## Abstract

We show that the photon distribution of a highly squeezed state exhibits oscillations.

© 1987 Optical Society of America

Full Article | PDF Article**Journal of the Optical Society of America B**- Vol. 4,
- Issue 10,
- pp. 1715-1722
- (1987)
- •https://doi.org/10.1364/JOSAB.4.001715

We show that the photon distribution of a highly squeezed state exhibits oscillations.

© 1987 Optical Society of America

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- For a review on squeezed states see, for example, D. F. Walls, Nature 306, 141–146 (1983); Nature 324, 210–211 (1986).

[Crossref] - For a review on gravitational wave detection, emphasizing the importance of quantum fluctuations and the role of squeezed states, see, for example, V. B. Braginsky and A. B. Manukin, Measurement of Weak Forces in Physics Experiments (U. Chicago Press, Chicago, Ill., 1977); C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge U. Press, Cambridge, 1981); see also the seminal articles by J. N. Hollenhorst, Phys. Rev. D 19, 1669–1679 (1979); C. M. Caves, Phys. Rev. D 23, 1693–1708 (1981).

[Crossref] - For a review on ring-laser gyroscopes see, for example, W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61–104 (1985); for the use of squeezed states in gyros see M. Marte and D. F. Walls (Department of Physics, University of Waikato, Hamilton, New Zealand), “Enhanced sensitivity of fiber optic rotation sensors with squeezed light” (advance communication).

[Crossref] - H. P. Yuen, Phys. Rev. A 13, 2226–2243 (1976); H. P. Yuen, in Quantum Optics, Experimental Gravity and Measurement Theory, P. Meystre and M. O. Scully, eds. (Plenum, New York, 1983), pp. 249–268.

[Crossref] - B. G. Levi, Phys. Today 39(3), 17–19 (1986); A. L. Robinson, Science 233, 280–281 (1986).

[Crossref] [PubMed] - R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409–2412 (1985); R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691–694 (1986); M. W. Maede, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161–163 (1987); L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520–2523 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198–201 (1987).

[Crossref] [PubMed] - J. A. Wheeler, Lett. Math. Phys. 10, 201–206 (1985); W. Schleich and J. A. Wheeler, Nature 326, 574–577 (1987); in Proceedings of the First International Conference on the Physics of Phase Space, W. W. Zachary, ed. (Springer-Verlag, New York, 1987); Verh. Dtsch. Phys. Ges. (VI) 22, Q15.3 (1987).

[Crossref] - Various mathematical techniques have been used to obtain the photon distribution Wm of a squeezed state, and rather complex expressions have been given, for example, in Ref. 4 and in D. Stoler, Phys. Rev. D 1, 3217–3219 (1970); Phys. Rev. D 4, 1925–1926 (1971); G. J. Milburn and D. F. Walls, Phys. Rev. A 27, 392–394 (1983); M. Nieto, in Frontiers in Nonequilibrium Statistical Mechanics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 287–307.

[Crossref] - Oscillations in the photon distribution of states of electromagnetic fields have also been pointed out in the context of the Jaynes–Cummings model [See, for example, P. Meystre, E. Geneux, A. Quattropani, and A. Faist, Nuovo Cimento B 25, 521–537 (1975)] and in the Rydberg maser [see, for example, P. Filipowicz, P. Meystre, G. Rempe, and H. Walther, Opt. Acta 32, 1105–1123 (1985); P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077–3087 (1986)]. In contrast to the first example, however, the modulations in the case of the squeezed state are time independent. Such oscillations have been recognized as a striking feature of a highly nonclassical, that is, a quantum-mechanical state, in M. Hillery, Phys. Lett. A 111, 409–411 (1985); Phys. Rev. A 31, 338–342 (1985); Phys. Rev. A 35, 725–732 (1987).

[Crossref] [PubMed] - A superposition of macroscopically distinguishable states such as coherent states of appropriate displacement can also lead to striking interference effects that reflect themselves in an oscillating probability distribution for the photocurrent, as is shown in B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13–16 (1986).

