High Harmonic Generation by Bright Squeezed Vacuum

The study investigates whether quantum light—specifically bright squeezed vacuum (BSV)—can drive high harmonic generation (HHG) in solids and how it compares to classical coherent light. HHG underpins attosecond science and extreme nonlinear optics but is typically described with classical driving fields, even though the matter response is quantum. The question is whether the unique photon statistics and sub-cycle electric-field fluctuations of BSV can enhance HHG efficiency, modify electron–hole dynamics, and extend access to intensity regimes typically limited by optical damage. The authors motivate the work by noting that HHG requires extreme intensities (∼1 TW/cm^2, ∼10^13 photons in tens of femtoseconds), historically inaccessible to engineered quantum states. BSV is an exception: it is a macroscopic quantum state with a broad photon-number distribution and strong sub-cycle fluctuations, predicted to enhance multiphoton processes and alter strong-field dynamics. The purpose is to demonstrate BSV-driven HHG in solids, quantify yield enhancements and power scalings versus classical light, and explore how BSV’s sparse high-intensity events allow probing regimes beyond conventional damage thresholds.

HHG was first demonstrated in gases and later in liquids and solids, enabling attosecond pulse generation and band-structure spectroscopy. While matter’s response is quantum, the driving field has largely been treated semiclassically. Recent work revealed quantum-optical signatures in HHG emission and in the driving field after interaction, but these studies used classical coherent drivers. Theoretical proposals suggested driving HHG with quantum light and specifically predicted for BSV extended plateaus, modified electron dynamics, selection rules, and squeezed XUV fields. However, generating quantum states at intensities sufficient for HHG has been challenging. BSV stands out as a macroscopic quantum state produced by unseeded parametric amplification, with demonstrated mean photon numbers up to 10^13 and superbunched photon statistics (high-order intensity correlation functions much larger than thermal light), known to enhance multiphoton processes such as perturbative harmonic generation and nonlinear electron emission. These works motivated testing BSV as a strong-field driver for HHG in solids.

Experimental: Two linearly polarized pump sources centered at 1.6 μm were used: (i) coherent pulses (70 fs FWHM) from a Ti:sapphire amplifier (800 nm, 45 fs, 1 kHz) with OPA (TOPAS Prime) and (ii) BSV pulses (25 fs FWHM) generated via high-gain parametric down-conversion in a 3-mm BBO (type-I, near-degenerate) pumped by the same Ti:sapphire system. The BSV bandwidth was narrowed by slight detuning from phase matching to achieve temporal single-mode; spatial single-moding was achieved by re-injecting BSV and pump into the BBO, amplifying primarily the lowest-diffraction mode. BSV spectra spanned ~150 nm around 1.6 μm. Residual 800 nm light was removed with dichroic and longpass filters. Pulse characterizations used SH-FROG, yielding near-transform-limited durations (70 fs coherent, 25 fs BSV). The BSV photon-number distribution was measured with a fast InGaAs photodiode, confirming single-mode statistics with mean ~2×10^12 photons per pulse and a tail up to ~2×10^13 photons (∼9 μJ).

Samples and detection: HHG was driven in x-cut 6 μm Mg:LiNbO3 and 1 μm amorphous Si, both on 1 mm fused silica substrates. The pump was focused to a 30 μm diameter spot (1/e^2) with an off-axis parabolic mirror. Post-sample, harmonics were dispersed by a MgF2 prism and refocused; 4th–7th harmonics were detected shot-by-shot with a UV-enhanced silicon APD. Bandpass filters for individual harmonics and an iris reduced background. For each condition, 200,000 synchronized single-shot records of the driver and harmonic signals were acquired, enabling joint photon-number statistics analyses.

Power-scaling and damage-threshold protocol: Power-scaling curves were obtained by analyzing shot-resolved harmonic counts versus instantaneous pump intensity. For coherent light, damage thresholds were identified by irreversible drops in harmonic yield when scanning intensity up and down over time and varying repetition rate down to 0.5 Hz to separate single-shot from cumulative damage.

Numerical simulations: HHG in LiNbO3 was modeled using 1D semiconductor Bloch equations along Γ–Z, including interband polarization and intraband currents. Band structure and transition dipole moments were derived from DFT with the direct bandgap set to 4.2 eV to match Mg:LiNbO3. A phenomenological decoherence time T2 = 0.5 optical cycles suppressed unphysical long recollisions while preserving harmonic structure. Ferroelectric spontaneous polarization was included via a first-order Stark shift of the bandgap, enabling even-order harmonics; saturation of the polarization parameter at high intensities was implemented. The HHG spectrum was obtained from Fourier transforms of interband and intraband currents. To simulate BSV driving, coherent-state HHG spectra were averaged over the Husimi Q-function distribution of BSV electric-field amplitudes (along the anti-squeezed axis), effectively integrating over the BSV field-amplitude fluctuations.

