Introduction

Quantum systems at optical frequencies are of significant interest in various applications including computing1, metrology2, and cryptography3. A crucial aspect is the ability to generate and characterize quantum states such as single photon sources4,5, entangled photon pairs6, and squeezed states of light7. Optical cavities play a crucial role by enabling enhanced control over light-matter coupling regimes8,9, facilitating the creation of complex non-Gaussian states10,11, and allowing the study of decoherence in quantum systems12,13.

Quantum tomography has become an essential tool to characterize quantum states12,14. However, conventional tomography techniques based on homodyne detection15—which relies on interference of propagating waves—are not suitable for measuring quantum states confined within a cavity—which are inherently stationary. Furthermore, extracting the quantum states from a cavity typically alters their quantum properties, making accurate measurements difficult. Although quantum filtering can estimate cavity quantum states, it is fundamentally limited by the detector efficiency16. Dynamical switching of the cavity can provide direct access to the cavity quantum state17,18,19, which requires precise control of the switching behavior and still suffers from the loss during the extraction.

To probe cavity quantum states, it is necessary to measure observables with sufficient sensitivity to detect subtle shifts induced by the cavity state, often involving just a few photons. For instance, neutral atoms have been used to probe cavity quantum states by detecting changes in the atom’s energy state as they interact with cavity photons12,20,21. Alternatively, microwave cavity systems have taken advantage of the strong Kerr nonlinearity of a Josephson junction to induce measurable changes in coupling rate with a readout cavity, proportional to the photon number of the quantum state13.

Achieving similar sensitivity at optical frequencies has proven to be very challenging due to inherently weak material nonlinearities. Nevertheless, nonlinear driven-dissipative systems such as optical parametric oscillators (OPOs) have revealed a wealth of intriguing non-classical behavior, including quantum states confined inside the cavity22 as well as out-coupled states of squeezed light2,23. In particular, studies have shown that macroscopic observables in similar multistable optical systems can be sensitive to fields at the single- and sub-photon levels24,25,26, opening new pathways for probing quantum optical systems.

Here, we propose a framework for quantum tomography of optical quantum states generated within a nonlinear cavity featuring multiple steady-states. Our method relies on mapping the initial field distribution of the cavity quantum state onto the statistics of the macroscopic cavity steady-states. The mapping is achieved by measuring the system’s sensitivity to an injected coherent field. Our approach enables the visualization of quasi-probability distribution of cavity quantum states in phase space representation. We demonstrate our concept in a degenerate optical parametric oscillator (DOPO) platform, and experimentally validate our method by reconstructing the Husimi Q function of a squeezed vacuum state generated inside the cavity. We further showcase the potential of our method by directly observing the phase-sensitive amplification of the quantum vacuum state inside the cavity under high gain pumping condition. Our nonlinear driven-dissipative system supports both generation and measurement of cavity quantum states, facilitating deeper understanding on intracavity phenomena such as non-classical state dynamics and quantum-to-classical transitions.

Results

Cavity state reconstruction protocol

A common method for visualizing the quasi-probability distribution of a quantum state in 2D phase space is through the Husimi Q function27, expressed as \(Q(\alpha,{\alpha }^{*})=\frac{1}{\pi }| \left\langle \alpha | \psi \right\rangle {| }^{2}\). This function evaluates the overlap \(\left\langle \alpha | \psi \right\rangle\) between a coherent state \(\left\vert \alpha \right\rangle\) and the quantum state of interest \(\left\vert \psi \right\rangle\) (see Supplementary Note 1 for further details).

Figure 1 illustrates the proposed method for measuring the overlap between a coherent state \(\left\vert \alpha \right\rangle\) (red signal) and an unknown cavity quantum state \(\left\vert \psi \right\rangle\) (green mode). The first step consists of coupling a coherent state into the cavity, in which the interaction with the quantum state leads to an increase in the mean electric field. The resulting state in the cavity can be described as a displacement D(α) of the cavity state \(\left\vert \psi \right\rangle\) proportional to the injected coherent state \(\left\vert \alpha \right\rangle\), as shown in panel I of Fig. 1a.

