We study the spatial properties of high-gain parametric down-conversion (PDC). From the Hamiltonian we find the Schmidt modes, apply the Bloch-Messiah reduction, and calculate analytically the measurable quantities, such as the angular distributions of photon numbers and photon-number correlations. Our approach shows that the Schmidt modes of PDC radiation can be considered the same as for the low-gain (biphoton) case while the Schmidt eigenvalues strongly depend on the parametric gain. Its validity is confirmed by comparison with several experimental results, obtained by us and by other groups.PACS numbers: 42.65. Lm, 42.65.Yj, 42.50.Lc Introduction. Currently, much interest is attracted to bright squeezed vacuum (BSV), a macroscopic nonclassical state of light that can be obtained via high-gain parametric down conversion (PDC). This is because BSV contains huge photon numbers and at the same time is strongly nonclassical, manifesting entanglement [1] and noise reduction below the standard quantum limit [2]. These features make it interesting for applications such as quantum imaging [3,4] and metrology [5], quantum optomechanics [6] and nonlinear optics with quantum light [7]. Besides, large photon-number correlations arising in BSV are richer than entanglement in two-photon light emitted via low-gain PDC [8].At the same time, theoretical description of BSV presents certain difficulties. In contrast to low-gain PDC, BSV generation cannot be described in the framework of the perturbation theory. The quantum state contains not only two-photon terms but also higher-order Fock components, and its calculation in the Schrödinger picture is difficult. Many recent theoretical investigations of such strong pumping regime are based on the concept of collective input/output modes introduced mostly to describe the spectral properties in the frequency domain [9][10][11][12][13]. In a high-gain regime it is convenient to calculate the observables in the Heisenberg picture. In this case the Schmidt-mode formalism used in the Schroedinger picture for the description of multimode two-photon light [14] is replaced by a similar procedure, called Bloch-Messiah reduction [15]. However beyond the perturbation approach the solution for the field operators was up to now obtained only numerically, usually through a set of integro-differential equations [12,13,[16][17][18].Here we present a fully analytical description of the angular properties of BSV. Our approach is based on the Bloch-Messiah reduction and allows one to obtain the evolution of the photon creation (annihilation) operators both for the Schmidt modes and for the plane-wave modes. After obtaining the evolution of these, we calculate analytically the angular distributions of the inten-

The phenomenon of strong laser field atomic stabilization is discussed. Earlier suggested models and mechanisms of stabilization are described: Λ- and V-type interference stabilization of Rydberg atoms, adiabatic (Kramers–Henneberger) and high-frequency stabilization of neutral atoms and negative ions, and so on. Both numerical and analytical approaches to the description of these phenomena are discussed. In this context, ab initio numerical solutions of the nonstationary Schrödinger equation, obtained by several groups of authors, are overviewed. Based on the most modern and recent solutions of this type, mechanisms of stabilization of a hydrogen atom are shown to vary with varying intensity and frequency of a laser field. Such an evolution and applicability condition of various stabilization mechanisms is described. Limitations arising due to relativistic effects are discussed. Existing experiments on strong-field stabilization are overviewed and their interpretation is considered.

Frolov, M. V.; Manakov, N. L.; Popov, A. M.; Tikhonova, O. V.; Volkova, E. A.; Silaev, A. A.; Vvedenskii, N. V.; and Starace, Anthony F., "Analytic theory of high-order-harmonic generation by an intense few-cycle laser pulse" (2012 We present a theoretical model for describing the interaction of an electron, weakly bound in a short-range potential, with an intense, few-cycle laser pulse. General definitions for the differential probability of abovethreshold ionization and for the yield of high-order-harmonic generation (HHG) are presented. For HHG we then derive detailed analytic expressions for the spectral density of generated radiation in terms of the key laser parameters, including the number N of optical cycles in the pulse and the carrier-envelope phase (CEP). In particular, in the tunneling approximation, we provide detailed derivations of the closed-form formulas presented briefly by M. V. Frolov et al. [Phys. Rev. A 83, 021405(R) (2011)], which were used to describe key features of HHG by both H and Xe atom targets in an intense, few-cycle laser pulse. We then provide a complete analysis of the dependence of the HHG spectrum on both N and the CEP φ of an N -cycle laser pulse. Most importantly, we show analytically that the structure of the HHG spectrum stems from interference between electron wave packets originating from electron ionization from neighboring half-cycles near the peak of the intensity envelope of the few-cycle laser pulse. Such interference is shown to be very sensitive to the CEP. The usual HHG spectrum for a monochromatic driving laser field (comprising harmonic peaks at odd multiples of the carrier frequency and spaced by twice the carrier frequency) is shown analytically to occur only in the limit of very large N , and begins to form, as N increases, in the energy region beyond the HHG plateau cutoff.

Bright squeezed vacuum, a macroscopic nonclassical state of light, can be obtained at the output of a strongly pumped nonseeded traveling-wave optical parametric amplifier (OPA). By constructing the OPA of two consecutive crystals separated by a large distance, we make the squeezed vacuum spatially single-mode without a significant decrease in the brightness or squeezing.

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