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Journal of the Optical Society of America B

From spectral matrix to sideband structure: exploring stereo multimodes

M. Martinelli

Published Apr 3, 2023• 3 citations
Publisher

Abstract

We recall the details involving the generation of light beams by optical cavities in order to explore the quantum properties that are revealed from the covariance matrices of the quadratures of the fields. This consideration is necessary to reveal the structure involving the sidebands of the resonant cavity modes, their reconstruction from the linearized Hamiltonian, and the fact that, for each output beam generated from an optical cavity observed under a defined electronic analysis frequency, we access a double space of modes involving the sidebands. The full measurement of those sidebands is revisited, and the distinct techniques are compared, showing their advantages, drawbacks, and perspectives.

DOI
10.1364/josab.482651

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We recall the details involving the generation of light beams by optical cavities in order to explore the quantum properties that are revealed from the covariance matrices of the quadratures of the fields. This consideration is necessary to reveal the structure involving the sidebands of the resonant cavity modes, their reconstruction from the linearized Hamiltonian, and the fact that, for each output beam generated from an optical cavity observed under a defined electronic analysis frequency, we access a double space of modes involving the sidebands. The full measurement of those sidebands is revisited, and the distinct techniques are compared, showing their advantages, drawbacks, and perspectives.
Interferometric techniques, combined with electronic signal processing, have provided powerful tools for the precise reconstruction of quantum states of the field. Nevertheless, in most cases the completeness of the measurement relies in strong assumptions about its symmetry. In the present work, I will show how the use of optical cavities as a tool for state reconstruction can provide a complete description of the state, relaxing a priori assumptions and revealing a broad distribution of entanglement among sidebands of different optical beams, as in the case of those generated by an optical parametric oscillator.
We revisit the problem of quantum state reconstruction of light beams from the photocurrent quantum noise. As is well known, but often overlooked, two longitudinal field modes contribute to each spectral component of the photocurrent (sideband modes). We show that spectral homodyne detection is intrinsically incapable of providing all the information needed for the full reconstruction of the two-mode spectral quantum state. Such a limitation is overcome by the technique of resonator detection. A detailed theoretical description and comparison of both methods is presented, as well as an experiment to measure the six-mode quantum state of pump-signal-idler beams of an optical parametric oscillator above the oscillation threshold.
The quantum noise in photocurrent fluctuations usually gives incomplete information about the quantum state of spectral sideband modes of bright light beams involved in the detection. Each frequency component of the noise spectrum corresponds to two sideband modes symmetrically located around the bright optical field. In the case of the usual homodyne detection, it limits the ability to recover discriminated information of each mode involved. We theoretically show that complete reconstruction of the two-mode quantum state can be obtained by using phase-locked (coherent) resonator detection, even for non-Gaussian states. We experimentally demonstrate the technique by measuring a two-mode displaced coherent state.
Proper characterization of quantum correlations in multimode optical quantum states is critical for applications in quantum information science. However, standard entanglement measurements can lead to incomplete state reconstruction and characterization. Here, we implement a resonator-based detection system that reveals entanglement between sideband modes of twin beams, achieving full tomography and retrieving often ignored quantum correlations. Unlike standard spectral measurements such as homodyne detection, resonator detection can independently address the sidebands of each beam, thereby accessing these hidden correlations. Additionally, we show how phase shifts between the carrier and the sideband modes of the involved fields redistribute information and modify the observation of entanglement for different witnesses. The ability of the resonant detection to independently address sideband modes of entangled states can contribute to enhancing the capacity for secure communication and quantum networking protocols.
In a previous paper [Phys. Rev. A 81, 063837 (2010)], I proposed a quantum model of the cavity electro-optic modulator, which can coherently couple an optical cavity mode to a microwave resonator mode and enable quantum operations on the two modes, including laser cooling of the microwave resonator, electro-optic entanglement, and backaction-evading optical measurement of a microwave quadrature. In this sequel, I focus on the quantum input-output relations between traveling optical and microwave fields coupled to the cavity electro-optic modulator. With red-sideband optical pumping, the relations are shown to resemble those of a beam splitter for the traveling fields, so that in the ideal case of zero parasitic loss and critical coupling, microwave photons can be coherently up converted to ``flying'' optical photons with unit efficiency, and vice versa. With blue-sideband pumping, the modulator acts as a nondegenerate parametric amplifier, which can generate two-mode squeezing and hybrid entangled photon pairs at optical and microwave frequencies. These fundamental operations provide a potential bridge between circuit quantum electrodynamics and quantum optics.
Spectral homodyne detection, a widely used technique for measuring quantum properties of light beams, cannot retrieve all the information needed to reconstruct the quantum state of spectral field modes. We show that full quantum state reconstruction can be achieved with the alternative measurement technique of resonator detection. We experimentally demonstrate this difference by engineering a quantum state with features that go undetected by homodyne detection but are clearly revealed by resonator detection.