The interaction between matter and squeezed light has mostly been treated within the approximation that the field-correlation time is small. Methods for treating squeezed light with more general correlations currently involve explicitly modeling the systems producing the light. We develop a general purpose input-output theory for a particular form of narrowband squeezed light, a squeezed wave-packet mode, that only concerns the statistics of the squeezed field and the shape of the wave packet. This formalism allows us to derive the input-output relations and the master equation. We also consider detecting the scattered field using photon counting and homodyne measurements, which necessitates the derivation of the stochastic master equation. The non-Markovian nature of the field manifests itself in the master equation as a coupled hierarchy of equations. We illustrate these with consequences for the decay and resonance fluorescence of two-level atoms in the presence of such fields.
The interaction between matter and squeezed light has mostly been treated within the approximation that the field-correlation time is small. Methods for treating squeezed light with more general correlations currently involve explicitly modeling the systems producing the light. We develop a general purpose input-output theory for a particular form of narrowband squeezed light, a squeezed wave-packet mode, that only concerns the statistics of the squeezed field and the shape of the wave packet. This formalism allows us to derive the input-output relations and the master equation. We also consider detecting the scattered field using photon counting and homodyne measurements, which necessitates the derivation of the stochastic master equation. The non-Markovian nature of the field manifests itself in the master equation as a coupled hierarchy of equations. We illustrate these with consequences for the decay and resonance fluorescence of two-level atoms in the presence of such fields.
Abstract We determine filtering and master equations for a quantum system interacting with wave packet of light in a continuous-mode squeezed number state. We formulate the problem of conditional evolution of a quantum system making use of model of repeated interactions and measurements. In this approach, the quantum system undergoes a sequence of interactions with an environment defined by a chain of harmonic oscillators. We assume that the environment is prepared in an entangled state being a discrete analogue of a continuous-mode squeezed number state. We present a derivation of a discrete stochastic dynamics that depends on the results of measurement performed on the field after its interaction with the system. In this paper, we consider a photon counting measurement scheme. By taking a continuous time limit, we finally obtain differential stochastic equations for the system. Analytical formulae for quantum trajectories and exclusive probability densities that allow to fully characterize the statistics of photons in the output field are given.
In this paper, photon-counting statistics for an output field of a single-photon wave packet interacting with a quantum system was derived. The problem was formulated using a collision model with a bidirectional field approximated by two sequences of qubits. It was shown that the collision model formulated for the light with the temporal correlations allows deriving formulae for quantum trajectories in an effective and intuitive way. Starting from the discrete in-time evolution of the composed system, we determined the conditional and unconditional evolution of the quantum system, provided a physical interpretation of quantum trajectories, and determined the analytical formulae for the exclusive probability densities for the reflected and transmitted fields. Finally, our results were applied to a two-level atom, and we also determined analytical expressions for probability densities of arrival times of the successive counts.
Quantum Brownian motion is the random motion of quantum particles suspended in a field (or an effective field) resulting from their collision with fast-moving modes in the field. It provides us with a fundamental model to understand various physical features concerning open systems in chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper, without either the Born-Markovian or rotating-wave approximation, we derive a master equation for a charged-Brownian particle in one dimension coupled with a thermal reservoir in electric fields. The effect of the reservoir and the electric fields is manifested as time-dependent coefficients and coherent terms, respectively, in the master equation. The two-photon correlation between the Brownian particle and the reservoir can induce nontrivial squeezing dynamics to the particle. We derive a current equation including the source from the driving fields, transient current from the system flowing into the environment, and the two-photon current caused by the non-rotating-wave term. The presented results then are compared with that given by the rotating-wave approximation in the weak-coupling limit, and these results are extended to a more general quantum network involving an arbitrary number of coupled-Brownian particles. The presented formalism might open a way to better understand exactly the non-Markovian quantum network.
The dynamics of a coherently driven two-level atom with parametric amplifier and coupled to a vacuum reservoir is analyzed. The combination of the master equation and the quantum Langevin equation is presented to study the quantum properties of light. By using these equations, we have determined the time evolution of the expectation values of the cavity mode and atomic operators. Moreover, with the aid of these results, the correlation properties of noise operators, and the large-time approximation scheme, we calculate the mean photon number, power spectrum, second-order correlation function, and quadrature variances for the cavity-mode light and fluorescence. It is found that the half-width of the power spectrum for the fluorescent light in the presence of a parametric amplifier increases, while it decreases for the cavity-mode light. Moreover, we have found the probability for the atom to be in the upper level in the presence of a parametric amplifier.
We develop a theoretical description of quantum wave mixing (QWM) in a cascaded waveguide-QED system of two superconducting qubits, where the probe is driven by an external coherent tone and by the resonance fluorescence of a strongly driven source qubit. Starting from the field correlation functions of the source emission, we derive an effective master-equation treatment for the probe and identify the regime in which the incident fluorescence is characterized by anomalous correlations. When the coherent Rayleigh component of the source spectrum is suppressed, the probe equations of motion become equivalent to those for a qubit driven by a coherent tone and broadband squeezed light, although the ligt emitted by the source is non-Gaussian and antibunched. This equivalence implies a selection rule for the peaks of the QWM spectrum, with a strong suppression of sidebands associated with processes involving an odd number of photons taken from the source field. Numerical simulations of the full cascaded two-qubit model for different ratios of radiative decay rates unambiguously confirm the participation of correlated photon pairs in QWM processes. The current research illustrates that the analysis of peak amplitudes can be used to probe photon statistics in the incident nonclassical field.
This dissertation concerns itself with the virtues and vices of weak measurements. Weak measurements are all around us, but this does not mean that one should manufacture weakness on all occasions. We critically evaluate two proposals that claim weak measurements provide a novel means of performing quantum state tomography, allegedly increasing tomographic efficacy and yielding foundational insights into the nature of quantum mechanics. We find weak measurements are not an essential ingredient for most of their advertised features. In contrast to this negative finding, we highlight an optimal tomographic scheme for which weak continuous measurements are the best known implementation, taking the opportunity to present this result in the language of quantum noise. Weak measurements are also used to describe continuous measurements and the associated unconditional and conditional master equations. Though this description has been presented before, we provide a new perspective by using tools and terminology from the discipline of quantum information and quantum computing. We do so by showing how the continuous-time evolution of these master equations arises from discretizing in time the interaction between a system and a probe field and by formulating quantum-circuit diagrams for the discretized evolution. We then reformulate this interaction by replacing the probe field with a bath of qubits, one for each discretized time segment, reproducing all of the standard quantum-optical master equations. This provides an economical formulation of the white-noise theory of continuous measurements, highlighting its fundamental underlying assumptions and limitations. White-noise theory is incapable of describing photon-counting measurements in the presence of thermal and squeezed noise. We accommodate such scenarios by considering an environment that includes traveling wave packets that are squeezed, deriving a hierarchy of equations similar to those used to describe traveling wave packets with fixed photon number. Squeezing introduces qualitatively different effects, however, complicating numerical solution of these hierarchies. We provide preliminary numerical analysis of the formalism and showcase its utility by calculating the resonance fluorescence of a two-level atom in squeezed vacuum with squeezing bandwidth narrower than the atomic linewidth, a regime previous techniques cannot explore.