Squeezed quadratures observed in degenerate optical parametric oscillator above threshold

Squeezed light is one of the most valuable resources in modern quantum technology, and a new theoretical study has now shown how to push its quality further than conventional limits allow, even in a regime where such light is notoriously difficult to produce. Writing in the open-access journal Results in Optics, Kokeb Belete Benti and Desalegn Ayehu of Addis Ababa University present a detailed quantum analysis of a degenerate optical parametric oscillator, or DOPO, operating above its oscillation threshold, with the signal field bathed in a squeezed vacuum reservoir and the pump field exposed to ordinary thermal noise. Their results demonstrate that deliberate reservoir engineering can redistribute quantum noise between the two quadratures of the light field, enhancing squeezing beyond the familiar 50 percent ceiling that constrains intracavity fields coupled to ordinary vacuum, and that this enhancement can survive the constant assault of thermal decoherence.

The system at the heart of the study is a workhorse of quantum optics. An optical parametric amplifier exploits parametric down-conversion, a nonlinear process in which a pump photon injected into a crystal splits into two lower-energy photons, the signal and the idler. In the degenerate case, signal and idler share the same frequency, polarization and direction, so the process is described by a single signal mode paired with the pump. Placing the nonlinear medium between highly reflective mirrors turns the amplifier into an oscillator, extending the interaction time and allowing the fields to build up. When the pump is weak, the signal field has zero mean amplitude and the device runs below threshold, a regime in which the quantum optics of the generated light has been studied exhaustively for decades. When the pump is strong enough that the parametric gain balances cavity loss, the system crosses threshold and the signal field acquires one of two stable states of equal intensity and opposite phase. Above threshold, the pump field itself becomes a dynamical participant in the quantum noise story, and it is precisely here that Benti and Ayehu focus their analysis.

The mathematical machinery begins with the interaction Hamiltonian for the parametric process in the rotating wave approximation, from which the authors derive quantum Langevin equations for the annihilation operators of the signal and pump modes. These coupled nonlinear equations describe the coherent flow of energy from pump to signal, the damping of each mode at its own rate, and the fluctuating forces injected by the reservoirs to which the modes are coupled. Because the equations cannot be solved exactly, the authors linearize them around the steady-state mean values, writing each operator as a large classical amplitude plus a small quantum fluctuation. The steady-state amplitudes follow from setting the time derivatives of the mean fields to zero, which yields the familiar above-threshold solution in which the normalized signal amplitude equals the square root of the excess pump strength. The linearized fluctuation dynamics are then solved exactly using the Laplace transform technique, giving closed-form expressions for how quantum noise propagates through the driven, dissipative oscillator.

Crucially, the two modes feel very different environments. The noise operators acting on the signal mode carry the fingerprints of a squeezed vacuum, characterized by a squeezing parameter r that appears in anomalous two-time correlation functions linking the creation and annihilation parts of the fluctuating force. An ordinary vacuum reservoir injects uncorrelated fluctuations with no preferred phase, and these fluctuations are what clamp the intracavity squeezing of a standard DOPO to 3 decibels, or 50 percent below the vacuum noise level. A squeezed vacuum, by contrast, has phase-sensitive correlations: by choosing the right quadrature, the injected noise can be biased downward, allowing the cavity field to quiet below what ordinary vacuum would ever permit. The pump mode, meanwhile, is coupled to a thermal reservoir with mean occupation number n, a minimal model of the unavoidable environmental heating, dissipation and decoherence that any realistic laboratory device suffers.

With the solutions in hand, the authors compute the quadrature variances of both intracavity fields. Quadrature operators are the optical analogues of position and momentum, and a field is squeezed when the uncertainty in one quadrature falls below the standard quantum limit while the conjugate quadrature absorbs the excess, in obedience to the Heisenberg principle. For the signal mode, the minus quadrature variance drops below the limit, with noise transferred into the plus quadrature, confirming genuine squeezing. The expressions reveal three competing influences: the injected squeezing, encoded in factors of exp(±2r); the nonlinear coupling strength, captured by the dimensionless parameter μ; and the relative damping ν, the ratio of the signal decay rate to the pump decay rate. The results show that squeezing improves as the relative damping increases, because a signal mode damped more strongly into the squeezed reservoir exchanges photons with it more effectively than the pump exchanges noise with the thermal bath. Perhaps most strikingly, near threshold the variance approaches exp(−2r)/2, meaning the squeezed reservoir lets the intracavity field surpass the 3-decibel barrier that bounds a vacuum-coupled oscillator.

