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arXiv:2508.09041 · Physical Review A

Finite-dimensional approximations of generalized squeezing

Sahel Ashhab, Felix Fischer, Davide Lonigro, Daniel Braak, Daniel Klaus Burgarth

Published Dec 15, 2025• 4 citations
arXivHTMLPDF

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.

DOI
10.1103/9vwp-f35c

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Abstract Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance, and is associated with operators involving nonlinear combinations of creation and annihilation operators. Here we study a class of operators of the form ξ ( a † ) k a l + ξ ∗ ( a † ) l a k + f ( a † a ) , which arise naturally in the analysis of higher-order quantum fluctuations. The operators are defined on the linear span of Fock states. We show that the essential self-adjointness of these operators depends on the asymptotics of the real-valued function f ( n ) at infinity. In particular, pure higher-order squeezing operators ( k ⩾ 3 , l = 0 , and f ( n ) = 0 ) are not essentially self-adjoint, but adding a properly chosen term f ( a † a ) , like a Kerr term, can have a regularizing effect and restore essential self-adjointness. In the non-self-adjoint regime, we compute the deficiency indices and classify all self-adjoint extensions. Our results provide a rigorous operator-theoretic foundation for modeling and interpreting higher-order squeezing in quantum optics, and reveal interesting connections with the Birkhoff–Trjitzinsky theory of asymptotic expansions for recurrence relations.

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

Sahel Ashhab, Mohammad Ayyash

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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Comment on ‘Properties and dynamics of generalized squeezed states’

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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.

arXiv:2507.12250•New Journal of Physics•Feb 1, 2026•1 citation

Time-evolution of squeezed coherent states of a generalized quantum parametric oscillator

Şirin A. Büyükaşık, Zehra Çayiç

Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.

Journal of Mathematical Physics•Jun 1, 2019•11 citations
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.
Abstract Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance, and is associated with operators involving nonlinear combinations of creation and annihilation operators. Here we study a class of operators of the form ξ ( a † ) k a l + ξ ∗ ( a † ) l a k + f ( a † a ) , which arise naturally in the analysis of higher-order quantum fluctuations. The operators are defined on the linear span of Fock states. We show that the essential self-adjointness of these operators depends on the asymptotics of the real-valued function f ( n ) at infinity. In particular, pure higher-order squeezing operators ( k ⩾ 3 , l = 0 , and f ( n ) = 0 ) are not essentially self-adjoint, but adding a properly chosen term f ( a † a ) , like a Kerr term, can have a regularizing effect and restore essential self-adjointness. In the non-self-adjoint regime, we compute the deficiency indices and classify all self-adjoint extensions. Our results provide a rigorous operator-theoretic foundation for modeling and interpreting higher-order squeezing in quantum optics, and reveal interesting connections with the Birkhoff–Trjitzinsky theory of asymptotic expansions for recurrence relations.
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 rrr. 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 rrr. 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.
When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian H as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian.In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs extension of the restriction of H to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from H itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with.As a practical demonstration of such results, we consider the quantum particle in the box, and we show that, for a wide class of bases (which include associated Legendre polynomials as a concrete example) the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed.
We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaussian functions and a family of wave packets based on the discretization of an arbitrary continuous momentum probability distribution. The problem of finiteness of the energy dispersion for the squeezed states in the infinite well is discussed. Finally, we perform the limit of large interval length and the semiclassical limit. As a supplementary general result, we show that an arbitrary physical quantity has a finite dispersion if and only if the wave function of a quantum system belongs to the domain of the corresponding self-adjoint operator. This can be regarded as a physical meaning of the domain of a self-adjoint operator.
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.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.