My sincere thanks also go to the current and former heads of the Department of Physics, Prof. However, due to the inevitable coupling of the mechanical resonator to its environment, the quantum behavior is masked by the thermal motion.

## Cavity optomechanics

### Optomechanical systems

The resonant frequency of the cavity mode is given by ωc = mlπc, where is the mode number. In cold atomic systems, the optomechanical single-photon coupling can be achieved on the order of the cavity decay rate.

### Optomechanical equations of motion

In floating dielectric systems, the cavity frequency depends on the position of the dielectric object in a sinusoidal pattern. The coupling with the environment acts as a channel for the decay of the cavity photons and the phonons on the mechanical resonator.

## Optomechanical effects

*Optomechanical bistability**Ground-state cooling of a mechanical resonator**Photon blockade**Phonon blockade*

Also, the overall detuning of the cavity changes in proportion to the displacement of the mechanical resonator. In the stationary state, the amplitude of the optical field within the cavity and the position of the mechanical resonator are given by .

## Outline of the thesis

We discuss the controllability of the bistable behavior of the average intracavity optical intensity in the optomechanical cavity depending on the system parameters provided by the feedback cavity. The last term represents the effect of the driving pump laser with frequency ωL and amplitudeεA=.

## Bistability of the intracavity field and mirror position

Next, we proceed to study the bistability behavior of the intracavity optical field in the optomechanical cavity in the presence of the atomic cavity. Lower decay rate of the atomic cavity ensures more feedback intensity to the optomechanical cavity.

## Summary

This chapter discusses the ground state cooling of a mechanical resonator in an optomechanical cavity induced by a quantum well, in the unresolved sideband regime. It is demonstrated that, even when the cavity is in the unresolved sideband regime, the effective interaction of the excitations and mechanical mode can return the system to an effective resolved sideband regime. The time evolution of the average phonon number in the mechanical resonator is studied using the quantum master equation.

3, year 2016, pages title: "Ground state cooling of micromechanical oscillators in the unresolved sideband regime induced by a quantum well"; authors: Bijita Sarma and Amarendra K. We investigate the aspect of ground state cooling of the mechanical oscillator in the unresolved sideband regime.

## Model and theory

The first term Hfree in Eq. 3.1) describes the free Hamiltonian of the system, given by Hfree=ωcc†c+ωdd†d+ωmb†b, where ωc, ωd and ωm are the resonant frequencies of the optical field of the cavity, QW excitons and mechanical oscillator, respectively. Since we are dealing with quasi-resonant coherent excitation, higher-lying QW exciton states are neglected. The optomechanical interaction between the cavity mode and the mechanical oscillator is described by the second term, Ho-m=gOMc†c¡.

In the frame rotating with the input laser frequency, ωL, we can obtain the Hamiltonian of the system as The time evolution of the system operators is given by the nonlinear Heisenberg-Langevin equations.

## Cooling of the mechanical resonator

The cooling rate of the mechanical resonator is given by A−=SF F(ωm)x2ZP F, while the heating rate is given by A+=SF F(−ωm)x2ZP F. The additional damping of the mechanical oscillator due to the optomechanical interaction is given by γOM=A− −A+= −2Im[Σ(ωm)], and the mechanical frequency shift with δωm= Re[Σ(ωm)]. This is possible due to the interaction of the high-Q QW with a mechanical mode through the cavity field.

The plots show that, for ground state cooling of the mechanical resonator, high values of bath phonon numbers are acceptable. This indicates ground state cooling of the resonator mode in the highly unresolved sideband regime.

## Summary

In this chapter, we discuss the phenomenon of photon blocking in optomechanical systems via destructive quantum interference between two-photon excitation paths. In the second part of this chapter, we discuss the photon blocking effect in an optomechanical cavity containing a degenerate optical parametric amplifier. Photon blocking by means of quantum interference in the cavity via parametric interactions”; by Bijita Sarma and Amarendra K.

1, year 2018, pages title: "Unconventional photon blockade in three-mode optomechanics"; authors: Bijita Sarma and Amarendra K. Photon blockade in an optomechanical system was recently studied, where due to the photon-phonon nonlinear interaction, realization of anti-bundled sub-Poissonian light was predicted [75].

## Photon blockade in a three-mode optomechanical cavity

### Model and theory

This Hamiltonian indicates a three-mode interaction between the two optical states and the mechanical state, where a photon from the state, a1, annihilates to create a photon in the state, a2, and a phonon in the mechanical state, b. In the reverse process, a photon from state a2 and a phonon in state b are annihilated by the mechanical resonator to create a photon in state a1. Here nth = 1/[exp(ωm/kBT)−1] indicates the thermal phonon number in the mechanical state at the temperature of the bath, T.

In addition to the master equation approach, optimal conditions for photon blockade can be determined in the following manner. The optimal condition for complete photon blockade corresponds to the case, when the probability of a photon in the state|200〉 is equal to zero.

