Decoherence of Quantum-Enhanced Timing Accuracy
Mankei Tsang
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125
(Dated: August 6, 2021)
Quantum enhancement of optical pulse timing accuracy is investigated in the Heisenberg picture. Effects of
optical loss, group-velocity dispersion, and Kerr nonlinearity on the position and momentum of an optical pulse
are studied via Heisenberg equations of motion. Using the developed formalism, the impact of decoherence by
optical loss on the use of adiabatic soliton control for beating the timing standard quantum limit [Tsang, Phys.
Rev. Lett. 97, 023902 (2006)] is analyzed theoretically and numerically. The analysis shows that an appreciable
enhancement can be achieved using current technology, despite an increase in timing jitter mainly due to the
Gordon-Haus effect. The decoherence effect of optical loss on the transmission of quantum-enhanced timing
information is also studied, in order to identify situations in which the enhancement is able to survive.
PACS numbers: 42.50.Dv, 42.65.Tg, 42.81.Dp
I. INTRODUCTION
It has been suggested that the use of correlated photons is
able to enhance the position accuracy of an optical pulse be-
yond the standard quantum limit, and the enhancement can be
useful for positioning and clock synchronization applications
[1]. Generation of two photons with the requisite correlation
has been demonstrated experimentally by Kuzucu et al. [2],
but in practice it is more desirable to produce as many corre-
lated photons as possible in order to obtain a higher accuracy.
To achieve quantum enhancement for a large number of pho-
tons, a scheme of adiabatically manipulating optical fiber soli-
tons has recently been proposed [3], opening up a viable route
of applying quantum enhancement to practical situations. The
analysis in Ref. [3] assumes that the optical fibers are loss-
less, so the Heisenberg limit [4] can be reached in principle.
In reality, however, the quantum noise associated with optical
loss increases the soliton timing jitter and limits the achievable
enhancement. Compared with the use of solitons for quadra-
ture squeezing [5], the adiabatic soliton control scheme poten-
tially suffers more from decoherence, because the soliton must
propagate for a longer distance to satisfy the adiabatic approx-
imation. The effect of loss on a similar scheme of soliton mo-
mentum squeezing has been studied by Fini and Hagelstein
[6], although they did not study the timing jitter evolution rel-
evant to the scheme in Ref. [3], and did not take into account
possible departure from the adiabatic approximation.
In this paper, the decoherence effect of optical loss on the
timing accuracy enhancement scheme proposed in Ref. [3] is
investigated in depth, in order to evaluate the performance of
the scheme in practice. Instead of approaching the problem in
the Schrödinger picture like prior work [1, 3, 6, 7], this paper
primarily utilizes Heisenberg equations of motion, since they
are able to account for dissipation and fluctuation in a more el-
egant way. For simplicity, scalar solitons, as opposed to vector
solitons studied in Ref. [3], are considered here. The theoret-
ical and numerical analyses show that, despite an increase in
timing jitter due to quantum noise and deviation from the adi-
abatic approximation, an appreciable enhancement can still be
achieved using a realistic setup.
The developed formalism is also used to study the propa-
gation of an optical pulse with quantum-enhanced timing ac-
curacy in a lossy, dispersive, and nonlinear medium, such as
an optical fiber, in order to identify situations in which the
enhancement can still survive. The effect of loss on many
correlated photons sent in as many channels has been inves-
tigated by Giovannetti et al. [1], but their analysis focuses on
a relatively small number of correlated photons and does not
include the effects of dispersion and nonlinearity.
This paper is organized as follows: Section II defines the
general theoretical framework, and derives the standard quan-
tum limits and Heisenberg limits on the variances of the pulse
position and momentum operators. Section III studies the evo-
lution of such operators in the presence of loss, group-velocity
dispersion, and Kerr nonlinearity, and determines the effect of
dissipation and fluctuation on the position and momentum un-
certainties. Section IV investigates theoretically and numeri-
cally the impact of optical loss on the adiabatic soliton con-
trol scheme using realistic parameters, while Sec. V studies
the decoherence effect on the transmission of the quantum-
enhanced timing information in various linear and nonlinear
systems.
II. THEORETICAL FRAMEWORK
A. Definition of pulse position and momentum operators
The positive-frequency electric field of a waveguide mode
at a certain longitudinal position can be defined as [8]
Ê(+)(t) = i
4πε0cn2S
ĉ(ω)e−iωt , (1)
where n is the refractive index, η is the real part of n, S is the
transverse area of the waveguide mode, and ĉ(ω) is the photon
annihilation operator. The annihilation operator is related to
the corresponding creation operator via the commutator [8],
[ĉ(ω), ĉ†(ω ′)] = δ (ω −ω ′). (2)
For a pulse with a slowly-varying envelope compared with the
optical frequency, the coefficient in front of the annihilation
operator can be assumed to be independent of frequency and
http://arxiv.org/abs/0704.0663v1
can be evaluated at the carrier frequency ω0, so that the elec-
tric field is proportional to the temporal envelope annihilation
operator Â(t),
Ê(+)(t) ∝ Â(t)e−iω0t , (3)
Â(t)≡ 1√
dω â(ω)e−iωt , (4)
â(ω)≡ ĉ(ω +ω0). (5)
The temporal envelope operator Â(t) and the spectral operator
â(ω) evidently also satisfy the following commutation rela-
tions with their corresponding creation operators,
[Â(t), Â†(t ′)] = δ (t − t ′), (6)
[â(ω), â†(ω ′)] = δ (ω −ω ′). (7)
The total photon number operator can be defined as
dt Â†(t)Â(t), (8)
and the pulse center position operator as [9]
T̂ ≡ 1
dt tÂ†(t)Â(t), (9)
where
is the average photon number. This definition uses 1/N as
the normalization coefficient, instead of the inverse photon
number operator N̂−1 used by Lai and Haus [10], in order
to express the position operator in terms of normally ordered
optical field operators that are easier to handle, as well as to
avoid the potential problem of applying N̂−1 on the vacuum
state. As long as the photon-number fluctuation is small, the
position operator naturally corresponds to the measurement of
the center position of the pulse intensity profile. An average
longitudinal momentum operator can be similarly defined,
Ω̂ ≡ 1
dω ω â†(ω)â(ω)
dt Â†(t)
Â(t). (11)
If the quantum state is close to a large-photon-number coher-
ent state, Â can be approximated as
+δ Â, with O(δ Â)≪
O(Â). Equations (9) and (11) then become the approximate
position and momentum operators defined by Haus and Lai
for solitons in a linearized approach [11]. The linearized ex-
pressions also describe how they can be accurately measured
in practice using balanced homodyne detection [11].
For simplicity, we shall hereafter assume that
= 0 and
= 0 [9]. In the systems considered in this paper, these
two quantities remain constant throughout propagation, if t
is regarded as the retarded time in the moving frame of the
optical pulse.
The commutator between the position and momentum op-
erators is
[T̂ ,Ω̂] =
. (12)
By the Heisenberg uncertainty principle,
[T̂ ,Ω̂]
. (13)
B. Derivation of standard quantum limits
The standard quantum limits and Heisenberg limits on
should be expressed in terms of the pulse
width ∆t, defined as
dt t2Â†(t)Â(t)
, (14)
and the bandwidth ∆ω ,
dω ω2â†(ω)â(ω)
dt Â†(t)
Â(t)
. (15)
To calculate the standard quantum limit on the position uncer-
tainty, consider the expansion
dω ω â†â
dω ′ ω ′â′†â′
, (16)
where we have written â= â(ω) and â′ = â(ω ′) as shorthands.
Rearranging the operators,
dω ω2â†â
dω ′ ωω ′â†â′†ââ′
. (17)
The first term on the right-hand side of Eq. (17) is proportional
to ∆ω2, while the second term contains a normally ordered
cross-spectral density. To derive the standard quantum limit,
we shall assume that the cross-spectral density satisfies the
factorization condition:
â†â′†ââ′
â†â
â′†â′
. (18)
This condition is always satisfied by any pure or mixed state
with only one excited optical mode, such as a coherent state
[12, 13]. The second term on the right-hand side of Eq. (17)
becomes
dω ′ ωω ′
â†â′†ââ′
, (19)
which is assumed to be zero, as per the convention of this
paper. Thus, the variance of Ω̂ is
, (20)
where the subscript “coh” denotes statistics of coherent fields
[12, 13] given by Eq. (18). By virtue of the Heisenberg uncer-
tainty principle given by Eq. (13), the standard quantum limit
on the position variance is hence
4N∆ω2
. (21)
This limit is applicable to any pure or mixed state, and is con-
sistent with the one suggested by Giovannetti et al. for Fock
states [1]. A very similar derivation of the limit for Fock states
and coherent states is also performed by Vaughan et al. [9].
