Quantum Optical Processes Research Articles

Stability diagram and growth rate of parametric resonances in Bose-Einstein condensates in one-dimensional optical lattices

A Bose-Einstein condensate in an optical lattice exhibits parametric resonances when the intensity of the lattice is periodically modulated in time. These resonances correspond to an exponential growth of the population of counter-propagating Bogoliubov excitations. A suitable linearization of the Gross-Pitaevskii (GP) equation is used to calculate the stability diagram and the growth rates of the unstable modes. The results agree with the ones extracted from time-dependent GP simulations, supporting our previous claim (M. Kraemer et al., Phys. Rev. A (2005) in press) concerning the key role of parametric resonances in the response observed by Stoeferle et al. (Phys. Rev. Lett. 92, 130403 (2004)) in the superfluid regime. The role of the seed excitations required to trigger the parametric amplification is discussed. The possible amplification of the quantum fluctuations present in the quasiparticle vacuum, beyond GP theory, is also addressed, finding interesting analogies with similar processes in nonlinear quantum optics and with the dynamic Casimir effect. Our results can be used in exploiting parametric instabilities for the purpose of spectroscopy, selective amplification of a particular excitation mode and for establishing a new type of thermometry.

Electromagnetic pulses from an oscillating high-finesse cavity: possible signatures for dynamic Casimir effect experiments

Vacuum field fluctuations exert radiation pressure on mirrors in quantum vacuum. For a pair of mirrors this effect is well known as the Casimir effect. When a single mirror is moving in vacuum, radiation pressure leads to a dissipative force which opposes itself to the mirror's motion. Accordingly the electromagnetic field does not remain in the vacuum state but photons are emitted by the mirror into vacuum. This paper describes the photon emission of a high-finesse cavity oscillating globally in quantum vacuum. Common features and differences with usual parametric processes in quantum optics are discussed. Interesting signatures of quantum radiation like pulse shaping and frequency up-conversion are predicted, which could be used to experimentally demonstrate motion-induced dissipative effects. The feasibility of an experimental realization is discussed at the end of the paper.

Quantum fluctuations of a vortex in an optical lattice.

Using a variational ansatz for the wave function of the Bose-Einstein condensate, we develop a quantum theory of vortices and quadrupole modes in a one-dimensional optical lattice. We study the coupling between the quadrupole modes and Kelvin modes, which turns out to be formally analogous to the theory of parametric processes in quantum optics. This leads to the possibility of squeezing vortices. We solve the quantum multimode problem for the Kelvin modes and quadrupole modes numerically and find properties that cannot be explained with a simple linear-response theory.

Quasimode theory of quantum optical processes in photonic band gap materials

This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.

Adiabatic Processes in Quantum Optics

This paper reviews the use of adiabatic approximations in quantum optics.The general principle is explained in terms of the Landau-Zener model and the recently developed stimulated Raman adiabatic passage scheme. The features characteristic of adiabatic evolution are extracted from these examples. Our recent work on adiabatic level preparation and cavity mode transfer of excitation is presented and discussed.

Theory of pseudomodes in quantum optical processes

This paper deals with non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in high Q cavities or photonic band gap materials. In cases such as the former, we show that the pseudo mode theory for single quantum reservoir excitations can be obtained by applying the Fano diagonalisation method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two and many discrete quasimodes are made. For a simple photonic band gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.

Non-Markovian Quantum Optical Processes and Pseudomodes

Non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, such as in high Q cavities, is treated using a quasimode description, and the pseudo mode theory for single quantum reservoir excitations is obtained via Fano diagonalisation. The atomic transitions are coupled to a discrete set of (cavity) quasimodes, which are also coupled to a continuum set of (external) quasimodes with slowly varying coupling constants. Each pseudomode corresponds to a cavity quasimode, and the original reservoir structure is obtained in expressions for the equivalent atom-true mode coupling constants. Cases of multiple excitation of the reservoir are now treatable via Markovian master equations for the atom-discrete quasimode system.

Coherent states of nonlinear algebras: applications to quantum optics

We present a general unified approach for finding the coherent states of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the non compact cases, to obtain the annihilation operator coherent states, by finding the canonical conjugates of these operators. Generalized coherent states, in the Perelomov sense also follow from this construction. This allows us to explicitly construct coherent states associated with various quantum optical systems.

Retrodiction in quantum optics

The usual formalism of quantum mechanics is predictive with state vectors being assigned on the basis of past preparation procedures. The retrodictive formalism, in which some state vectors are assigned on the basis of future measurement results is less usual, although it has been known for some time. Although at first sight this formalism may appear to be anticausal and subject to paradoxes, if used carefully it still predicts the same measurable correlations as the more usual approach despite different assignments of state vectors to states in the interval between preparation and measurement of the system. Here we use a procedure which involves retrodiction to examine some quantum optical processes, including a recent state truncation process. The collapse of the state vector occurs at a different time from that for the usual approach underlining the difficulty in regarding the collapse as a real physical process.

