The simplest open, nonlinear optical system consists of a single bosonic mode with nonlinear self-interactions coupled to driving fields and reservoirs. In this chapter, we will examine a number of systems of this type using the methods that have been developed in the preceding chapters. Systems of this type can exhibit a variety of interesting physical effects. For example, in the case of a driven nonlinear oscillator, there can be more than one steady state (bistability), and quantum statistical effects can manifest themselves by affecting the stability of these states. Other quantum statistical effects have also been predicted in a variety of related systems – such as photon anti-bunching, squeezing and changes to spectra. In general, the size of these effects scales inversely with the size of the system. This ‘system size’ refers to a threshold photon number, a number of atoms, or some similar quantity. The nature of the steady states themselves can become less than straightforward. Nonequilibrium dissipative systems, of which the systems considered in this chapter are examples, have parameters that describe the energy input to the system, and their steady states depend on these parameters. When driven far from equilibrium, such systems can exhibit bifurcations in their steady states. In more complex cases, this eventually leads to periodic oscillations and chaos. Devices like this can be realized with multiple types of nonlinearity, ranging from nonlinear dielectrics through to cavity QED (with near-resonant atoms), cavity optomechanics (with nanomechanical oscillators) and circuit QED (with superconducting Josephson junctions). We will start with cases where all the interactions are with the reservoirs, which may be linear or nonlinear, while the cavity itself is harmonic in its response.
The simplest open, nonlinear optical system consists of a single bosonic mode with nonlinear self-interactions coupled to driving fields and reservoirs. In this chapter, we will examine a number of systems of this type using the methods that have been developed in the preceding chapters. Systems of this type can exhibit a variety of interesting physical effects. For example, in the case of a driven nonlinear oscillator, there can be more than one steady state (bistability), and quantum statistical effects can manifest themselves by affecting the stability of these states. Other quantum statistical effects have also been predicted in a variety of related systems – such as photon anti-bunching, squeezing and changes to spectra. In general, the size of these effects scales inversely with the size of the system. This ‘system size’ refers to a threshold photon number, a number of atoms, or some similar quantity. The nature of the steady states themselves can become less than straightforward. Nonequilibrium dissipative systems, of which the systems considered in this chapter are examples, have parameters that describe the energy input to the system, and their steady states depend on these parameters. When driven far from equilibrium, such systems can exhibit bifurcations in their steady states. In more complex cases, this eventually leads to periodic oscillations and chaos. Devices like this can be realized with multiple types of nonlinearity, ranging from nonlinear dielectrics through to cavity QED (with near-resonant atoms), cavity optomechanics (with nanomechanical oscillators) and circuit QED (with superconducting Josephson junctions). We will start with cases where all the interactions are with the reservoirs, which may be linear or nonlinear, while the cavity itself is harmonic in its response.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
The optical parametric oscillator is one of the most well-characterized devices in nonlinear optics, with many useful applications, especially as a frequency converter and as a wide-band amplifier. Novel discoveries made with these devices in the quantum domain include demonstrations of large amounts of both single-mode and two-mode squeezing, and, in the two-mode case, significant quantum intensity correlations between the two modes and quadrature correlations, which provided the first experimental demonstration of the original Einstein–Podolsky–Rosen (EPR) paradox. In this chapter, a systematic theory is developed of quantum fluctuations in the degenerate parametric oscillator. The quantum statistical effects produced by the degenerate parametric amplifier and oscillator have been intensely studied. As we have seen, the amount of transient squeezing produced in an ideal lossless parametric amplifier was calculated using asymptotic methods by Crouch and Braunstein. It was found to scale inversely with the square root of the initial pump photon number. When the system is put into a cavity, and driving and dissipation are introduced, we find that the steady state of a driven degenerate parametric oscillator bifurcates. There is a single steady state at low driving strengths, but, as the driving field increases, a threshold value is reached and the single state splits and forms two stable states. Below the threshold, a linearized analysis of squeezing in the quantum quadratures is possible; while above this threshold, or critical point, the tunneling time between the two states due to quantum fluctuations can be calculated.