Owing to Fourier duality of position and momentum in the
slowly-varying envelope regime, the standard quantum limit
on the momentum can be derived in the same way. The vari-
ance of T̂ , assuming coherent-field statistics, is
, (22)
and the standard quantum limit on the momentum variance is
4N∆t2
. (23)
C. Derivation of Heisenberg limits
To derive the Heisenberg limit on the position uncertainty,
one needs an absolute upper bound on the momentum uncer-
tainty
. Consider the following non-negative quantity
proportional to the coherence bandwidth squared,
dω ′ (ω −ω ′)2
â†â′†ââ′
≥ 0. (24)
This quantity is non-negative because (ω − ω ′)2 is non-
negative and
â†â′†ââ′
is also non-negative [13]. It can be
rewritten as
dω ′ (ω −ω ′)2
â†â′†ââ′
dω ′ (ω −ω ′)2
â†ââ′†â′
, (25)
and expanded as
dω ′ (ω2 +ω ′2 − 2ωω ′)â†ââ′†â′
dω ω2â†â
− 2N2
≥ 0. (26)
Here we shall approximate N̂ with N, and neglect any photon-
number fluctuation. This approximation is exact for Fock
states, and acceptable for any quantum state with a small
photon-number fluctuation, such as a large-photon-number
coherent state. We then obtain the following approximate in-
equality,
≤ ∆ω2. (27)
With the Heisenberg uncertainty principle given by Eq. (13)
and the upper bound on
given by Eq. (27), one can then
obtain the Heisenberg limit on the uncertainty of T̂ :
4N2∆ω2
. (28)
Equation (28) is again consistent with the Heisenberg limit
suggested by Giovannetti et al. [1], although the derivation
here shows that it is not only valid for Fock states but also
correct to the first order for any quantum state with a small
photon-number fluctuation.
The Heisenberg limit on
is similar,
4N2∆t2
. (29)
A more exact derivation of the Heisenberg limits is given in
Appendix A, where the inverse photon-number operator N̂−1
is used instead of 1/N in the definitions of T̂ , Ω̂, ∆ω , and
∆t. The difference between the approximate Heisenberg limits
derived here and the exact Heisenberg limits in Appendix A is
negligible for small photon-number fluctuations.
III. OPTICAL PULSE PROPAGATION IN THE
HEISENBERG PICTURE
The classical nonlinear Schrödinger equation that describes
the propagation of pulses in a lossy, dispersive, and nonlinear
medium, such as an optical fiber, is given by [14]
−κ |A|2A−
A, (30)
where t is the retarded time coordinate in the frame of the
moving pulse, β is the group-velocity dispersion coefficient,
κ is the normalized Kerr coefficient, and α is the loss coeffi-
cient, all of which may depend on z. The phenomenological
quantized version that preserves the commutator between Â
and Â† is [5]
∂ 2Â
−κÂ†ÂÂ−
Â+ iŝ. (31)
Â ≡ Â(z, t) is the pulse envelope annihilation operator in the
Heisenberg picture, and ŝ is the Langevin noise operator, sat-
isfying the commutation relation
[ŝ(z, t), ŝ†(z′, t ′)] = αδ (z− z′)δ (t − t ′). (32)
Rewriting the position and momentum operators in Eqs. (9)
and (11) in the Heisenberg picture as T̂ (z) and Ω̂(z) in terms of
Â(z, t), differenting them with respect to z, and using Eq. (31),
their equations of motion can be derived,
dT̂ (z)
= β (z)Ω̂(z)+ ŜT (z), (33)
dΩ̂(z)
= ŜΩ(z), (34)
where ŜT and ŜΩ are position and momentum noise operators
defined as
ŜT (z)≡
dt tŝ†(z, t)Â(z, t)+H. c., (35)
ŜΩ(z)≡
dt ŝ†(z, t)
Â(z, t)+H. c., (36)
and H. c. denotes Hermitian conjugate. If the noise reservoir
is assumed to be in the vacuum state, the noise operators have
the following statistical properties, as shown in Appendix B,
ŜT (z)
ŜΩ(z)
= 0, (37)
ŜT (z)ŜT (z
α(z)∆t2(z)
δ (z− z′), (38)
ŜΩ(z)ŜΩ(z
α(z)∆ω2(z)
δ (z− z′), (39)
ŜT (z)ŜΩ(z
′)+ ŜΩ(z)ŜT (z
α(z)C(z)
δ (z− z′), (40)
where C(z) is the pulse chirp factor, defined as
C(z)≡
dt Â†(z, t)
Â(z, t)
The average position 〈T̂ (z)〉 and average momentum 〈Ω̂(z)〉
are constant and assumed to be zero throughout propagation.
The variance of Ω̂ is then
Ω̂2(z)
Ω̂2(0)
α(z′)∆ω2(z′)
N(z′)
, (42)
while the variance of T̂ is more complicated due to the pres-
ence of dispersion,
T̂ 2(z)
T̂ 2(0)
T̂ (0)Ω̂(0)+ Ω̂(0)T̂ (0)
dz′β (z′)
Ω̂2(0)
dz′β (z′)
α(z′)∆t2(z′)
N(z′)
dz′β (z′)
α(z′′)C(z′′)
N(z′′)
dz′β (z′)
dz′′β (z′′)
∫ z′′
dz′′′
α(z′′′)∆ω2(z′′′)
N(z′′′)
. (43)
Equation (43) is the central result of this paper. It is similar
to that derived by Haus for optical solitons using a linearized
approach [15], but Eq. (43) is valid for arbitrary loss, arbi-
trary dispersion profile β (z), and arbitrary evolution of pulse
width ∆t(z), chirp C(z), and bandwidth ∆ω(z), so that it is
able to describe the effect of loss on the quantum enhance-
ment scheme proposed in Ref. [3]. The first term on the right-
hand side of Eq. (43) is the initial quantum fluctuation, while
the second and third term on the right-hand side describe the
quantum dispersion effect [16]. In an ideal scenario described
in Ref. [3],
T̂ 2(z)
remains constant if the net dispersion
′β (z′) is zero and quantum dispersion is compensated.
With loss, however, noise introduces a diffusive jitter given
by the fourth term on the right-hand side of Eq. (43),
T̂ 2(z)
α(z′)∆t2(z′)
N(z′)
, (44)
a less well-known chirp-induced jitter given by the fifth term,
T̂ 2(z)
dz′β (z′)
α(z′′)C(z′′)
N(z′′)
, (45)
and also the Gordon-Haus timing jitter [17] given by the sixth
term,
T̂ 2(z)
dz′β (z′)
dz′′β (z′′)
∫ z′′
dz′′′
α(z′′′)∆ω2(z′′′)
N(z′′′)
. (46)
In most cases considered here, N ≫ 1,
≪ ∆t2,
≪ ∆ω2, so one can use the classical nonlinear
Schrödinger equation, Eq. (30), to predict the evolution of
∆t(z), C(z), and ∆ω(z) accurately. The evolution of
T̂ 2(z)
can subsequently be calculated analytically or numerically us-
ing Eq. (43) and the classical evolution of ∆t(z), C(z), and
∆ω(z), analogous to the linearized approach [11, 15].
It is worth noting that the chirp-induced jitter, Eq. (45), de-
pends on the cross-correlation between the position and mo-
mentum noise in Eq. (40), so it can be positive as well as neg-
ative, but the sum of the three sources of jitter must obviously
remain positive.
IV. EFFECT OF LOSS ON ADIABATIC SOLITON
CONTROL
A. Review of the ideal case
Consider the scheme proposed in Ref. [3] and depicted in
Fig. 1. Assume that the dispersion coefficient of the first fiber
β (z) is negative and its magnitude increases along the fiber
slowly compared with the soliton period. The classical soliton
solution of Eq. (30), assuming adiabatic change in parameters
FIG. 1: (Color online) Schematic (not-to-scale) of the adiabatic soli-
ton control scheme. An optical pulse is coupled into a dispersion-
increasing fiber of length L with a negative dispersion coefficient β ,
followed by a much shorter dispersion-compensating fiber of length
L′ with a positive dispersion coefficient β ′.
