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Physical Review A

Photon-number distribution of squeezed states: A graphical treatment

Swapan Kumar Mandal

Published Jul 1, 1998• 5 citations
Publisher

Abstract

A graphical method is used to obtain the quasiphoton number and hence the photon-number distribution (PND) for squeezed states of real parameters. The existing results on the PND of a squeezed state are compared with the results obtained by the present method. The recent experiment on the PND agrees well with the giant oscillation reported in the present paper.

DOI
10.1103/physreva.58.752

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A graphical method is used to obtain the quasiphoton number and hence the photon-number distribution (PND) for squeezed states of real parameters. The existing results on the PND of a squeezed state are compared with the results obtained by the present method. The recent experiment on the PND agrees well with the giant oscillation reported in the present paper.
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.
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.
We study the photon-number distribution in squeezed states of a single-mode radiation field. A U(1)-invariant squeezing criterion is compared and contrasted with a more restrictive criterion, with the help of suggestive geometric representations. The U(1) invariance of the photon-number distribution in a squeezed coherent state, with arbitrary complex squeeze and displacement parameters, is explicitly demonstrated. The behavior of the photon-number distribution for a representative value of the displacement and various values of the squeeze parameter is numerically investigated. A new kind of giant oscillation riding as an envelope over more rapid oscillations in this distribution is demonstrated.
Phase properties of squeezed number states and squeezed thermal states are studied. Exact analytical formulae for phase distributions based on different phase approaches are derived and illustrated graphically. It is shown that the phase quasiprobability distribution P (W) ( theta ) associated with the Wigner function does not depend on the photon number of the initial number state and has the same form for both kinds of squeezed states under consideration while the Pegg-Barnett phase distribution and the phase quasiprobability distributions associated with the Q function and the Glauber-Sudarshan P function, approach P (W) ( theta ) in the limit of highly excited states.