[Crossref] [PubMed] - W. Schleich and J. A. Wheeler, “Interference in phase space in diatomic molecules,” presented at the John Klauder Seminar, AT&T Bell Laboratories, Murray Hill, New Jersey, June 12, 1987.

- Cited in C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), p. 304.

- W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

- See, for example, M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

- In general, the minor and major axes of the error ellipse make an angle θ relative to the coordinate axes of x and p. However, a rotation1 of the coordinate axes suffices to reduce the angle θ to the zero value presupposed in this paper. This rotation induces a rescaling of Re α and Im α.

- See, for example, the reviews by M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, Usp. Fiz. Nauk. 139, 587–619 (1983) [Sov. Phys. Usp. 26,311–327 (1983)]; L. Cohen, in Frontiers of Nonequilibrium Statistical Physics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 97–117.

[Crossref] - M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974–3987 (1986); Y. Yamamoto, S. Machida, and O. Nilsson, Phys. Rev. A 34, 4025–4042 (1986).

[Crossref] [PubMed] - D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J., 1951).

- G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939).

- When we take real and imaginary parts ofHm(z)=2zHm-1(z)-2(m-1)Hm-2(z),we find the following recurrence relations for Re Hm(z) and Im Hm(z):Re Hm=2(zr Re Hm-1-zi Im Hm-1)-2(m-1)Re Hm-2,Im Hm=2(zr Im Hm-1+zi Re Hm-1)-2(m-1)Im Hm-2.Since H0(z) = 1 and H1(z) = 2z the initial conditions for the iteration areRe H0=1,Im H0=0,Re H1=2zr,Im H1=2zi.

- M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

- For an instructive geometrical discussion of this broadening see G. Leuchs, in Frontiers of Nonequilibrium Statistical Physics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 329–360.

[Crossref] - W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966).

- For discussion of an improved Stirling formula see, for example, C. Leubner, Eur. J. Phys. 6, 299–301 (1985), and references therein. See also W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957), Vol. I, p. 64, problem (25).

[Crossref]

B. G. Levi, Phys. Today 39(3), 17–19 (1986); A. L. Robinson, Science 233, 280–281 (1986).

[Crossref]
[PubMed]

A superposition of macroscopically distinguishable states such as coherent states of appropriate displacement can also lead to striking interference effects that reflect themselves in an oscillating probability distribution for the photocurrent, as is shown in B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13–16 (1986).

[Crossref]
[PubMed]

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974–3987 (1986); Y. Yamamoto, S. Machida, and O. Nilsson, Phys. Rev. A 34, 4025–4042 (1986).

[Crossref]
[PubMed]

For discussion of an improved Stirling formula see, for example, C. Leubner, Eur. J. Phys. 6, 299–301 (1985), and references therein. See also W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957), Vol. I, p. 64, problem (25).

[Crossref]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409–2412 (1985); R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691–694 (1986); M. W. Maede, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161–163 (1987); L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520–2523 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198–201 (1987).

[Crossref]
[PubMed]

J. A. Wheeler, Lett. Math. Phys. 10, 201–206 (1985); W. Schleich and J. A. Wheeler, Nature 326, 574–577 (1987); in Proceedings of the First International Conference on the Physics of Phase Space, W. W. Zachary, ed. (Springer-Verlag, New York, 1987); Verh. Dtsch. Phys. Ges. (VI) 22, Q15.3 (1987).

[Crossref]

For a review on ring-laser gyroscopes see, for example, W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61–104 (1985); for the use of squeezed states in gyros see M. Marte and D. F. Walls (Department of Physics, University of Waikato, Hamilton, New Zealand), “Enhanced sensitivity of fiber optic rotation sensors with squeezed light” (advance communication).

[Crossref]

See, for example, the reviews by M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, Usp. Fiz. Nauk. 139, 587–619 (1983) [Sov. Phys. Usp. 26,311–327 (1983)]; L. Cohen, in Frontiers of Nonequilibrium Statistical Physics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 97–117.

[Crossref]

For a review on squeezed states see, for example, D. F. Walls, Nature 306, 141–146 (1983); Nature 324, 210–211 (1986).

