Kerr-induced non-Gaussianity of a bright ultrafast quantum state

Abstract

Characterizing macroscopic quantum states of light is a frontier challenge at the interface of quantum optics and high-intensity photonics. Here, we demonstrate the first, to our knowledge, direct phase-space tomography of a bright, ultrafast quantum state. We adapt a single-shot f-2f interferometer to sample the Husimi Q-function of a 25 fs bright squeezed vacuum (BSV) pulse (mean photon number N of order 10 to the twelfth) subject to Kerr nonlinearity. Our measurement reveals a clear transformation from a Gaussian distribution with pronounced quadrature antisqueezing to a characteristic S-shaped profile as the intensity-dependent nonlinear phase increases. Crucially, we show theoretically that despite optical loss, the state does not degrade into classically modulated light but is rigorously described as a statistical mixture of squeezed coherent states. While global Wigner negativity is masked by technical noise, our results show theoretically that the constituent states individually retain strong Wigner-negative features. This work bridges quantum optics and high-intensity photonics, establishing a platform for diagnosing bright quantum resources.

1. Introduction

The generation and control of macroscopic quantum states of light — pulses containing trillions of photons yet exhibiting nonclassical correlations — is a rapidly emerging frontier in photonics. These "bright" quantum states promise to enhance sensitivity in ultrafast spectroscopy and field-resolved metrology. However, a fundamental barrier impedes progress in this regime: the lack of detection techniques capable of resolving the phase-space distribution of such bright, broadband light. While methods such as double-homodyne detection and optical parametric amplification (OPA)-based tomography provide accurate reconstructions of the Husimi or Wigner functions for microscopic and mesoscopic states, they are fundamentally restricted by detector saturation at extreme macroscopic intensities (photon numbers of order 10 to the twelfth). Furthermore, applying standard homodyne techniques to unseeded macroscopic sources is often fundamentally hindered by the absence of a phase-locked local oscillator at the generation wavelength. Consequently, the phase-space structure of bright quantum light has remained largely inaccessible to direct experimental characterization.

Here, we bridge this gap between quantum and ultrafast technology by demonstrating direct phase-space tomography of a macroscopic quantum state. To overcome the saturation barrier and the lack of a native phase reference, we re-purpose the single-shot f-2f interferometer — a standard tool in ultrafast physics for carrier-envelope phase (CEP) metrology. By utilizing perturbative second-harmonic generation, this technique maps the macroscopic fundamental field to standard spectrometer arrays without saturation — an application recently investigated for extracting phase correlations in bright twin beams. This enables the direct single-shot mapping of the macroscopic Husimi Q-function for ultrafast pulses with mean photon numbers of order 10 to the twelfth, providing a detailed map of their amplitude and phase statistics without the need for attenuation or probabilistic post-selection.

To demonstrate the resolving power of this tomographic method, we apply it to a stringent test case: the deterministic generation of bright non-Gaussian states. The best-known non-Gaussian states are the single-photon state, the Schrödinger cat and superposition states, and the Gottesman–Kitaev–Preskill state. While non-Gaussian states are indispensable for universal quantum computing and sensing, they are typically produced in the faint regime via heralding. Deterministically generating them at high brightness requires extreme nonlinearities and has remained elusive. We solve this by exploiting bright squeezed vacuum (BSV), a macroscopic non-classical state generated from a strongly pumped unseeded optical parametric amplifier. Recently, we achieved single-mode BSV with the mean photon number of order 10 to the twelfth confined in a 25 fs pulse. Single-shot measurements on single-mode BSV demonstrated its binary (0 or pi) phase, although the quadrature-space distribution was not investigated. Now, we perform the Husimi-function tomography of single-mode BSV. By subjecting this state to a Kerr nonlinearity, we induce a non-Gaussian evolution in phase space, which serves as the ideal target to validate our tomographic capability.

The physical mechanism driving this transformation is the Kerr, or third-order, nonlinearity, which induces a nonlinear phase proportional to the peak intensity: the Kerr phase equals the nonlinear refractive index times the intensity times the length of the nonlinear material times the frequency, divided by the speed of light in vacuum. This effect leads to self-phase modulation, which is the workhorse for nonlinear spectral broadening and pulse compression. It also plays a crucial role in soliton dynamics. In quantum optics, weak Kerr nonlinearity is known to squeeze coherent states, which can be used to generate continuous-variable entanglement and polarization squeezing. However, a further increase in interaction strength is predicted to shear the state into a "banana" profile, exhibiting the Wigner-function negativity. While observing this effect with coherent states typically requires extreme interaction parameters, theoretical works suggest that for BSV, even a weak Kerr nonlinearity can induce strong non-Gaussianity. This state has been predicted so far only theoretically but has never been observed in an experiment.

