Quantum electromagnetic X-waves
Alessandro Ciattoni1, 2, ∗ and Claudio Conti3, †
Consiglio Nazionale delle Ricerche, CASTI Regional Lab 67010 L’Aquila, Italy
Dipartimento di Fisica, Universitá dell’Aquila, 67010 L’Aquila, Italy
Research Center ”Enrico Fermi” and SOFT INFM-CNR,
Universitá La Sapienza, Piazzale Aldo Moro 2, 00185 Rome, Italy
(Dated: November 1, 2018)
We show that two distinct quantum states of the electromagnetic field can be associated to a
classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference
between the two states is of pure quantum mechanical origin since they are internally entangled and
disentangled, respectively and can be generated by different linear or nonlinear processes. Detec-
tion and generation of Schrödinger-cat states comprising two entangled X-waves and their possible
applications are discussed.
PACS numbers: 03.75.Lm, 05.30.Jp, 42.65.Jx
INTRODUCTION
Electromagnetic X waves and, more generally, “localized waves” [1] are propagation-invariant solutions of classical
Maxwell equations in vacuum [2, 3, 4], dielectric media [5], plasmas [6], optically nonlinear materials [7, 8, 9, 10] and
periodic structures [11, 12, 13, 14]. In recent years these waves have attracted a considerable interest since, despite at
a first glance they are non-physical (for example some of them apparently travel superluminally and carry an infinite
amount of energy), propagation-invariant fields support several applications like image transmission, reconstruction
and telecommunications (for a recent review see Ref.[1]).
Since the first observation of localized optical X waves [15], the fundamental implications of rigidly travelling spatio-
temporal correlations have been recognized. Classically, these waves can be generated by means of linear devices (like
axicons) [15] and, recently, their experimental feasibility has been also proved through nonlinear optical processes
[16, 17]. Within the quantum framework, analytical techniques borrowed from the subject of localized waves have
allowed some authors to consider highly localized states for the photon-wavefunction [18], whereas the role of quantum
X waves in nonlinear optical process (limited to the paraxial regime) were analyzed in [19].
In this paper we consider the issue of “constructing a quantum state of the electromagnetic field whose classical
counterpart is an X-wave satisfying the full set of Maxwell equations in vacuum”. More precisely, we investigate the
possibility of introducing various quantum states on which the mean value of the electric field operator coincides with
a prescribed classical vector X wave. We show that the solution of this problem is not-trivial by finding two different
quantum states characterized by the above property. Even if from a classical perspective they describe the same
entity, these two states essentially differ in the degree of entanglement of their internal modal structure so that both
a “disentangled” and an “entangled” quantum description can be provided of an arbitrary classical X wave. In this
respect one can investigate a method for discerning between the two possibilities and, correspondingly, for recognizing
which kind of quantum X wave is produced by a given mechanism (e.g. axicons or nonlinear processes). In addition,
we show that some X wave states can be considered as multi-mode, multi-dimensional Schrödinger cats, exhibiting
the relevant feature of being propagation invariant; the corresponding macroscopic quantum content is controlled by
the velocity of the X waves and can be maintained for long distances.
QUANTUM DESCRIPTION OF X WAVES
Within a classical formulation, the x-component of the electric field of a radially symmetric linearly polarized X
wave can be written as (extension to more general X waves solutions [1] is trivial)
Ex(r, t) = ℜ
dkf(k)J0(sin θ0kr⊥)e
ik cos θ0
cos θ0
where r⊥ =
x2 + y2 and ℜ stands for the real part, whereas the longitudinal component is simply expressed in
terms of the transversal components. (see e.g. [4]) This kind of classical waves propagates along the z−axis without
distortion and it is fully characterized by its spectrum f(k) and its velocity c/ cos θ0. Physically, such an X wave
http://arxiv.org/abs/0704.0442v1
arises as the superposition of various plane-waves at different frequencies whose wave-vectors are all inclined at the
same angle θ0 with the propagation direction z. In order to give a quantum description of X waves, we construct