[Crossref]

H. P. Yuen, Phys. Rev. A 13, 2226–2243 (1976); H. P. Yuen, in Quantum Optics, Experimental Gravity and Measurement Theory, P. Meystre and M. O. Scully, eds. (Plenum, New York, 1983), pp. 249–268.

[Crossref]

Oscillations in the photon distribution of states of electromagnetic fields have also been pointed out in the context of the Jaynes–Cummings model [See, for example, P. Meystre, E. Geneux, A. Quattropani, and A. Faist, Nuovo Cimento B 25, 521–537 (1975)] and in the Rydberg maser [see, for example, P. Filipowicz, P. Meystre, G. Rempe, and H. Walther, Opt. Acta 32, 1105–1123 (1985); P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077–3087 (1986)]. In contrast to the first example, however, the modulations in the case of the squeezed state are time independent. Such oscillations have been recognized as a striking feature of a highly nonclassical, that is, a quantum-mechanical state, in M. Hillery, Phys. Lett. A 111, 409–411 (1985); Phys. Rev. A 31, 338–342 (1985); Phys. Rev. A 35, 725–732 (1987).

[Crossref]
[PubMed]

Various mathematical techniques have been used to obtain the photon distribution Wm of a squeezed state, and rather complex expressions have been given, for example, in Ref. 4 and in D. Stoler, Phys. Rev. D 1, 3217–3219 (1970); Phys. Rev. D 4, 1925–1926 (1971); G. J. Milburn and D. F. Walls, Phys. Rev. A 27, 392–394 (1983); M. Nieto, in Frontiers in Nonequilibrium Statistical Mechanics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 287–307.

[Crossref]

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J., 1951).

For a review on gravitational wave detection, emphasizing the importance of quantum fluctuations and the role of squeezed states, see, for example, V. B. Braginsky and A. B. Manukin, Measurement of Weak Forces in Physics Experiments (U. Chicago Press, Chicago, Ill., 1977); C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge U. Press, Cambridge, 1981); see also the seminal articles by J. N. Hollenhorst, Phys. Rev. D 19, 1669–1679 (1979); C. M. Caves, Phys. Rev. D 23, 1693–1708 (1981).

[Crossref]

For a review on ring-laser gyroscopes see, for example, W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61–104 (1985); for the use of squeezed states in gyros see M. Marte and D. F. Walls (Department of Physics, University of Waikato, Hamilton, New Zealand), “Enhanced sensitivity of fiber optic rotation sensors with squeezed light” (advance communication).

[Crossref]

[Crossref]
[PubMed]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409–2412 (1985); R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691–694 (1986); M. W. Maede, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161–163 (1987); L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520–2523 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198–201 (1987).

[Crossref]
[PubMed]

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974–3987 (1986); Y. Yamamoto, S. Machida, and O. Nilsson, Phys. Rev. A 34, 4025–4042 (1986).

[Crossref]
[PubMed]

See, for example, M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

For discussion of an improved Stirling formula see, for example, C. Leubner, Eur. J. Phys. 6, 299–301 (1985), and references therein. See also W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957), Vol. I, p. 64, problem (25).

[Crossref]

For an instructive geometrical discussion of this broadening see G. Leuchs, in Frontiers of Nonequilibrium Statistical Physics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 329–360.

[Crossref]

B. G. Levi, Phys. Today 39(3), 17–19 (1986); A. L. Robinson, Science 233, 280–281 (1986).

[Crossref]
[PubMed]

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966).

[Crossref]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

Cited in C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), p. 304.

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

[Crossref]

W. Schleich and J. A. Wheeler, “Interference in phase space in diatomic molecules,” presented at the John Klauder Seminar, AT&T Bell Laboratories, Murray Hill, New Jersey, June 12, 1987.

[Crossref]

[Crossref]

[Crossref]
[PubMed]

A superposition of macroscopically distinguishable states such as coherent states of appropriate displacement can also lead to striking interference effects that reflect themselves in an oscillating probability distribution for the photocurrent, as is shown in B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13–16 (1986).