β (z) and N(z), is [18]
A(z, t) = A0(z)sech
dz′|A0(z′)|2
, (47)
A0(z) =
2τ(z)
, τ(z) =
2|β (z)|
κN(z)
. (48)
The adiabatic approximation is satisfied when
β (z)
dβ (z)/dz
dN(z)/dz
≪ Λ, (49)
where Λ is the soliton period,
Λ(z)≡
τ2(z)
|β (z)|
. (50)
The root-mean-square pulse width ∆t(z) and bandwidth
∆ω(z) then become
∆t(z) =
τ(z) =
|β (z)|
κN(z)
, (51)
∆ω(z) =
3τ(z)
κN(z)
|β (z)| . (52)
The bandwidth ∆ω(z) is thus reduced in the first fiber. If the
second fiber has a positive dispersion coefficient β ′ so that
the net dispersion is zero (
0 dzβ (z)+β
′L′ = 0), the quantum
dispersion effect given by the second and third term on the
right-hand side of Eq. (43) can be eliminated. Furthermore, if
β ′ has a very large magnitude compared with β (z) so that the
second fiber can be very short compared with the first fiber, the
effective nonlinearity experienced by the pulse in the second
fiber can be neglected, and ∆ω(z) remains essentially constant
in the second fiber. In the lossless case, the final timing jitter
T̂ 2(L+L′)
is therefore the same as the input
T̂ 2(0)
, but
∆ω(L+L′) has been reduced and the standard quantum limit
T̂ 2(L+L′)
, Eq. (21), is raised. Provided that the initial
timing jitter of a laser pulse obeys the coherent-field statistics
given by Eq. (22), the final timing jitter is
T̂ 2(L+L′)
T̂ 2(0)
∆t2(0)
β 2(0)
, (53)
while the final standard quantum limit is
T̂ 2(L+L′)
4N∆ω2(L+L′)
3β 2(L)
. (54)
A timing jitter squeezing ratio, analagous to the squeezing ra-
tio defined by Haus and Lai [11], can be defined as
T̂ 2(L+L′)
T̂ 2(L+L′)
β 2(0)
β 2(L)
. (55)
The factor of π2/9 arises because the initial jitter for a sech
pulse shape is slightly higher than the standard quantum limit
given by Eq. (21) in terms of the bandwidth. As long as β (L)
at the end of the first fiber is significantly larger than the initial
value, the timing jitter becomes lower than the raised standard
quantum limit, R becomes smaller than 1, and quantum en-
hancement of position accuracy is accomplished. This semi-
classical analysis is valid in all practical cases, where N ≫ 1,
R ≫ 1/N,
≪ ∆t2,
≪ ∆ω2, and is consistent with
the analysis of exact quantum soliton theory in Ref. [3]. R
is related to the quantum enhancement factor γ defined in
Ref. [3] by R = 1/γ2. The semiclassical analysis is no longer
valid when R is close to the Heisenberg limit 1/N, but as the
next sections will show, owing to decoherence effects, it is ex-
tremely difficult for the enhancement to get anywhere close to
the Heisenberg limit.
B. Numerical analysis of a realistic case
To investigate the impact of noise and the validity of the
adiabatic approximation in practice, a numerical evaluation of
∆t(z), C(z), ∆ω(z), and
T̂ 2(z)
, using Eqs. (30) and (43) and
realistic parameters, is necessary. β (z) is assumed to have the
following profile used in Ref. [19],
β (z) =
−12.75 ps2/km
1+(L− z)/Lβ
. (56)
Lβ = 1 km is used here instead of the Lβ = 1/12 km used
in Ref. [19], in order to satisfy the adiabatic approximation
for a longer pulse in this example. Other fiber parameters are
α = 0.4 dB/km, n2 = 2.6×10−16 cm2/W, Aeff = 30 µm2 [19],
λ0 = 1550 nm, ω0 = 2πc/λ0, so that κ = h̄ω0(ω0n2/cAeff).
L is assumed to be 2 km. A dispersion-compensating fiber
with β ′ = 127.5 ps2/km, α = 0.4 dB/km, n2 = 2.7× 10−16
cm2/W, Aeff = 15 µm2 [20], and L′ = 110 m is used in the
numerical analysis as the second fiber. The classical nonlin-
ear Schrödinger equation, Eq. (30), is numerically solved us-
ing the Fourier split-step method [14]. An initial sech soliton
pulse with τ(0) = 1 ps, N(0) = 1.9×107, and an initial energy
of 2.4 pJ is assumed.
Figure 2 plots the numerical evolution of pulse intensity and
spectrum in the two fibers. As expected, the bandwidth is nar-
rowed in the first fiber and remains approximately constant
in the second (z > 2000 m), owing to the latter’s relative short
length. Figure 3 plots the evolution of pulse width ∆t(z), chirp
−8 −6
−4 −2 0 2 4 6 8
z (m)
|A(z, t)|2
t (ps)
1.5 0
z (m)
|a(z, ω)|2
ω (THz)
FIG. 2: (Color online) Numerical evolution of pulse intensity (top)
and spectrum (bottom). The denser plots for z > 2000 m indicate
pulse propagation in the second fiber. The color codes are in the
same arbitrary units as the heights of the plots.
C(z), and bandwidth ∆ω(z), compared with the adiabatic ap-
proximation, Eqs. (51) and (52). The adiabatic approximation
is evidently not exact, and the pulse acquires a chirp due to
excess dispersion in the first fiber, leading to slight refocusing
in the second fiber. The bandwidth is reduced by a factor of
2.2, as opposed to the ideal factor of 3.6.
Figure 4 plots the evolution of the diffusive jitter given by
Eq. (44), the chirp-induced jitter given by Eq. (45), and the
Gordon-Haus jitter given by Eq. (46). It can be seen that al-
though the Gordon-Haus jitter increases much more quickly
than the other jitter components in the first fiber, the for-
mer drops abruptly in the second fiber (z > 2000 m) due
to the opposite dispersion. This kind of Gordon-Haus jit-
ter reduction by dispersion management is well known [21].
The chirp-induced jitter component drops below zero in the
second fiber, but as noted before, the total noise jitter re-
mains positive. The final jitter values are numerically deter-
mined to be
T̂ 2(L+L′)
= 0.71
T̂ 2(0)
T̂ 2(L+L′)
−0.93
T̂ 2(0)
, and
T̂ 2(L+L′)
= 1.42
T̂ 2(0)
, result-
ing in a total jitter of
T̂ 2(L+L′)
T̂ 2(0)
T̂ 2(L+L′)
T̂ 2(L+L′)
T̂ 2(L+L′)
= 2.19
T̂ 2(0)
. (57)
The final squeezing ratio is hence
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Pulse Width ∆t(z)
Numerical
Adiabatic Approximation
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Chirp C(z)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
z (m)
Bandwidth ∆ω(z)
FIG. 3: (Color online) Evolution of pulse width ∆t(z) (top), chirp
C(z) (center), and bandwidth ∆ω (bottom), compared with the adia-
batic approximation (dash). Plots of ∆t and ∆ω are normalized with
respect to their initial values, respectively.
T̂ 2(L+L′)
T̂ 2(L+L′)
T̂ 2(L+L′)
T̂ 2(0)
N(L+L′)
∆ω2(L+L′)
∆ω2(0)
= 0.42 =−3.8 dB. (58)
Despite taking into account the increased timing jitter and the
non-ideal bandwidth narrowing, a squeezing ratio of −3.8 dB
is predicted by the numerical analysis, suggesting that one
should be able to observe the quantum enhancement experi-
mentally using current technology.
