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arXiv:1908.10314 · Quantum

Engineering Schrödinger cat states with a photonic even-parity detector

Guillaume Thekkadath, Bryn A. Bell, Ian A. Walmsley, Alexander I. Lvovsky

Published Mar 2, 2020• 44 citations
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Abstract

When two equal photon-number states are combined on a balanced beam splitter, both output ports of the beam splitter contain only even numbers of photons. Consider the time-reversal of this interference phenomenon: the probability that a pair of photon-number-resolving detectors at the output ports of a beam splitter both detect the same number of photons depends on the overlap between the input state of the beam splitter and a state containing only even photon numbers. Here, we propose using this even-parity detection to engineer quantum states containing only even photon-number terms. As an example, we demonstrate the ability to prepare superpositions of two coherent states with opposite amplitudes, i.e. two-component Schrödinger cat states. Our scheme can prepare cat states of arbitrary size with nearly perfect fidelity. Moreover, we investigate engineering more complex even-parity states such as four-component cat states by iteratively applying our even-parity detector.

DOI
10.22331/q-2020-03-02-239

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When two equal photon-number states are combined on a balanced beam splitter, both output ports of the beam splitter contain only even numbers of photons. Consider the time-reversal of this interference phenomenon: the probability that a pair of photon-number-resolving detectors at the output ports of a beam splitter both detect the same number of photons depends on the overlap between the input state of the beam splitter and a state containing only even photon numbers. Here, we propose using this even-parity detection to engineer quantum states containing only even photon-number terms. As an example, we demonstrate the ability to prepare superpositions of two coherent states with opposite amplitudes, i.e. two-component Schrödinger cat states. Our scheme can prepare cat states of arbitrary size with nearly perfect fidelity. Moreover, we investigate engineering more complex even-parity states such as four-component cat states by iteratively applying our even-parity detector.
We present an algorithm of quantum engineering of large-amplitude [Formula: see text] high-fidelity [Formula: see text] even/odd Schrödinger cat states (SCSs) using a single mode squeezed vacuum (SMSV) state as resource. Set of [Formula: see text] beam splitters (BSs) with arbitrary transmittance and reflectance coefficients sequentially following each other acts as a hub that redirects a multiphoton state into the measuring modes simultaneously measured by photon number resolving (PNR) detectors. We show that the multiphoton state splitting guarantees significant increase of the success probability of the SCSs generator compared to its implementation in a single PNR detector version and imposes less requirements on ideal PNR detectors. We prove that the fidelity of the output SCSs and its success probability are in conflict with each other (which can be quantified) in a scheme with ineffective PNR detectors, especially when subtracting large (say, [Formula: see text]) number of photons, i.e., increasing the fidelity to perfect values leads to a sharp decrease in the success probability. In general, the strategy of subtracting up to [Formula: see text] photons from initial SMSV in setup with two BSs is acceptable for achieving sufficiently high values of the fidelity and success probability at the output of the generator of the SCSs of amplitude [Formula: see text] with two inefficient PNR detectors.
Passing a photon number state through a balanced beam splitter will produce an entangled state in which the phases of the two output beams are highly correlated. This entangled state can be viewed as a generalized form of a Schr\"odinger cat state where there is an equal probability amplitude for all possible phases. We show that Bell's inequality can be violated using this entangled state and two distant measuring devices that consist of a single-photon interferometer with a Kerr medium in one path, a set of single-photon detectors, and postselection based on a homodyne measurement. These entangled states are sensitive to photon loss and a violation of Bell's inequality requires either that the losses are inherently small or that their effects have been minimized using linear optics techniques [Micuda et al., Phys. Rev. Lett. 109, 180503 (2012)]. Somewhat surprisingly, the use of the fair sampling assumption is not required for a violation of Bell's inequality despite the use of postselection if the measurements are made in the correct order.
We propose an optical scheme to generate a superposition of coherent states with enhanced size adopting an interferometric setting at the single-photon level currently available in the laboratory. Our scheme employs a nondegenerate optical parametric amplifier together with two beam splitters so that the detection of single photons at the output conditionally implements the desired superposition of second-order photonic operations. We analyze our proposed scheme by considering realistic on-off photodetectors with nonideal efficiency in heralding the success of conditional events. A high-quality performance of our scheme is demonstrated in view of various criteria such as quantum fidelity, mean output energy, and measure of quantum interference.
Coherent state superpositions, also known as Schrödinger cat states, are widely recognized as promising resources in quantum information, quantum metrology, as well as fundamental tests. These states are hard to produce deterministically and most schemes for their probabilistic generation can only attain amplitudes too small for practical use. This is for example the case for photon-subtracted squeezed vacuum (PSSV), which can be used to approximate cat states of amplitude no larger than y = 1.5 if the fidelity is to be maintained above 95%. One way to reach larger amplitudes is to start with pairs of small cats and then to interfere them on a balanced beam splitter. The projective measurement of one of the outputs is used to herald a larger cat resulting from the constructive interference of the initial states. The scheme proposed here uses the projection |x = 0〉〈x = 0| as the heralding condition. Homodyning is proposed, as opposed to photon counting, because homodyne detection has high a quantum efficiency, and - as demonstrated in the paper - can be tuned to increase the success probability of the amplification without heavily compromising the output's fidelity.
We present an algorithm of quantum engineering of large-amplitude>5 high-fidelity>0.99 even/odd Schrodinger cat states (SCSs) using a single mode squeezed vacuum (SMSV) state as resource. Set of k beam splitters (BSs) with arbitrary transmittance and reflectance coefficients sequentially following each other acts as a hub that redirects a multiphoton state into the measuring modes simultaneously measured by photon number resolving (PNR) detectors. We show that the multiphoton state splitting guarantees significant increase of the success probability of the cat state generator compared to its implementation in a single PNR detector version and imposes less requirements on ideal PNR detectors. We prove that the fidelity of the output SCSs and its success probability are in conflict with each other (which can be quantified) in a scheme with ineffective PNR detectors, especially when subtracting large (say, 100) number of photons, i.e., increasing the fidelity to perfect values leads to a sharp decrease in the success probability. In general, the strategy of subtracting up to 20 photons from initial SMSV in setup with two BSs is acceptable for achieving sufficiently high values of the fidelity and success probability at the output of the generator of the SCSs of amplitude <3 with two inefficient PNR detectors.
We proposed and analyzed a scheme to generate large-size Schr\"odinger cat states based on linear operations of Fock states, squeezed vacuum states, and conditional measurements. By conducting conditional measurements via photon number detectors, two unbalanced small-amplitude Schr\"odinger kitten states combined by a beam splitter can be amplified to a large-size cat state with the same parity. According to simulation results, two Schr\"odinger odd kitten states with amplitudes of $|\ensuremath{\beta}|=1.06$ and $|\ensuremath{\beta}|=1.11$ generated from one-photon-subtracted 3 dB squeezed vacuum states, are amplified to an odd cat state of $|\ensuremath{\beta}|=1.73$ with a fidelity of F=99F=99%F=99. A large-size Schr\"odinger odd cat state with $|\ensuremath{\beta}|=2.51$ and F=97.30F=97.30%F=97.30 is predicted when 5.91 dB squeezed vacuum states are employed. According to the analysis on the impacts of experimental imperfections in practice, Schr\"odinger odd cat states of $|\ensuremath{\beta}|>2$ are available. A feasible configuration based on a quantum frequency comb is developed to realize the large-size cat state generation scheme we proposed.