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Physical Review A

Coherently stimulated parametric down-conversion, phase effects, and quantum-optical interferometry

Richard J. Birrittella, Anna Gura, Christopher C. Gerry

Published May 5, 2015• 14 citations
Publisher

Abstract

We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.

DOI
10.1103/physreva.91.053801

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We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
We study multiphoton quantum-interference effects at a beam splitter and its connection to the prospect of attaining interferometric phase-shift measurements with noise levels below the standard quantum limit. Specifically, we consider the mixing of the most classical states of light coherent states with the most nonclassical states of light number states at a 50 : 50 beam splitter. Multiphoton quantum-interference effects from mixing photon-number states of small photon numbers with coherent states of arbitrary amplitudes are dramatic even at the level of a single photon. For input vacuum and coherent states, the joint photon-number distribution after the beam splitter is unimodal, a product of Poisson distributions for each of the output modes but with the input of a single photon, the original distribution is symmetrically bifurcated into a bimodal distribution. With a two-photon-number state mixed with a coherent state, a trimodal distribution is obtained, etc. The obtained distributions are shown to be structured so as to be conducive for approaching Heisenberg-limited sensitivities in photon-number parity-based interferometry. We show that mixing a coherent state with even a single photon results in a significant reduction in noise over that of the shot-noise limit. Finally, based on the results of mixing coherent light with single photons, we consider the mixing coherent light with the squeezed vacuum and the squeezed one-photon states and find the latter yields higher sensitivity in phase-shift measurements for the same squeeze parameter owing to the absence of the vacuum state.
The fields of quantum and nonlinear optics have given rise to a variety of nonclassical states of light that have been proven to surpass certain limitations set by classical physics. Namely, certain squeezed and entangled states have been shown to beat the shot-noise limit when making precision phase measurements in interferometry, as well as write lithographic patterns that are smaller than classically allowed by the Rayleigh diffraction limit. Additionally, single-photon sources and entangled photon pairs have given rise to provably secure quantum key distribution for cryptography. Producing these quantum states of light has proven a difficult task. Nonlinear crystals, when pumped by a laser, produce pairs of single photons via the process of spontaneous parametric down conversion (SPDC). This process is mediated by the second order nonlinear susceptibility of the material. When pumped in a high gain regime, these crystals give rise to optical parametric amplification, which is a viable source of squeezed light. The vast majority of research in this area has focused on crystals that are seeded by vacuum in their two modes. This dissertation concerns the field of quantum nonlinear optics. It is an investigation into the processes that occur when nonlinear materials interact with the electromagnetic field on the single photon level. I have focused on seeding nonlinear crystals with quantum states of light, including single photons and entangled states. This process results in various states directly applicable to interferometry, imaging, and cryptography. Another application investigated is an absolute radiance measurement via stimulated parametric down conversion resulting from non-vacuum seeding of a nonlinear crystal. Additionally, other nonlinear processes, including four-wave mixing, nonlinear magneto-optical effects and coherent population trapping in warm atomic vapor involving quantum states of light are investigated. The process of seeding third-order nonlinear interactions, such as in atomic vapors, gives rise to a variety of interesting, nonclassical phenomena such as entangled image transfer and nonlocal imaging. Strong analogies between SPDC and four-wave mixing are drawn. I also experimentally show an all optical pi-only phase shift of one light beam via another in warm Cesium vapor.
Parametric down-conversion is a second-order nonlinear optical process annihilating a pump photon and creating a pair of photons in the signal and idler modes. Then, by using two parametric down-converters and introducing a path indistinguishability for the two generated idler modes, a quantum coherence between two conjugate signal beams can be induced. Such a double spontaneous or stimulated parametric down-conversion scheme has been used to demonstrate quantum spectroscopy and imaging with undetected idler photons via measuring one-photon interference between their correlated signal beams. Recently, we considered another quantum optical measurement scheme utilizing W-type tripartite entangled signal photons that can be generated by employing three spontaneous parametric down-conversion crystals and by inducing coherences or path-indistinguishabilities between their correlated idler beams and between quantum vacuum fields. Here, we consider an extended triple stimulated parametric down-conversion scheme for quantum optical measurement of sample properties with undetected idler and photons. Noting the real effect of vacuum field indistinguishability on the fringe visibility as well as the role of zero-point field energy in the interferometry, we show that this scheme is an ideal and efficient way to create a coherent state of W-type entangled signal photons. We anticipate that this scheme would be of critical use in further developing quantum optical measurements in spectroscopy and microscopy with undetected photons.
