Information-theoretic perspective on energy conservation in high harmonic generation

The use of energy conservation arguments is ubiquitous in understanding the process of high harmonic generation, yet a complete quantum optical description of exact photon number exchange remained elusive. Here, we solve this gap in description by introducing the energy conserving subspace in high harmonic generation in which many photons of the driving field are absorbed to generate a single photon of higher energy. The presented solution to energy conservation in quantum optical high harmonic generation naturally results in highly entangled states of light with non-classical properties in their marginals and photon statistics. This new technique can be seen as an information-theoretic approach to the problem of photon exchange between field modes, providing a new kind of selection rule imposed on the quantum optical state by the structure of the Hilbert space. In addition to providing the quantum state satisfying exact energy conservation, it allows to explain recent experimental results for quantum state engineering of optical cat states.

💡 Research Summary

The manuscript addresses a long‑standing gap in the quantum‑optical description of high‑harmonic generation (HHG): how to enforce exact energy conservation at the level of photon numbers. While classical and semi‑classical models routinely invoke energy‑conservation arguments (e.g., “many infrared photons are absorbed to emit a single harmonic photon”), a fully quantum state that respects this constraint has been missing. The authors fill this void by introducing the concept of an “energy‑conserving subspace” for HHG.

Starting from the total Hilbert space of all field modes, they define projectors Π(N ω) onto subspaces of fixed total energy N ω, where each mode q carries photon energy ω_q = q ω. This construction is illustrated first with second‑harmonic generation (SHG) as a two‑mode example, then generalized to the multi‑mode HHG scenario where many odd harmonics up to a cutoff q_c are involved. By projecting the naïve product‑of‑coherent‑states output |ψ⟩ = |α+δα⟩⊗∏_q|χ_q⟩ onto Π(N ω), they obtain a normalized state

|Ψ_N⟩ ∝ ∑{n_q=0}^{⌊N/q⌋} c{N−q n_q, n_q} |N−q n_q, n_q⟩,

with coefficients c_{i,j} given by the overlap of the original coherent amplitudes with photon‑number states. This state is intrinsically entangled between the fundamental infrared (IR) mode and the harmonic mode(s); the photon number in the IR field is no longer independent but constrained by the number of generated harmonic photons via N = n_IR + q n_harmonic.

The authors explore the physical consequences of this constraint. The marginal photon‑number distribution of the IR mode, P_IR(n₁), becomes a discrete set of values satisfying n₁ = N − q n_q, in stark contrast to the Poissonian distribution of an unconstrained coherent state. Numerical examples for N = 14, 15 with q = 3 illustrate how individual energy subspaces miss specific photon numbers, while the combined distribution over all N approaches the original Poisson shape.

To demonstrate non‑classicality, two standard diagnostics are employed. First, the Wigner function of the reduced IR state is computed as a weighted sum of single‑photon‑number Wigner functions, W_{|n⟩⟨n|}(β) = (−1)^n π e^{−|β|²} L_n(2|β|²). For subspaces N = 3, 8, 15 the Wigner functions exhibit clear negativities, a hallmark of quantum behavior. Second, the Mandel Q‑parameter is evaluated; all examined subspaces yield Q < 0, indicating sub‑Poissonian statistics. As N grows, Q approaches zero, reflecting the gradual convergence toward classical statistics, yet the non‑classical signature persists for each fixed‑energy sector.

A particularly compelling application is the explanation of recent experiments that conditionally generate optical cat states via HHG. In those experiments, detection of a harmonic photon post‑selects the IR field into a specific energy subspace Π(N ω). The resulting IR state, being the projection |Ψ_N⟩, naturally possesses the Wigner negativities and sub‑Poissonian statistics observed experimentally, thereby providing a unified theoretical framework for the creation of high‑photon‑number cat states without invoking ad‑hoc assumptions.

The paper concludes by emphasizing the generality of the energy‑conserving subspace approach. Any nonlinear optical process that converts p photons of frequency ω₁ into q photons of frequency ω₂ can be treated by defining appropriate projectors onto fixed total energy, leading inevitably to entangled, non‑classical output states. This perspective bridges the gap between photon‑number conservation and energy conservation, offering a new information‑theoretic selection rule that could guide the design of quantum‑light sources, quantum‑information protocols, and engineered metamaterials.

Overall, the work delivers a rigorous, mathematically transparent method to embed exact energy conservation into the quantum description of HHG, reveals the inevitable emergence of entanglement and non‑classicality, and connects directly to experimentally relevant state‑engineering schemes.