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Acta Physica Sinica

Effect of imperfect experimental condition on generation of Schrödinger cat state

Nana Zhang, Shujing Li, Yan Hong-Mei, 何亚亚 HE Ya-ya, Wang Hai

Published Jan 1, 2018• 3 citations
Publisher

Abstract

Schrödinger cat state is an important non-classical state, and it can be used in quantum teleportation, quantum computation and quantum repeater. Schrödinger cat state is usually obtained experimentally by subtracting one photon from a squeezed-vacuum state. The fidelity between a photon-subtracted squeezed state and a cat state can be very high under suitable parameters. However, the quality of the generated state will be affected by the imperfect experimental conditions. In this paper, the effect of imperfect experimental conditions on the generation of cat state is theoretically calculated and analyzed. The input squeezed-vacuum field is represented by Weyl characteristic function, which contains the fluctuation variance of the squeezed and amplified noises. The characteristic function of generated state is obtained by using the transmission matrix of beam splitter and the measurement operator of single-photon detector. We acquire the expression of Wigner function of generated state by the Fourier transform of the Weyl characteristic function. The fidelity is calculated by using the formula F=1/π∫d2ζC1(ζ)C|cat->(ζ), where C1(ζ) and C|cat->(ζ) represent Weyl characteristic function of the generated state and the Schrodinger cat state, respectively. The imperfection of the input squeezed state, the imperfection of the single-photon detector and the loss of the balanced homodyne detection are included in our theoretical model. We calculate the Wigner function at the phase-space origin W(0) and the fidelity in terms of different experimental parameters. The results show that the fidelity and negativity of W(0) decrease with squeezing purity decreasing. A pure squeezed-vacuum state is composed of even photon number states. In the case of impure squeezing, some odd photon number states appear in the photon number distribution. After subtracting one photon from the impure squeezing state, the generated state consists of not only odd photon number state but also even photon states, which degrades the fidelity of the generated state. The lower squeezing purity is required to meet the demand for W(0)<0 under the condition of higher squeezing degree. There is an optimal squeezing degree to maximize the fidelity of generated state with impure squeezing. The use of inefficient on-ff single-photon detector and the loss of the balanced homodyne detection will further reduce the fidelity of the generated state. Under the practical experimental condition:squeezing degree s=-3 dB, the squeezing purity μ=99% and the quantum efficiency of balanced homodyne detection η=98%, the fidelity of generated state can reach 0.88 with using a commercially available on-off single-photon detector. This work can provide theoretical guidance for generating a high-quality Schrödinger cat state.