[Crossref]
[PubMed]

Various mathematical techniques have been used to obtain the photon distribution Wm of a squeezed state, and rather complex expressions have been given, for example, in Ref. 4 and in D. Stoler, Phys. Rev. D 1, 3217–3219 (1970); Phys. Rev. D 4, 1925–1926 (1971); G. J. Milburn and D. F. Walls, Phys. Rev. A 27, 392–394 (1983); M. Nieto, in Frontiers in Nonequilibrium Statistical Mechanics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 287–307.

[Crossref]

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939).

[Crossref]
[PubMed]

[Crossref]

J. A. Wheeler, Lett. Math. Phys. 10, 201–206 (1985); W. Schleich and J. A. Wheeler, Nature 326, 574–577 (1987); in Proceedings of the First International Conference on the Physics of Phase Space, W. W. Zachary, ed. (Springer-Verlag, New York, 1987); Verh. Dtsch. Phys. Ges. (VI) 22, Q15.3 (1987).

[Crossref]

W. Schleich and J. A. Wheeler, “Interference in phase space in diatomic molecules,” presented at the John Klauder Seminar, AT&T Bell Laboratories, Murray Hill, New Jersey, June 12, 1987.

[Crossref]

[Crossref]
[PubMed]

[Crossref]

A superposition of macroscopically distinguishable states such as coherent states of appropriate displacement can also lead to striking interference effects that reflect themselves in an oscillating probability distribution for the photocurrent, as is shown in B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13–16 (1986).

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]

[Crossref]

D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J., 1951).

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939).

When we take real and imaginary parts ofHm(z)=2zHm-1(z)-2(m-1)Hm-2(z),we find the following recurrence relations for Re Hm(z) and Im Hm(z):Re Hm=2(zr Re Hm-1-zi Im Hm-1)-2(m-1)Re Hm-2,Im Hm=2(zr Im Hm-1+zi Re Hm-1)-2(m-1)Im Hm-2.Since H0(z) = 1 and H1(z) = 2z the initial conditions for the iteration areRe H0=1,Im H0=0,Re H1=2zr,Im H1=2zi.

For an instructive geometrical discussion of this broadening see G. Leuchs, in Frontiers of Nonequilibrium Statistical Physics, G. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 329–360.

[Crossref]

W. Schleich and J. A. Wheeler, “Interference in phase space in diatomic molecules,” presented at the John Klauder Seminar, AT&T Bell Laboratories, Murray Hill, New Jersey, June 12, 1987.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

In general, the minor and major axes of the error ellipse make an angle θ relative to the coordinate axes of x and p. However, a rotation1 of the coordinate axes suffices to reduce the angle θ to the zero value presupposed in this paper. This rotation induces a rescaling of Re α and Im α.

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(a) Relation between the squeeze parameter, ^{−1/4} exp(−^{2}/2), ^{−1/4} exp(−^{2}/2).

Probability, _{m}_{m}^{2}^{m}^{−}^{α}^{2} associated with a coherent state (field oscillator subjected to a pure displacement; _{av} = ^{2}). Curves that are farther forward display the newly predicted oscillations in the probability distribution of excitation. As the squeezing becomes extreme (_{m}

Overlap in phase space as a determinant of the photon-count probability of a squeezed state. A single mode of the electromagnetic field in a number state is equivalent to a harmonic oscillator. In suitably normalized coordinate and momentum variables, _{m}_{m}_{+1} do not overlap at all, overlap tangentially, or overlap in two diamond-shaped areas, depending on whether the excitation ^{2}. As a result, the photon-count probability is small (for ^{2}) or large (for ^{2}). In the case of the two symmetrically located areas of overlap, however, the field oscillator moves to the right in one and to the left in the other. The total probability amplitude
_{m}_{m}^{2}. No simpler illustration presents itself for interference in phase space.