This chapter concludes the discussion about temporal instabilities and self-oscillations, considering the instabilities that arise in the systems analyzed in Chapter 13, with the exception of optical bistability, which has been discussed separately in Chapter 11. Section 25.1 focuses on the laser with an injected signal. As already mentioned in Section 13.1, in this system self-pulsing instabilities are rather ubiquitous, and on exploring the parameter space one meets an extremely rich variety of oscillatory behaviors. For example, under extremely high-gain conditions, breathing, spiking and chaos have been identified over various ranges of the driving field strength [294]. In this section we show some results that arise under parametric conditions that are more accessible experimentally. We illustrate especially the coexistence, for the same parametric values, of different oscillatory behaviors that are reached by starting from different initial conditions. On the other hand, Section 25.2 deals with a laser with a saturable absorber. In this system even the full scenario of stationary intensity solutions in the single-mode model is very complex. For the sake of simplicity, in this section we focus on a simpler rate-equation model that is obtained from the single-mode model of Eqs. (13.13)–(13.17) by adiabatically eliminating the atomic polarization of both active and passive medium. As in Chapter 13, we assume that the atomic transition frequencies are equal and also coincide with a cavity frequency. In this way all the variables in the model can be assumed real, and the linearization around the stationary solution described in Section 13.2 leads to a cubic eigenvalue equation. The spontaneous oscillations which arise from this instability are called repetitive passive Q-switching in the literature. Section 25.3 considers the stability of the stationary solutions of the degenerate optical parametric oscillator described in Section 13.4 and outlines the scenario of period doubling and chaos that arises under appropriate parametric conditions.
The nonlinearity is essential for creation of non-classical states of the cavity or mechanical resonator such as squeezed or cat states. A microwave cavity can be made nonlinear by, for instance, adding Josephson junctions. The mechanical resonator is inherently nonlinear. The radiation pressure interaction between cavity and mechanical resonator is also inherently nonlinear but typically under strong drive of the cavity interaction can be linearized. However, if the optomechanical system is in the strong coupling regime nonlinear quantum effects become observable. These three cases provide the motivation for our studies. We start with backaction analysis of the classical regime of a dc SQUID (superconducting quantum interference device) with an embedded mechanical resonator. Then, we perform quantum analysis to understand how the asymmetry of two Josephson junctions influences the coupling strength of optomechanical interactions such as radiation pressure, cross-Kerr and single-photon beam splitter. Using variational method and self-consistent harmonic approximation we estimate the influence of the radiation pressure coupling and Kerr nonlinearity of the cavity on the effective frequency and dissipation of the cavity. The second part of this thesis focuses on the nonlinearity of the mechanical resonator coupled to the optical/microwave cavity. We study in details the nonlinear optomechanical response and draw the response map to summarize all results of the overcoupled and undercoupled cavity for the red and blue sidebands additionally to the weak and strong drive powers. In the end, w e confirm the developed theory by using it to numerically fit the experimental results.
In this thesis we have focused on the study of various systems which are presently widely studied in different areas of quantum optics and quantum information sciences. These, for example, include the coupled system of photonic waveguides which are known to be highly efficient in manipulating the flow of light. The Hamiltonian describing the evolution of field mode in coupled waveguides is effectively identical to the well-known tight binding Hamiltonian used in solid state physics. The advantage of waveguide system is the possibility to control various interactions by design and their low decoherence rate. The excellent stability offered by coupled waveguides has led to the observation of many key coherent effects such as quantum walk, Bloch oscillation, and discrete Talbot effect. For example, Bloch oscillations have been investigated in coupled waveguides using coherent beam of light. We wanted to inquire whether coherent phenomena such as Bloch oscillations can be possible with incoherent single photon sources. We discovered that Bloch oscillations are indeed possible with single photons provided we prepare single photons in a W state. Moreover, coupled waveguides also find applications in the field of quantum information processing. Since entanglement plays a prominent role in all these applications, it is important to understand the entanglement dynamics in these structures. We considered the case of squeezed input in one of the waveguide and showed that one can generate entanglement between the waveguide modes. We further continued our work on the entanglement generation in coupled waveguides by incorporating the effect of loss in the waveguide structure for the squeezed and photon number input states. We considered relevant experimental parameters and showed that waveguide structures are reasonably robust against the effect of loss. Another system which has attracted a great deal of interest is the optomechanical system. We consider an optomechanical system where an optical cavity mode is coupled to the square of the position of a mechanical oscillator. The optomechanical system can then be regarded as a quantum optical spring, i.e., a spring whose spring constant depends on the quantum state of another system. In particular, we consider the situation where the field inside the cavity is in a coherent state and the oscillator is prepared in its ground state. The quantized nature of the field produces new features in the optomechanical system.
Current research in single-atom cavity quantum electrodynamics largely emphasizes the input-output properties of strongly coupled systems, from normal mode splitting to photon blockade. But over the last decade, experiments have, with few exceptions, focused on relatively weak driving conditions. This thesis concentrates on a range of quantum nonlinear phenomena in the strong driving regime. In particular, I discuss the observation of random-telegraph phase switching in the light transmitted through a Fabry-Perot resonator containing one strongly coupled atom and 10-100 photons, confirming long-standing predictions of a phenomenon known as single-atom phase bistability. These results highlight the relevance of cavity quantum electrodynamics in the development of attojoule nanophotonic logic and signal processing. In addition, I consider a general class of bifurcation phenomena that are manifest within this physical setting. Here, focus is placed on the investigation of quantum-classical correspondence near semiclassical bifurcation points.