Fig. 1: Measuring quantum states generated inside nonlinear optical cavities.
figure 1

a Reconstructing cavity quantum states leveraging the sensitivity of a multistable system to weak coherent fields. The injected coherent field \(\left\vert \alpha \right\rangle\) displaces the cavity state \(\left\vert \psi \right\rangle\), changing the probability p of measuring certain steady-state. We measure the change in the probability p to reconstruct the cavity state \(\left\vert \psi \right\rangle\). b Visualizing cavity quantum states by measuring the Husimi Q function. The sensitivity ∂p/∂b gives Qθ, which is the marginal distribution of the Husimi Q function. Qθ for different θ is measured to reconstruct the full 2D Q function.

Full size image

To measure the overlap \(\left\langle \alpha | \psi \right\rangle\), we leverage the sensitivity of a multistable system to its initial conditions. Specifically, we consider a DOPO pumped above threshold, in which two degenerate steady-states α(0) (purple signal) and α(1) (orange signal) with phases 0 and π exist28. Because the parametric amplification in DOPO is phase-sensitive, the initial cavity field distribution can be mapped on the statistics of the measured output steady-states (panel II of Fig. 1a)2,7,25. Hence, when the initial condition of the system is the quantum vacuum state \(\left\vert 0\right\rangle\), one observes an equal probability p = p(α(1)) and 1 − p = p(α(0)) of measuring either state (i.e., p = 0.5)28. By measuring the intensity of the interference between the out-coupled OPO signal and the local oscillator (LO), we extract the phase of the OPO signal. Unlike standard homodyne detection, which requires precise calibration of the LO amplitude and phase to resolve continuous quadrature values, our protocol only distinguishes between two phase states (i.e., 0 or π phase). We then count the number of high-intensity outcomes to determine the probability distribution. Because the outcome is binary and insensitive to the exact intensity level, we do not require exact calibration of the amplitude of the LO. Moreover, we set the phase of the LO to maximize the contrast between low and high level signals by moving a piezo-driven delay stage. Details about the experimental setup can be found in the “Methods” section.

On the other hand, when displacing the cavity state with a coherent state, the initial field distribution is displaced (e.g., shift to the right side in panel II of Fig. 1a). This shift is directly translated in the measured state probability (e.g., p > 0.5 in panel III of Fig. 1a)25. Notably, the displacement also results in a change in the overlap between the quantum vacuum state and the cavity state, which can be described as \(\left\langle 0| D(\alpha )| \psi \right\rangle\) (panel I of Fig. 1a). Therefore, we can analytically describe the probability of measuring certain steady-state as a function of the amplitude of the coherent field injected into the cavity. In general, when we displace the cavity state \(\left\vert \psi \right\rangle\) along the axis rotated by θ with respect to Re(α) and amplify along the rotated axis, the probability p as a function of the coherent field b and the rotation angle θ can be written as:

$$p(b| \theta )=\int_{-D(b)}^{\infty }\frac{1}{\pi }| \left\langle \alpha | {\psi }_{\theta }\right\rangle {| }^{2}d\alpha,$$
(1)

where \(\left\vert {\psi }_{\theta }\right\rangle\) is the 1D field distribution defined along the rotated axis. To avoid confusion between integration variable and the probe signal we inject to the cavity, we use \(\left\vert b\right\rangle\) in Eq. (1) to define the probe signal. For the rest of the discussion, we will refer to the injected probe field b as a “bias field”, as this field biases the probability distribution of measuring a certain steady-state toward the value corresponding to the bias’ phase. Derivation of Eq. (1) can be found in the “Methods” section, and the dependence of phase-sensitive amplification behavior on the phase of the pump and the bias can be found in the Supplementary Note 2.

Equation (1) shows that the probability p is determined by the cumulative distribution of the cavity quantum state for α ≥ −D(b) (or equivalently the orange area of the displaced quantum state in panel II of Fig. 1a). It becomes evident that the sensitivity of the probability to the bias, ∂p/∂b, can be used to reconstruct the marginal distribution of the Husimi Q function at given θ, which we denote as Qθ.

