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arXiv:1210.0289 · Physical Review A

Analysis of the time-energy entanglement of down-converted photon pairs by correlated single-photon interference

Changliang Ren, Holger Friedrich Hofmann

Published Oct 15, 2012• 10 citations
arXivHTMLPDF

Abstract

The time-energy entanglement of down-converted photon pairs is particularly difficult to characterize because direct measurements of photon arrival times are limited by the temporal resolution of photon detection. Here, we explore an alternative possibility of characterizing the temporal coherence of two-photon wave functions using single-photon interference with weak coherent light. Specifically, this method makes use of the fact that down conversion generates a coherent superposition of vacuum and two-photon states, so that the coincidence count rates for photon pairs after interference with weak coherent light are given by a superposition of the two-photon wave function from the down conversion with the product wave function defined by the weak coherent references. By observing the frequency-dependent interference pattern, it is possible to reconstruct the amplitudes and the phases of the two-photon wave function within the bandwidth of the reference pulses used in the single-photon interferences. The correlated single-photon interferences therefore provide a direct map of the entangled two-photon wave function generated in the down-conversion process.

DOI
10.1103/physreva.86.043823

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The time-energy entanglement of down-converted photon pairs is particularly difficult to characterize because direct measurements of photon arrival times are limited by the temporal resolution of photon detection. Here, we explore an alternative possibility of characterizing the temporal coherence of two-photon wave functions using single-photon interference with weak coherent light. Specifically, this method makes use of the fact that down conversion generates a coherent superposition of vacuum and two-photon states, so that the coincidence count rates for photon pairs after interference with weak coherent light are given by a superposition of the two-photon wave function from the down conversion with the product wave function defined by the weak coherent references. By observing the frequency-dependent interference pattern, it is possible to reconstruct the amplitudes and the phases of the two-photon wave function within the bandwidth of the reference pulses used in the single-photon interferences. The correlated single-photon interferences therefore provide a direct map of the entangled two-photon wave function generated in the down-conversion process.
To fully utilize the energy-time degree of freedom of photons for optical quantum-information processes, it is necessary to control and characterize the temporal quantum states of the photons at extremely short time scales. For measurements of the temporal coherence of the quantum states beyond the time resolution of available detectors, two-photon interference with a photon in a short-time reference pulse may be a viable alternative. In this paper, we derive the temporal measurement operators for the bunching statistics of a single-photon input state with a photon from a weak coherent reference pulse. It is shown that the effects of the pulse shape of the reference pulse can be expressed in terms of a spectral filter selecting the bandwidth within which the measurement can be treated as an ideal projection on eigenstates of time. For full quantum tomography, temporal coherence can be determined by using superpositions of reference pulses at two different times. Moreover, energy-time entanglement can be evaluated based on the two-by-two entanglement observed in the coherences between pairs of detection times.
We experimentally demonstrate amplitude and phase modulation of a time-energy entangled two-photon wave function. The entangled photons are produced by spontaneous parametric down-conversion, spectrally dispersed in an prism compressor, modulated in amplitude and/or phase, and detected in coincidence by sum-frequency generation. First, we present a Fourier optical analysis of the optical setup yielding an analytic expression for the resulting field distribution at the exit plane of the shaping apparatus. We then introduce amplitude and/or phase shaping and present results which can only be obtained through a combination of the two. Specifically, we use a shaper-based interferometer to measure the two-photon interference of an almost bandwidth-limited two-photon wave function.
We experimentally demonstrate amplitude and phase modulation of a time-energy entangled two-photon wave function. The entangled photons are produced by spontaneous parametric down-conversion, spectrally dispersed in an prism compressor, modulated in amplitude and/or phase, and detected in coincidence by sum-frequency generation. First, we present a Fourier optical analysis of the optical setup yielding an analytic expression for the resulting field distribution at the exit plane of the shaping apparatus. We then introduce amplitude and/or phase shaping and present results which can only be obtained through a combination of the two. Specifically, we use a shaper-based interferometer to measure the two-photon interference of an almost bandwidth-limited two-photon wave function.
We demonstrate a technique for characterizing two-photon quantum states based on joint temporal correlation measurements using time-resolved single-photon detection by femtosecond up-conversion. We measure for the first time the joint temporal density of a two-photon entangled state, showing clearly the time anticorrelation of the coincident-frequency entangled photon pair generated by ultrafast spontaneous parametric down-conversion under extended phase-matching conditions. The new technique enables us to manipulate the frequency entanglement by varying the down-conversion pump bandwidth to produce a nearly unentangled two-photon state that is expected to yield a heralded single-photon state with a purity of 0.88. The time-domain correlation technique complements existing frequency-domain measurement methods for a more complete characterization of photonic entanglement.
We experimentally demonstrate shaping of the two-photon wave function of entangled-photon pairs, utilizing coherent pulse-shaping techniques. By performing spectral-phase manipulations we tailor the second-order correlation function of the photons exactly like a coherent ultrashort pulse. To observe the shaping we perform sum-frequency generation with an ultrahigh flux of entangled photons. At the appropriate conditions, sum-frequency generation performs as a coincidence detector with an ultrashort response time (approximately 100 fs), enabling a direct observation of the two-photon wave function. This property also enables us to demonstrate background-free, high-visibility two-photon interference oscillations.
We show that a weak projective measurement of photon arrival time can be realized by controllable two-photon interferences with photons from short-time reference pulses at a polarization beam splitter. The weak value of the projector on the arrival time defined by the reference pulse can be obtained from the coincidence rates conditioned by a specific output measurement. If the weak measurement is followed by a measurement of frequency, the coincidence counts reveal the complete temporal coherence of the single-photon wave function. Significantly, the weak values of the input state can also be obtained at higher measurement strengths, so that correlations between weak measurements on separate photons can be observed and evaluated without difficulty. The method can thus be used to directly observe the nonclassical statistics of time-energy entangled photons.