The probability _{m}^{2} = 6 of the displacement parameter and the same squeezing, that is,

The probability _{m}_{m}

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$$\psi (x)={\pi}^{-1/4}\hspace{0.17em}\text{exp}(-{x}^{2}/2),$$

$${\psi}_{\text{sq}}(x)={\pi}^{-1/4}{s}^{1/4}\hspace{0.17em}\text{exp}[-s{(x-{2}^{1/2}\alpha )}^{2}/2].$$

$$\widehat{E}(t,\mathbf{r})=\chi (\mathbf{r})\hspace{0.17em}(\widehat{a}{e}^{-i\omega t}+{\widehat{a}}^{\u2020}{e}^{i\omega t}).$$

$$\widehat{b}\mid {\psi}_{\beta}\u3009=\beta \mid {\psi}_{\beta}\u3009,$$

$$\widehat{b}=\rho \widehat{a}+\nu {\widehat{a}}^{\u2020},$$

$$\mid \rho {\mid}^{2}-\mid \nu {\mid}^{2}=1.$$

$$\widehat{x}\equiv {2}^{-1/2}(\widehat{a}+{\widehat{a}}^{\u2020})$$

$$\widehat{p}\equiv \frac{1}{i}{2}^{-1/2}(\widehat{a}-{\widehat{a}}^{\u2020}).$$

$$\widehat{E}(t,\mathbf{r})=\sqrt{2}\chi (\mathbf{r})\hspace{0.17em}[\widehat{x}\hspace{0.17em}\text{cos}(\omega t)+\widehat{p}\hspace{0.17em}\text{sin}(\omega t)].$$

$${2}^{-1/2}[(\rho +\nu )\widehat{x}+i(\rho -\nu )\widehat{p}]\hspace{0.17em}{\psi}_{\beta}(x)=\beta {\psi}_{\beta}(x).$$

$${\psi}_{\beta}(x)={\mathcal{N}}_{\beta}\hspace{0.17em}\text{exp}\left(\frac{\sqrt{2}\beta}{\rho -\nu}x-\frac{1}{2}\frac{\rho +\nu}{\rho -\nu}{x}^{2}\right)$$

$$\frac{\rho +\nu}{\rho -\nu}\equiv s\ge 0.$$

$$\frac{\rho}{\nu}=\frac{s+1}{s-1}\equiv c,$$

$${\psi}_{\text{sq}}(x)={\mathcal{N}}_{\text{sq}}\hspace{0.17em}\text{exp}\left(\frac{\sqrt{2}\beta}{\nu (c-1)}x-\frac{1}{2}\frac{c+1}{c-1}{x}^{2}\right).$$

$$\beta =\rho \alpha +\nu {\alpha}^{*}.$$

$$\beta =\nu \hspace{0.17em}(c\alpha +{\alpha}^{*}).$$

$${\psi}_{\text{sq}}(x)={\mathcal{N}}_{\text{sq}}\hspace{0.17em}\text{exp}\left(-\frac{1}{2}\frac{c+1}{c-1}{x}^{2}+\sqrt{2}\frac{c\alpha +{\alpha}^{*}}{c-1}x\right),$$

$${\mathcal{N}}_{\text{sq}}={\pi}^{-1/4}{\left(\frac{c+1}{c-1}\right)}^{1/4}\hspace{0.17em}\text{exp}\left[-\frac{c+1}{c-1}\hspace{0.17em}{(\text{Re}\hspace{0.17em}\alpha )}^{2}\right]$$

$$\mid {\psi}_{\text{sq}}(x){\mid}^{2}={\pi}^{-1/2}{\left(\frac{c+1}{c-1}\right)}^{1/2}\hspace{0.17em}\text{exp}\left[-\frac{c+1}{c-1}\hspace{0.17em}{(x-\sqrt{2}\hspace{0.17em}\text{Re}\hspace{0.17em}\alpha )}^{2}\right]$$

$$\mid {\varphi}_{\text{sq}}(p){\mid}^{2}={\pi}^{-1/2}{\left(\frac{c-1}{c+1}\right)}^{1/2}\hspace{0.17em}\text{exp}\left[-\frac{c-1}{c+1}{(p-\sqrt{2}\hspace{0.17em}\text{Im}\hspace{0.17em}\alpha )}^{2}\right].$$