The protocol for reconstructing the Husimi Q function of the cavity quantum state is depicted in Fig. 1b. The procedure starts from measuring the change in the probability p by sweeping the bias field b at given θ. The sensitivity ∂p/∂b of the bias–probability curve at a given θ provides the marginal distribution Qθ (panel I of Fig. 1b). This procedure is repeated to collect marginal distributions at different θ by rotating the pump phase (panel II of Fig. 1b). Finally, an inverse Radon transform is applied to reconstruct the full 2D Husimi Q function29. See Supplementary Note 2 for the detailed discussion of the reconstruction protocol. We also discuss the potential applications of using non-coherent bias fields in the measurement protocol in the Supplementary Note 2.

Measuring cavity squeezed vacuum state

We first apply the quantum tomography protocol described in the previous section to measure cavity squeezing in a DOPO. A simplified schematic of our experimental setup is shown in Fig. 2a, in which the DOPO consists of a nonlinear bow-tie cavity25. In order to prepare and reconstruct different cavity quantum states, we implemented rapid control over both the amplitude Ap and phase θp of the pump (green signal). To measure ∂p/∂b under different bias conditions, we have independent control of the amplitude Ab, phase θb, and injection time tdelay of the bias field b (red signal). Details of the experimental setup can be found in the “Methods” section as well as in the Supplementary Note 4.

Fig. 2: Measuring cavity squeezing.
figure 2

a Schematic of the experimental setup. b Phase-sensitive amplification in nonlinear drive-dissipative systems. When the DOPO is pumped below threshold (Ppump < Pth), the noise for a certain quadrature can be reduced below the standard quantum limit. c Measurement protocol for a cavity squeezed vacuum state. Pump and bias are controlled to both generate and measure the cavity squeezed vacuum state. d Bias–probability curves showing phase-sensitive amplification. e Reconstructed marginal Q function Qθ. Each curve is drawn by calculating the sensitivity from the bias–probability curve. f Reconstructed 2D Q function. Solid (dashed) black line represents the contour line at 1/e for squeezed vacuum state (analytical solution of quantum vacuum state). Black shaded area corresponds to the measurement uncertainty in the contour line at 1/e for the squeezed vacuum state.

Full size image

As shown in Fig. 2b, the noise of the initial quantum vacuum state undergoes phase dependent amplification. The pump is given along θ = 0° to amplify the quadrature noise along Re(α). For the quadrature orthogonal to the initial pump phase (i.e., θ = 90°), de-amplification occurs, resulting in quadrature noise below the standard quantum limit30. We follow the protocol in Fig. 2c to generate and measure the cavity squeezed vacuum state. First, the DOPO is pumped below threshold Pth to create a squeezed vacuum state \(S\left\vert 0\right\rangle\). Next, the bias field is introduced to displace the squeezed vacuum state (\(D(b)S\left\vert 0\right\rangle\)) and the pump power is simultaneously increased above threshold to amplify the displaced state. Steady-state outcomes are recorded to obtain the system’s probability. This is repeated with varying the amplitude of the bias field to establish the bias–probability relationship for a given pump phase. Finally, we measure the bias–probability relationship at different θ, by introducing an additional rotation of the pump phase when injecting the bias, allowing us to reconstruct the 2D Husimi Q function.

The measured sensitivity of the probability to the bias at different θ is shown in the panel I of Fig. 2d. As θ gets closer to 90°, the probability becomes more sensitive to the bias field, showing that the phase-sensitive amplification happens inside the DOPO. In other words, a sharper transition is a result of a smaller cavity field amplitude along that quadrature. We also fitted the experimental data with the theoretically predicted bias–probability curve which follows a Gauss error function, showing a good agreement with the data. Details of the theoretical model of the bias–probability relationship can be found in the “Methods” section. The panel II of Fig. 2d highlights the amplification (purple curve) and deamplification (red curve) of the initial quantum vacuum state along different pump phase. Figure 2e shows the computed Qθ functions for the two major quadratures. Comparing it to the Qθ function of the quantum vacuum state, we observe squeezing and anti-squeezing of the Qθ function at θ ~ 90° and θ ~ 0°, respectively.

