Abstract

Errors in the recent article, “Quantum optics with particles of light,” are discussed. “Dispersed states” resulting from linear optics are simply coherent states, and have no interesting quantum statistics.

© Optical Society of America

A recent article [1] introduced the notion of dispersed states, which arise from a dispersive linear propagation with coherent-state inputs. Kozlov’s claim was that these states had interesting quantum statistics, including an observable reduction of quantum fluctuations. That article [1] contains errors, providing an opportunity to clarify some basic quantum properties.

The model describes linear propagation with lowest-order dispersion. The and operators evolve as free-particle coordinates, as discussed extensively for the soliton case [2, 3]. The “non-conventional perspective” discussed by Kozlov deals with the collective coordinates of a group of photons. This perspective brings about interesting insight, and has been discussed in several papers by Hagelstein [3, 4].

A major theme in quantum optics is the generation of non-classical field statistics by nonlinear systems. The association between nonlinearity and strange statistics is so strong, that a claim of squeezing in a linear system seems counter-intuitive. This is for good reason. We have shown that the fluctuations of the relevant operator = tc cos(ψ) + tc1 sin(ψ) are equal to (not below) the standard quantum limit. Further, the dispersed state is simply a coherent state.

It is easy to show mathematically that a linear system evolves coherent states into coherent states. For the generalized Hamiltonian [which includes Kozlov’s system as a special case]

Ĥ=dτdτE(τ,τ)ϕ̂(τ)ϕ̂(τ)

an initial coherent state will remain a coherent state at every position z. That is, if

ϕ̂(τ)ψ(z)=ϕ0(τ,z)ψ(z),
iħddzϕ0(τ,z)=dτE(τ,τ)ϕ0(τ,z),

then |ψ(z)〉 is an exact coherent-state solution of the Schrödinger equation, as confirmed with a few steps of algebra. This is unlike the evolution of squeezing systems, whose states cannot satisfy Eq. (2) for all z. Since it is a coherent state, the dispersed state has no special quantum properties—for important physical reasons (see [5, pp. 192,217] and [6, Sec. 9.2]) the coherent-state is considered the most “ordinary” field state.

In the discussion surrounding [1, Eq. 19], Kozlov seems to argue that a dispersed state has smaller fluctuations than a coherent state (2) with the same mean field (and thus has fluctuations below the standard quantum limit defined by the coherent state). Since the dispersed state is a coherent state this is clearly not so.

To clarify this, we have calculated the coherent-state expectations by the usual method: by rearranging the operators into normally-ordered expressions, and then replacing the field operator with the classical field in the expectation, ϕ̂(τ)ψ(z)〉 = ϕ 0(τ,z)ψ(z)〉. Naturally, we must include the correlations (perhaps neglected by Kozlov), which are essential to the reduction of fluctuations. For example,

X̂P̂=:X̂P̂:indτϕ̂(τ)τϕ̂τ.
X̂P̂=X̂P̂indτϕ0*(τ,z)ττϕ0(τ,z).

Instead of “〈2cohout ≈ 1,” we find that for zZD and ψ=12arccot(-z/2Zd ) specified by Kozlov, 〈2cohout equals the right hand side of [1, Eq. 18]. But then,

Δq̂2cohout=Δq̂2mindispersed1

Thus the fluctuations are reduced as z increases, but the standard quantum limit also reduces, and so the fluctuations are always at, not below, this limit. Since the dispersed state is a coherent state, the equality (6) of their fluctuations is obvious. We have calculated it explicitly to show the consistency of our interpretation.

Kozlov proposes the linearized operator ∆ as a variable with observable quantum field fluctuations. As he points out, the uncertainty of this projection decreases with z, but this is a property of the classical projection function. For any projection of the field, = ∫ dτf * (τ)ϕ̂(τ) + h.a., the coherent-state fluctuation is

α(τ,z)r̂2α(τ,z)=α(τ,z)r̂α(τ,z)2+dτf(τ)2.

The uncertainty is simply equal to the “energy” of the projection function f. Kozlov’s projection functon fq has decreasing energy with z due to the cancellations between correlated x and p components. In terms of a physical heterodyne measurement, this corresponds to using a (non-sinusoidal) local oscillator with diminishing energy. Naturally, the signal is reduced, but not because of any quantum properties.