Thermal noise erodes this advantage, but not as quickly as one might fear. As the thermal occupation number rises, the variance climbs and the squeezing progressively degrades, eventually vanishing for large enough n. Yet the analysis shows a clear window in which the squeezed reservoir dominates the dynamics: for modest thermal occupation, considerable squeezing survives because the phase-biased input field counterbalances the extra photons contributed by the heat bath. The authors also identify a saturation effect: beyond a certain reservoir squeezing strength, further increases in r no longer reduce the variance, because the thermal contribution becomes the residual floor. The same story plays out for the pump mode, which in a below-threshold oscillator would never squeeze at all. Above threshold, the pump field also develops squeezing in its minus quadrature, again growing with the relative damping rate and again resisting thermal decoherence thanks to the reservoir-engineered signal.

The analysis then moves from the fields trapped inside the cavity to the light that actually escapes, which is what experiments measure. Using the input-output formalism, the authors derive the spectrum of squeezing, a frequency-resolved measure of the noise reduction in the output field. Here the above-threshold DOPO displays its signature double-peaked spectrum. For the output signal mode, injecting a squeezed vacuum does something remarkable: it shifts the frequency of maximum squeezing away from exact resonance, whereas with a plain vacuum reservoir the deepest noise suppression sits at resonance, powered by the strong pump-signal interaction there. The squeezing parameter deepens the noise reduction overall, and the thermal bath fills in the noise floor as temperature rises. The relative damping rate acts as a bandwidth knob: increasing ν widens the separation between the two spectral peaks, allowing the system to deliver useful squeezing across a broader range of frequencies, at the cost of a slightly shallower optimum.

The output pump mode tells a complementary tale. Its squeezing spectrum is likewise double-peaked, but the optimum remains pinned at resonance regardless of the injected squeezed field. Increasing the reservoir squeezing produces dramatic gains, with the calculations showing near-perfect squeezing of the output pump field at a squeezing parameter of r = 2. Thermal photons still degrade the effect, yet substantial squeezing persists even at nonzero occupation numbers because the input squeezed field compensates the decoherence. Interestingly, the optimum pump squeezing decreases as the relative damping rate increases, the mirror image of the signal behavior: stronger signal-reservoir coupling populates more squeezed photons at resonance, and the resulting rise in mean photon number slightly dilutes the pump-mode noise reduction. Together, these trends give experimentalists a tunable trade-off between depth and bandwidth of squeezing in the two output channels.

The authors emphasize that their findings elevate reservoir engineering from a supplementary trick to a primary control lever in parametric oscillators. Earlier work had either focused on squeezing below threshold, where record 15-decibel noise reductions have been demonstrated in the output, or on schemes such as squeezed lasing, in which an oscillator’s vacuum reservoir is replaced by a squeezed vacuum generated by a second oscillator. The present study is distinguished by treating the combined and competing effects of squeezed-vacuum injection, thermal decoherence and strong nonlinear interaction simultaneously in the above-threshold regime, where few analyses have ventured. The practical implications reach well beyond the cavity: squeezed light underpins gravitational-wave detectors that push interferometric sensitivity past the shot-noise limit, continuous-variable quantum communication networks, distributed quantum sensing, and precision measurement platforms that exploit noise reduction in one quadrature to extract signals classical light would bury.

The study also maps a path for follow-up research. The authors propose extending the same model to richer quantum correlations, including entanglement, quantum discord and quantum steering, in a doubly damped DOPO, and to hybrid platforms such as optomechanical and magnomechanical systems, where reservoir engineering could grant additional control over nonclassical states. In the nearer term, the message for experimental quantum optics is concrete: choose the damping rates of the pump and signal modes deliberately, matching them to whether the application demands the deepest possible squeezing in a narrow band or moderate squeezing across a wide one, and let an injected squeezed vacuum absorb the punishment that thermal environments inflict. As quantum sensors move from laboratory benches into atmospheric monitoring, medical diagnostics and navigation, the ability to manufacture robust, decoherence-resistant squeezed light above the oscillation threshold may prove one of the quiet enablers of the coming generation of quantum technology.