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Physical Review A

Improved measurement of multimode squeezed light via an eigenmode approach

Ryan S. Bennink, Robert W. Boyd

Published Nov 22, 2002• 61 citations
Publisher

Abstract

We analyze the output of a degenerate optical parametric amplifier using a different approach to understanding and characterizing multimode squeezed light. This approach is based on the concept of eigenmodes of the squeezing and predicts the mode structure of the local oscillator that should be used in order to measure the smallest quadrature noise. Although the importance of mode matching is well known, we find that typical experimental setups are suboptimal and we predict that in such cases squeezed light measurements stand to be improved noticeably with appropriate shaping (in space and/or time) of the local oscillator field in accordance with the results of the theory. Under conditions of negligible or small phase mismatch, the optimal local oscillator pulse duration is found to be approximately equal to the geometric mean of the pump pulse duration and nonlinear response time. Also, we find that the effective number of squeezed modes can be altered by varying the pump pulse duration and/or phase-matching parameters.

DOI
10.1103/physreva.66.053815

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Summary form only given. We find that the quantum statistics of multimode squeezed fields are easily obtained and simply understood via an eigenmode description. The eigenmode description of squeezed light is based on our finding that the squeezing matrix /spl Gamma/ can always be diagonalized by a physical transformation of the fields - either a change of basis or linear unitary evolution. Each eigenmode possesses single-mode squeezing statistics, and the corresponding eigenvalue is the effective squeezing parameter for that mode. The squeezing eigenmodes are completely uncoupled from one another (at the quantum level) and represent independent channels (pixels) useful for carrying noise-sensitive data. We show how an eigenmode description of a squeezed field allows one to easily calculate noise statistics of the field upon diffraction and/or propagation through any sequence of lossless linear optical elements. We further show that the eigenmodes define the shape of the local oscillator which must be used in order to obtain the smallest quantum noise in balanced homodyne or heterodyne measurements. Finally, we use the theory to model pulsed optical parametric downconversion in a /spl chi//sup (2)/ medium with a classical pump field of arbitrary bandwidth, allowing for the possibility of phase and/or group velocity mismatch.

