Victor Kozlov's reply

doi:10.1364/OE.10.000157

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Victor Kozlov's reply

Victor Kozlov »View Author Affiliations

Optics Express, Vol. 10, Issue 3, pp. 157-158 (2002)
http://dx.doi.org/10.1364/OE.10.000157

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Abstract

We prove formal identity of the dispersed states and the two-photon coherent states. As the last fall in the class of squeezed states and are non-classical, so are the dispersed states. A state which exhibits squeezing and/or non-classicality should not be called coherent as suggested in the Comment.

Momentum and position operators evolve in a dispersive dielectric as

\hat{P} (z) = {\hat{P}}_{0} and \hat{X} (z) = {\hat{X}}_{0} + (\frac{k ″ z}{n_{0}}) {\hat{P}}_{0} .

(1)

For notations and definitions see Ref. [2

V.V. Kozlov, “Quantum optics with particles of light,” Opt. Express 8, 688–693 (2001).http://www.opticsexpress.org/oearchive/source/34146.htm [CrossRef] [PubMed]

]. As an observation, the same equations can be generated by the Heisenberg equation of motion (with differentiation with respect to z instead of t) with the “free-particle” Hamiltonian in the form of Ĥ_D = (k″/2ħn ₀)P̂² .

In order to build a link to the two-photon coherent states, see Ref. [3

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976). [CrossRef]

], we introduce a set of normalized non-Hermitian operators ŷ and ŷ^† as

\hat{y} \equiv \frac{1}{\sqrt{2}} (\frac{1}{t_{c}} \hat{X} + i \frac{t_{c}}{ħ} \hat{P}) and {\hat{y}}^{†} \equiv \frac{1}{\sqrt{2}} (\frac{1}{t_{c}} \hat{X} - i \frac{t_{c}}{ħ} \hat{P}),

(2)

and allow the constant t_c be arbitrary for the time being. Since [X̂, P̂] = iħ, one finds [ŷ,ŷ^†]= 1. By inverting definitions (2) we can write Ĥ_D in terms of ŷ and ŷ^†. So,

{\hat{H}}_{D} = \frac{1}{2} ħ Ω (\hat{y} {\hat{y}}^{†} + {\hat{y}}^{†} \hat{y} - \hat{y} \hat{y} - {\hat{y}}^{†} {\hat{y}}^{†}) with Ω \equiv \frac{k ″}{2 ħ t_{c}^{2} n_{0}} .

(3)

The last two terms are responsible for squeezing. Evolution equations for ŷ and ŷ^† emerge when eqns. (1) are substituted in definitions (2) or they can be deduced from the Heisenberg equation of motion. Both methods yield

\hat{y} (z) = μ \hat{y} (0) + ν {\hat{y}}^{†} (0) and {\hat{y}}^{†} (z) = ν^{*} \hat{y} (0) + μ^{*} {\hat{y}}^{†} (0),

(4)

with μ ≡ 1 - iΩz and ν ≡ iΩz. Since |μ|² - |ν|² =1 implies [ŷ(z),ŷ^†(z)] = 1, the transformation (4) is unitary. We derive from the Hamiltonian eqn. (3) a unitary disperse operator

{\hat{S}}_{D} \equiv \exp [\frac{i}{2} Ω z (\hat{y} {\hat{y}}^{†} + {\hat{y}}^{†} \hat{y} - \hat{y} \hat{y} - {\hat{y}}^{†} {\hat{y}}^{†})],

(5)

and define the dispersed state by letting Ŝ_D act on the coherent state |ϕ〉:

∣ Ω, ϕ 〉 \equiv {\hat{S}}_{D} ∣ ϕ 〉 .