Figure 2f shows the reconstructed 2D Husimi Q function of the cavity squeezed vacuum state, with a squeezing ratio of  ~1.2 ± 0.6 dB along θ = 90° obtained directly from the Qθ functions. A squeezing of  ~1.8 ± 0.7 dB is estimated when accounting for experimental limitations such as the finite rise time of the pump signal, which pre-amplifies the squeezed vacuum state prior to the measurement. Detailed discussion on how including finite rise time of the pump signal allows us to get better estimation can be found in the Supplementary Note 5.

To achieve maximum squeezing, which is known to be 3 dB for cavity squeezed vacuum state31, the pump power must be just below the threshold during the quantum state preparation. In our current setup, we set the pump power at  ~80% of the threshold to provide a sufficient buffer and avoid amplification while preparing the squeezed vacuum state. Future improvement in using a low-noise laser and high bandwidth voltage signal, should bring us closer to the theoretical squeezing limit of 3 dB.

Observing intracavity dynamics

Next, we use our reconstruction protocol to visualize the dynamics of the cavity quantum state. Figure 3a illustrates how the initial quantum vacuum state evolves when the DOPO is pumped above the threshold. Stochastic trajectories show the time evolution of the DOPO signal amplitude, which are obtained by solving multiple times the stochastic differential equations (SDEs) derived from the original Hamiltonian (see Supplementary Note 2 for further details). The cavity state undergoes two different stages before reaching steady-state—a mixed state of two coherent states (see rightmost panel in Fig. 3a). During the first stage, the parametric gain of the pump linearly amplifies the quantum vacuum state, broadening the distribution along one quadrature, i.e., Re(α). As the photon number inside the cavity increases to the level where nonlinear absorption occurs, part of the signal field is converted back into the pump field. This marks the onset of the second stage, where the nonlinear absorption term becomes apparent, and the broadened Gaussian distribution bifurcates into two sidelobes. Once the nonlinear absorption and the linear amplification reach equilibrium, we observe a mixed state of coherent states with an equal probability.

Fig. 3: Visualizing the cavity dynamics of the DOPO.
figure 3

a Dynamics of the quantum vacuum state in the DOPO. Stochastic trajectories (lower panel) and Q function (upper panel) visualize the amplification and bifurcation of the cavity state. b Measurement protocol for an amplified quantum vacuum state. A is an operator that maps the initial quantum vacuum state to its amplified state. c Bias–probability curves at different bias injection time measured during the linear amplification stage. d Amplification of the quantum vacuum state described as broadening of the marginal Q function Qθ=0. e Asymmetric amplification of the quantum vacuum state. σb is the standard deviation of the sensitivity curve (i.e., ∂p/∂b). Faster increase of σb along the horizontal, which is aligned to the pump phase, shows that the quantum vacuum state amplifies along the pump phase.

Full size image

Using the reconstruction protocol, we can track the evolution of the Husimi Q function, which manifests itself in the time dependence of the sensitivity of the probability to the bias field. Figure 3b illustrates the measurement protocol to observe the amplification dynamics of the DOPO. The DOPO is pumped above threshold at t = 0, amplifying the quantum vacuum state. After a time delay tdelay, the bias is introduced to measure the amplified state. By repeating the measurement for different delays, we can observe the cavity dynamics of the DOPO. In our experiment, we focus on the linear amplification stage (dashed box in Fig. 3a) due to the finite extinction ratio of the bias amplitude modulator. Because a stronger bias field is required to displace and probe a cavity state at a later stage, the finite extinction ratio of our system can unintentionally displace the cavity state even before the intended injection of the bias. Further discussions on how finite extinction ratio of the bias can affect the bias–probability relationship can be found in the Supplementary Note 5.

Figure 3c shows the measured bias-probability curves for different delays. It is evident that the system becomes less sensitive to the bias with increasing delay. Calculating the sensitivity of the bias–probability curves, we can visualize the broadening of the marginal distribution of the Husimi Q function along the pump phase Qθ=0 as Fig. 3d. As we expect from Fig. 3c, the amplified quantum vacuum state becomes less sensitive to the bias field as we need stronger bias field to fully displace it. We also measure the bias sensitivity of the amplified quantum vacuum at different pump phases (Fig. 3e). We can clearly see the asymmetric phase-sensitive amplification along different pump phases. The additional amplification along the orthogonal axis is introduced because of the slow rise time of the voltage signal that controls the pump amplitude modulator. The measurement of bias–probability curves at different pump phases and the detailed discussion on phase-sensitive amplification in Fig. 3e can be found in Supplementary Note 5.

