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arXiv:2404.05237 · Physical Review A

Spatiotemporal effects in heralded state preparation

Filippus Stefanus Roux

Published Jul 26, 2024• 1 citation
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Abstract

Heralding, which is often used for preparing quantum optical states, is studied to determine the effects of the spatiotemporal properties of the process. Incorporating all the spatiotemporal degrees of freedom, we follow a Wigner functional approach to consider cases where these states are prepared to have Wigner functionals with negative regions, being suitable resources for quantum information technologies. General expressions are derived for single-photon-subtracted and single-photon-added states. As examples, we consider the photon-subtracted squeezed vacuum state, the photon-added coherent state, the photon-added thermal state, and the photon-added squeezed vacuum state. The Wigner functional approach reveals the importance of the spatiotemporal transformations imposed by the experimental conditions.

DOI
10.1103/physreva.110.013714

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Heralding, which is often used for preparing quantum optical states, is studied to determine the effects of the spatiotemporal properties of the process. Incorporating all the spatiotemporal degrees of freedom, we follow a Wigner functional approach to consider cases where these states are prepared to have Wigner functionals with negative regions, being suitable resources for quantum information technologies. General expressions are derived for single-photon-subtracted and single-photon-added states. As examples, we consider the photon-subtracted squeezed vacuum state, the photon-added coherent state, the photon-added thermal state, and the photon-added squeezed vacuum state. The Wigner functional approach reveals the importance of the spatiotemporal transformations imposed by the experimental conditions.
Optical cat states, the nonclassical superposition of two quasi-classical coherent states, serve as a basis for gedankenexperiments 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 cat states 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 cat states at rates exceeding $2.3\ifmmode\times\else\texttimes\fi{}{10}^{5}$ counts per second. Our most highly squeezed vacuum input state shows \ensuremath{-}8.9 dB squeezing, but such squeezing entails some degradation---in this case, 15.1 dB antisqueezing. Even so, our approach enables us to synthesize a state with a maximum cat amplitude of $|\ensuremath{\alpha}|\ensuremath{\approx}1.77$ whose Wigner distribution still shows pronounced negative parts. Our experimental implementation with controlled photon addition demonstrates a powerful and robust building block for advanced quantum state engineering and shows that heralded photon addition can be controlled well and performed at high rates.
An analysis of the homodyne tomography process that is often used to determine the Wigner functions of quantum optical states is performed to consider the effects of the spatiotemporal degrees of freedom. The homodyne tomography process removes those parts of the input state that are not associated with the mode of the local oscillator by tracing out those degrees of freedom. Using a functional approach to incorporate all the spatiotemporal degrees of freedom, we find that this reduction in the degrees of freedom introduces distortions in the observed Wigner function. The analysis also shows how the homodyne tomography process introduces a scaling caused by the efficiency and a resolution that depends on the strength of the local oscillator. As examples, we consider coherent states, Fock states, and squeezed vacuum states.
We discuss nonclassical properties of single-photon subtracted squeezed vacuum states in terms of the sub-Poissonian statistics and the negativity of the Wigner function. We derive a compact expression for the Wigner function from which we find the region of phase space where Wigner function is negative. We find an upper bound on the squeezing parameter for the state to exhibit sub-Poissonian statistics. We then study the effect of decoherence on the single-photon subtracted squeezed states. We present results for two different models of decoherence, viz. amplitude decay model and the phase diffusion model. In each case we give analytical results for the time evolution of the state. We discuss the loss of nonclassicality as a result of decoherence. We show through the study of their phase-space properties how these states decay to vacuum due to the decay of photons. We show that phase damping leads to very slow decoherence, slower than the photon-number decay.
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.
Based on the expressions of Wigner functions in Fock-spaces,we reconstruct the Wigner functions of photon-added and-subtracted states of squeezed vacuum states.Negativities of these Wigner functions numerically indicate that these states are really non-classical quantum states.We also discuss how these negativities depend on the photon number k added to and subtraced from the squeezed vacuum states.The negativities are greater when k is odd numbers than those when k is even numbers.The result provides a theoretical foundation for measuring these states.