Parametric down-conversion

For second harmonic generation (SHG), the tensor is y(2)(—2co co, co) (useful for frequency doubling and parametric down-conversion) while for the linear electrooptic or Pockels71 effect the tensor is y(2)(— co co, 0) (useful for Q-switching of lasers, for phase or amplitude modulators, and for beam deflectors) for optical rectification the tensor is y 2>(0 00, —co) for frequency mixing the tensor is y(2)(— co3 oolr co2) (useful for frequency up-converters, optical parametric oscillators, and spectroscopy). 

ON THE ADVANCED WAVE MODEL OF PARAMETRIC DOWN-CONVERSION... 

Keywords Parametric down-conversion, conditional preparation, quantum imaging... 

On the advanced wave model of parametric down-conversion... 

The interaction Hamiltonian of parametric down-conversion is given by [Ou 1989]... 

Figure 1. Photograph of the light emitted in type-II parametric down-conversion (false colours). The polarization-entangled photons emerge along the directions of the intersection between the green rings and are selected by placing small holes there...

The most widely used source for the polarization-entangled photons today utilizes the process of spontaneous parametric down-conversion in nonlinear optical crystals [Kwiat 1995], A typical picture of the emerging radiation is shown in Fig. 1. 

Figure 3 The experimental setup. A type II Spontaneous parametric down-conversion is used both to produce the ancilla pair (in the spatial modes <23 and a4) and to produce the two input qubits (in the spatial modes ai and 0,2). In this case initial entanglement polarization is not desired, and it is destroyed by making the photons go through polarization filters which prepare the required input state. Half-wave plates have been placed in the photon paths in order to rotate the polarization compensators are able to nullify the birefringence effects of the non-linear crystal and of the polarizing beam splitters. Overlap of the wavepackets at the PBSs is assured through spatial and spectral filtering.

In the original KLM scheme, the fundamental element is not the CNOT operation, but the more or less equivalent nonlinear sign-shift (NS) operation from which the two-qubit conditional sign flip gate can be constructed. Similar to the accessibility of the CNOT operation, universal quantum computation becomes possible with such a two-qubit gate [Barenco 1995 (a) Sleator 1995], Recently, our group has experimentally demonstrated the NS operation using photons produced via parametric down-conversion. In contrast to the KLM scheme, our method to observe the NS operates in the polarization basis and therefore does not require interferometric phase stability. 

Figure 7. Experimental setup for the demonstration of nonlinear sign-shift (NS) operation using double-pass parametric down-conversion. Photon pairs created from first pass are used for the input of NS operation and pairs from the second pass are used for the triggered single photon source as ancilla. Successful operation is identified through four-fold coincidence counts between all four detectors.

While parametric down-conversion techniques have recently been used to generate multi-photon states [Waks 2004], it remains experimentally challenging to implement schemes that allow for simultaneous control over both photon number and spatio-temporal properties of the pulse. 

In this section we give a brief description of the generalized scheme of NOPO. As an entangler we consider the combination of two processes in a triply resonant cavity, namely, the type-II parametric down-conversion in y ( 2)-medium and polarization mixing between subharmonics in lossless symmetric quarter-wave plate. The Hamiltonian describing intracavity interactions is... 

Approximate versions of the translational EPR state, wherein the -function correlations are replaced by finite-width (Gaussian) distributions, have been shown to characterize the quadratures of the two optical-field outputs of parametric down-conversion, or of a fiber interferometer with Kerr nonlinearity. Such states allow for various schemes of continuous-variable quantum information processing such as quantum teleportation [Braunstein 1998 (b) Furu-sawa 1998] or quantum cryptography [Silberhorn 2002], A similar state has also been predicted and realized using collective spins of large atomic samples [Polzik 1999 Julsgaard 2001]. It has been shown that if suitable interaction schemes can be realized, continuous-variable quantum states of the original EPR type could even serve for quantum computation. 

For the second harmonic generation, we find that above a certain input intensity a dynamics reminiscent of a competitive, multi-wave mixing process occurs the pump field is mostly reflected, revealing a novel type of optical limiting behavior, while forward a nd b ackward g eneration i s g enerally b alanced. W e a Iso s tudy t he case of parametric down-conversion, where an intense second harmonic signal is injected in order to control a much weaker fundamental beam. Our results reveal the onset of a new process that has no counterpart in bulk materials both transmission and reflection display an unexpected, unusual, resonance-like effect as functions of input second harmonic power. 

A technique for measuring the quantum efficiency of a photon counting detector without a calibrated reference detector is described in [301, 356, 357, 358, 423, 536]. The technique is based on the generation of photon pairs - or entangled photons" - by parametric down-conversion. The principle is shown in Fig. 6.28. 

FIGURE 7 Schematic illustration of the waves used in parametric down-conversion. Radiation at co is supplied, and radiation at C02 and 0)2 is generated in the nonlinear interaction. 

FIGURE 8 Off-axis phase-matching diagram for parametric down-conversion. Direction of arrows indicates direction of propagation of the various waves. 

In the absence of pump depletion, the amplitudes of the waves generated in a parametric down-conversion process are given by... 

Parametric down-conversion can also be used to amplify radiation at C02 or 0)3. In this arrangement radiation is supplied at both the pump wavelength and the lower-frequency wave to be amplified, which is termed the signal. The process is similar to difference-frequency mixing. 

Another example of a second-order non-linear process is optical parametric down-conversion. In optical parametric down-conversion, a photon from an input pump (p) wave at frequency lower frequencies ffl and coi. These are termed the signal (s) and idler (i) frequencies, with the convention cOg < coj. Within a photon picture, the OPO can be thought of as a photon splitter. However, it should be made clear that a photon is not really cut into two it is the non-linear action with the incoming high-frequency wave with the NLO material that gives rise to the generation of two new low-frequency waves. 

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