C. Potential improvements
As shown in the previous section, the Gordon-Haus effect
contributes the largest amount of noise in the soliton control
scheme, despite its partial reduction by dispersion manage-
ment. Its magnitude at the end of the first fiber can be esti-
mated roughly as
T̂ 2(L)
T̂ 2(L)
(αL). (59)
As the length of the first fiber must be at least a few times
longer than the soliton period Λ for the adiabatic approxima-
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Diffusive Jitter
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Chirp-Induced Jitter
0 200 400 600 800 1000 1200 1400 1600 1800 2000
z (m)
Gordon-Haus Jitter
FIG. 4: (Color online) Evolution of diffusive jitter (top), chirp-
induced jitter (center), and Gordon-Haus jitter (bottom). All plots
are normalized with respect to the initial jitter
T̂ 2(0)
tion to hold and for the bandwidth to be significantly reduced,
L/Λ is approximately fixed, and the Gordon-Haus jitter can
be reduced only if a figure of merit,
FOM ≡ 1
, (60)
is enhanced. Since this is a rough order-of-magnitude esti-
mate, a representative value of Λ, say at z = L, can be used.
The figure of merit suggests that the performance of the soli-
ton control scheme can be improved by reducing the pulse
width, increasing the overall dispersion coefficient, or reduc-
ing the loss coefficient.
Reducing the pulse width is the most convenient way of
obtaining better enhancement, as the adiabatic bandwidth re-
duction can be achieved over a shorter distance with less loss
of photons. For example, using τ(0) = 500 fs, L = 1 km,
Lβ = 0.3 km, L
′ = 44 m, and otherwise the same parame-
ters as in Sec. IV B, the squeezing ratio becomes −6.0 dB,
while using τ(0) = 200 fs, L = 500 m, Lβ = 1/12 km, and
L′ = 16.2 m gives a squeezing ratio of −7.3 dB. The shorter
pulse width, however, significantly enhances higher-order dis-
persive and nonlinear effects. Raman scattering, in particular,
contributes additional quantum noise because of coupling to
optical phonons [22]. It is beyond the scope of this paper to
investigate these higher-order effects, so a more conservative
pulse width of 1 ps is used in the preceding section. A larger
overall dispersion coefficient, on the other hand, means that
more photons or a higher nonlinearity are required for a soli-
ton to form, so the Raman effect may also become more sig-
nificant with a larger dispersion coefficient. The Raman effect
can be reduced by cooling the fiber and reducing the number
of thermal phonons [22], if it becomes a significant problem.
Further advance in optical fiber technology should be able
to increase the figure of merit by reducing loss, since the spe-
cialty fibers assumed in Sec. IV B have a higher loss than usual
transmission fibers by a factor of two. Using α = 0.2 dB/km
instead of 0.4 dB/km in Sec. IV B, for instance, reduces the
squeezing ratio to −4.7 dB. Spectral filtering or frequency-
dependent gain [23] provides another way of controlling the
Gordon-Haus effect, although it adds another level of com-
plexity to the experimental setup, and it is beyond the scope
of this paper to investigate how the frequency-dependentdissi-
pation or amplification might help the quantum enhancement
scheme. Finally, the design of the setup assumed in Sec. IV B
is not fully optimized, and further optimization of parame-
ters, fiber dispersion profiles, and bandwidth narrowing strat-
egy should be able to improve the enhancement.
V. EFFECT OF LOSS ON THE TRANSMISSION OF
QUANTUM-ENHANCED TIMING INFORMATION
Provided that quantum enhancement of pulse position ac-
curacy can be achieved, the information still needs to be
transmitted through unavoidably lossy channels. It is hence
an important question to ask how loss affects the quantum-
enhanced information in optical information transmission sys-
tems. Equation (43) governs the general evolution of the tim-
ing jitter under the effects of loss, dispersion, and nonlinearity,
but in order to estimate the relative magnitude of the decoher-
ence effects and gain more insight into the decoherence pro-
cesses, in this section Eq. (43) is explicitly solved for various
systems and compared with the standard quantum limit.
A. Linear non-dispersive systems
Without dispersion, the timing jitter increases only due to
the diffusive component
T̂ 2(z)
. An analytic expression for
T̂ 2(z)
can then be derived from Eq. (43), as ∆t(z) and ∆ω(z)
remain constant,
T̂ 2(z)
T̂ 2(0)
∆t2(0)
(1− e−αz). (61)
If the initial variance obeys coherent-field statistics, that is,
T̂ 2(0)
= ∆t2(0)/N(0) according to Eq. (22), the subsequent
jitter is
T̂ 2(z)
∆t2(0)
, (62)
and obeys the same coherent-field statistics but for the re-
duced photon number N(z). This is consistent with intu-
ition. On the other hand, in the high loss limit (αz ≫ 1), the
term ∆t2(0)/N(z) is likely to dominate over the initial jitter
T̂ 2(0)
, so in most cases the position of a significantly atten-
uated pulse relaxes to coherent-field statistics independent of
its initial fluctuation. This justifies the assumption in Sec. IV
that a laser pulse exiting a laser cavity has such statistics, re-
gardless of the quantum properties of the pulse inside the cav-
Equation (61) can be renormalized as
R(z)≡
T̂ 2(z)
T̂ 2(z)
= R(0)e−αz + 4∆t2(0)∆ω2(0)(1− e−αz). (63)
≪ ∆t2 and
≪ ∆ω2, classical theory pre-
dicts that 4∆t2∆ω2 ≈ 1. Equation (63) then suggests that the
relative increase in timing jitter is independent of the initial
squeezing ratio R(0). This is nevertheless not true in general,
as ∆t may depend on both ∆ω and R when the classical theory
fails. In Appendix C, the exact dependence of ∆t on ∆ω and R
is calculated for a specific multiphoton state with a Gaussian
pulse shape called the jointly Guassian state. The expression
4∆t2∆ω2 is given by
4∆t2∆ω2 =
(1− 1/N)2
1− 1/(NR)
, (64)
which results in the following exact expression for an initial
jointly Gaussian state,
R(z) = R(0)e−αz +
(1− 1/N)2
1− 1/(NR)
(1− e−αz). (65)
For a large photon number (N ≫ 1) and moderate enhance-
ment (1 ≥ R ≫ 1/N), 4∆t2∆ω2 ≈ 1, as classical theory would
predict for a Gaussian pulse. In this regime, the quantum-
enhanced information is just as sensitive to loss as standard-
quantum-limited information. When R gets close to the
Heisenberg limit 1/N, however, ∆t∆ω approaches infinity.
This is because maximal coincident-frequency correlations
are required to achieve the Heisenberg limit [1], but heuris-
tically speaking, if the photons have exactly the same mo-
mentum, they must have infinite uncertainties in their rela-
tive positions, leading to an infinite pulse width ∆t. Owing to
the abrupt increase in 4∆t2∆ω2 when R approaches 1/N, the
quantum enhancement becomes much more sensitive to loss.
In the Heisenberg limit of R → 1/N, ∆t → ∞, any loss com-
pletely detroys the timing accuracy and leads to an infinite
jitter, according to Eq. (65).
B. Linear dispersive systems
If the system is lossy, dispersive, but linear, it is not difficult
to show that
∆t2(z) = ∆t2(0)+C(0)
dz′β (z′)
+∆ω2(0)
dz′β (z′)
, (66)
C(z) =C(0)+ 2∆ω2(0)
dz′β (z′), (67)
∆ω2(z) = ∆ω2(0). (68)
The following result can then be obtained from Eq. (43) after
some algebra,
T̂ 2(z)
T̂ 2(0)
T̂ (0)Ω̂(0)+ Ω̂(0)T̂ (0)
dz′β (z′)
Ω̂2(0)
dz′β (z′)
∆t2(z)
(1− e−αz).
This result is similar to that in the previous section, except for
the presence of quantum dispersion and the dispersive spread
of the pulse width ∆t(z) that leads to increased jitter. With ini-
tially coherent-field statistics,
T̂ 2(0)
Ω̂2(0)
are given
by Eqs. (22) and (20), respectively, while by similar argu-
ments, the coherent-field statistics for
T̂ Ω̂+ Ω̂T̂
T̂ Ω̂+ Ω̂T̂
. (70)
This leads to the following position variance for a pulse with
initially coherent-field statistics,
T̂ 2(z)
∆t2(z)
, (71)
which still maintains the coherent-field statistics for the dis-
persed pulse width and the reduced photon number. In the
high loss limit (αz ≫ 1), the coherent-field statistics is again
approached regardless of the initial conditions.
