One Complex Number May Track How Quantum Information Flows From the Big Bang to the Lab

Physicists have long known that the early universe and a laser-driven crystal share a deep mathematical kinship. Both manufacture pairs of quantum excitations out of empty space: one spawns correlated modes of primordial perturbations as spacetime expands, the other conjures twin photons in a nonlinear optical crystal. A new theoretical study now compresses the entire information-theoretic bookkeeping of such pair creation into a single complex number, offering what its authors describe as a compact diagnostic tool for quantum correlations in any time-dependent two-mode bosonic system.

The work, published in The European Physical Journal C by Shi-Cheng Liu, Lei-Hua Liu, Bichu Li, Yuebing Zhou and Hai-Qing Zhang, builds on the Bogoliubov description of pair production, a framework that dates back to Parker’s classic analyses of quantized fields in expanding universes. When translational symmetry is preserved, excitations are produced in pairs of opposite momenta, and the quantum state takes the form of a two-mode squeezed state. This squeezed-state picture has been central to discussions of how cosmological fluctuations acquire classical behavior, and it underlies the same mathematics in two-photon quantum optics and continuous-variable quantum information.

Conventionally, the dynamics of such a state is tracked by two real variables: a squeezing amplitude and a squeezing phase, or equivalently a pair of Bogoliubov coefficients. These parametrizations are complete but cumbersome. The squeezing phase can oscillate rapidly in time-dependent backgrounds, and when different models are compared, the two-mode state and its reduced density matrix are often reconstructed separately for each case before information measures are evaluated. The new paper sidesteps this by defining the Bogoliubov ratio as the quotient of the pair-production coefficient by the free-evolution coefficient. Because the coefficients satisfy the bosonic normalization condition, the modulus of this ratio is strictly less than one, and the entire normalized two-mode state can be written as a geometric series over paired number states with the ratio as its single complex parameter.

The crucial dynamical result is that for a general time-dependent quadratic two-mode Hamiltonian, the Schrödinger equation reduces to a closed complex Riccati equation for this ratio. One function in the Hamiltonian controls pair creation and annihilation, while the other generates the phase rotation of the mode pair; their competition determines both the magnitude and phase of the ratio. Once the ratio is solved, the reduced-state spectrum follows from its squared modulus alone. Writing the ratio in polar form further exposes how pair production drives radial growth, whereas frequency rotation and detuning act through the phase and can suppress coherent squeezing accumulation.

After one partner mode is traced out, the phase of the ratio drops out entirely. The eigenvalues of the one-mode reduced density matrix form a geometric distribution determined by the squared modulus of the ratio. From this single real quantity, the purity, linear entropy, Rényi-2 entropy and von Neumann entropy of the reduced state follow directly through standard definitions. The authors emphasize that these Gaussian results are mathematically equivalent to those obtained from the standard density-matrix or covariance-matrix formalism; the advantage of the ratio representation is dynamical rather than definitional. It separates the model-dependent evolution of one complex group coordinate from the universal conversion to information-theoretic quantities, so that after solving one Riccati equation, no reduced density matrix needs to be reconstructed or diagonalized independently for each physical realization.

The framework also extends exactly beyond Gaussian states. Because the two-mode Hamiltonian conserves the number difference between the partner modes, the evolution separates into invariant sectors of the underlying SU(1,1) algebra. Starting from a lowest-weight Fock state with a conserved number difference, the same ratio equation governs the dynamics, while the reduced spectrum becomes a negative-binomial distribution determined by the squared modulus of the ratio together with the sector label. These non-Gaussian squeezed Fock states cannot be represented by a covariance matrix alone unless the non-Gaussian family and its label are specified. The Gaussian formulas are recovered continuously when the sector label vanishes. The authors are careful to state the boundary of the method: for a generic non-Gaussian state with independent amplitudes, a single ratio is not a complete state coordinate, and higher-order correlation data must also be evolved.

