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Table Of Integrals Series And Products

Squeezed states of the electromagnetic field have been generated by nondegenerate four-wave mixing due to Na atoms in an optical cavity. The optical noise in the cavity, comprised of primarily vacuum fluctuations and a small component of spontaneous emission from the pumped Na atoms, is amplified in one quadrature of the optical field and deamplified in the other quadrature. These quadrature components are measured with a balanced homodyne detector. The total noise level in the deamplified quadrature drops below the vacuum noise level.

Observation of squeezed states generated by four-wave mixing in an optical cavity

Nearly a century after Einstein first predicted the existence of gravitational waves, a global network of Earth-based gravitational wave observatories1, 2, 3, 4 is seeking to directly detect this faint radiation using precision laser interferometry. Photon shot noise, due to the quantum nature of light, imposes a fundamental limit on the attometre-level sensitivity of the kilometre-scale Michelson interferometers deployed for this task. Here, we inject squeezed states to improve the performance of one of the detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) beyond the quantum noise limit, most notably in the frequency region down to 150 Hz, critically important for several astrophysical sources, with no deterioration of performance observed at any frequency. With the injection of squeezed states, this LIGO detector demonstrated the best broadband sensitivity to gravitational waves ever achieved, with important implications for observing the gravitational-wave Universe with unprecedented sensitivity.

Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light

https://physicstoday.scitation.org/doi/pdf/10.1063/1.1510285

Quantum Optics in Phase Space

The Jaynes–Cummings Model

The squeezing effect arising in the interacting qubit-oscillator system is studied with the presence of a parametric oscillator in the Rabi model. To solve the system, we employ the generalized rotating wave approximation which works well in the wide range of coupling strength as well as detuning. The analytically derived approximate energy spectrum portrays good agreement with the numerically determined spectrum of the Hamiltonian. For the initial state of the bipartite system, the dynamical evolution of the reduced density matrix corresponding to the oscillator is obtained by partial tracing over the qubit degree of freedom. The oscillator’s reduced density matrix yields the nonnegative phase space quasi-probability distribution known as Husimi Q-function which is utilized to compute the quadrature variance. It is noticed that the squeezing produced in the oscillator sector gets reduced with increasing coupling strength. We demonstrate the role of the parametric term to obtain adequate squeezing in the strong coupling regime. We also study the revival–collapse phenomenon and the generation of squeezed coherent state.

Squeezing in the quantum Rabi model with parametric nonlinearity

Quantum coherence and quantum correlations are studied in the strongly interacting system composed of two qubits and an oscillator with the presence of a parametric medium. To analytically solve the system, we employ the adiabatic approximation approach. It assumes each qubit’s characteristic frequency is substantially lower than the oscillator frequency. To validate our approximation, a good agreement between the calculated energy spectrum of the Hamiltonian with its numerical result is presented. The time evolution of the reduced density matrices of the two-qubit and the oscillator subsystems are computed from the tripartite initial state. Starting with a factorized two-qubit initial state, the quasi-periodicity in the revival and collapse phenomenon that occurs in the two-qubit population inversion is studied. Based on the measure of relative entropy of coherence, we investigate the quantum coherence and its explicit dependence on the parametric term both for the two-qubit and the individual qubit subsystems by adopting different choices of the initial states. Similarly, the existence of quantum correlations is demonstrated by studying the geometric discord and concurrence. Besides, by numerically minimizing the Hilbert–Schmidt distance, the dynamically produced near maximally entangled states are reconstructed. The reconstructed states are observed to be nearly pure generalized Bell states. Furthermore, utilizing the oscillator density matrix, the quadrature variance and phase-space distribution of the associated Husimi Q -function are computed in the minimum entropy regime and conclude that the obtained nearly pure evolved state is a squeezed coherent state.

Quantum coherence, correlations and nonclassical states in the two-qubit Rabi model with parametric oscillator

Quantum coherence and quantum correlations are studied in the strongly interacting system composed of two qubits and an oscillator with the presence of a parametric medium. To analytically solve the system, we employ the adiabatic approximation approach. It assumes each qubit's characteristic frequency is substantially lower than the oscillator frequency. To validate our approximation, a good agreement between the calculated energy spectrum of the Hamiltonian with its numerical result is presented. The time evolution of the reduced density matrices of the two-qubit and the oscillator subsystems are computed from the tripartite initial state. Starting with a factorized two-qubit initial state, the quasi-periodicity in the revival and collapse phenomenon that occurs in the two-qubit population inversion is studied. Based on the measure of relative entropy of coherence, we investigate the quantum coherence and its explicit dependence on the parametric term both for the two-qubit and the individual qubit subsystems by adopting different choices of the initial states. Similarly, the existence of quantum correlations is demonstrated by studying the geometric discord and concurrence. Besides, by numerically minimizing the Hilbert-Schmidt distance, the dynamically produced near maximally entangled states are reconstructed. The reconstructed states are observed to be nearly pure generalized Bell states. Furthermore, utilizing the oscillator density matrix, the quadrature variance and phase-space distribution of the associated Husimi $Q$-function are computed in the minimum entropy regime and conclude that the obtained nearly pure evolved state is a squeezed coherent state.

Various approaches have been used in the literature for eliminating nonresonant levels in atomic systems and deriving eﬀective Hamiltonians. Important among these are elimination techniques at the level of probability amplitudes, operator techniques to project the dynamics on to the subspace of resonant levels, Green’s function techniques, the James’ eﬀective Hamiltonian approach, etc. None of the previous approaches is suitable for deriving eﬀective Hamiltonians in intracavity situations. However, the James’ approach does work in the case of only two-photon transitions in a cavity. A generalization of the James’ approach works in the case of three-photon transitions in a cavity, but only under Raman-like resonant conditions. Another important approach for adiabatic elimination is based on an adaptation of the Markov approximation well-known in the theory of system-bath interactions. However, this approach has not been shown to work in intracavity situations. In this paper, we present a method of adiabatic elimination for atoms inside cavities in the presence of multiphoton transitions. We work in the Heisenberg picture, and our approach has the advantage that it allows one to derive eﬀective Hamiltonians even when Raman-like resonance conditions do not hold.

Prosenjit Maity

Papers

Squeezing in the quantum Rabi model with parametric nonlinearity

Quantum coherence, correlations and nonclassical states in the two-qubit Rabi model with parametric oscillator

Quantum coherence, correlations and nonclassical states in the two-qubit Rabi model with parametric oscillator

Adiabatic elimination in the presence of multiphoton transitions in atoms inside a cavity

Implementation of a holonomic 3-qubit gate using Rydberg superatoms in a microwave cavity