## 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.

© 2002 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 *X̂* and
*P̂* 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 *q̂* =
*t*_{c}*P̂* cos(*ψ*) +
${t}_{c}^{-1}$
*X̂* 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]

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

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,

Instead of “〈*q̂*^{2}${\u3009}_{\text{coh}}^{\text{out}}$ ≈ 1,” we find that for *z* ≫
*Z*_{D} and $\psi =\frac{1}{2}$arccot(-*z*/2*Z*_{d} ) specified by
Kozlov, 〈*q̂*^{2}${\u3009}_{\text{coh}}^{\text{out}}$ equals the right hand side of [1, Eq.
18]. But then,

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 ∆*q̂* 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, *r̂* = ∫
*dτf*
^{*} (*τ*)*ϕ̂*(*τ*) + h.a., the coherent-state fluctuation
is

The uncertainty is simply equal to the “energy” of the projection function *f*. Kozlov’s
projection functon *f*_{q} 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 *X̂* and *P̂* 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.