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arXiv:2207.10210 · Physical Review A

Homodyne measurement with a Schrödinger cat state as a local oscillator

Joshua Combes, Austin P. Lund

Published Dec 12, 2022• 6 citations
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Abstract

Homodyne measurements are a widely used quantum measurement. Using a coherent state of large amplitude as the local oscillator, it can be shown that the quantum homodyne measurement limits to a field quadrature measurement. In this work, we give an example of a general idea: injecting nonclassical states as a local oscillator can lead to nonclassical measurements. Specifically, we consider injecting a superposition of coherent states, a Schr\"odinger cat state, as a local oscillator. We derive the Kraus operators and the positive operator-valued measure in this situation.

DOI
10.1103/physreva.106.063706

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Homodyne measurements are a widely used quantum measurement. Using a coherent state of large amplitude as the local oscillator, it can be shown that the quantum homodyne measurement limits to a field quadrature measurement. In this work, we give an example of a general idea: injecting nonclassical states as a local oscillator can lead to nonclassical measurements. Specifically, we consider injecting a superposition of coherent states, a Schr"odinger cat state, as a local oscillator. We derive the Kraus operators and the positive operator-valued measure in this situation.
Homodyne measurement is a crucial tool widely used to address continuous variables for bosonic quantum systems. While an ideal homodyne detection provides a powerful analysis, e.g., to effectively measure quadrature amplitudes of light in quantum optics, it relies on the use of a strong reference field, the so-called local oscillator, typically in a coherent state. Such a strong coherent local oscillator may not be readily available, particularly for a massive quantum system like a Bose-Einstein condensate, posing a substantial challenge in dealing with continuous variables appropriately. It is necessary to establish a practical framework that includes the effects of nonideal local oscillators for a rigorous assessment of various quantum tests and applications. We here develop entanglement criteria beyond a Gaussian regime applicable for this realistic homodyne measurement that do not require assumptions on the state of local oscillators. We discuss the working conditions of homodyne detection to effectively detect non-Gaussian quantum entanglement under various states of local oscillators.
We consider the generation of Schrödinger cat states using a quantum measurement-induced logical gate where entanglement between the input state of the target oscillator and the Fock state of the ancillary system produced by the quantum non-demolition entangling C^Z operation is combined with the homodyne measurement. We utilize the semiclassical approach to construct both the input-output mapping of the field variables in the phase space and the wave function of the output state. This approach is found to predict that the state at the gate output can be represented by a minimally disturbed cat-like state which is a superposition of two copies of the initial state symmetrically displaced by momentum variable. For the target oscillator prepared in the coherent state, we show that the fidelity between the exact solution for the gate output state and the “perfect” Schrödinger cat reconstructed from the semiclassical theory can reach high values exceeding 0.99.
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 derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.