Our tomographic measurement successfully resolves these dynamics, revealing a clear transformation from the initial Gaussian distribution to a non-Gaussian S-shaped profile — a fingerprint of the intensity-dependent phase shift. While the global negativity of the Wigner function is washed out by the inevitable optical losses of the macroscopic setup, our measured Husimi function provides robust evidence of the deterministic Kerr phase shift. We show theoretically that the decoherence affecting the macroscopic BSV allows it to be represented as a mixture of squeezed coherent states. Despite the loss, the constituent states of this mixture retain their large photon number and sensitivity to Kerr nonlinearity. The observed S-shape, therefore, indicates a macroscopic envelope consistent with a mixture of Wigner-negative states, marking the first step toward quantum applications, such as strong-field quantum optics, that require high photon fluxes.

2. Experiment

In the experiment, we generate temporally and spatially single-mode BSV using an unseeded two-stage optical parametric amplifier (OPA 1 and OPA 2), comprising two 3 mm BBO crystals pumped by a Ti–Sa laser system (805 nm, 35 fs, 1 kHz). This allows us to obtain a 25 fs BSV centered at 1.6 micrometres. Next, we send both the BSV and the OPA pump to the two-color interferometer, where we separate and combine them by dichroic mirrors DM1 and DM2 to attenuate the OPA pump by a neutral-density filter (ND), rotate its polarization to be aligned with the BSV polarization and set the optical delay between the BSV pulse and the OPA pump pulse. Finally, we focus both beams using a calcium fluoride lens into a 1.5 mm thick slab of fused silica, which imprints the Kerr phase on the BSV. The fused silica slab is placed at different positions after the focus, marked as (1) far from the focus, (2) at an intermediate position, and (3) at the focus, so that by moving it from position (1) to position (3), we increase the peak intensity in the slab.

Afterwards, using a 0.5 mm thick periodically poled lithium niobate (PPLN), we generate the second harmonic of BSV, which interferes with the attenuated OPA pump. As there is a delay between the second-harmonic (SH) pulse and the OPA pump pulse, we observe fringes in the spectra, which are recorded by the single-shot CCD-based spectrometer. The effect of cross-phase modulation between BSV and the attenuated OPA pump is excluded because of the temporal delay between the pulses. This configuration effectively serves as a single-shot f-2f interferometer. The OPA pump pulse is phase-locked to BSV and its SH because BSV is generated in a phase-sensitive manner. Because the generated SH field remains in the macroscopic photon-number regime, the linear interferometric readout effectively operates as a classical macroscopic field tracker. To process the data, we apply a discrete Fourier transform to each of the 12,000 single-shot spectral interferograms. This allows us to isolate the phase-dependent interference term (the AC peak) from the unmodulated background, extracting the classical c-number components: the single-shot SH amplitude and relative phase. Assuming the SH is generated in the perturbative, undepleted-pump regime, the fundamental BSV amplitude scales as the square root of the second-harmonic amplitude, while the fundamental phase is rigorously mapped as half the second-harmonic phase. By mapping these polar coordinates to the Cartesian optical quadratures — the amplitude times the cosine and the sine of the phase — we continuously sample the joint probability distribution. Thus, we directly measure the macroscopic Husimi Q-function, which in standard double-homodyne detection is obtained by simultaneously measuring both quadratures after a beamsplitter.

3. The S-shaped Husimi function

The measured Husimi functions of BSV after the Kerr nonlinear interaction are as follows. When the fused silica slab is far from the focus (position 1 of the slab), we observe predominantly amplitude fluctuations, typical for BSV. While the extraction of single-shot amplitude and phase distributions is currently being explored for twin beams, to the best of our knowledge, this is the first measurement of the Husimi function for a bright quantum state in general and for BSV in particular. Near the center of the polar plot (for small field amplitudes), the histogram exhibits an area with no detectable signal. As detailed in Supplement 1, this is because of the dark noise floor of the spectrometer, corresponding to a root-mean-square of 1000 photons per pixel. Because the SH photon number drops quadratically with the fundamental amplitude, low-amplitude fluctuations fall below the fundamental detection threshold of the discrete Fourier transform. Crucially, this detection limit simply truncates our sampling of low-amplitude events without artificially biasing the reconstructed S-shaped shear of the macroscopic, high-amplitude tails. As the Kerr interaction gets stronger (position 2 of the slab), an additional nonlinear phase is observed for larger amplitudes, leading to an S-shaped Husimi function. Finally, if the sample is at the focus (position 3 of the slab), a much stronger Kerr phase is imprinted onto the original state.