Two concrete applications illustrate the common mechanism. The first is cosmological perturbations, where an evolving background determines the effective frequency and the pair-production coupling. In the parametrization adopted, the effective frequency contains an inverse-square contribution controlled by a parameter that encodes the background expansion and the effective mass of the perturbation, while the coupling represents the conformal expansion rate in the de Sitter limit. Numerical integration of the Riccati equation over a wide range of conformal time shows that the ratio is initially negligible and grows as the endpoint is approached, producing a monotonic decrease in one-mode purity and corresponding increases in all three entropy measures. At the final slice, the calculation yields a ratio modulus of 0.447, a purity of 0.667, a linear entropy of 0.333, a Rényi-2 entropy of 0.405 and a von Neumann entropy of 0.625. This joint behavior is the standard signature of two-mode cosmological squeezing: the complete pair state remains pure, while either member becomes mixed after its partner is traced out.

The comparison with an ideal resonant benchmark is striking. When the free phase-rotation term is removed while the same cosmological squeezing drive is retained, the ratio approaches unity almost exactly, the purity collapses to roughly two parts per million, and the von Neumann entropy climbs above thirteen. The strong separation from the cosmological curves quantifies how much the free phase rotation and the background-dependent mode dynamics suppress coherent squeezing accumulation. As the background parameter increases, the effective frequency softens earlier, the phase rotation weakens, and the pair-production term produces a larger amplitude; the purity at the final slice drops from 0.581 for one parameter choice to 0.063 for another, while the von Neumann entropy rises from 0.786 to 3.075. The enhancement of squeezing and reduced-state entropy by a nontrivial background term agrees with established inflationary squeezing analyses and recent quantum-information studies of cosmological perturbations with modified propagation dynamics.

The second application is a chirped-pulse optical parametric amplifier, a laboratory realization with a finite Gaussian pump envelope and a linearly increasing detuning. In the resonant constant-coupling limit, the Riccati equation reproduces the standard two-mode squeezing solution exactly, with the ratio equal to the hyperbolic tangent of the dimensionless time, providing an analytic consistency test of the sign, normalization and information-theoretic conversion. For the physical Gaussian chirped pulse, pair production becomes efficient near the pulse center, where the ratio rises sharply and all information measures change over the same interval. At later times, the Gaussian envelope suppresses the gain while the detuning sweep drives the system away from resonance. At the final time, the Gaussian chirped case gives a ratio modulus of 0.9923 and a Rényi-2 entropy of 4.861, whereas removing the chirp gives 0.9963 and 5.606; with constant coupling, the chirped case and the standard benchmark give 11.383 and 19.413 respectively. These controlled comparisons show separately that the finite envelope limits the interaction duration and that the detuning sweep further reduces the accumulated squeezing, consistent with pulsed parametric amplification and chirped quasi-phase-matched down-conversion experiments.

Despite their different microscopic origins, the cosmological background and the optical drive enter the construction only through the two model-dependent functions of the quadratic Hamiltonian, and both are governed by the same ratio equation and reduced-state formulas. The same structural correspondence also appears in the dynamical Casimir effect and in superconducting-circuit realizations of parametrically generated photon pairs, suggesting the framework may extend to other time-dependent quadratic bosonic systems provided the evolved state retains a normalized paired Bogoliubov representation. The authors stress that the Bogoliubov-ratio framework should be understood as a compact dynamical parametrization rather than a new definition of Gaussian information measures; in the vacuum pair sector, the ratio, density-matrix and covariance-matrix descriptions contain the same information. Its real value lies in isolating the universal information content of two-mode pair production from the physical mechanism responsible for the evolution, a separation that turns quantum-information diagnostics of systems ranging from the inflationary universe to a pulsed laser amplifier into the solution of a single complex differential equation.

Subject of Research: A Bogoliubov-ratio framework for quantum-information diagnostics of time-dependent two-mode Boson System

Article Title: A Bogoliubov-ratio framework for quantum-information diagnostics of time-dependent two-mode Boson System

Article References: Liu, S.-C., Liu, L.-H., Li, B., Zhou, Y., & Zhang, H.-Q. (2026). A Bogoliubov-ratio framework for quantum-information diagnostics of time-dependent two-mode Boson System. The European Physical Journal C, 86(10), Article 1170. https://doi.org/10.1140/epjc/s10052-026-16459-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16459-5

Keywords: Bogoliubov-ratio, framework, quantum-information, diagnostics, time-dependent, two-mode, Boson, System, scientific research