space of the problem. In particular, we varied three main system parameters, the strength of the Kerr nonlinearity *, three-mode coupling coefficient 6, and the detuning between laser frequency and the optical mode resonance Δ to identify the possible working regimes for single-photon generation. Broadly speaking, we found that these system parameters can be chosen for a wide range of values that can result in unconventional photon blockade and hence support single-photon detection. As an example, we noticed that the condition 6(2) (0) < 1 necessary for antibunched light can be met by fixing

46 */6 = 0.345 for 0.25^ ≤ Δ ≤ 1.5^ (^ being the unit of the problem is the optical leakage rate). Also, we pointed out that as the three-mode coupling is enhanced the minimum value of 6(2) (0) shifted towards lager detunings. This indicated that the 6 parameter can work as a probe to tune the best working value for the single-photon source. When we compared our results with the conventional photon blockade (that typically relies on establishing strong light-matter interaction), we concluded that the single photons can be generated in our system even with weak Kerr nonlinearities if three-mode coupling 6 is properly optimized. Even though in this thesis we have focused on the UPB and advocated how it is experimentally less challenging for the single-photon generation as compared to other methods, we would like to point out that the UPB has its comparative disadvantages. As an example, we mention the assumption of weak drive generating on average two photons. This provides limitations as even if the laser has a mean less than 2 photons, there is still a chance that once in a while the source can emit more than two photons (say 3 for instance). In which case UPB may cease to function as a single-photon source as the premise of using destructive interference among different pathways to 2-photon emission does not help if there are more than two photons in our system. Due to the richness of the hybrid system considered in this thesis, there can be several possible directions in which this work can be extended. For instance, the two-phonon re- striction made to simplify the analytic analysis can be relaxed. Such relaxation will require the formulation of the entire problem numerically by applying Monte-Carlo simulation methods based on quantum trajectories. Ref. [22] has done some work along these lines but many aspects (for example, stabilization of the code with increasing phonon number, temporal dynamics of the system at the level of individual quantum trajectories, etc.) are still unanswered. Another interesting direction that will make this study even more realis- tic is the inclusion of the mechanical losses in both analytic and numerical results. These mechanical losses will enter in our model through mechanical damping as well as through

47 the non-zero temperature of the mechanical reservoir. Furthermore, in our model, we assumed the strength of the Kerr nonlinearity (characterized by *) is a fixed parameter. However, due to mechanical oscillations, the Kerr medium can also be compressed and hence may lead to a position-dependent nonlinear strength *(G). In what ways then such a position dependence of * will alter the results in this work? We leave these and other related questions as the future explorations of this thesis.

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