To theoretically describe the large-scale structure of the measured Husimi function for a single-mode BSV under Kerr nonlinearity, it is sufficient to use the classical Liouville equation. In this regime, the Kerr nonlinearity reduces to a phase-space rotation by an angle proportional to the squared distance from the origin, i.e., to the peak intensity. This amplitude-dependent shear converts the initially elongated Gaussian Husimi function of BSV into the characteristic S-shaped profile. While the experimental distributions for the two weaker settings agree very well with the theoretical ones, there is a noticeable disagreement in the case of strong Kerr interaction. In contrast to the theoretical distribution, for the measured Husimi function the spiral tails are cut, and the probability of large amplitudes is suppressed. This observation indicates that at this high intensity, the desired Kerr effect is competing with other nonlinear processes, such as those related to nonlinear spectral broadening (supercontinuum generation), Raman nonlinearity, and non-degenerate four-wave mixing. Direct spectral evidence of these competing processes — including spectral broadening, the emergence of spectral anti-correlations, and a sharp cut-off in the photon-number distribution for the same setting — is provided in Supplement 1. Because these nonlinear effects are especially pronounced for high intensities, they deplete the high-photon-number components of the BSV state more than the rest.

4. Bright squeezed vacuum as a mixture of squeezed states

The S-shaped Husimi function is direct evidence of the Kerr nonlinearity acting on the macroscopic state of light. For a pure state, this nonlinear phase would lead to a Wigner-function negativity. However, BSV is not a pure state: its purity degrades even under minute optical losses, which inevitably occur on optical elements and through additional nonlinear-optical processes accompanying optical parametric amplification.

The purity of a squeezed vacuum state can be calculated as the inverse product of its maximal and minimal quadrature uncertainties. From this definition, we see that losses on the order of one divided by the mean photon number will reduce its purity by about a factor of two. Practically, it is impossible to manage such tiny losses for a photon number of order 10 to the twelfth; therefore, BSV of such brightness is never a pure quantum state. At the same time, under linear losses, the minimal variance of the BSV quadratures remains below the vacuum level, so that the state is still non-classical.

Using the fact that BSV remains a Gaussian state of light even after linear losses, although with reduced quadrature squeezing, one can show that after loss, BSV becomes a squeezed thermal state, which can be also seen as a mixture of squeezed coherent states: the density operator of the BSV is the integral over the complex displacement of a statistical weight times the projector onto a squeezed coherent state of that displacement and squeezing parameter. The statistical weight is a two-dimensional Gaussian in the real and imaginary parts of the displacement, whose widths are the number of thermal photons scaled up and down by the exponential of twice the squeezing parameter.

The initial mean photon number of BSV and the amount of optical losses are mapped to the number of thermal photons, related to the optical losses, and the effective squeezing parameter, which is determined by the initial squeezing and the loss. In the asymptotic limit of large photon numbers, this relation simplifies to the effective squeezing parameter being half the initial one plus a quarter of the natural logarithm of the reciprocal of the loss fraction minus one.

As a result, after losses, BSV becomes a mixture of pure squeezed coherent states stochastically displaced mostly in the amplitude-quadrature direction, all squeezed along the same quadrature, which for large displacements approximately corresponds to phase squeezing. The effective squeezing parameter defines the area of quantum coherence in the phase space. Meanwhile, displacement acts as a stochastic variable responsible for the quantum decoherence of BSV. This stochasticity washes out the negativity of the Wigner function of the non-Gaussian bright state. However, as we theoretically show further, those pure states in the mixture that are squeezed along the phase quadrature become strongly non-Gaussian as a result of the Kerr effect, and under reasonable loss, their Wigner functions maintain negativity even for rather high photon numbers.

5. Phase-squeezed states and the Kerr effect

As the mixed BSV state is composed of pure squeezed states stochastically displaced predominantly in the antisqueezed quadrature direction, we further address the Kerr evolution of these displaced states. In the limit of weak Kerr nonlinearity, a coherent state of light experiences predominantly quadrature squeezing with an extremely low level of Wigner-function negativity. At the same time, some components of the statistical mixture, namely, those resembling phase-squeezed states, are expected to have a very pronounced Wigner-function negativity under the same weak action of Kerr nonlinearity that leads to the quadrature squeezing of the coherent state. We compare the simulated effect of the Kerr nonlinearity on the Wigner functions of a coherent state and a phase-squeezed state with the same displacement parameter. The considered level of squeezing is 8 dB, corresponding to one photon. The accumulated nonlinear Kerr phase — twice the interaction constant times the interaction time times the squared displacement — is assumed to be the same as in our experiment with BSV, a value of 0.6, as estimated from the strongest-interaction measurement; however, the photon number of the simulated state is considerably lower, 200 photons, and an accordingly higher interaction constant is assumed.