a quantum state of the electromagnetic field through the requirement that the quantum mean value of the electric
field operator on this state yields a propagation invariant wave of the kind of Eq.(1) and we show that this can be
done in two different manners. In order to fix the notation, we start from the standard expression of the electric
field operator in Heisenberg representation, Ê(r, t) =
h̄c|k|
2ε0L3
iâk,s e
i(k·r−c|k|t)
ek,s + h.c., where L is the edge of
the quantization cubic box (see e.g. [20, 21]), k = (2π/L)(nxex + nyey + nzez) (where nx, ny and nz are integers),
ek,s are the pair (s = 1, 2) of polarization unit vectors associated to the mode k and âk,s are the standard photon
annihilation operators. Bearing in mind that the mean value of the electric field operator on a coherent state is a
plane wave, let us consider the quantum states
|X〉 =
D̂k,s[∆θ0(k)β(k, s)]|0〉 (2)
where D̂k,s(α) is the displacement operator which, acting on the vacuum state |0〉, produces the coherent state
|α〉k,s (associated to the mode k, s) according to the relation D̂k,s(α)|0〉 = |α〉k,s [20]. Here β(k, s) is a complex
weight function to be determined below and, in order to deal with the discreteness of the k vectors, we have defined
∆θ0(k) = 1 if k ∈ C(θ0, δθ) and ∆θ0(k) = 0 elsewhere, where C(θ0, δθ) is the portion of space comprised between the
two coaxial cones of aperture angles θ0 − δθ/2 and θ0 + δθ/2 and whose common axis coincides with the z−axis. The
presence of the function ∆θ0(k) assures that the state |X〉 of Eq.(2) is the tensor product of coherent states whose
modes have wave vectors globally lying, for δθ ≪ θ0, on a cone with aperture angle θ0 so that the relation kz = η|k⊥|
with η = 1/ tan θ0 holds for each wave vector. The mean value of the electric field operator on the state |X〉 is given
〈X |Ê|X〉 = ℜ
∆θ0(k)
2h̄c|k|
iβ(k, s) ei(k·r−c|k|t)ek,s
 (3)
from which we envisage that, for L → ∞ and δθ ≪ θ0, 〈X |Ê(r, t)|X〉 → E(x, y, z−V t), which is a genuine propagation-
invariant vector field travelling along the z−axis. In order to prove this assertion we note that the limit is different
from zero only if β is dependent on δθ and L and it is given by
β(k, s) =
(2π)3
2h̄c|k|L3
fs(k), (4)
where k⊥ = kxex + kyey, fs(k) are two arbitrary function of k and the various coefficients are chosen for later
convenience. Substituting Eq.(4) into Eq.(3) and performing the limits L → +∞ and δθ → 0 by means of the rule
(2π)3
d3k and the relation limδθ→0 ∆θ0(k)/δθ = δ(θ − θ0) we obtain
〈X |Ê|X〉 = ℜ
dφ [f1(k)ek1 + f2(k)ek2] e
i(k·r−ckt)
where, after defining k = k[sin θ0(cosφex + sinφey) + cos θ0ez], polar coordinates has been introduced for the k
integration. Note that, inside the integral of Eq.(5), the relation k·r−ckt = k[(x cosφ sin θ0+y sinφ sin θ0)+cos θ0(z−
ct/ cos θ0)] holds, so that the field in Eq.(5) describes a wave travelling undistorted along the z−axis with velocity
V = c/ cos θ0, as expected. Note that, strictly speaking, this rigorous nondiffracting behavior is a consequence of the
limit δθ → 0 which is an idealization of the realistic condition δθ ≪ θ0, δθ representing the experimental uncertainty
of the wave vectors smeared around the selected conical surface. Since δθ can be experimentally chosen much smaller
than θ0, the actual fields are nearly nondiffracting objects, their departure from ideal fields being experimentally
tunable.
A convenient choice for the modes polarization unit vectors is given by ek1 = cos θ0(cosφex + sinφey) −
sin θ0ez and ek2 = − sinφex + cosφey so that a linearly polarized wave along the x−direction is obtained
by letting f2 = −f1 tanφ cos θ0 with f1 = (cosφ/ cos θ0)[f(k)/k2] from which we get f1(k)ek1 + f2(k)ek2 =
f(k) (ex − cosφ tan θ0ez) /k2 where the factor k−2 has been added for later convenience. Inserting this expression
into Eq.(5), we obtain
〈X |Ê|X〉 = ℜ
dkf(k)
J0 (sin θ0kr⊥) ex −
x tan θ0
J1 (sin θ0kr⊥) ez
eik cos θ0(z−V t) (6)
where Jn(ξ) is the Bessel function of the first kind of order n and the relation
dφeih(x cosφ+y sinφ) = 2πJ0(hr⊥)
has been exploited. The expectation value of the electric field in Eqs.(6) describes a linearly polarized X-waves (see
Eq.(1)) which is an exact solution of Maxwell equations in vacuum (see Ref.[4]). Note that the longitudinal component
Ez is due to the full non-paraxial character of the exact approach we are considering and it is negligible for small
aperture angles θ0.