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Multimode squeezed light is a key resource for quantum technologies, enabling applications in boson sampling, quantum networks, and optical quantum computation [1],[2]. Parametric down-conversion in synchronously pumped optical parametric oscillators (SPOPOs) offers a versatile platform for generating multimode squeezed states across the time-frequency domain. In these systems, supermodes manifest as independent quantum states in the temporal domain and as frequency combs in the spectral domain [3]. Precise control of these states is crucial for quantum applications and requires an experimental setup capable of fine-tuning multiple parameters to optimize squeezing properties. In this work, we present an experimental setup for generating and characterizing multimode vacuum squeezed states using a SPOPO and a balanced homodyne detection scheme. Two spatial light modulators (SLMs) placed on the pump and local oscillator (LO) beams provide full control over their spectral properties, including amplitude and phase shaping. This flexibility allows us to optimize the experimental conditions and investigate squeezing dependence on key parameters such as the LO spectral detuning and bandwidth. By introducing a programmable$\pi$-phase shift to the LO via the SLM, we selectively probe the multimode structure, distinguishing between even and odd supermodes. Additionally, the SLM on the pump can introduce an arbitrary chirp, further enhancing control over the system. Our setup, schematized in Figure 1a, is based on a commercial mode-locked Yb-fiber laser operating at 93MHz, emitting pulses centered at 1035nm with a 10nm bandwidth. A portion of the radiation is frequency-doubled to 517.5nm via an LBO crystal to serve as the pump for the SPOPO, while the remaining part is used as the LO for homodyne detection after transmission through the SPOPO cavity, ensuring high beam quality. The SPOPO cavity is frequency-locked to the laser via a Pound-Drever-Hall technique, ensuring stable and optimized operation. Figure 1b presents representative measurements of squeezing levels as a function of the LO spectral bandwidth, showing how its symmetric or antisymmetric phase influences the detection of different supermodes and, consequently, the squeezing level depending on their structure. The measurements were performed using lithium niobate (LNB) nonlinear crystals of 3.0mm and 1.5mm lengths. Our setup allows for future investigations of dispersion effects on squeezing, thus far only theoretically predicted [4], by introducing chirped mirrors for dispersion compensation. Moreover, different nonlinear crystals, such as lithium niobate, beta barium borate (BBO), and lithium triborate (LBO) of various lengths, will be tested to explore the multimode properties of the squeezed states. These investigations will provide insights into the experimental conditions required to achieve single-mode squeezing, crucial for applications such as Gaussian Boson Sampling, with results to be presented at the conference.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 23, 2025•0 citations
We analyze the output of a degenerate optical parametric amplifier using a different approach to understanding and characterizing multimode squeezed light. This approach is based on the concept of eigenmodes of the squeezing and predicts the mode structure of the local oscillator that should be used in order to measure the smallest quadrature noise. Although the importance of mode matching is well known, we find that typical experimental setups are suboptimal and we predict that in such cases squeezed light measurements stand to be improved noticeably with appropriate shaping (in space and/or time) of the local oscillator field in accordance with the results of the theory. Under conditions of negligible or small phase mismatch, the optimal local oscillator pulse duration is found to be approximately equal to the geometric mean of the pump pulse duration and nonlinear response time. Also, we find that the effective number of squeezed modes can be altered by varying the pump pulse duration and/or phase-matching parameters.
Summary form only given. We find that the quantum statistics of multimode squeezed fields are easily obtained and simply understood via an eigenmode description. The eigenmode description of squeezed light is based on our finding that the squeezing matrix /spl Gamma/ can always be diagonalized by a physical transformation of the fields - either a change of basis or linear unitary evolution. Each eigenmode possesses single-mode squeezing statistics, and the corresponding eigenvalue is the effective squeezing parameter for that mode. The squeezing eigenmodes are completely uncoupled from one another (at the quantum level) and represent independent channels (pixels) useful for carrying noise-sensitive data. We show how an eigenmode description of a squeezed field allows one to easily calculate noise statistics of the field upon diffraction and/or propagation through any sequence of lossless linear optical elements. We further show that the eigenmodes define the shape of the local oscillator which must be used in order to obtain the smallest quantum noise in balanced homodyne or heterodyne measurements. Finally, we use the theory to model pulsed optical parametric downconversion in a /spl chi//sup (2)/ medium with a classical pump field of arbitrary bandwidth, allowing for the possibility of phase and/or group velocity mismatch.
Nowadays, the realization of quantum computations and communications based on continuous variables has attracted a significant attention due to a substantial expansion of the system dimensionality. The main progress in this area is attributed to the implementation of multimode systems based on squeezed states of light. One of the simplest ways to generate such states relies upon their producing in a single-pass optical parametric amplifier (OPA) using ultrafast pumping. However, for homodyne detection of such multimode states, the profile of the local oscillator (LO) must perfectly match the profile of the measured mode. Usually, this is not the case; therefore a proper treatment of multimode squeezing is required. In this work, we study both theoretically and experimentally the multimode squeezed light generated in type-0 and type-II OPA. We characterize such sources and investigate the degree of squeezing in dependence on the LO spectral profile, employing a pulse shaping technique. The theoretical analysis is performed using the Schmidt-mode theory. This work might have a significant impact on the realization of multimode quantum protocols
