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Physical Review A

Photon number distribution of detuned squeezed states

Mordechay Y. Zahler, Y. Ben Aryeh

Published Jun 1, 1991• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.43.6368

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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.
Making use of the phase state theory introduced by Pegg and Barnett,the phase probability distributions of the excited squeezed vacuum states were investigated.And then the influence of the phase parameter,the squeezing parameter and the excited photon number on the phase probability distributions were studied with the aid of numerical method.It is shown that these probability distributions of the states are regulated by the phase parameter and the squeezing parameter mainly.The variety of phase parameter causes the probability distributions exhibiting peak and vale shape.The peak and vale values are determined by the squeezing parameter.Furthermore,the phase probability distributions are influenced by the excited photon number in a certain extent.
Using the phase state and phase operator introduced by Pegg and Barnett,the phase probability distributions of the photon-subtracted squeezed vacuum states were investigated.Numerical results indicate that the phase probability distribution of the photon-subtracted squeezed vacuum states are mainly regulated by the phase parameter and the squeezing parameter.The phase parameter causes the probability distributions exhibiting peak and vale shape,and peak value is determined by the squeezing parameter.Furthermore,the phase probability distributions are influenced by the photon-subtracted number in a certain extent.
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.
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.
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.