For an initial jointly Gaussian quantum state, on the other
hand, the normalized version of Eq. (69) is
R(z) =
R(0)+
4∆t2(0)∆ω2(0)+ 4ζ 2
(1− e−αz), (72)
where ζ is the normalized effective propagation distance,
ζ ≡ ∆ω2(0)
dz′β (z′), (73)
and 4∆t2(0)∆ω2(0) is given by Eq. (64) evaluated at z= 0. As
long as the loss is moderate so that e−αz ≫ 1−e−αz, quantum
dispersion, given by the term proportional ζ 2/R(0), becomes
the dominant effect and overwhelms the initial enhancement
when ζ exceeds R(0)/2.
If the net dispersion
′β (z′) is zero, both quantum and
classical dispersion are eliminated, and the jitter growth be-
comes identical to that in a non-dispersive and linear system
given by Eq. (61).
C. Soliton-like systems
The previous sections show that coherent-field statistics is
maintained in a linear system, but as Sec. IV clearly shows,
non-trivial statistics can arise from the quantum dynamics of
a nonlinear system. The complex evolution of ∆t(z), C(z), and
∆ω(z) in general prevents one from solving Eq. (43) explic-
itly, except for special cases such as solitons.
If the dispersion is constant and the pulse propagates in the
fiber as a soliton, C(z) is zero, while ∆t(z) and ∆ω(z) can be
regarded as constant if
≪∆t2 and
≪∆ω2 through-
out propagation. Equation (43) can then be solved explicitly,
T̂ 2(z)
T̂ 2(0)
Ω̂2(0)
β 2z2 +
∆t2(0)
(eαz − 1)
2β 2∆ω2(0)
eαz − 1
, (74)
where
T̂ (0)Ω̂(0)+ Ω̂(0)T̂ (0)
is asssumed to be zero for
simplicity. If 4
T̂ 2(0)
Ω̂2(0)
= (π2/9)[1/N2(0)] is also
assumed for a soliton pulse for simplicity, Eq. (74) can be
normalized to give
R(z) = R(0)e−αz +
e−αz +
(eαz − 1)
eαz − 1
. (75)
In the low loss regime with αΛ ≪ 1 and αz ≪ 1, Eq. (75) can
be further simplified,
R(z)≈ R(0)+
(αz). (76)
Quantum dispersion is again the dominant effect in this
regime, while decoherence effects are much smaller, by a fac-
tor of αz approximately.
Even if the net dispersion is zero and quantum dispersion is
compensated, the Gordon-Haus effect cannot be completely
eliminated by dispersion management in the presence of non-
linearity and may become significant, as the numerical anal-
ysis in Sec. IV B shows. An order-of-magnitude estimate of
Gordon-Haus jitter can be performed by considering soliton
propagation in a constant negative dispersion fiber, just as
in the previous case, followed by a dispersion-compensating
fiber of length L′ with positive dispersion coefficient β ′. If
L′ is short, the effective nonlinearity experienced by the pulse
in the second fiber can be neglected, and ∆ω(z) can be re-
garded as constant. Assuming that β L+β ′L′ = 0, the integral
in Eq. (46) can be solved to give the Gordon-Haus jitter,
T̂ 2(L+L′)
≈ α∆ω
6N(0)
β 2L2(L+L′)
≈ α∆ω
6N(0)
β 2L3. (77)
The normalized contribution to the squeezing ratio is therefore
T̂ 2(L+L′)
T̂ 2(L+L′)
(αL). (78)
Compared with the Gordon-Haus jitter at the end of the first
fiber given by the last term of Eq. (76), dispersion manage-
ment cuts the jitter by half, but the expression maintains its
functional dependence on the parameters of the first fiber.
This estimate also justifies the use of Eq. (59) to estimate the
Gordon-Haus jitter at the end of the two fibers in Sec. IV C. To
minimize the impact of Gordon-Haus jitter on the quantum-
enhanced timing accuracy in a dispersion-managed soliton
system, the condition
L3 ≪ 54
R (79)
is required.
VI. CONCLUSION
In conclusion, the decoherence effect by optical loss on
adiabatic soliton control and on the transmission of quantum-
enhanced timing information has been extensively studied. It
is found that an appreciable enhancement can still be achieved
by the soliton scheme using current technology, despite an in-
crease of timing jitter due to the presence of loss. It is also
found that the quantum-enhanced timing accuracy should be
much lower than the Heisenberg-limited accuracy to avoid in-
creased sensitivity to photon loss during transmission, and the
net dispersion in the transmission system should be minimized
in order to reduce quantum dispersion and the Gordon-Haus
effect.
Although the most important pulse propagation effects have
been considered in this analysis, higher-order effects, such as
third-order dispersion, self-steepening, and Raman scattering
[14] might provide further adverse impact on the quantum en-
hancement if the optical pulse is ultrashort. In particular, the
inelastic Raman scattering process is expected to be a signifi-
cant source of decoherence for ultrashort pulses [22]. It is be-
yond the scope of this paper to investigate these higher-order
effects, but they should be of minor importance for picosec-
ond pulses and the propagation distances considered in this
paper.
Finally, it is worth noting that while this paper focuses on
optical pulses, the developed formalism is equally valid for
describing the transverse position and momentum of optical
beams [24] and the center-of-mass variables of Bose-Einstein
condensates [9]. Decoherence by loss of particles in those
systems can be studied using the formalism developed in this
paper and parameters specific to those systems.
VII. ACKNOWLEDGMENTS
This work is financially supported by the DARPA Center
for Optofluidic Integration and the National Science Founda-
tion through the Center for the Science and Engineering of
Materials (DMR-0520565).
APPENDIX A: DERIVATION OF EXACT HEISENBERG
LIMITS
An exact Heisenberg limit can be derived if the inverse
photon-number operator N̂−1 is used instead of 1/N in the
definitions of T̂ , Ω̂, ∆t, and ∆ω in Eqs. (9), (11), (14), and
(15), just as in Refs. [9] and [10],
T̂ ′ ≡ N̂−1
dt tÂ†Â, (A1)
Ω̂′ ≡ N̂−1
dω ω â†â, (A2)
∆t ′ ≡
dt t2Â†Â
, (A3)
∆ω ′ ≡
dω ω2â†â
. (A4)
These operators are well defined as long as the quantum state
has zero vacuum-state component (〈0|ρ̂|0〉 = 0). Starting
from the Heisenberg uncertainty principle for T̂ ′ and Ω̂′,
T̂ ′2
, (A5)
and the inequality
dω ′ (ω −ω ′)2â†â′†ââ′
≥ 0, (A6)
one can obtain the exact inequality
≤ ∆ω ′2, (A7)
and the exact Heisenberg limit for the new position operator,
T̂ ′2
4∆ω ′2
. (A8)
The difference between Eqs. (28) and (A8) is negligible for
small photon-number fluctuations. The exact Heisenberg limit
is similar.