Optical parametric amplifiers (OPAs) configured as phase-sensitive amplifiers (PSAs) can be used for noiseless optical image amplification, generation of non-classical states of light, and, in particular, for multimode squeezed light generation. For all these applications and for effective use of the quantum properties of the multimode PSA, we need to know its independently squeezed (or amplified) eigenmodes. First we present the quantum theory of a spatially multimode traveling-wave PSA pumped by a high-power pump beam with arbitrary spatial profile. To quantitatively identify the Green's function of the PSA, we have developed a semi-analytical coupled-mode-theory of the PSA using 2D Hermite-Gaussian mode expansions of the signal- and pump-beam spatial distributions. By using Green's functions of the classical OPA, we calculate the normallyordered quadrature correlators at its output, which provide complete quantum description of the phase-sensitive OPA and enable determination of its independently squeezed eigenmodes. We find the number of the supported eigenmodes and their shapes for a spatially broadband frequency-degenerate optical parametric amplifier with elliptical Gaussian pump. We conclude by discussing our recent extensions of the coupled-mode-theory to study higher-order pump modes, compact representation of the PSA eigenmodes (especially convenient for experiments), and effect of non-zero phase mismatch on the PSA eigenmodes. We expect that the results from our model can be used for optimum mode matching in phase-sensitive image amplification and multimode squeezing generation.
Multimode squeezed light is an increasingly popular tool in photonic quantum technologies, including sensing, imaging, and computation. Meanwhile, the existing methods of its characterization are technically complicated, which reduces the level of squeezing, and mostly deal with a single mode at a time. Here, for the first time, to the best of our knowledge, we employ optical parametric amplification to characterize multiple squeezing eigenmodes simultaneously. We retrieve the shapes and squeezing degrees of all modes at once through direct detection followed by modal decomposition. This method is tolerant to inefficient detection and does not require a local oscillator. For a spectrally and spatially multimode squeezed vacuum, we characterize eight strongest spatial modes, obtaining squeezing and anti-squeezing values of up to −5.2 ± 0.2 dB and 8.6 ± 0.3 dB, respectively, despite the 50% detection loss. This work, being the first exploration of an optical parametric amplifier’s multimode capability for squeezing detection, paves the way for the real-time detection of multimode squeezing.
We report pulsed squeezed-light generation by means of an optical parametric downconverter that is pumped by the second harmonic of a mode-locked Q-switched laser. Using the fundamental beam of the laser as a local oscillator, we observe 2-dB squeezing for a parametric gain of 2.0. This local oscillator, however, is nonoptimal because of its spatiotemporal mode mismatch with the squeezed mode generated by the downconverter. We describe an experiment in which a matched local oscillator is generated with the use of an optical parametric amplifier that is pumped by the same laser as is the downconverter. In this case, 2-dB squeezing is observed for a parametric gain of 1.5. The present experimental setup is limited by the uncontrollable phase fluctuations that prohibit us from making squeezing measurements at higher parametric gains.
Multimode squeezed light is a key resource for quantum technologies, enabling applications in boson sampling, quantum networks, and optical quantum computation [1],[2]. Parametric down-conversion in synchronously pumped optical parametric oscillators (SPOPOs) offers a versatile platform for generating multimode squeezed states across the time-frequency domain. In these systems, supermodes manifest as independent quantum states in the temporal domain and as frequency combs in the spectral domain [3]. Precise control of these states is crucial for quantum applications and requires an experimental setup capable of fine-tuning multiple parameters to optimize squeezing properties. In this work, we present an experimental setup for generating and characterizing multimode vacuum squeezed states using a SPOPO and a balanced homodyne detection scheme. Two spatial light modulators (SLMs) placed on the pump and local oscillator (LO) beams provide full control over their spectral properties, including amplitude and phase shaping. This flexibility allows us to optimize the experimental conditions and investigate squeezing dependence on key parameters such as the LO spectral detuning and bandwidth. By introducing a programmableπ\piπ-phase shift to the LO via the SLM, we selectively probe the multimode structure, distinguishing between even and odd supermodes. Additionally, the SLM on the pump can introduce an arbitrary chirp, further enhancing control over the system. Our setup, schematized in Figure 1a, is based on a commercial mode-locked Yb-fiber laser operating at 93MHz, emitting pulses centered at 1035nm with a 10nm bandwidth. A portion of the radiation is frequency-doubled to 517.5nm via an LBO crystal to serve as the pump for the SPOPO, while the remaining part is used as the LO for homodyne detection after transmission through the SPOPO cavity, ensuring high beam quality. The SPOPO cavity is frequency-locked to the laser via a Pound-Drever-Hall technique, ensuring stable and optimized operation. Figure 1b presents representative measurements of squeezing levels as a function of the LO spectral bandwidth, showing how its symmetric or antisymmetric phase influences the detection of different supermodes and, consequently, the squeezing level depending on their structure. The measurements were performed using lithium niobate (LNB) nonlinear crystals of 3.0mm and 1.5mm lengths. Our setup allows for future investigations of dispersion effects on squeezing, thus far only theoretically predicted [4], by introducing chirped mirrors for dispersion compensation. Moreover, different nonlinear crystals, such as lithium niobate, beta barium borate (BBO), and lithium triborate (LBO) of various lengths, will be tested to explore the multimode properties of the squeezed states. These investigations will provide insights into the experimental conditions required to achieve single-mode squeezing, crucial for applications such as Gaussian Boson Sampling, with results to be presented at the conference.