Discussion

The presented framework leverages the phase-sensitive amplification in a nonlinear driven-dissipative system to generate a quantum state, and maps its field distribution onto the statistics of macroscopic cavity steady-states. This allows for generation and measurement of quantum states in a single optical cavity.

Our current experimental setup is limited to measuring quantum states that have a low photon number, such as squeezed vacuum state or linearly amplified quantum vacuum state. This limitation originates from two main factors: (1) the relatively low extinction ratio of the bias field; and (2) the slow rise time of the pump and bias signal due to the low bandwidth of the voltage amplifier that drives the electro-optic modulators. Further details on the origin of these experimental limitations are discussed in Supplementary Note 4. We anticipate that integrated platforms could significantly improve both the response time and the extinction ratio32.

With a higher extinction ratio of the bias field, displacement of cavity states with a greater number of photons would become possible. This is crucial for observing the bifurcation process, where the distribution transitions from a Gaussian to Gaussian mixture (i.e., timeline between the second and third Husimi Q functions in the upper panel of Fig. 3a). Numerical experiments reconstructing the dynamics of the bifurcation is discussed in Supplementary Note 3.

Reducing the rise time of the optical signals and increasing the lifetime of the cavity would open the avenue for capturing the DOPO dynamics with a better time resolution. While short rise time of the optical signal would also allow fast switching between different dynamical phases, longer cavity lifetime allows each dynamical phase to remain longer, thus enabling detailed reconstruction of intracavity dynamics. For instance, this would allow real-time observation of the generation and collapse of the squeezed vacuum state (see Supplementary Note 3 for the corresponding numerical experiment). This would provide valuable insights into exotic quantum-optical phenomena in nonlinear driven-dissipative systems, including soliton generation33, optical bistability25,28, and Kerr frequency combs34.

While the Husimi Q function contains complete information about the quantum state, it is indeed a “blurred” version of the Wigner function due to intrinsic convolution with a Gaussian function. Several mathematical algorithms have been developed to reconstruct the Wigner function from the Husimi Q function35,36,37, and their reconstruction performance depends on the signal-to-noise ratio of the Husimi Q function. Our reconstruction protocol acquires high sensitivity during the phase-sensitive amplification. In addition, the measurement is less sensitive to noise as the information we extract from the OPO steady state is the binary phase information with only two possible outcomes (0 or π phase). Therefore, we expect the high signal-to-noise of our framework would allow high-fidelity reconstruction of the Wigner function from the experimentally measured Husimi Q function.

While both concepts—parametric amplification and displacement of quantum states—have been utilized to reconstruct quadrature probability distributions from the measured intensity distributions without photon number resolving detectors38, our framework differs in that it measures the steady-state probability distribution of the cavity phase, which encodes the field distribution of the displaced cavity quantum state. Beyond the DOPO platform proposed here, which exhibits bistability through phase-sensitive parametric amplification, our framework can be generalized to other bistable systems with parametric amplification, such as Kerr-nonlinearity based OPOs39.

In conclusion, we have presented a framework for observing the dynamics of the quantum states generated inside optical cavities, experimentally realized in a DOPO system. We demonstrated a full reconstruction of a cavity squeezed vacuum state. In addition, we visualized the phase-sensitive amplification of the quantum vacuum state. Our method offers a robust approach for exploring cavity quantum phenomena in nonlinear driven-dissipative systems.

Methods

Experimental setup

Here, we briefly introduce our experimental setup. More detailed description of the setup can be found in the Supplementary Note 4.

Laser

For the laser system, a Toptica FemtoFiber ultra 780 was customized to provide dual-wavelength femtosecond pulses, with 780 nm and 1560 nm. A total of 780 nm signal is used to pump the OPO, and 1560 nm signal is split into two different lines, one is the bias field and the other is the local oscillator (LO) that is used to measure the phase of the OPO steady-state. Pump and bias field have the same polarization state, and they are orthogonal to the polarization of the LO signal.