$$\mathrm{\Delta}x={\left(\frac{c-1}{c+1}\right)}^{1/2},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\Delta}p={\left(\frac{c+1}{c-1}\right)}^{1/2}.$$

$${2}^{-1/2}[(c+1)\widehat{x}+i(c-1)\widehat{p}]\hspace{0.17em}{\psi}_{\text{sq}}(x)=(c\alpha +{\alpha}^{*})\hspace{0.17em}{\psi}_{\text{sq}}(x).$$

$${P}_{\text{sq}}(x,p)={\pi}^{-1}\hspace{0.17em}\text{exp}[-s{(x-\sqrt{2}\hspace{0.17em}\text{Re}\hspace{0.17em}\alpha )}^{2}-\frac{1}{s}{(p-\sqrt{2}\hspace{0.17em}\text{Im}\hspace{0.17em}\alpha )}^{2}].$$

$${W}_{m}={\left|{\int}_{-\infty}^{\infty}\text{d}x{u}_{m}(x){\psi}_{\text{sq}}(x)\right|}^{2}.$$

$${u}_{m}(x)={\pi}^{-1/4}{({2}^{m}m!)}^{-1/2}{H}_{m}(x)\hspace{0.17em}\text{exp}[-(\xbd){x}^{2}]$$

$${W}_{m}={\left(\frac{{c}^{2}-1}{{c}^{2}}\right)}^{1/2}\frac{\mid c{\mid}^{-m}}{{2}^{m}m!}\{{[\text{Re}\hspace{0.17em}{H}_{m}(z)]}^{2}+{[\text{Im}\hspace{0.17em}{H}_{m}(z)]}^{2}\}\text{exp}\left\{-[1+\frac{1}{c}\text{cos}(2\phi )]\mid \alpha {\mid}^{2}\right\},$$

$$z(\mid \alpha \mid ,\phi ,c)\equiv {z}_{r}+i{z}_{i}=\mid \alpha \mid \frac{c{e}^{i\phi}+{e}^{-i\phi}}{{(2c)}^{1/2}}.$$

$${W}_{m}={\left(\frac{{c}^{2}-1}{{c}^{2}}\right)}^{1/2}\frac{{c}^{-m}}{{2}^{m}m!}{{H}_{m}}^{2}\hspace{0.17em}\left(\frac{c+1}{\sqrt{2c}}\alpha \right)\text{exp}\left(-\frac{c+1}{c}{\alpha}^{2}\right),$$

$$\widehat{a}{\psi}_{\text{coh}}=\alpha {\psi}_{\text{coh}},$$

$${W}_{m}=\frac{{({\alpha}^{2})}^{m}}{m!}{e}^{-{\alpha}^{2}},$$

$$z\left(\mid \alpha \mid ,\phi -\frac{\pi}{2},-c\right)=z(\mid \alpha \mid ,\phi ,+c)$$

$$\left(-\frac{1}{c}\right)\text{cos}\left[2\left(\phi -\frac{\pi}{2}\right)\right]=\frac{1}{c}\text{cos}(2\phi ).$$

$${W}_{m}={(4\pi )}^{1/2}{\left(\frac{{c}^{2}-1}{c}\right)}^{1/2}\hspace{0.17em}\text{exp}\left[-\left(m+\frac{1}{2}\right)\text{ln}\hspace{0.17em}c\right]\times \hspace{0.17em}{\left[\frac{{t}_{m}}{\frac{{(c+1)}^{2}}{2c}{\alpha}^{2}-2\left(m+\frac{1}{2}\right)}\right]}^{1/2}\times \hspace{0.17em}{[\text{Ai}({t}_{m})]}^{2}\hspace{0.17em}\text{exp}\left(\frac{{c}^{2}-1}{2c}{\alpha}^{2}\right),$$

$${t}_{m}=-{\left[\frac{3}{2}{\int}_{\frac{c+1}{\sqrt{2c}}\alpha}^{{\xi}_{m}}\text{d}x{p}_{m}(x)\right]}^{2/3}$$