• First experimental observation of HHG in solids driven by quantum light (BSV). BSV-generated harmonics (4th–7th) in LiNbO3 are enhanced by factors of about 5–15 compared to coherent light at the same mean intensity (~2 TW/cm^2), despite BSV’s shorter pulse duration.
• The LiNbO3 spectrum shows both odd and even harmonics; 6th and 7th harmonics lie above the direct bandgap (~4.2 eV). Simulations based on SBEs with BSV statistics qualitatively reproduce the enhancement trend and its harmonic-order dependence.
• Joint photon-number statistics differ starkly between coherent and BSV driving. With coherent driving, pump and 6th-harmonic photon numbers are narrowly Gaussian-distributed and shift with mean pump photon number. With BSV, the pump distribution is broad with a maximum at zero; the 6th-harmonic distribution inherits a large width and correlates with the pump, allowing extraction of power dependence from a single joint-distribution measurement at a fixed mean.
• Power scaling in LiNbO3: For 4th and 5th harmonics (below bandgap), coherent excitation shows perturbative scaling at low intensity with deviations below ~1 TW/cm^2, indicating onset of non-perturbative dynamics. BSV maintains perturbative scaling over a broader range, deviating near ~2 TW/cm^2. For 6th and 7th harmonics (above bandgap), both drivers are non-perturbative; however, scaling exponents and their intensity dependence differ. At low intensities the 6th harmonic scales with an exponent ~3.4 for both drivers, whereas the 7th scales with ~2.0 (coherent) versus ~3.4 (BSV). Two plateaus in scaling above ~1.5 TW/cm^2 (notably for 4th and 7th) are visible with BSV, likely due to quantum path interference; these features are obscured with coherent light due to damage limits.
• In amorphous Si (odd harmonics only), coherent excitation yields non-perturbative scaling exponents N_5ω ∝ N_L^2.8 and N_7ω ∝ N_L^2.7, limited by damage above ~2 TW/cm^2. With BSV, the 5th harmonic initially follows perturbative scaling (∝ N_L^5) then saturates above ~2 TW/cm^2, indicative of strong depletion; the 7th harmonic remains non-perturbative but exhibits a larger exponent than with coherent driving, suggesting competition between multiphoton and tunneling channels.
• Interpretation: Superbunched BSV photon statistics enhance n-photon processes by g_BSV^(n), extending the perturbative regime’s range and increasing apparent exponents at low intensities for high-order processes. For above-bandgap harmonics with effective scaling exponents near 3, the statistical enhancement is much smaller than for n=5, consistent with the observed smaller BSV-vs-coherent enhancement at higher orders.
• Damage tolerance: BSV enables HHG studies in solids beyond 10 TW/cm^2 without damage, whereas coherent pulses damage samples around ~2–3 TW/cm^2 depending on repetition rate. The broad BSV photon-number distribution yields sparse high-intensity events that drive HHG while the average remains below damage threshold, permitting sample relaxation between events.

The results demonstrate that bright squeezed vacuum can efficiently drive HHG in solids, addressing the core question of whether quantum light can access and enhance strong-field phenomena. Compared to coherent light of the same mean intensity, BSV increases harmonic yields and modifies power scaling behavior, consistent with enhancements expected from superbunched photon statistics. The joint photon-number distributions underline how BSV’s broad fluctuations map onto electron–hole dynamics, allowing single-measurement access to power dependence and potentially to band-structure information via correlations.

BSV’s sparse high-intensity shots enable probing regimes typically inaccessible due to optical damage with classical drivers at high repetition rates, revealing non-perturbative features such as scaling plateaus and saturation due to valence-band depletion. While coherent sources could emulate sparsity by extreme repetition-rate reduction or active shot-to-shot modulation, this adds complexity and reduces data throughput; BSV provides an intrinsic, practical route.

Numerical simulations incorporating BSV statistics and ferroelectric effects capture the main experimental trends, supporting the interpretation that photon statistics play a central role in boosting multiphoton channels below the bandgap and modulating non-perturbative dynamics above the bandgap. Deviations between simulation and experiment at higher intensities and for the 7th harmonic highlight model simplifications and suggest richer band-structure and many-body physics at play.

This work reports the first observation of non-perturbative HHG in solids driven by bright squeezed vacuum, establishing a pathway for quantum-light-driven strong-field physics. BSV enhances harmonic yields relative to classical coherent light, extends the accessible intensity range beyond conventional damage thresholds, and enables observation of features such as scaling plateaus and saturation in amorphous Si. The approach suggests a new spectroscopy modality leveraging photon-number statistics—extreme nonlinear quantum spectroscopy—and paves the way for sub-cycle quantum noise engineering to control HHG. The demonstrated BSV driver intensities are compatible with HHG in gases, liquids, and wide-bandgap solids, potentially reaching the XUV. Future efforts may exploit BSV’s giant field uncertainty and dense photon-pair content to observe quantum interference and many-body correlations in solids. Moreover, since HHG proceeds on sub-cycle timescales, advancing beyond single-mode quantum-optical descriptions will be important to fully capture the role of quantum fluctuations in extreme nonlinear optics.

• The semiconductor Bloch model uses a simplified band structure (effectively a two-band, single conduction band picture) and a 1D Γ–Z orientation, which can deviate from real materials, especially at higher harmonic orders and intensities.
• A phenomenological decoherence time (T2 = 0.5 cycles) is imposed to regularize recollisions; results may depend on this choice at some level.
• The spontaneous polarization model includes a saturation term and manual bandgap setting to match Mg:LiNbO3; these approximations can affect quantitative accuracy.
• Experimental comparison between coherent and BSV pulses involves different pulse durations (70 fs vs 25 fs), which is accounted for in intensity estimates but could still influence dynamics.
• Damage-threshold characterization indicates strong dependence on repetition rate; comprehensive lifetime and cumulative heating effects are not exhaustively explored.
• The analysis assumes temporally and spatially single-mode BSV; residual multimode content or alignment drifts could affect photon-number statistics and coupling to the sample.