OPO

The OPO consists of a MgO:PPLN crystal (MSHG1550-0.5-1, Covesion Ltd., United Kingdom) in a free-space bow tie optical cavity. The crystal is placed on the oven mount (PV10, Covesion Ltd., United Kingdom), which is connected to the temperature controller (OC2, Covesion Ltd., United Kingdom). The cavity length is matched to a single round trip of the laser pulse, and stabilized by detecting the power of the out-coupled sum-frequency mixed signal between the pump and the OPO signal40. The out-coupled sum-frequency mixed signal passes a narrow-band pass filter, which gives a linear change in the detected signal power as the cavity length deviates from the degenerate mode. A photodetector measures this error signal and a proportional-integral-derivative controller yields a voltage feedback to the piezoelectric actuator (PA44M3KW, Thorlabs, USA) attached to one of the cavity mirrors. To collect the statistics of the OPO steady-state and measure the probability, we modulate the pump and the bias signal at 10 kHz using a lithium niobate electro-optic modulators (EO-AM-NR-C1, EO-AM-NR-C3, Thorlabs, USA for amplitude modulation, and EO-PM-NR-C1, EO-PM-NR-C3, Thorlabs, USA for phase modulation). The cavity lifetime is calculated from the loss of the OPO signal from each cavity mirror, which gives  ~300 ns.

Preparing bias signal

To match the temporal alignment between pump and bias pulses, we use an optical delay line with a linear translational stage. Additional precision could be achieved by using an electronically controlled motorized actuator (PIA25, Thorlabs, USA). To attenuate the bias signal before entering to the cavity, the bias field first passes a series of neutral density filters (NDC-100S-4, Thorlabs, USA). Then we use a combination of waveplates and polarizers to further attenuate the bias field. The last waveplate is mounted on a motorized precision rotation stage (KPRM1E/M, Thorlabs, USA) which can be electronically controlled.

Measuring bias field

Because the bias pulse entering the cavity contains less than a few photons, we use the following protocol to measure the amplitude of the bias field. First, we place the power meter after the last polarizer and set the waveplate angle that gives the maximum bias power P0. Then the bias power at waveplate angle ϕ would be written as \(P={P}_{0}{\cos }^{2}(2(\phi -{\phi }_{0}))\), where ϕ0 is the angle that gives the maximum power. We rotate the angle of the waveplate to sweep the bias field and measure the bias–probability curve. After passing the last polarizer, we calculate the actual bias power that interacts inside the crystal, considering all the possible power loss happening at each optical component before it reaches at the crystal. The loss of each element is obtained either from our previous measurements25 or the manufacturer. Then we calculate the bias field \(b=\frac{{E}_{{{{{\rm{bias,mean}}}}}}}{{T}_{{{{{\rm{rt}}}}}}}\sqrt{\frac{\epsilon V}{\hslash {\omega }_{0}}}\). Ebias,mean is the mean electric field of the bias, Trt is the round trip time of the cavity, ϵ is the electric permittivity of the MgO:PPLN crystal, V is the mode volume, and ω0 is the angular frequency of the bias field. We use pulse duration of 190 ns, beam waist of 10 μm to calculate Ebias,mean and V25.

Although we initially estimated the bias field by measuring the power loss at each optical element, more precise calibration of the bias field can be achieved by measuring the bias–probability relationship of the quantum vacuum state. Details of the calibration method can be found in the Supplementary Note 4.

Measuring the statistics of the OPO steady-states

The sampled OPO signal and the LO are combined in a standard interferometry setup to measure the phase of the OPO steady-state. The polarization states of the OPO signal and the LO are matched before they are combined with the beam splitter cube. We add an additional delay stage on the LO line, using the same module that we used to control the temporal alignment between the pump and the bias pulses. Using the one output arm of the beam splitter, we check the temporal alignment between the OPO signal and the LO by observing the inference pattern with the camera (CMLN-13S2M-CS, Edmund Optics, USA). For the other arm, we measure the interference pattern with the photodetector (PDA50B2, Thorlabs, USA). The detected signal was recorded on the oscilloscope, collecting either 1000 (Fig. 2) or 10,000 OPO steady-states (Fig. 3) to measure the probability at a given bias field. The probability is defined as the number of steady-states with phase 0 rad divided by the number of total steady-states measured with the oscilloscope41.