(6)

We can also define operators Ŷ ≡ Ŝ_Dŷ

{\hat{S}}_{D}^{†}

and Ŷ^† ≡ Ŝ_Dŷ^†

{\hat{S}}_{D}^{†}

It can be seen by inspection that (i) the pair (ŷ, ŷ^†) is formally equivalent to the pair of photon creation and annihilation operators, (â, â^†); (ii) the “disperse” Hamiltonian (3) — to the squeeze Hamiltonian; (iii) the disperse operator (5) — to the squeeze operator; (iv) the dispersed state (6) — to the two-photon coherent state; (v) the pair (Ŷ, Ŷ^†) — to the pair of pseudo-annihilation and creation operators in the formalism of the two-photon coherent states, (Â, Â^†) (notations following Ref. [4

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

]); etc., by induction. We now repeat the proof of squeezing given in Ref. [2

V.V. Kozlov, “Quantum optics with particles of light,” Opt. Express 8, 688–693 (2001).http://www.opticsexpress.org/oearchive/source/34146.htm [CrossRef] [PubMed]

]. We introduce quadrature amplitude

\hat{M} \equiv {\hat{y}}^{†} e^{i ψ} + \hat{y} e^{- i ψ}

(7)

together with its conjugate partner M̂_⊥ = i(ŷ^† e ^iψ-ŷ e ^-iψ) such that the two obey commutator [M̂, M̂_⊥]= 2i. For a pulse of Gaussian shape ϕ(0, τ)= (n ₀/π ^1/2 τ_p )^1/2 exp(-τ ²/2

τ_{p}^{2}

) from Ref. [2

V.V. Kozlov, “Quantum optics with particles of light,” Opt. Express 8, 688–693 (2001).http://www.opticsexpress.org/oearchive/source/34146.htm [CrossRef] [PubMed]

] and with t_c = τ_p / √n ₀ we get 〈ϕ|(∆M̂)²|ϕ〉 = 1 = 〈ϕ|(∆M̂_⊥)²|ϕ〉 for the two variances in the coherent state. Product of the variances minimizes the Heisenberg uncertainty relation. The state is called squeezed when the quantum noise in one quadrature is reduced below the minimum uncertainty level. Using definitions of Ŷ and Ŷ^† and relations (5) and (6) we obtain for the variance of M̂ in the dispersed state:

〈 Ω, ϕ ∣ {(Δ \hat{M})}^{2} ∣ Ω, ϕ 〉 = \cos^{2} δ + {[(\frac{z}{Z_{D}}) \cos δ + \sin δ]}^{2} .

(8)

where Ω is redefined in terms of dispersion length Z _D via relation 2ħΩ =

Z_{D}^{- 1}

. With

ψ = ψ_{min} = \frac{1}{2} arcctg (\frac{- z}{2 Z_{D}})

the variance (8) is minimal and copies the main result eqn. (18) in the paper Ref. [2

V.V. Kozlov, “Quantum optics with particles of light,” Opt. Express 8, 688–693 (2001).http://www.opticsexpress.org/oearchive/source/34146.htm [CrossRef] [PubMed]

]. Observation of 〈Ω,ϕ|(∆M̂)²|Ω,ϕ〉=exp(-2R) < 1 for R>0 completes the proof of squeezing. Here, sinh(R) ≡ z/2Z _D.

Now, non-classicality. We use the optical equivalence theorem to write for any state |〉 in the Glauber-Sudarshan P-representation:

〈 : {(Δ \hat{M})}^{2} : 〉 = \int {(Δ M)}^{2} P (ϕ) d (Re ϕ) d (Im ϕ),

(9)

where the colon stands for the normally ordered operator product. Here (∆M)² is the c-number obtained from :M̂² : by replacing ϕ̂ by ϕ + υ and correspondingly, ϕ̂^† by ϕ ^*+υ ^*. P(ϕ) is a phase-space density and d(Re ϕ)d(lm ϕ) is a surface element of the phase space. With the linearized definitions of P̂ and X̂ in eqns. (2) and then via eqns. (2) in eqn. (7) we get 〈:(∆M̂)²:〉 = 〈(∆M̂)²〉-1. We can now write (9) as

〈 {(Δ \hat{M})}^{2} 〉 = 1 + \int {(Δ M)}^{2} P (ϕ) d (Re ϕ) d (Im ϕ) .