Let us consider the state of the electromagnetic field given by
|XE〉 =
∆θ0(k)Φ(k, s)D̂k,s [α(k, s)] |0〉 (7)
where α(k, s) and Φ(k, s) are arbitrary complex function, the second one being constrained by the normalization
conditions 〈XE |XE〉 = 1. Note that the very presence of the function ∆θ0(k) in Eq.(7) implies that modes are collected
in such a way that their wave vectors almost lies, in the limit δθ ≪ θ0, on the surface of the cone characterizing the
spectrum of an arbitrary classical X wave and, therefore |XE〉 is expected to describe a quantum X wave. In order to
prove this assertion we note that
〈XE |âks|XE〉 = ∆θ0(k)α(k, s)Q(k, s) (8)
where Q(k, s) = |Φ(k, s)|2 + Φ(k, s)e− 12 |α(k,s)|2
k′ 6=k,s′ 6=s
∆θ0(k
′)Φ∗(k′, s′)e−
|α(k′,s′)|2 so that the mean value of the
electric field on |XE〉 turns out to be
〈XE |Ê|XE〉 = ℜ
∆θ0(k)G(k, s)e
i(k·r−c|k|t)
 (9)
where G(k, s) = i
2h̄c|k|
α(k, s)Q(k, s). Comparing Eq.(9) with Eq.(3) and noting that their structures are identical,
we deduce that, in the limit δθ → 0 and L → ∞, the state |XE〉 describes a quantum X wave. It is worth noting
that this state is completely different from the |X〉 of Eq.(2) and it is obtained as a linear combination of the states
D̂k,s[α(k, s)]|0〉 each corresponding to a coherent state in the mode (k, s), all other modes being in the vacuum state.
This implies that the state |XE〉 arises from an entangled superposition of modes in coherent states, where with
“entanglement” we mean that the state cannot be expressed as a product of kets containing different modes. We
therefore conclude that a prescribed classical X wave can be represented by two different quantum states |X〉 and
|XE〉 exhibiting an internal disentangled and entangled modal structure, respectively whose quantum difference is
expected to play a fundamental role during the measurement process. As a matter of fact, a realistic measurement
device sensible to a bounded set S of wave vectors k affects the states |X〉 and |XE〉 in a dramatically different way
after the measurement since a measurement carried out on |X〉 leaves the modes k /∈ S unaltered whereas the same
measurement on |X〉E profoundly changes the structure of the modes k /∈ S causing, as usual, an irreversible loss of
information and notably of the propagation invariance.
GENERATION AND DETECTION OF QUANTUM X WAVES
The same difference between the two proposed quantum X waves also raises the problem of discerning which one
(i.e. disentangled or entangled) is obtained from a prescribed mechanism capable of generating a classical X wave.
Since the disentangled X wave is just the product of coherent states with different frequencies but with the same
axicon angle, |X〉 is well expected to be produced by a simple linear device, namely an axicon (as in [15]), exposed
to a coherent non-monochromatic source like a mode-locked laser. Conversely the entangled X wave is expected
when considering nonlinear processes since it is well known that frequency-conversion or parametric processes are
accompanied by tight phase-matching condition able to provide the spectral structure characterizing a classical X
wave, as addressed by various authors with reference to various kind of nonlinearities (as in [22, 23, 24, 25]).
A viable scheme for detecting entanglement on a generated X-waves is offered by homodyne detection that select a
specific mode, eventually including a prism or a grating spatially separating the various angular frequencies. This cor-
respond to consider the projection of the state |XE〉 onto a specific mode (at frequency ω1 and within a normalization
constant)
|XE〉projected = |α1〉+ΦE |0〉 (10)
where ΦE will in general be dependent on the specific X-wave or on its axicon angle. Using homodyne detection (see
[26]) and tuning the local field phase θ it is possible to unveil fringes in the probability distribution of the detected
current. As an example, for θ = − arg(α1) + π/2, the probability of detecting the value x of the output current
|〈x|α〉|2 is an oscillating function:
|〈x|α〉|2 = e
1 + |ΦE |2 + 2|ΦE | cos(
2|α1|x− argΦE)
. (11)
Note that the same reasoning with the state |X〉 yields ΦE = 0 so that oscillations in |〈x|α〉|2 do not appear if the
original state is not entangled. More general results can be obtained by considering beam-splitters and generic phases
θ, but their mathematical description is cumbersome and will be reported elsewhere.