APPENDIX B: NOISE STATISTICS
In this section the expression
ŜT (z)ŜT (z
in Eq. (38) is
calculated. The derivations of
ŜΩ(z)ŜΩ(z
in Eq. (39) and
ŜT (z)ŜΩ(z
′)+ ŜΩ(z)ŜT (z
in Eq. (40) are similar. Substi-
tuting Eq. (35) into Eq. (38) gives
ŜT (z)ŜT (z
dt ′ tt ′
ŝ†Âŝ′†Â′
ŝ†ÂÂ′†ŝ′
Â†ŝÂ′†ŝ′
Â†ŝŝ′†Â′
, (B1)
where N = N(z), N′ = N(z′), ŝ = ŝ(z, t), Â = Â(z, t), ŝ′ =
ŝ(z′, t ′), and Â′ = Â(z′, t ′). If the noise reservoir is in the vac-
uum state, ŝ|0reservoir〉= 〈0reservoir|ŝ† = 0, so only the last term
in Eq. (B1) is non-zero,
ŜT (z)ŜT (z
dt ′ tt ′
Â†ŝŝ′†Â′
dt ′ tt ′
Â†Â′
αδ (z− z′)δ (t − t ′)+
Â†ŝ′† ŝÂ′
δ (z− z′)
dt ′ tt ′
Â†ŝ′†ŝÂ′
. (B2)
The first term on the right-hand side of Eq. (B2) is the desired
result, while the second term can be rewritten as
dt ′ tt ′
Â†ŝ′†ŝÂ′
dt ′ tt ′
Â†, ŝ′†
ŝ, Â′
. (B3)
If the system is linear, the commutator between ŝ and Â is
always zero [13], but because ŝ does not commute with Â†
and Â is coupled to Â† by the nonlinear term in Eq. (31), ŝ
may fail to commute with Â. That said, it can be argued that
the optical field operator must always commute with future
noise operators due to causality and the infinitesimally short
memory of ŝ,
Â†, ŝ′†
= 0 if z < z′, (B4)
ŝ, Â′
= 0 if z > z′, (B5)
so Eq. (B3) can be non-zero only at z = z′. The commutator
between ŝ and Â at z = z′ due to the parametric coupling of Â
and Â† can be estimated by a perturbative technique. Consider
an integral form of Eq. (31) with the nonlinear term and the
Langevin noise term only,
Â(z+∆z) = Â(z)+
∫ z+∆z
iκÂ†(z′)Â(z′)Â(z′)+ ŝ(z′)
and Â†(z′) given by the Hermitian conjugate of Eq. (B6),
Â†(z′) = Â†(z)+
−iκÂ†(z′′)Â†(z′′)Â(z′′)+ ŝ†(z′′)
The commutator between ŝ and Â at z+∆z becomes
[ŝ(z+∆z), Â(z+∆z)]
∫ z+∆z
ŝ(z+∆z), Â†(z′)
Â(z′)Â(z′). (B8)
ŝ(z+∆z) commutes with Â(z′) because z+∆z > z′, while it
fails to commute with Â†(z′) because Â†(z′) given by Eq. (B7)
depends explicitly on ŝ†. Thus, in the leading order of ∆z,
[ŝ(z+∆z), Â(z+∆z)]
∫ z+∆z
ŝ(z+∆z), ŝ†(z′′)
Â(z′)Â(z′), (B9)
which approaches 0 in the limit of ∆z → 0. Hence ŝ commutes
with Â at z= z′, and the position noise is given only by the first
term on the right-hand side of Eq. (B2), resulting in Eq. (38).
APPENDIX C: THE JOINTLY GAUSSIAN STATE
A Fock state can be expressed as [13, 25]
dω1 . . .
dωN φ(ω1, . . . ,ωN)|ω1, . . . ,ωN〉,
dt1 . . .
dtN ψ(t1, . . . , tN)|t1, . . . , tN〉, (C1)
where the spectral and temporal eigenstates are given by
|ω1, . . . ,ωN〉 ≡
â†(ω1) . . . â†(ωN)|0〉, (C2)
|t1, . . . , tN〉 ≡
Â†(t1) . . . Â
†(tN)|0〉. (C3)
Theses states are eigenstates of the following operators rele-
vant to our purpose,
Ω̂|ω1, . . . ,ωN〉=
|ω1, . . . ,ωN〉,
T̂ |t1, . . . , tN〉=
|t1, . . . , tN〉, (C5)
dω ω2â†â|ω1, . . . ,ωN〉=
|ω1, . . . ,ωN〉,
dt t2Â†Â|t1, . . . , tN〉=
|t1, . . . , tN〉. (C7)
φ(ω1, . . . ,ωN) is the spectral multiphoton probability ampli-
tude, and it is related to the temporal probability amplitude
ψ(t1, . . . , tN) by the N-dimensional Fourier transform in the
slowly-varying envelope regime. Both amplitudes should also
satisfy normalization and boson symmetry. To study temporal
quantum enhancement, it is convenient to define the probabil-
ity amplitude as a jointly Gaussian function [25],
φ(ω1, . . . ,ωN) =C exp
, (C8)
ψ(t1, . . . , tN) =C′ exp
−N2B2
, (C9)
where B and b are arbitrary and real constants, and C and C′
are normalization constants. Explicit expressions for
, ∆ω2, and ∆t2 can be obtained using Eqs. (C4)-(C7) and
Appendix B of Ref. [25],
= B2, (C10)
4N2B2
, (C11)
∆ω2 = B2 +
b2, (C12)
∆t2 =
4N2B2
. (C13)
In the limit of b → 0,
reaches the Heisenberg limit,
4N2∆ω2
, (C14)
and the quantum state can be written as a state of photons with
maximal coincident-frequency correlations,
|N〉 ∝
dω exp
|ω , . . . ,ω〉. (C15)
On the other hand, when B2 = b2/N,
is at the standard
quantum limit,
4N∆ω2
, (C16)
the quantum state has only one excited Gaussian mode [25],
|N〉 ∝
dω1 . . .
|ω1, . . . ,ωN〉
dω exp
â†(ω)
|0〉, (C17)
and therefore also satisfies the coherent-field statistics [12,
13]. These limits and the corresponding quantum states are
consistent with those suggested in Ref. [1]. With Eqs. (C12)
and (C13), the pulse width ∆t can be determined explicitly in
terms of ∆ω and the squeezing ratio R = ∆ω2/(NB2),
∆t2 =
(1− 1/N)2
1− 1/(NR)
. (C18)
[1] V. Giovannetti, S. Lloyd, and L. Maccone, Nature (London)
412, 417 (2001); V. Giovannetti, S. Lloyd, and L. Maccone,
Phys. Rev. A 65, 022309 (2002).
[2] V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong,
Phys. Rev. Lett. 88, 183602 (2002); O. Kuzucu, M. Fiorentino,
M. A. Albota, F. N. C. Wong, and F. X. Kärtner, Phys. Rev. Lett.
94, 083601 (2005).
[3] M. Tsang, Phys. Rev. Lett. 97, 023902 (2006).
[4] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96,
010401 (2006); V. Giovannetti, S. Lloyd, and L. Maccone, Sci-
ence 306, 1330 (2004).
[5] S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby,
Phys. Rev. Lett. 58, 1841 (1987); P. D. Drummond and S. J.
Carter, J. Opt. Soc. Am. B 4, 1565 (1987); M. J. Potasek and
B. Yurke, Phys. Rev. A 35, 3974 (1987); M. J. Potasek and
B. Yurke, ibid. 38, 1335 (1988); H. A. Haus, Electromagnetic
Noise and Quantum Optical Measurements (Springer, Berlin,
2000).
[6] J. M. Fini and P. L. Hagelstein, Phys. Rev. A 66, 033818 (2002).
[7] P. L. Hagelstein, Phys. Rev. A 54, 2426 (1996); J. M. Fini, P. L.
Hagelstein, and H. A. Haus, ibid. 60, 2442 (1999).
[8] B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992);
R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, ibid. 52,
4823 (1995).
[9] T. Vaughan, P. Drummond, and G. Leuchs, Phys. Rev. A 75,
033617 (2007).
[10] Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989).
[11] H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[12] U. M. Titulaer and R. J. Glauber, Phys. Rev. 140, 676 (1965);
U. M. Titulaer and R. J. Glauber, ibid. 145 1041 (1966).
[13] L. Mandel and E. Wolf, Optical Coherence and Quantum Op-
tics (Cambridge University Press, Cambridge, 1995).
[14] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San
Diego, 2001).
[15] H. A. Haus, J. Opt. Soc. Am. B 8, 1122 (1991).
[16] Y. Lai and H. A. Haus, Phys. Rev. A 40, 854 (1989).
[17] J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
[18] H. H. Kuehl, Opt. Lett. 5, 709 (1988).
[19] V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov,
P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A.
Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A.
N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, J.
Lightwave Technol. 9, 561 (1991).
[20] L. Grüner-Nielsen, M. Wandel, P. Kristensen, C. Jørgensen,
L. V. Jørgensen, B. Edvold, B. Pálsdóttir, and D. Jakobsen, J.
Lightwave Technol. 23, 3566 (2005).
[21] N. J. Smith, W. Forysiak, and N. J. Doran, Electron. Lett. 32,
2085 (1996).
[22] F. X. Kärtner, D. J. Dougherty, H. A. Haus, and E. P. Ippen,
J. Opt. Soc. Am. B 11, 1267 (1994); J. F. Corney and P. D.
Drummond, ibid. 18, 153 (2001).
[23] A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett.
16, 1841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17,
31 (1992).
[24] S. M. Barnett, C. Fabre, and A. Maı̂tre, Eur. Phys. J. D 22,
513 (2003); N. Treps, U. Andersen, B. Buchler, P. K. Lam, A.
Maı̂tre, H.-A. Bachor, and C. Fabre, Phys. Rev. Lett. 88, 203601
(2002); N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Ba-
chor, P. K. Lam, Science 301, 940 (2003).