(10)

If the state |〉 is squeezed, as the dispersed state from eqn. (6), the integral in the right side of eqn. (10) is negative. Since (∆M̂)² is real and non-negative, it is the quasi-probability density P(ϕ) which must take on negative values. A classical state of light is one in which P(ϕ) is a probability density. Since P(ϕ) takes negative values for the dispersed states, the states are by definition (see Ref. [4

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

]) non-classical.

In conclusion, the above derivations show that the dispersed states are squeezed and non-classical. It is in contrast to Ref. [1

J. M. Fini, “Comment on: Quantum optics with particles of light,” Opt. Express 10, 155 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-3-688 [CrossRef] [PubMed]

] where the states were found to be coherent. We attribute the difference in the two conclusions to the fact that Ref. [1

J. M. Fini, “Comment on: Quantum optics with particles of light,” Opt. Express 10, 155 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-3-688 [CrossRef] [PubMed]

] deals with the photon creation and annihilation operators while the effect of squeezing shows up in the momentum-position observables Ref. [2

V.V. Kozlov, “Quantum optics with particles of light,” Opt. Express 8, 688–693 (2001).http://www.opticsexpress.org/oearchive/source/34146.htm [CrossRef] [PubMed]

]. We recommend to an interested reader Ref. [5

V. V. Kozlov, “Squeezing light in a linear dispersive dielectric,” in Conference Proceedings, ICSSUR 2001, Boston, Massachusetts U.S. http://www.bu.edu/qil/icssur2001.html

] where the derivations presented above will be given in great detail. Further developments will be reported in Ref. [6

V. V. Kozlov, “A quantum-mechanical calculation of frequency and timing jitters for optical pulses in dispersive fibers with losses,” Opt. Lett., accepted (2002). [CrossRef]

References and links

1.	J. M. Fini, “Comment on: Quantum optics with particles of light,” Opt. Express 10, 155 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-3-688 [CrossRef] [PubMed]
2.	V.V. Kozlov, “Quantum optics with particles of light,” Opt. Express 8, 688–693 (2001).http://www.opticsexpress.org/oearchive/source/34146.htm [CrossRef] [PubMed]
3.	H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976). [CrossRef]
4.	L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).
5.	V. V. Kozlov, “Squeezing light in a linear dispersive dielectric,” in Conference Proceedings, ICSSUR 2001, Boston, Massachusetts U.S. http://www.bu.edu/qil/icssur2001.html
6.	V. V. Kozlov, “A quantum-mechanical calculation of frequency and timing jitters for optical pulses in dispersive fibers with losses,” Opt. Lett., accepted (2002). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(270.6570) Quantum optics : Squeezed states

ToC Category:
Research Papers

History
Original Manuscript: January 18, 2002
Revised Manuscript: January 31, 2002
Published: February 11, 2002

Citation
Victor Kozlov, "Victor Kozlov's reply," Opt. Express 10, 157-158 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-3-157

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References

J. M. Fini, ?Comment on: Quantum optics with particles of light,? Opt. Express 10, 155 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-3-155"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-3-155</a> [CrossRef] [PubMed]
V. V. Kozlov, ?Quantum optics with particles of light,? Opt. Express 8, 688-693 (2001). <a href="http://www.opticsexpress.org/oearchive/source/34146.htm">http://www.opticsexpress.org/oearchive/source/34146.htm</a> [CrossRef] [PubMed]
H. P. Yuen, ?Two-photon coherent states of the radiation field,? Phys. Rev. A 13, 2226-2243 (1976). [CrossRef]
L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).
V. V. Kozlov, ?Squeezing light in a linear dispersive dielectric,? in Conference Proceedings, ICSSUR 2001, Boston, Massachusetts U.S. <a href="http://www.bu.edu/qil/icssur2001.html">http://www.bu.edu/qil/icssur2001.html</a>
V. V. Kozlov, ?A quantum-mechanical calculation of frequency and timing jitters for optical pulses in dispersive fibers with losses,? Opt. Lett. accepted (2002). [CrossRef]

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