Finite-dimensional approximations of generalized squeezing

TL;DR

The paper addresses the challenge of simulating generalized $n$-photon squeezing in finite-dimensional Fock spaces, revealing parity-dependent dynamics that depend on whether the truncation size is even or odd. It identifies the root cause as the non-essential self-adjointness of the higher-order squeezing Hamiltonian $\hat{H}_n = i\big[(\hat{a}^\dagger)^n - \hat{a}^n\big]$ for $n\ge 3$, which leads to two distinct self-adjoint extensions tied to even and odd truncations. Through spectral analysis and rigorous operator-theoretic arguments (Kato approximation), the authors show that finite-dimensional truncations converge to the respective even or odd extensions, explaining the observed numerical parity dependence. They further demonstrate that introducing a Kerr regularization term can restore essential self-adjointness, yielding unique, parity-independent dynamics and enabling physically meaningful extrapolations to infinite dimensions. These results have practical implications for modeling generalized squeezing in nonlinear quantum optics and for interpreting experiments where multi-photon processes and nonlinearity are present.

Abstract

We show unexpected behaviour in simulations of generalized squeezing performed with finite-dimensional truncations of the Fock space: even for extremely large dimension of the state space, the results depend on whether the truncation dimension is even or odd. This situation raises the question whether the simulation results are physically meaningful. We demonstrate that, in fact, the two truncation schemes correspond to two well-defined, distinct unitary evolutions whose generators are defined on different subsets of the infinite-dimensional Fock space. This is a consequence of the fact that the generalized squeezing Hamiltonian is not self-adjoint on states with finite excitations, but possesses multiple self-adjoint extensions. Furthermore, we present results on the spectrum of the squeezing Hamiltonians corresponding to even and odd truncation size that elucidate the properties of the two different self-adjoint extensions corresponding to the even and odd truncation scheme. To make the squeezing operator applicable to a physical system, we must regularize it by other terms that depend on the specifics of the experimental implementation. We show that the addition of a Kerr interaction term in the Hamiltonian leads to uniquely converging simulations, with no dependence on the parity of the truncation size, and demonstrate that the Kerr term indeed renders the Hamiltonian self-adjoint and thus physically interpretable.

Figures (11)

• Figure 1: Average photon number $\left\langle \hat{a}^{\dagger} \hat{a} \right\rangle$ for the state $\hat{U}^{(N)}_n(r)\left| 0 \right\rangle$ as a function of the squeezing parameter $r$. The red, green, cyan and blue lines correspond, respectively, to $N=1000$, 1001, $10^4$ and $10^4+1$. Although not immediately visible in the figure, the $N=1001$ and $10^4+1$ simulation results in the cases $n=5,6$, as well as the $N=1000$ and $10^4$ simulation results, agree with each other. The maximum height of the odd-$N$ oscillations is 0.4 and 0.09 for $n=5$ and 6, respectively.
• Figure 2: Eigenvalues of $\hat{H}_n$ in the middle of the spectrum for $n=1$, 2, 3 and 4. Here we plot the few lowest non-negative eigenvalues in each case. In each panel, we plot the eigenvalues for two truncation sizes together. The two truncation sizes are $N=1000$ (blue squares) and 1001 (red circles). The green lines are fits of the form $E=\alpha j^\gamma$ with fitting parameters $\gamma=1.001$, 1.217, 1.590 and 2.035 for $n=1$, 2, 3 and 4, respectively.
• Figure 3: Average photon number $\left\langle \hat{a}^{\dagger} \hat{a} \right\rangle$ for the state $\hat{U}_3^{(N)}(r)\left| 0 \right\rangle$ as a function of the squeezing parameter $r$ with a quadratic Kerr term of varying strength ($K$). The red, green, cyan and blue lines correspond, respectively, to $N=1000$, 1001, $10^4$ and $10^4+1$. The dynamics becomes independent of the truncation size when the largest Kerr (diagonal) matrix element in the Hamiltonian becomes larger than the largest squeezing (off-diagonal) matrix element.
• Figure 4: Ten smallest positive eigenvalues of $\hat{H}_n$ as functions of truncation size $N$ for $n=1$, 2, 3 and 4. For all the data points, we chose even values of $N$.
• Figure 5: Smallest positive eigenvalue of $\hat{H}_n$ as a function of truncation size $N$ for $n=1$, 2, 3 and 4. As in Fig. \ref{['Fig:SpectrumSmallestTenEigenvaluesVsTruncationSize']}, we use only even values of $N$ in this figure. The insets show the same data in a log-log plot that demonstrates the asymptotic value of the data if extrapolated to $N\rightarrow\infty$. The orange lines are straight-line fits showing the quality of the agreement.

Theorems & Definitions (4)

• Proposition 4.1: FischerInPreparation
• Theorem 4.2
• proof
• Proposition 4.3: FischerInPreparation