In addition to the considered X waves, we discuss here the interesting possibility of turning a disentangled X wave
into an entangled one by letting the former (appropriately generated by an axicon) to pass through a nonlinear Kerr
medium. In order to avoid mathematical complications, we consider the typical spectrum of a broad band mode-locked
laser containing M harmonics ωm = mω0 (m = 1, 2, ...,M) and we focus on the case where each mode is in a coherent
state corresponding to a plane wave propagating with the conical angle θ0. The disentangled X wave generated by
the axicon is formed by the modes whose wave vectors are such that km = mω0/c with kz = km/(1 + η
2) and reads
|X〉 = |α1〉1|α2〉2...|αM 〉M where αm denotes the complex parameter of the coherent state with angular frequency
ωm. When travelling through a Kerr medium the various modes induce cross and self-phase modulation and the
relevant and well-known interaction Hamiltonian corresponds to that of a nonlinear oscillator (see [26]) and reads
ĤI = χ(h̄/2)
m,p n̂mn̂p, where n̂m = â
mâm is the photon number operator of the mode m. If t is the interaction
time, the output state from the crystal |Xout〉 = e−
ĤI t|X〉 is given by
|Xout〉 = e−
|αr |
n1,n2,...,nM=0
m,p=0
q=1,2,...,M
|nq〉. (12)
Note that, for t = 4π/χ the output state coincides with the incoming one whereas for t = 2π/χ the state |Xout〉 is
obtained from |X〉 after the replacement αm → −αm and the resulting disentangled X wave state can be denoted
as | − X〉. More interesting is the situation for t = π/χ since, defining N =
m nm and exploiting the relation
e−i(π/2)N
= [e−iπ/4 + (−1)N eiπ/4]/
2, we obtain obtain
|Xout〉 =
e−iπ/4|X〉+ eiπ/4| −X〉
. (13)
which is an entangled superposition of two disentangled quantum X waves travelling at the same velocity. The
key point is here that the interaction time t coincides with the time spent by the classical X wave to pass through
the nonlinear medium or t = D cos θ0/c, D being the nonlinear medium length. As for coherent states, the states
periodically evolve with temporal period 4π/χ and with spatial period 4πc/(cos θ0χ). As a consequence, by simply
acting on the velocity of the X-wave (e.g. by varying the axicon angle) it is possible to fine-tune the degree of
entanglement of the two macroscopic classical X-waves, and even to switching from classical states to Schrödinger-cat
states. This is a not-trivial outcome of the interplay between the classical interference process supporting X waves and
the entanglement of coherent states. In addition, X waves travel at a velocity which is different from the other linear
waves so that, by using coincidences measurements, it is straightforward to distinguish these states from a statistical
mixture, and hence to point out their purely quantum properties as described above.
CONCLUSIONS
In conclusion we have shown that classical electromagnetic X-waves may hide different degrees of quantum entan-
glement, depending on their generation mechanism (exploiting a linear, a nonlinear medium or both of them). In this
respect their velocity (i.e. the axicon angle) plays a prominent role. Other interesting consequences of the X-waves
structure arise when dealing with interferometric setup, as for example that considered in [27]. If a nonlinear medium
is placed in one or both arms of an interferometer, the output state is a “progressive undistorted squeezed vacuum”
or entangled superposition of classical X waves so that any detection scheme is affected by the axicon angle, and
tuning among various quantum states can be attained, as it will be detailed elsewhere. The quantum properties of
X-waves can hence be exploited for free-space quantum communications, highly sensible interferometers or quantum
computing.
We acknowledge fruitful discussions with E. Del Re, S. Trillo, B. Crosignani and P. Di Porto.
∗ Electronic address: alessandro.ciattoni@aquila.infn.it
† Electronic address: claudio.conti@phys.uniroma1.it
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mailto:alessandro.ciattoni@aquila.infn.it
mailto:claudio.conti@phys.uniroma1.it
http://arxiv.org/abs/quant-ph/0409130
	Introduction
	Quantum Description of X waves
	Generation and Detection of quantum X Waves
	Conclusions
	Acknowledgments
	References

  We show that two distinct quantum states of the electromagnetic field can be
associated to a classical vector X wave or a propagation-invariant solution of
Maxwell equations. The difference between the two states is of pure quantum
mechanical origin since they are internally entangled and disentangled,
respectively and can be generated by different linear or nonlinear processes.
Detection and generation of Schr\"odinger-cat states comprising two entangled
X-waves and their possible applications are discussed.

Introduction
	Quantum Description of X waves
	Generation and Detection of quantum X Waves
	Conclusions
	Acknowledgments
	References
