Auto-and cross-correlation functions of a one-atom laser in a regime of strong coupling

The spectral properties of an incoherently pumped one-atom laser operating in a regime of strong coupling are studied. The spectrum of the field outgoing from the cavity, the spectrum of resonance fluorescence of an atom, and the spectrum of the atom—field cross-correlation function are numerically calculated. Relations between the parameters of the system for which spectral lines corresponding to eigenstates of the Jaynes—Cummings Hamiltonian are resolved are found. A method for determining the widths of resonances from the spectrum of the atom—field cross-correlation function is proposed, and a procedure for measuring this function using spectroscopy of intensity fluctuations is described.     

Single-atom laser: Coherent and nonclassical effects in the regime of a strong atom-field correlation

Based on the approximation of strong correlations between an atom and an intracavity field, which implies the equal probabilities of finding the atom in the ground state and n photons in the field and of finding the atom in an excited state and n?1 photons in the field, it is shown that the conditional states of a field generated by a single-atom laser are described by the diagonal part of the generalized coherent Mittag-Leffler state. The quasi-distributions P and Q of the intracavity-field probability amplitude are found, and the boundedness of the Glauber function on a segment is shown. The possibility of inversionless lasing is demonstrated, and the absence of a lasing threshold is found for some region of parameters. The regimes of generation of the amplitude-squeezed states of the field are studied and the parameters of the system providing the maximum squeezing are determined. It is shown that the atom-field states are entangled at weak pump intensities.     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Quantum statistical properties of one-atom lasers

Properties of a laser formed by a high-Q cavity and a single incoherently excited atom are studied on the basis of analytical solutions obtained for the quasi-probability P-function. Expressions for the boundaries separating various operating regimes of a one-atom laser (below threshold, above threshold, and self-quenching) are found. It is shown that lasing without inversion and lasing with the decay rate of the excited state exceeding the pump intensity are possible in the system. The degree of coherence of laser radiation is studied.     

Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some disintegration condition which combines properties of spaces and curves, the Boyd and spirality indices.We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps.These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from in terms of their symbols.     

Fredholmness of singular integral operators with discrete subexponentialgroups of shifts on Lebesgue spaces

The paper is devoted to an application of a general local method of studying the Fredholmness of nonlocal bounded linear operators to Banach algebras of singular integral operators with piecewise continuous coefficients and discrete subexponential groups of piecewise smooth shifts acting topologically freely on composed contours. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)