Reconstruction protocol

In this section, we describe some mathematical background on our Husimi Q function reconstruction protocol by measuring the sensitivity of the bias–probability relationship.

Husimi Q function

The Husimi Q function gives the probability of observing a certain coherent state \(\left\vert \alpha \right\rangle\) for a given quantum state \(\left\vert \psi \right\rangle\):

$$Q(\alpha,{\alpha }^{*}) =\frac{1}{\pi }\left\langle \alpha | \rho | \alpha \right\rangle \\ =\frac{1}{\pi }| \left\langle \alpha | \psi \right\rangle {| }^{2}.$$
(2)

The Husimi Q function can be either reconstructed with a standard homodyne detection setup that combines the unknown quantum state \(\left\vert \psi \right\rangle\) with a reference coherent state \(\left\vert \alpha \right\rangle\)42 or directly measured by using a heterodyne setup43, which is also known as a double homodyne setup. We rewrite Eq. (2) using the displacement operator D(α):

$$Q(\alpha,{\alpha }^{*}) \equiv \frac{1}{\pi }| \left\langle \alpha | \psi \right\rangle {| }^{2}\\ =\frac{1}{\pi }| \left\langle 0| D(-\alpha )| \psi \right\rangle {| }^{2}.$$
(3)

Equation (3) shows that measuring the Husimi Q function is equivalent to measuring the overlap between the displaced quantum state \(D(-\alpha )\left\vert \psi \right\rangle\) and quantum vacuum state \(\left\vert 0\right\rangle\).

Displacement operator

The stochastic differential equation of the OPO signal defined along the real quadrature of the phase space X can be written as below25:

$$\dot{X}=(\lambda -1)X+\sqrt{2}b+\sqrt{\lambda }{\eta }_{X}.$$
(4)

Here, ηX is the Gaussian noise. Because the derivative of the drift term \((\lambda -1)X+\sqrt{2}b\) with respect to X is positive when the OPO is pumped above threshold (i.e., λ > 1), the sign of the drift term at the time when the bias field is injected determines whether X will grow exponentially toward positive or negative side. Therefore, the direction of the amplification can be determined by the sign of the drift term. The critical point X0 with zero drift happens at \({X}_{0}=-\sqrt{2}b/(\lambda -1)\). X0 = 0 for an unbiased OPO, meaning that the amplification behavior of the cavity quantum state \(\left\vert \psi \right\rangle\) in the biased OPO is identical to that of displaced quantum state \(D(b)\left\vert \psi \right\rangle\) in the unbiased OPO, where the displacement operator D(b) defined along the X quadrature is written as:

$$D(b)\equiv \frac{\sqrt{2}b}{\lambda -1}.$$
(5)

Reconstructing marginal distribution of the Husimi Q function

After the bias field displaces the quantum state, the OPO signal is amplified until it reaches one of the steady-states. As we described in the previous section, the original quantum state \(\left\vert \psi \right\rangle\) in the biased OPO shows the same dynamics as \(D(b)\left\vert \psi \right\rangle\) in the unbiased OPO system. The parametric amplification process is phase-sensitive, so X > 0 (or equivalently Re(α) > 0) part of the displaced quantum state gives the probability p of measuring a certain steady-state, which can be analytically described as:

$$p(b| \theta )\equiv \int_{\alpha=0}^{\alpha=\infty }| {\psi }_{\theta,D(b)}(\alpha ){| }^{2}d\alpha,$$
(6)

where \(\left\vert {\psi }_{\theta,D(b)}\right\rangle \equiv D(b)\left\vert {\psi }_{\theta }\right\rangle\). Because the phase-sensitive amplification happens along one quadrature, we consider the marginal distribution \(\left\vert {\psi }_{\theta }\right\rangle\). Using basic algebra,