We study the application of photon-subtracted two-mode squeezed vacuum states (TMSVS), where identical photon numbers are subtracted from each beam, to the problem of quantum optical interferometry with the photon-number parity measurements scheme. Previously, Anisimov et al. [Phys. Rev. Lett.104, 103602 (2010)] have studied the prospect of using the TMSVS for parity measurements based interferometry. However, the joint photon number distribution of this state before beam splitting is thermal-like in each mode, meaning that its statistics are super-Poissonian and that the most probable state of the field is the double vacuum and that the average photon number is low. Furthermore, with these states the sensitivity of the phase-shift measurement is extremely sensitive to the size of the phase shift to be measured for large average photon numbers. The simultaneous subtraction of 1, 2, or 3 photons from each mode has the effects of increasing the average photon numbers of each mode and rendering the statistics nearly Poissonian and in some cases sub-Poissonian. We show that phase uncertainties for such states are less sensitive to the phase shift itself at large average photon numbers. We also show that the photon-subtracted TMSVS also lead to increased resolution.
By implementing a parametric down-conversion process with a strong signal field injection, we demonstrate that frequency down-conversion from pump photons to idler photons can be a coherent process. Contrary to a common misconception, we show that the process can be free of quantum noise. With an interference experiment, we demonstrate that coherence is preserved in the conversion process. This technique could lead to a high-fidelity quantum state transfer from a high-frequency photon to a low-frequency photon and connect a missing link in quantum networks. Coherent and efficient nonlinear interaction and frequency conversion are of great interest in many areas of quantum optics. Traditionally, the low efficiency of Raman scattering is improved by a high-finesse optical resonator or stimulated Raman conversion. It was recently found that the atomic spin wave initially built through electromagnetically induced transparency or a weak Raman process can actively enhance the Raman frequency conversion. An experimental demonstration of an efficient Raman conversion scheme with coherent feedback of both pump and Stokes fields is presented. The temporal profile of the generated Raman pulse shows that the coherence time of the atomic spin wave is $\sim$1.8 ms. A laser-like power threshold is observed and its low threshold is attributed to the long coherence time of the atomic spin wave. The mechanism of the conversion enhancement process is discussed and the conversion efficiency of a single pass of the beams is compared with that of double passes. Finally, a beat signal is observed between the two Stokes fields and its Fourier transform shows that the frequency difference is caused by the AC Stark effect. Precision phase measurement is traditionally restricted by the standard quantum limit. However, this limit is not as fundamental as the Heisenberg limit and can be circumvented by use of nonclassical quantum states and structure modification of the interferometers. Several examples of nonlinear interferometers are proposed and implemented. The wave propagation equations for the nonlinear interferometers are solved. The interference fringes are measured and compared with that of linear interferometers. The first nonlinear interferometer presented is based on second harmonic generation and degenerate parametric down-conversion. The second nonlinear interferometer is composed of two parametric amplifiers. The idea of nonlinear beam splitters is introduced as an analogy to traditional beam splitters. We show that a nonlinear interferometer can be built alternatively by using only one parametric amplifier. Type II phase-matched crystals can be used to increase the amplification factor. An interferometer based on a Raman amplifier is analyzed for its application to sensitive magnetic field measurement.
We propose a novel quantum interferometer based on two stimulated Parametric Down-Conversion (PDC) crystals seeded by weak coherent idler fields to investigate the relationship between the nonclassicality of the signal photon state and the fidelity of the conjugate idler photon state. We show that the purity of the signal photon state is tightly bounded by the entanglement between the signal photon and its conjugate idler photon, i.e., the fidelity of the idler state in the bipartite system, resulting in a novel complementarity relation for wave-particle duality. Moreover, we find that the nonclassical measure of the signal photon state is identical to the fidelity of the conjugate idler photon states prepared in each crystal. In our novel interferometer, the input beam splitter of the conventional SU(2) interferometer is replaced with two seeded SU(1,1) PDC systems that emit only a single photon at once from two crystals, thus producing an entangled superposition state where the conjugate idler photon states act as path detectors with controllable fidelity. We demonstrate that the presented quantum interferometer consisting of two stimulated PDC crystals positioned in parallel is an ideal setup for investigating and quantitatively understanding the complementarity relation of wave-particle duality.