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arXiv:2507.12250 · New Journal of Physics

Comment on ‘Properties and dynamics of generalized squeezed states’

Rubén Gordillo-Hachuel, Ricardo Puebla

Published Feb 1, 2026• 1 citation

Abstract

Abstract A recent article (Ashhab and Ayyash 2025 New J. Phys. 27 , 054104) has reported unexpected oscillatory dynamics in generalized squeezed states of order higher than two as their squeezing parameter increases. This behaviour, observed through numerical simulations using truncated bosonic annihilation and creation operators, appeared in several properties of these states, including their average photon number. The authors argued that these oscillations reflect a genuine physical effect. Here, however, we demonstrate that the observed oscillatory behaviour is a consequence of numerical artefacts. A numerical analysis reveals that the oscillations are highly sensitive to the truncation of the Fock basis, indicating a lack of convergence. This is further supported by a theoretical analysis of the Taylor series of the average photon number, suggesting that these generalized squeezed states contain infinite energy after a finite value of the squeezing parameter. Finally, we provide an analytical proof that the average photon number of any generalized squeezed state is a non-decreasing function, thereby ruling out the possibility of intrinsic oscillatory dynamics. We hope these results help clarify the origin of the reported oscillations and highlight the special care required when dealing with high-order squeezing states.

DOI: 10.1088/1367-2630/ae4379

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Reply to Comment on "Properties and dynamics of generalized squeezed states"

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In our paper [1], our numerical simulations showed that, unlike displacement and conventional squeezing, higher-order squeezing exhibits oscillatory dynamics. Subsequently, Gordillo and Puebla pointed out that simulation results depend on whether the size of the state space in the simulations is even or odd [2]. Using additional derivations, they argued that the oscillatory dynamics is unphysical and that the photon number must increase monotonically as a function of the squeezing parameter $r$. We agree with the observation of an even-odd parity dependence in the simulations. We independently noticed the same feature in our simulations after the publication of Ref. [1]. This observation led us to perform a more detailed investigation of the numerical simulation and mathematical aspects of the generalized squeezing problem. Our new findings were reported in Ref. [3]. Further analysis was reported in Ref. [4]. Our conclusion is that the generalized squeezing operator is physically not well defined but can be made well defined when combined with additional information about the physical system under study. We demonstrated this point in the case where we include an additional nonlinear interaction term in the Hamiltonian. We disagree with the claim that the photon number must be a monotonically increasing function of $r$. This claim contradicts the mathematically rigorous results of Ref. [4]. Furthermore, we show that the oscillatory behaviour persists in two closely related, well-behaved models.

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Abstract A recent article (Ashhab and Ayyash 2025 New J. Phys. 27 , 054104) has reported unexpected oscillatory dynamics in generalized squeezed states of order higher than two as their squeezing parameter increases. This behaviour, observed through numerical simulations using truncated bosonic annihilation and creation operators, appeared in several properties of these states, including their average photon number. The authors argued that these oscillations reflect a genuine physical effect. Here, however, we demonstrate that the observed oscillatory behaviour is a consequence of numerical artefacts. A numerical analysis reveals that the oscillations are highly sensitive to the truncation of the Fock basis, indicating a lack of convergence. This is further supported by a theoretical analysis of the Taylor series of the average photon number, suggesting that these generalized squeezed states contain infinite energy after a finite value of the squeezing parameter. Finally, we provide an analytical proof that the average photon number of any generalized squeezed state is a non-decreasing function, thereby ruling out the possibility of intrinsic oscillatory dynamics. We hope these results help clarify the origin of the reported oscillations and highlight the special care required when dealing with high-order squeezing states.

In our paper [1], our numerical simulations showed that, unlike displacement and conventional squeezing, higher-order squeezing exhibits oscillatory dynamics. Subsequently, Gordillo and Puebla pointed out that simulation results depend on whether the size of the state space in the simulations is even or odd [2]. Using additional derivations, they argued that the oscillatory dynamics is unphysical and that the photon number must increase monotonically as a function of the squeezing parameter

r

. We agree with the observation of an even-odd parity dependence in the simulations. We independently noticed the same feature in our simulations after the publication of Ref. [1]. This observation led us to perform a more detailed investigation of the numerical simulation and mathematical aspects of the generalized squeezing problem. Our new findings were reported in Ref. [3]. Further analysis was reported in Ref. [4]. Our conclusion is that the generalized squeezing operator is physically not well defined but can be made well defined when combined with additional information about the physical system under study. We demonstrated this point in the case where we include an additional nonlinear interaction term in the Hamiltonian. We disagree with the claim that the photon number must be a monotonically increasing function of

r

. This claim contradicts the mathematically rigorous results of Ref. [4]. Furthermore, we show that the oscillatory behaviour persists in two closely related, well-behaved models.

Abstract We analyse the properties and dynamics of generalized squeezed states. We find that, in stark contrast to displacement and two-photon squeezing, higher-order squeezing leads to oscillatory dynamics. The state is squeezed in the initial stages of the dynamics but the squeezing reverses at later stages, and the state reverts almost completely back to the initial state. We analyse various quantities to verify that the oscillatory dynamics is physical and not a mathematical artefact. We also show that the maximum squeezing diminishes with increasing squeezing order, rendering the squeezing mechanism increasingly ineffective. Our results provide important rules that can help guide the development of more effective higher-order squeezing techniques.

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.

We generalize the generalized-squeezing problem to include fractional values of the squeezing order

n

. This approach allows us to determine the locations of critical points at which qualitative changes in behaviour occur and accurately predict the behaviour at these critical points, which are challenging for conventional computational methods. Based on our numerical calculations, we identify with a high degree of confidence the point at which the spectrum turns from continuous to discrete and the point at which oscillations turn from having asymptotically infinite amplitudes to having finite amplitudes. Furthermore, we numerically investigate the behaviour in the large

n

regime and provide an intuitive explanation for the numerical results.

A quantum-mechanical treatment is developed, by using the Lie Group methods, for the properties of the detuned degenerate, vacuum, and excited squeezed-states photon number distribution. For a vacuum squeezed state, the detuning decreases the average photon number without changing the semichaotic distribution. In addition, an oscillatory behavior of the photon pair probability as a function of the detuning parameter appears below threshold. For an excited squeezed state, the photon number distribution, including its oscillatory characteristic at resonance, is strongly affected by the detuning. In addition to a continuous relaxation of the oscillations toward a Poissonian-like shape, there is also a very sensitive movement of the oscillations' peaks, depending on a detuning produced phase.

From the normally ordered form of the density operator of a squeezed coherent state(SCS),we directly derive the compact expression of the SCS’s photon-number distribution(PND).Besides the known oscillation characteristics,we find that the PND is a periodic function with a period of π and extremely sensitive to phase.If the squeezing is strong enough,and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values,different oscillation behaviours in PND for even and odd photon numbers appear,respectively.