One can define a field with operators related to and that do show squeezing. This was known even before the cited quantum optics discussions. However, it is clear that these operators are not photon operators of the system, and so if there is squeezing, it is not optical squeezing. This is in contrast to the soliton case. Nonlinearity of the fiber in soliton propagation allows incommensurate evolution of classical and quantum properties of the field [2] discussed at length in [7].

References and links

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

2. H. A. Haus and Y. Lai. Quantum theory of soliton squeezing: a linearized approach.J. Opt. Soc. Am. B , 7,386 (1990). [CrossRef]  

3. P. L. Hagelstein. Application of a photon configuration-space model to soliton propagation in a fiber Phys.Rev. A. , 54, 2426 (1996). [CrossRef]  

4. J. M. Fini, P. L. Hagelstein, and H. A. Haus. Configuration-space quantum-soliton model including loss and gain.Phys. Rev. A , 57,4842 (1998). [CrossRef]  

5. Claude Cohen-Tannoudji. Atom-Photon Interactions. (New York, Wiley, 1992).

6. H. A. Haus. Electromagnetic noise and quantum optical measurements(New York, Springer, 2000).

7. J. M. Fini and P. L. Hagelstein. Momentum squeezing of quantum optical pulses. Submitted to Phys.Rev. A.

8. V. V. Kozlov. private communication, dated September 17, 2001.

References

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  1. V. V. Kozlov. "Quantum optics with particles of light," Opt. Express 8, 688 (2001), <a href="http://www.opticsexpress.org/oearchive/source/34146.htm">http://www.opticsexpress.org/oearchive/source/34146.htm</a>
    [CrossRef] [PubMed]
  2. H. A. Haus and Y. Lai, ?Quantum theory of soliton squeezing: a linearized approach,? J. Opt. Soc. Am. B 7, 386 (1990).
    [CrossRef]
  3. P. L. Hagelstein. ?Application of a photon configuration-space model to soliton propagation in a fiber,? Phys. Rev. A 54, 2426 (1996).
    [CrossRef]
  4. J. M. Fini, P. L. Hagelstein, and H. A. Haus, ?Configuration-space quantum-soliton model including loss and gain,? Phys. Rev. A 57, 4842 (1998).
    [CrossRef]
  5. Claude Cohen-Tannoudji. Atom-Photon Interactions, (New York, Wiley, 1992).
  6. H. A. Haus, Electromagnetic noise and quantum optical measurements, (New York, Springer, 2000).
  7. J. M. Fini and P. L. Hagelstein. ?Momentum squeezing of quantum optical pulses,? Submitted to Phys. Rev. A.
  8. V. V. Kozlov, Private communication, dated September 17, 2001.

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Phys. Rev. A (2)

P. L. Hagelstein. ?Application of a photon configuration-space model to soliton propagation in a fiber,? Phys. Rev. A 54, 2426 (1996).
[CrossRef]

J. M. Fini, P. L. Hagelstein, and H. A. Haus, ?Configuration-space quantum-soliton model including loss and gain,? Phys. Rev. A 57, 4842 (1998).
[CrossRef]

Phys. Rev. A. (1)

J. M. Fini and P. L. Hagelstein. ?Momentum squeezing of quantum optical pulses,? Submitted to Phys. Rev. A.

Other (3)

V. V. Kozlov, Private communication, dated September 17, 2001.

Claude Cohen-Tannoudji. Atom-Photon Interactions, (New York, Wiley, 1992).

H. A. Haus, Electromagnetic noise and quantum optical measurements, (New York, Springer, 2000).

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Equations (7)

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H ̂ = d τ d τ E ( τ , τ ) ϕ ̂ ( τ ) ϕ ̂ ( τ )
ϕ ̂ ( τ ) ψ ( z ) = ϕ 0 ( τ , z ) ψ ( z ) ,
i ħ d d z ϕ 0 ( τ , z ) = d τ E ( τ , τ ) ϕ 0 ( τ , z ) ,
X ̂ P ̂ = : X ̂ P ̂ : i n d τ ϕ ̂ ( τ ) τ ϕ ̂ τ .
X ̂ P ̂ = X ̂ P ̂ i n d τ ϕ 0 * ( τ , z ) τ τ ϕ 0 ( τ , z ) .
Δ q ̂ 2 coh out = Δ q ̂ 2 min dispersed 1
α ( τ , z ) r ̂ 2 α ( τ , z ) = α ( τ , z ) r ̂ α ( τ , z ) 2 + d τ f ( τ ) 2 .

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