[25] M. Tsang, “Relationship between resolution enhancement and
multiphoton absorption rate in quantum lithography,” e-print
quant-ph/0607114 (to appear in Phys. Rev. A).
http://arxiv.org/abs/quant-ph/0607114

  Quantum enhancement of optical pulse timing accuracy is investigated in the
Heisenberg picture. Effects of optical loss, group-velocity dispersion, and
Kerr nonlinearity on the position and momentum of an optical pulse are studied
via Heisenberg equations of motion. Using the developed formalism, the impact
of decoherence by optical loss on the use of adiabatic soliton control for
beating the timing standard quantum limit [Tsang, Phys. Rev. Lett. 97, 023902
(2006)] is analyzed theoretically and numerically. The analysis shows that an
appreciable enhancement can be achieved using current technology, despite an
increase in timing jitter mainly due to the Gordon-Haus effect. The decoherence
effect of optical loss on the transmission of quantum-enhanced timing
information is also studied, in order to identify situations in which the
enhancement is able to survive.

Decoherence of Quantum-Enhanced Timing Accuracy
Mankei Tsang
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125
(Dated: August 6, 2021)
Quantum enhancement of optical pulse timing accuracy is investigated in the Heisenberg picture. Effects of
optical loss, group-velocity dispersion, and Kerr nonlinearity on the position and momentum of an optical pulse
are studied via Heisenberg equations of motion. Using the developed formalism, the impact of decoherence by
optical loss on the use of adiabatic soliton control for beating the timing standard quantum limit [Tsang, Phys.
Rev. Lett. 97, 023902 (2006)] is analyzed theoretically and numerically. The analysis shows that an appreciable
enhancement can be achieved using current technology, despite an increase in timing jitter mainly due to the
Gordon-Haus effect. The decoherence effect of optical loss on the transmission of quantum-enhanced timing
information is also studied, in order to identify situations in which the enhancement is able to survive.
PACS numbers: 42.50.Dv, 42.65.Tg, 42.81.Dp
I. INTRODUCTION
It has been suggested that the use of correlated photons is
able to enhance the position accuracy of an optical pulse be-
yond the standard quantum limit, and the enhancement can be
useful for positioning and clock synchronization applications
[1]. Generation of two photons with the requisite correlation
has been demonstrated experimentally by Kuzucu et al. [2],
but in practice it is more desirable to produce as many corre-
lated photons as possible in order to obtain a higher accuracy.
To achieve quantum enhancement for a large number of pho-
tons, a scheme of adiabatically manipulating optical fiber soli-
tons has recently been proposed [3], opening up a viable route
of applying quantum enhancement to practical situations. The
analysis in Ref. [3] assumes that the optical fibers are loss-
less, so the Heisenberg limit [4] can be reached in principle.
In reality, however, the quantum noise associated with optical
loss increases the soliton timing jitter and limits the achievable
enhancement. Compared with the use of solitons for quadra-
ture squeezing [5], the adiabatic soliton control scheme poten-
tially suffers more from decoherence, because the soliton must
propagate for a longer distance to satisfy the adiabatic approx-
imation. The effect of loss on a similar scheme of soliton mo-
mentum squeezing has been studied by Fini and Hagelstein
[6], although they did not study the timing jitter evolution rel-
evant to the scheme in Ref. [3], and did not take into account
possible departure from the adiabatic approximation.
In this paper, the decoherence effect of optical loss on the
timing accuracy enhancement scheme proposed in Ref. [3] is
investigated in depth, in order to evaluate the performance of
the scheme in practice. Instead of approaching the problem in
the Schrödinger picture like prior work [1, 3, 6, 7], this paper
primarily utilizes Heisenberg equations of motion, since they
are able to account for dissipation and fluctuation in a more el-
egant way. For simplicity, scalar solitons, as opposed to vector
solitons studied in Ref. [3], are considered here. The theoret-
ical and numerical analyses show that, despite an increase in
timing jitter due to quantum noise and deviation from the adi-
abatic approximation, an appreciable enhancement can still be
achieved using a realistic setup.
The developed formalism is also used to study the propa-
gation of an optical pulse with quantum-enhanced timing ac-
curacy in a lossy, dispersive, and nonlinear medium, such as
an optical fiber, in order to identify situations in which the
enhancement can still survive. The effect of loss on many
correlated photons sent in as many channels has been inves-
tigated by Giovannetti et al. [1], but their analysis focuses on
a relatively small number of correlated photons and does not
include the effects of dispersion and nonlinearity.
This paper is organized as follows: Section II defines the
general theoretical framework, and derives the standard quan-
tum limits and Heisenberg limits on the variances of the pulse
position and momentum operators. Section III studies the evo-
lution of such operators in the presence of loss, group-velocity
dispersion, and Kerr nonlinearity, and determines the effect of
dissipation and fluctuation on the position and momentum un-
certainties. Section IV investigates theoretically and numeri-
cally the impact of optical loss on the adiabatic soliton con-
trol scheme using realistic parameters, while Sec. V studies
the decoherence effect on the transmission of the quantum-
enhanced timing information in various linear and nonlinear
systems.
II. THEORETICAL FRAMEWORK
A. Definition of pulse position and momentum operators
The positive-frequency electric field of a waveguide mode
at a certain longitudinal position can be defined as [8]
Ê(+)(t) = i
4πε0cn2S
ĉ(ω)e−iωt , (1)
where n is the refractive index, η is the real part of n, S is the
transverse area of the waveguide mode, and ĉ(ω) is the photon
annihilation operator. The annihilation operator is related to
the corresponding creation operator via the commutator [8],
[ĉ(ω), ĉ†(ω ′)] = δ (ω −ω ′). (2)
For a pulse with a slowly-varying envelope compared with the
optical frequency, the coefficient in front of the annihilation
operator can be assumed to be independent of frequency and
http://arxiv.org/abs/0704.0663v1
can be evaluated at the carrier frequency ω0, so that the elec-
tric field is proportional to the temporal envelope annihilation
operator Â(t),
Ê(+)(t) ∝ Â(t)e−iω0t , (3)
Â(t)≡ 1√
dω â(ω)e−iωt , (4)
â(ω)≡ ĉ(ω +ω0). (5)
The temporal envelope operator Â(t) and the spectral operator
â(ω) evidently also satisfy the following commutation rela-
tions with their corresponding creation operators,
[Â(t), Â†(t ′)] = δ (t − t ′), (6)
[â(ω), â†(ω ′)] = δ (ω −ω ′). (7)
The total photon number operator can be defined as
dt Â†(t)Â(t), (8)
and the pulse center position operator as [9]
T̂ ≡ 1
dt tÂ†(t)Â(t), (9)
where
is the average photon number. This definition uses 1/N as
the normalization coefficient, instead of the inverse photon
number operator N̂−1 used by Lai and Haus [10], in order
to express the position operator in terms of normally ordered
optical field operators that are easier to handle, as well as to
avoid the potential problem of applying N̂−1 on the vacuum
state. As long as the photon-number fluctuation is small, the
position operator naturally corresponds to the measurement of
the center position of the pulse intensity profile. An average
longitudinal momentum operator can be similarly defined,
Ω̂ ≡ 1
dω ω â†(ω)â(ω)
dt Â†(t)
Â(t). (11)
If the quantum state is close to a large-photon-number coher-
ent state, Â can be approximated as
+δ Â, with O(δ Â)≪
O(Â). Equations (9) and (11) then become the approximate
position and momentum operators defined by Haus and Lai
for solitons in a linearized approach [11]. The linearized ex-
pressions also describe how they can be accurately measured
in practice using balanced homodyne detection [11].
For simplicity, we shall hereafter assume that
= 0 and
= 0 [9]. In the systems considered in this paper, these
two quantities remain constant throughout propagation, if t
is regarded as the retarded time in the moving frame of the
optical pulse.
The commutator between the position and momentum op-
erators is
[T̂ ,Ω̂] =
. (12)
By the Heisenberg uncertainty principle,
[T̂ ,Ω̂]
. (13)
B. Derivation of standard quantum limits
The standard quantum limits and Heisenberg limits on
should be expressed in terms of the pulse
width ∆t, defined as
dt t2Â†(t)Â(t)
, (14)
and the bandwidth ∆ω ,
dω ω2â†(ω)â(ω)
dt Â†(t)
Â(t)
. (15)
To calculate the standard quantum limit on the position uncer-
tainty, consider the expansion
dω ω â†â
dω ′ ω ′â′†â′
, (16)
where we have written â= â(ω) and â′ = â(ω ′) as shorthands.