### Results

Therefore, it is clear that the ambient thermal phonon population has an undesired effect on the perception of photon blockade. So far, we have not considered the effect of pure dephasing-induced decoherences in our analysis. Pure dephasing can arise from the instability of the laser drive, or coupling of the cavity modes with other mechanical modes, and this can have a confounding effect on polarization, linewidth, transmittance and photon statistics [202].

Therefore, in the following we analyze the effect of pure dephasing on the anti-beam properties of the cavity photons. The effects of pure dephasing can be modeled by adding another Lindblad term in the form Lp(ρ)= γ2p P.

Summary

## Photon blockade in an optomechanical cavity with a degenerate optical

### Model and theory

If we consider the basis of the Fock state, |m,n〉in Hilbert space, where mandnars are the number of photons or phonons, the state of the system can be expressed as [81]:. In terms of probability coefficients, the second-order correlation function for the optical mode can be written as:. The steady state solution can be found by solving the coupled equations for the coefficients.

The second (2b†ρb−bb†ρ−ρbb†), are the Liouvillian operators for the optical and mechanical modes respectively. We also derive an analytical expression for the second-order correlation function with zero lag time and compare it with the numerical solution.

### Results

As expected, a strong photon blockade occurs near the red detuning with∆≈κ, and occurs at G/κ≈4.85×10−5 as optimized exactly in Eq. Here also, a strong antibunching occurs near the detuned blue regime with Δ≈ -2κ again atG/κ while there is no photon blockade in the red detuned regime in this case. For ∆≈κ in the red detonation regime, g(2)(0) exhibits strong sub-Poissonian quantum statistics at a phase of θ/π≈ -0.42.

Now we analyze the effect of pure dephasing on the counterbeam properties of the cavity photons. The effects of pure dephasing can be modeled by solving the master equation after adding another Lindblad term of the form Lp(ρ)=γ2p(2a†aρa†a−(a†a)2ρ−).

### Summary

In this chapter we show that phonon blockade can be achieved in a system of two weakly nonlinear mechanical resonators coupled by a Coulomb interaction. 93] The realization of phonon blockade in this system requires strong Kerr-type nonlinearity to obtain an anharmonic energy level. Other than this, here we show that phonon blockade can be realized in a weakly nonlinear mechanical resonator by coupling it to another weakly nonlinear mechanical resonator via Coulomb interaction [226-228].

Although the nonlinearities in mechanical resonators are weak, due to the presence of quantum interference paths, the system may exhibit phonon blocking. The detection of phonon blockade by measuring photon correlations in the presence of an optomechanical interaction is also discussed.

## Model and Hamiltonian

Here Hfree is the free Hamiltonian of the two mechanical resonators, Hnlis the Hamiltonian describing the Kerr nonlinearity, U, in both mechanical resonators and Hco represents the Coulomb interaction Hamiltonian of the two charged mechanical oscillators. In the Coulomb interaction Hamiltonian, Hco denotes the electrostatic constant, d is the equilibrium separation of the two charged oscillators without any interaction between them, and x1 and x2 are the minor oscillations of the two mechanical oscillators from their equilibrium positions. Here, the first term is a constant term and the second is a linear term that can be included in the definition of the equilibrium positions.

The last term consists of two parts: one part refers to the small frequency shift of the original frequencies and can be neglected by re-normalizing the mechanical frequencies, and the other part is the coupling term between the oscillators. The charge contained by the electrodes is given by q1=C1V1 and q2= −C2V2, where Cj is the capacitance of the bias gate on the resonator Mj.

## Phonon blockade with a single drive

The coefficientsCi j's can be obtained by solving the Schrödinger equation,iddt|ψ〉=Heff|ψ〉, where,Heff=H0m−iγ. 2b2 is the non-Hermitian Hamiltonian that includes the damping of the mechanical oscillators. Following an iterative method prescribed by Bambaet al. related to photon blockade in coupled photonic molecules [81], in the limit of weakΩ1, at steady state the optimal parameters are obtained as follows: . 5.7) The limit for the coupling,J, in this case is that the value ofJ must be greater than γ/p.

It is observed that phonon blocking in the weakly nonlinear regime can be achieved for optimal conditions. Therefore, it is evident that the ambient thermal population has an undesirable effect on the observation of phonon blocking.

## Phonon blockade with two drives

To obtain non-trivial solutions for C11, C00 and C02, the determinant of the coefficient matrix must be zero, which gives rise to a quadratic equation inζe−iφ:. The solutions of the quadratic equation are given by: 5.18), it can be seen that for specific values of the parameters U, J and ∆ the optimal values of ζ and φ could be obtained and there are two optimal values of ζ and φ for a specific set of system parameters. Therefore, by using the additional pump, we can choose the optimal values of the amplitude and the phase of the second drive for different coupling strengths and tuning in the system.

It is observed that phonon blockade can be obtained at ∆=0.5γ, which is consistent with ∆opt, as predicted by analytical calculations. Next, we discuss the variation of the second-order correlation function with finite time delay, g(2).

Measurement of phonon blockade via photon correlations

Summary