$$p(b| \theta ) \equiv \int_{-D(b)}^{\infty }| {\psi }_{\theta }(\alpha ){| }^{2}d\alpha \\ ={\left\langle {\psi }_{\theta }| {\psi }_{\theta }\right\rangle }_{[-D(b),\infty ]}\\ ={\left\langle {\psi }_{\theta }| I| {\psi }_{\theta }\right\rangle }_{[-D(b),\infty ]}\\ =\frac{1}{\pi }\int| \left\langle {\psi }_{\theta }| \alpha \right\rangle {| }_{[-D(b),\infty ]}^{2}\,{d}^{2}\alpha \\ =\int_{-D(b)}^{\infty }\int_{-\infty }^{\infty }Q\,\,{\mbox{dIm}}({\alpha }_{\theta }){\mbox{dRe}}\,({\alpha }_{\theta }).$$
(7)

We introduced the identity operator \(I=\frac{1}{\pi }\int\left\vert \alpha \right\rangle \left\langle \alpha \right\vert {{\mbox{d}}}^{2}\alpha\), to expand the representation from a specific coordinate to the 2D phase space. The final expression represents the same probability but now sums over the possible coherent states in the region of interest. Therefore, by taking the derivative with respect to D(b), we can reconstruct the marginal distribution Qθ.

Stochastic differential equations

Until now, we have focused on using the displacement operator to understand how the bias field b affects the probability p. In this section, we provide another approach, which solves the SDE of the OPO system to calculate the bias–probability relationship. When a time-dependent bias field b(τ) is introduced, the OPO dynamics at the linear amplification stage can be written as:

$$\dot{X}=(\lambda -1)X+\sqrt{2}b(\tau )+\sqrt{\lambda }{\eta }_{X}.$$
(8)

With Z = Xe−(λ−1)τ,

$$Z(\tau )=Z(0)+\int_{0}^{\tau }{e}^{-(\lambda -1)\tau {\prime} }[\sqrt{2}b(\tau {\prime} )+\sqrt{\lambda }{\eta }_{X}(\tau {\prime} )]d\tau {\prime} .$$
(9)

Calculating the positive area of the probability distribution of Z(τ), we can calculate the probability p. When a constant bias field b0 is injected at τ = τ0,

$$P[Z(\tau )\ge 0]=P\left[f(\tau )\ge -\frac{\sqrt{2}{b}_{0}}{\sqrt{\lambda }}\frac{{e}^{-(\lambda -1){\tau }_{0}}-{e}^{-(\lambda -1)\tau }}{\lambda -1}\right],$$
(10)
$$f(\tau )\equiv \int_{0}^{\tau }{e}^{-(\lambda -1)\tau {\prime} }{\eta }_{X}(\tau {\prime} )d\tau {\prime}$$
(11)

Because X(τ) is the Wiener process (linear superposition of Gaussian noises), f(τ) is a Gaussian with mean 〈f(τ)〉 = 0, and variance \(\langle \,{f}^{2}(\tau )\rangle=\frac{1-{e}^{-2(\lambda -1)\tau }}{2(\lambda -1)}\). Therefore,

$$P[Z(\tau )\ge 0]=\int_{-{f}_{0}(\tau )}^{\infty }\frac{1}{\sqrt{2\pi \langle \,{f}^{2}(\tau )\rangle }}{e}^{-\frac{{f}^{2}}{2\langle \,{f}^{2}(\tau )\rangle }}df,$$
(12)

where \({f}_{0}(\tau )\equiv \frac{\sqrt{2}{b}_{0}}{\sqrt{\lambda }}\frac{{e}^{-(\lambda -1){\tau }_{0}}-{e}^{-(\lambda -1)\tau }}{\lambda -1}\). As we measure the probability p after several cavity cycles, which corresponds to the time τ when e−(λ−1)τ → 0+, so the probability p becomes

$$p\equiv P[Z(\tau )\ge 0]=\frac{1}{2}\left(1+\,{\mbox{erf}}\,\left[\frac{\sqrt{2}{b}_{0}{e}^{-(\lambda -1){\tau }_{0}}}{\sqrt{\lambda }\sqrt{\lambda -1}}\right]\right),$$
(13)

where erf is the Gauss error function. In the Supplementary Information 3 and 5, we discuss the SDEs for time-dependent λ(τ) and b(τ).