$${t}_{m}={\left[\frac{3}{2}{\int}_{{\xi}_{m}}^{\frac{c+1}{\sqrt{2c}}\alpha}\text{d}x{\overline{p}}_{m}(x)\right]}^{2/3}.$$

$${p}_{m}(x)=\sqrt{2(m+\xbd)-{x}^{2}}$$

$${\overline{p}}_{m}(x)=\sqrt{{x}^{2}-2(m+\xbd)}.$$

$${W}_{m}={(4\pi \u220a)}^{1/2}\hspace{0.17em}\text{exp}[\u220a({\alpha}^{2}-m-\xbd)]\times {\left[\frac{{t}_{m}}{{\alpha}^{2}-(m+\xbd)}\right]}^{1/2}{[\text{Ai}({t}_{m})]}^{2},$$

$${t}_{m}=-{\left[\frac{3}{2}{\int}_{\sqrt{2}\alpha}^{{\xi}_{m}}\text{d}x\hspace{0.17em}{\text{p}}_{m}(x)\right]}^{2/3};$$

$${t}_{m}={\left[\frac{3}{2}{\int}_{{\xi}_{m}}^{\sqrt{2}\alpha}\text{d}x{\overline{p}}_{m}(x)\right]}^{2/3}.$$

$$\text{Ai}({t}_{m})\cong \frac{1}{2}{\pi}^{-1/2}{{t}_{m}}^{-1/4}\hspace{0.17em}\text{exp}\left(-\frac{2}{3}{{t}_{m}}^{3/2}\right),$$

$${W}_{m}={\left(\frac{\u220a}{4\pi}\right)}^{1/2}\frac{\text{exp}[\u220a({\alpha}^{2}-m-\xbd]}{{({\alpha}^{2}-m-\xbd)}^{1/2}}\text{exp}(-{\overline{\varphi}}_{m}).$$

$${\overline{\varphi}}_{m}=2{\int}_{\sqrt{2}\alpha}^{{\xi}_{m}}\text{d}x{\overline{p}}_{m}(x)=2\left\{\alpha {[{\alpha}^{2}-(m+\xbd)]}^{1/2}-(m+\xbd)\text{ln}\left({\frac{\alpha +[{\alpha}^{2}-m-\xbd]}{{(m+\xbd)}^{1/2}}}^{1/2}\right)\right\}.$$

$$\text{Ai}(-\mid {t}_{m}\mid )={\pi}^{-1/2}\mid {t}_{m}{\mid}^{-1/4}\hspace{0.17em}\text{sin}\left(\frac{2}{3}\mid {t}_{m}{\mid}^{3/2}+\frac{\pi}{4}\right),$$

$${W}_{m}=4{\mathcal{A}}_{m}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}{\varphi}_{m}.$$

$${\mathcal{A}}_{m}={\left(\frac{\u220a}{4\pi}\right)}^{1/2}\frac{\text{exp}\hspace{0.17em}[\u220a({\alpha}^{2}-m-\xbd)]}{{(m+\xbd-{\alpha}^{2})}^{\xbd}},$$

$${\varphi}_{m}={\int}_{\sqrt{2}\alpha}^{{\xi}_{m}}\text{d}x{p}_{m}(x)-\frac{\pi}{4}=(m+\xbd)\text{arctan}\frac{\sqrt{m+\xbd-{\alpha}^{2}}}{\alpha}-\alpha \sqrt{m+\xbd-{\alpha}^{2}}-\frac{\pi}{4}.$$

$${\varphi}_{m}\cong \frac{2}{3}\frac{{(m+\xbd-{\alpha}^{2})}^{3/2}}{\alpha}-\frac{\pi}{4},$$

$$m+\xbd={\alpha}^{2}+{\left[\frac{3}{2}\left({\varphi}_{m}+\frac{\pi}{4}\right)\alpha \right]}^{2/3}.$$