DOI
10.7498/aps.67.20180381

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Schrödinger cat state is an important non-classical state, and it can be used in quantum teleportation, quantum computation and quantum repeater. Schrödinger cat state is usually obtained experimentally by subtracting one photon from a squeezed-vacuum state. The fidelity between a photon-subtracted squeezed state and a cat state can be very high under suitable parameters. However, the quality of the generated state will be affected by the imperfect experimental conditions. In this paper, the effect of imperfect experimental conditions on the generation of cat state is theoretically calculated and analyzed. The input squeezed-vacuum field is represented by Weyl characteristic function, which contains the fluctuation variance of the squeezed and amplified noises. The characteristic function of generated state is obtained by using the transmission matrix of beam splitter and the measurement operator of single-photon detector. We acquire the expression of Wigner function of generated state by the Fourier transform of the Weyl characteristic function. The fidelity is calculated by using the formula F=1/π∫d2ζC1(ζ)C|cat->(ζ), where C1(ζ) and C|cat->(ζ) represent Weyl characteristic function of the generated state and the Schrodinger cat state, respectively. The imperfection of the input squeezed state, the imperfection of the single-photon detector and the loss of the balanced homodyne detection are included in our theoretical model. We calculate the Wigner function at the phase-space origin W(0) and the fidelity in terms of different experimental parameters. The results show that the fidelity and negativity of W(0) decrease with squeezing purity decreasing. A pure squeezed-vacuum state is composed of even photon number states. In the case of impure squeezing, some odd photon number states appear in the photon number distribution. After subtracting one photon from the impure squeezing state, the generated state consists of not only odd photon number state but also even photon states, which degrades the fidelity of the generated state. The lower squeezing purity is required to meet the demand for W(0)<0 under the condition of higher squeezing degree. There is an optimal squeezing degree to maximize the fidelity of generated state with impure squeezing. The use of inefficient on-ff single-photon detector and the loss of the balanced homodyne detection will further reduce the fidelity of the generated state. Under the practical experimental condition:squeezing degree s=-3 dB, the squeezing purity μ=99% and the quantum efficiency of balanced homodyne detection η=98%, the fidelity of generated state can reach 0.88 with using a commercially available on-off single-photon detector. This work can provide theoretical guidance for generating a high-quality Schrödinger cat state.
A quantitative analysis is conducted on the impacts of experimental imperfections in the input state, the detector properties, and their interactions on photon-subtracted squeezed vacuum states in terms of a quantum non-Gaussian character witness and Wigner function. Limitations of the non-classicality and quantum non-Gaussian characteristic of Schrödinger kitten states are identified and addressed. The detrimental effects of a photon-number detector on the generation of odd Schrödinger kitten states at near-infrared wavelengths (∼860 nm) and telecommunication wavelengths (∼1550 nm) are presented and analysed. This analysis demonstrates that the high dark count probability of telecommunication-wavelength photon-number detectors significantly undermines the negativity of the Wigner function in Schrödinger kitten state generation experiments. For a one-photon-subtracted squeezed vacuum state at ∼1550 nm, an avalanche photodiode-based photon-number-resolving detector provides no significant advantage over a non-photon-number-resolving detector when imperfections, such as dark count probability and inefficiency, are taken into account.
We have generated and measured an approximation of an optical Schrödinger cat state by photon subtraction from a squeezed state. Using single-photon avalanche photodiode detectors and photon-number-resolving transition edge sensors, we were able to extract Wigner distributions for one or two photons subtracted from the squeezed state, resulting in both an odd and even cat state. The one-photon-subtracted Wigner distribution shows measured negative features (without any correction), indicating quantum character. The Wigner distribution of the two-photon-subtracted state is negative after correcting for homodyne losses. A full statistical analysis is under way to quantify the magnitude and fidelity of the cat states generated.
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.
Light is a perfect candidate to carry out quantum information for applications to quantum computation, communication and metrology. Information can be encoded using observables with discrete spectrum (or DV for discrete variable encoding) as well as with continuous spectrum observables (or CV for continuous variable encoding). One of the most used states to encode a CV qubit is the so called optical Schrödinger cat state, i.e. a superposition of two coherent states with the same amplitude and opposite phases. Approximation of these states, named Schrödinger kittens, have been generated in an heralded fashion by subtraction of one or two photons from a degenerate single mode squeezed vacuum states. Experiments have been conducted in continuous wave regime generally using optical parametric oscillators as well as in the pulsed regime. The quality of Schrödinger kittens generated with this strategy, however, suffers from the impurity of the squeezing in the wave-packet of the heralded state. In the CW regime, it is shown theoretically and experimentally that a narrowband filtering of the heralding photon increase the purity of thegenerated state. We extend this study so as to describe also the pulsed regime where the squeezed vacuum state is in general spectrally multimode. We calculate and numerically simulate the state generated when subtracting a photon from an arbitrary multimode pulsed squeezed vacuum state. Inaddition, we investigate its homodyne detection by considering an arbitrary pulsed local oscillator. The fidelity between the heralded state and a perfect single mode photon subtracted squeezed state decreases when the number of modes excited in the multimode squeezed vacuum increases. We show how to increase this fidelity by using a narrowband filter in the heralding path. This study quantifies the effect of spectral filtering on the quality of the generated state using a multimode approach that can be applied to a wide variety of situations. We believe that it will have a major impact on realisation of non-Gaussian states in an heralded fashion.
Optical "Schrödinger cat" states, the non-classical superposition of two quasi-classical coherent states, serve as a basis for gedanken experiments testing quantum physics on mesoscopic scales and are increasingly recognized as a resource for quantum information processing. Here, we report the first experimental realization of optical "Schrödinger cats" by adding a photon to a squeezed vacuum state, so far only photon-subtraction protocols have been realized. Photon-addition gives us the advantage of using heralded signal photons as experimental triggers, and we can generate "Schrödinger cats" at rates exceeding 8.5×1048.5 \times 10^48.5×104 counts per second; at least one order of magnitude higher than all previously reported realizations. Wigner distributions with pronounced negative parts are demonstrated at down to -8.89 dB squeezing, even when the initial squeezed vacuum input state has low purity. Benchmarking against such a degraded squeezed input state we report a maximum fidelity of more than 80% with a maximum cat amplitude of ∣α∣≈1.66|α| \approx 1.66∣α∣≈1.66. Our experiment uses photon-addition from pairs, one of those photons is used for monitoring, giving us enhanced control; moreover the pair production rates are high and should allow for repeated application of photon-addition via repeat-stages.
Measurement-induced nonlinear operation in linear optics is a promising technique to realize quantum computation and communications. Such an operation has been demonstrated via the generation of photon-subtracted squeezed states, namely, Schrödinger-cat states. Nielsen and Mølmer have proposed an efficient scheme to generate the cat states with larger amplitudes relative to previous experimental results. We present a simple experimental implementation of their scheme and analyze it while considering experimental imperfections. Our study indicates that the current photon detection technology meets the conditions necessary for their operation.