Hyper-Raman scattering in semiconductors: A quantum optical process in the strong-coupling regime

We present a theory of hyper-Raman scattering in semiconductors. The theory provides a microscopic description of this process intimately tied to the quantization of the electromagnetic field. The exciton-photon coupling is treated nonperturbatively to include polariton effects and the Coulomb interaction between electrons is treated beyond the Hartree-Fock approximation of semiconductor Bloch equations. The theory demonstrates the role of the quantum-mechanical zero-point motion of excitons in the optical decay of the virtually excited biexcitons into two final polaritons or a longitudinal exciton and a polariton. We calculate the steady-state emission spectrum and present numerical results for CuCl.

Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera

A single-photon-sensitive intensified charge-coupled-device (ICCD) camera has been used to simultaneously detect, over a broad area, degenerate and nondegenerate photon pairs generated by the quantum-optical process of spontaneous parametric down-conversion. We have developed a new method for determining the quantum fourth- order correlations in spatially extended detection systems such as this one. Our technique reveals the expected phase-matching-induced spa- tial correlations in a 2-f Fourier-transform system.

Hyper Raman Scattering in Microcavity Quantum Wells: A Quantum Optical Process in the Strong Coupling Regime

We analyze the hyper Raman scattering process in a quantum well grown inside a semiconductor microcavity and present a microscopic calculation of emission spectra as a function of excitation energy, detuning, and angles. The exciton-photon coupling is treated non-perturbatively to include polariton effects and the Coulomb interaction between electrons is treated beyond the mean field approximation. The observation of hyper Raman scattering in semiconductor microcavities would represent an important step for the realization of non-classical optical states based on excitons. We find that the polaritons emitted in this scattering process are strongly correlated. This correlation,can be efficiently transferred to output photons, resulting in emission of two correlated non-classical light beams. As a probe of the quantum correlation we calculate the output spectrum of fluctuations in the intensity difference of the two beams generated in the hyper Raman process.

Hyper Raman Scattering in Microcavity Quantum Wells: A Quantum Optical Process in the Strong Coupling Regime

Hyper Raman Scattering in Microcavity Quantum Wells: A Quantum Optical Process in the Strong Coupling Regime

Neoclassical radiation theory as an integral part of the Monte Carlo wave-function method

We analyze the so-called Monte Carlo wave-function (MCWF) method from a conceptual point of view. This method has been recently introduced as a technique to simulate dissipative processes in quantum optics. For the case where dissipation consists of spontaneous emission, we find that the coherent decay part of the MCWF method is identical to neoclassical radiation theory. This unexpected reappearance of the neoclassical theory suggests the identification of the MCWF coherent decay with classical radiation reaction. It leads to an alternative interpretation of the MCWF method instead of the usual one which is based upon quantum measurmeent theory. We give a derivation of the MCWF method, illustrated by simple Feynman diagrams, in which the appearance of radiation reaction is shown to be a natural feature of the coherent decay part.

Lie-algebra methods in quantum optics: The Liouville-space formulation.

Lie-algebra methods for investigating quantum optical systems are presented within the framework of the Liouville-space formulation. The generalized decomposition formulas for exponential functions of the generators of su(2) and su(1,1) Lie algebras are derived and their expectation values are calculated for typical states in quantum optics. The general procedure for using Lie algebras in the Liouville space to treat quantum optical processes is given in terms of generalized decomposition formulas and their use is demonstrated by calculating the absorption line shape and photon echo signal in a localized electron-phonon system. It is also shown that the photon-counting probability can be calculated by using the su(1,1) Lie algebra and that the electron-counting probability can be calculated by using the su(2) Lie algebra. The su(1,1) Lie algebra is also used to investigate a quantum-nondemolition measurement of photon number in the four-wave-mixing model.

SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics

An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time-evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.

Wave-function approach to dissipative processes in quantum optics.

A novel treatment of dissipation of energy from a ``small'' quantum system to a reservoir is presented. We replace the usual master equation for the small-system density matrix by a wave-function evolution including a stochastic element. This wave-function approach provides new insight and it allows calculations on problems which would otherwise be exceedingly complicated. The approach is applied here to a two- or three-level atom coupled to a laser field and to the vacuum modes of the quantized electromagnetic field.

An algebraic approach to quadratic parametric processes

A Lie algebraic approach to quadratic parametric processes in quantum optics and quantum acoustics is presented. In this approach the Heisenberg-Weyl and symplectic dynamical algebras are used to obtain a general exact solution for the time evolution operator. The solution is then applied to describe quantum mechanically the parametric process of backward-wave echo generation in dielectrics.