$$\u220a\lesssim {\left(\frac{8}{9\pi}\right)}^{2/3}{\alpha}^{-2/3}$$

$${H}_{m}(y)={2}^{m/2}\hspace{0.17em}{e}^{{y}^{2}/2}U[-(m+\xbd);{2}^{1/2}y].$$

$$U[-(m+\xbd);{2}^{1/2}y]={2}^{(m+\xbd)/2}\mathrm{\Gamma}\left(\frac{m+\xbd}{2}\right)\hspace{0.17em}{(m+\xbd)}^{1/4}\times {\left[\frac{{t}_{m}}{{y}^{2}-2(m+\xbd)}\right]}^{1/4}\hspace{0.17em}Ai({t}_{m}),$$

$${t}_{m}=-{\left[\frac{3}{2}{\int}_{y}^{{\xi}_{m}}\text{d}x{p}_{m}(x)\right]}^{2/3}$$

$${t}_{m}={\left[\frac{3}{2}{\int}_{{\xi}_{m}}^{y}\text{d}x{\overline{p}}_{m}(x)\right]}^{2/3},$$

$${p}_{m}(x)=\sqrt{2(m+\xbd)-{x}^{2}}$$

$${\overline{p}}_{m}(x)=\sqrt{{x}^{2}-2(m+\xbd)}.$$

$${H}_{m}(y)={2}^{1/4}{e}^{{y}^{2}/2}{(m+\xbd)}^{1/4}{2}^{m}\mathrm{\Gamma}\left(\frac{m+\xbd}{2}\right)\times {\left[\frac{{t}_{m}}{{y}^{2}-2(m+\xbd)}\right]}^{1/4}\text{Ai}({t}_{m}).$$

$$m!\cong {(2\pi )}^{1/2}\frac{{(m+\xbd)}^{m+\xbd}}{{e}^{m+\xbd}},$$

$$\begin{array}{c}\frac{{2}^{m}}{m!}{\left[\mathrm{\Gamma}\left(\frac{m+\xbd}{2}\right)\right]}^{2}\cong {\left(\frac{2\pi}{m+\xbd}\right)}^{1/2}\frac{{e}^{1/2}}{{\left(1+\frac{1}{2m}\right)}^{m}}\\ \cong {\left(\frac{2\pi}{m+\xbd}\right)}^{1/2},\end{array}$$

$${2}^{m/2}{(m+\xbd)}^{1/4}\mathrm{\Gamma}\left(\frac{m+\xbd}{2}\right)\cong {(2\pi )}^{1/4}{(m!)}^{1/2}.$$

$${H}_{m}(y)={(4\pi )}^{1/4}{({2}^{m}m!)}^{1/2}\times {\left[\frac{{t}_{m}}{{y}^{2}-2(m+\xbd)}\right]}^{1/4}Ai({t}_{m}){e}^{{y}^{2}/2},$$

$${H}_{m}(z)=2z{H}_{m-1}(z)-2(m-1){H}_{m-2}(z),$$

$$\begin{array}{l}\text{Re}\hspace{0.17em}{H}_{m}=2({z}_{r}\hspace{0.17em}\text{Re}\hspace{0.17em}{H}_{m-1}-{z}_{i}\hspace{0.17em}\text{Im}\hspace{0.17em}{H}_{m-1})-2(m-1)\text{Re}\hspace{0.17em}{H}_{m-2},\\ \text{Im}\hspace{0.17em}{H}_{m}=2({z}_{r}\hspace{0.17em}\text{Im}\hspace{0.17em}{H}_{m-1}+{z}_{i}\hspace{0.17em}\text{Re}\hspace{0.17em}{H}_{m-1})-2(m-1)\text{Im}\hspace{0.17em}{H}_{m-2}.\end{array}$$

$$\begin{array}{ll}\text{Re}\hspace{0.17em}{H}_{0}=1,\hfill & \text{Im}\hspace{0.17em}{H}_{0}=0,\hfill \\ \text{Re}\hspace{0.17em}{H}_{1}=2{z}_{r},\hfill & \text{Im}\hspace{0.17em}{H}_{1}=2{z}_{i}.\hfill \end{array}$$

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