The dependence of the volume of the Wigner-function negativity of an initially coherent and an initially phase-squeezed state on the number of photons, for a fixed nonlinear phase of 0.6, is as follows. In both cases, the lossless case is compared with 5 per cent losses before the Kerr effect and another 5 per cent losses after it. As the displacement increases, the value of the Kerr nonlinearity is assumed to drop, to keep the nonlinear phase constant. This leads to an exponential reduction of the Wigner negativity with increasing photon number. At the same time, even a moderate (8 dB) phase squeezing of the state amplifies the Wigner negativity by several orders of magnitude. Moreover, it improves the exponential decay rate of the Wigner negativity with the number of photons by more than a factor of 10. This allows us to expect substantial Wigner negativity for thousands of displacement photons for phase-squeezed light, while in the case considered for the coherent pump, the negativity gets vanishingly small for about a hundred photons.

The reason for this behavior is that a phase-squeezed state has reduced phase noise and amplified amplitude noise, which makes it more sensitive to the intensity-dependent Kerr phase. Different parts of the Wigner function along the radius experience different nonlinear phases, resulting in a non-Gaussian shape and, as a consequence, in negativity.

6. Discussion

We have experimentally realized the first tomographic characterization of a macroscopic quantum state. Using a single-shot f-2f interferometer, we sampled the Husimi function of bright squeezed vacuum (BSV), reconstructing its phase-space distribution. Upon subjecting the state to Kerr nonlinearity, the measured Husimi function acquired a characteristic S-shape. This specific deformation, a phase-space fingerprint of the Kerr interaction, serves as the statistical evidence of the non-Gaussian dynamics. Under moderate Kerr interaction, the photon-number statistics of the BSV remained unchanged, and the state remained single-mode, confirming that the observed reshaping is purely due to the deterministic nonlinear phase.

At stronger interaction strengths, achieved by tighter focusing, the measured Husimi function deviated from the single-mode Liouville simulation. This regime coincided with the appearance of higher-order frequency modes and modified photon statistics. We attribute these deviations to the onset of additional nonlinear processes, such as four-wave mixing and spectral broadening, which naturally occur at such high intensities. This observation establishes the operational boundaries for pure Kerr-induced non-Gaussianity in bright pulses.

The observed S-shaped Husimi profile provides direct experimental evidence of the nonlinear phase shift required to generate non-Gaussianity. Ideally, for a pure quantum state, this phase-space curvature would be accompanied by regions of Wigner function negativity. However, the global purity of a macroscopic state such as BSV is intrinsically sensitive to optical loss. In our setup, the reflectivities of uncoated optical surfaces and multiple mirrors lead to an 18 per cent optical loss. The purity of our BSV is therefore of order one part in a million. While this level of decoherence is bound to mask the global Wigner negativity, the persisting non-Gaussian shape of the Husimi function confirms that the deterministic Kerr phase has been effectively imprinted on the state.

Crucially, this decoherence does not erase the quantum nature of the state. Despite the limited phase-space resolution of the measurement, explicitly quantified by the instrumental phase-noise budget detailed in Supplement 1, the macroscopic S-shape can be entirely captured by classical statistical dynamics. Indeed, a classical stochastic field prepared with an identical initial phase-space distribution would exhibit the same bulk deformation. However, the observed Husimi-function shape provides direct experimental evidence of the deterministic Kerr dynamics acting on the state. The quantum nature of the state lies not in this shape itself, but in the non-classical properties of the underlying statistical mixture. As detailed in Section 4, the lossy BSV is formally decomposed into a statistical mixture of squeezed coherent states — an exact analytical representation valid for any arbitrary linear loss. Unlike a classical chaotic field, whose constituent states are standard coherent states, the constituents of this quantum mixture maintain strong quadrature squeezing. This inherent squeezing drives their theoretical individual evolution into non-Gaussian states with rigorous Wigner negativity under the Kerr interaction. This adds to the ongoing discussion regarding the extent to which BSV can be treated as a quantum superposition. Our model reveals that the constituent states of this mixture retain their quantum coherence and high sensitivity to nonlinearity. Consequently, the observed S-shape acts as the macroscopic envelope for an ensemble of pure quantum states, each of them locally exhibiting the Wigner negativity required for non-classical applications.

To fully harness these features for applications requiring global negativity, future work must focus on state distillation to isolate the pure squeezed coherent components. Adopting purification protocols for Gaussian states and developing new schemes based on strong-field light–matter interaction presents a clear avenue for research. This work establishes the necessary metrological and theoretical framework to explore these quantum dynamics beyond the statistical limit.

(Figures, the reference list and Supplement 1 are omitted for length; the complete text is at the source.)