Rearranging the operators,
dω ω2â†â
dω ′ ωω ′â†â′†ââ′
. (17)
The first term on the right-hand side of Eq. (17) is proportional
to ∆ω2, while the second term contains a normally ordered
cross-spectral density. To derive the standard quantum limit,
we shall assume that the cross-spectral density satisfies the
factorization condition:
â†â′†ââ′
â†â
â′†â′
. (18)
This condition is always satisfied by any pure or mixed state
with only one excited optical mode, such as a coherent state
[12, 13]. The second term on the right-hand side of Eq. (17)
becomes
dω ′ ωω ′
â†â′†ââ′
, (19)
which is assumed to be zero, as per the convention of this
paper. Thus, the variance of Ω̂ is
, (20)
where the subscript “coh” denotes statistics of coherent fields
[12, 13] given by Eq. (18). By virtue of the Heisenberg uncer-
tainty principle given by Eq. (13), the standard quantum limit
on the position variance is hence
4N∆ω2
. (21)
This limit is applicable to any pure or mixed state, and is con-
sistent with the one suggested by Giovannetti et al. for Fock
states [1]. A very similar derivation of the limit for Fock states
and coherent states is also performed by Vaughan et al. [9].
Owing to Fourier duality of position and momentum in the
slowly-varying envelope regime, the standard quantum limit
on the momentum can be derived in the same way. The vari-
ance of T̂ , assuming coherent-field statistics, is
, (22)
and the standard quantum limit on the momentum variance is
4N∆t2
. (23)
C. Derivation of Heisenberg limits
To derive the Heisenberg limit on the position uncertainty,
one needs an absolute upper bound on the momentum uncer-
tainty
. Consider the following non-negative quantity
proportional to the coherence bandwidth squared,
dω ′ (ω −ω ′)2
â†â′†ââ′
≥ 0. (24)
This quantity is non-negative because (ω − ω ′)2 is non-
negative and
â†â′†ââ′
is also non-negative [13]. It can be
rewritten as
dω ′ (ω −ω ′)2
â†â′†ââ′
dω ′ (ω −ω ′)2
â†ââ′†â′
, (25)
and expanded as
dω ′ (ω2 +ω ′2 − 2ωω ′)â†ââ′†â′
dω ω2â†â
− 2N2
≥ 0. (26)
Here we shall approximate N̂ with N, and neglect any photon-
number fluctuation. This approximation is exact for Fock
states, and acceptable for any quantum state with a small
photon-number fluctuation, such as a large-photon-number
coherent state. We then obtain the following approximate in-
equality,
≤ ∆ω2. (27)
With the Heisenberg uncertainty principle given by Eq. (13)
and the upper bound on
given by Eq. (27), one can then
obtain the Heisenberg limit on the uncertainty of T̂ :
4N2∆ω2
. (28)
Equation (28) is again consistent with the Heisenberg limit
suggested by Giovannetti et al. [1], although the derivation
here shows that it is not only valid for Fock states but also
correct to the first order for any quantum state with a small
photon-number fluctuation.
The Heisenberg limit on
is similar,
4N2∆t2
. (29)
A more exact derivation of the Heisenberg limits is given in
Appendix A, where the inverse photon-number operator N̂−1
is used instead of 1/N in the definitions of T̂ , Ω̂, ∆ω , and
∆t. The difference between the approximate Heisenberg limits
derived here and the exact Heisenberg limits in Appendix A is
negligible for small photon-number fluctuations.
III. OPTICAL PULSE PROPAGATION IN THE
HEISENBERG PICTURE
The classical nonlinear Schrödinger equation that describes
the propagation of pulses in a lossy, dispersive, and nonlinear
medium, such as an optical fiber, is given by [14]
−κ |A|2A−
A, (30)
where t is the retarded time coordinate in the frame of the
moving pulse, β is the group-velocity dispersion coefficient,
κ is the normalized Kerr coefficient, and α is the loss coeffi-
cient, all of which may depend on z. The phenomenological
quantized version that preserves the commutator between Â
and Â† is [5]
∂ 2Â
−κÂ†ÂÂ−
Â+ iŝ. (31)
Â ≡ Â(z, t) is the pulse envelope annihilation operator in the
Heisenberg picture, and ŝ is the Langevin noise operator, sat-
isfying the commutation relation
[ŝ(z, t), ŝ†(z′, t ′)] = αδ (z− z′)δ (t − t ′). (32)
Rewriting the position and momentum operators in Eqs. (9)
and (11) in the Heisenberg picture as T̂ (z) and Ω̂(z) in terms of
Â(z, t), differenting them with respect to z, and using Eq. (31),
their equations of motion can be derived,
dT̂ (z)
= β (z)Ω̂(z)+ ŜT (z), (33)
dΩ̂(z)
= ŜΩ(z), (34)
where ŜT and ŜΩ are position and momentum noise operators
defined as
ŜT (z)≡
dt tŝ†(z, t)Â(z, t)+H. c., (35)
ŜΩ(z)≡
dt ŝ†(z, t)
Â(z, t)+H. c., (36)
and H. c. denotes Hermitian conjugate. If the noise reservoir
is assumed to be in the vacuum state, the noise operators have
the following statistical properties, as shown in Appendix B,
ŜT (z)
ŜΩ(z)
= 0, (37)
ŜT (z)ŜT (z
α(z)∆t2(z)
δ (z− z′), (38)
ŜΩ(z)ŜΩ(z
α(z)∆ω2(z)
δ (z− z′), (39)
ŜT (z)ŜΩ(z
′)+ ŜΩ(z)ŜT (z
α(z)C(z)
δ (z− z′), (40)
where C(z) is the pulse chirp factor, defined as
C(z)≡
dt Â†(z, t)
Â(z, t)
The average position 〈T̂ (z)〉 and average momentum 〈Ω̂(z)〉
are constant and assumed to be zero throughout propagation.
The variance of Ω̂ is then
Ω̂2(z)
Ω̂2(0)
α(z′)∆ω2(z′)
N(z′)
, (42)
while the variance of T̂ is more complicated due to the pres-
ence of dispersion,
T̂ 2(z)
T̂ 2(0)
T̂ (0)Ω̂(0)+ Ω̂(0)T̂ (0)
dz′β (z′)
Ω̂2(0)
dz′β (z′)
α(z′)∆t2(z′)
N(z′)
dz′β (z′)
α(z′′)C(z′′)
N(z′′)
dz′β (z′)
dz′′β (z′′)
∫ z′′
dz′′′
α(z′′′)∆ω2(z′′′)
N(z′′′)
. (43)
Equation (43) is the central result of this paper. It is similar
to that derived by Haus for optical solitons using a linearized
approach [15], but Eq. (43) is valid for arbitrary loss, arbi-
trary dispersion profile β (z), and arbitrary evolution of pulse
width ∆t(z), chirp C(z), and bandwidth ∆ω(z), so that it is
able to describe the effect of loss on the quantum enhance-
ment scheme proposed in Ref. [3]. The first term on the right-
hand side of Eq. (43) is the initial quantum fluctuation, while
the second and third term on the right-hand side describe the
quantum dispersion effect [16]. In an ideal scenario described
in Ref. [3],
T̂ 2(z)
remains constant if the net dispersion
′β (z′) is zero and quantum dispersion is compensated.
With loss, however, noise introduces a diffusive jitter given
by the fourth term on the right-hand side of Eq. (43),
T̂ 2(z)
α(z′)∆t2(z′)
N(z′)
, (44)
a less well-known chirp-induced jitter given by the fifth term,
T̂ 2(z)
dz′β (z′)
α(z′′)C(z′′)
N(z′′)
, (45)
and also the Gordon-Haus timing jitter [17] given by the sixth
term,
T̂ 2(z)
dz′β (z′)
dz′′β (z′′)
∫ z′′
dz′′′
α(z′′′)∆ω2(z′′′)
N(z′′′)
. (46)
In most cases considered here, N ≫ 1,
≪ ∆t2,
≪ ∆ω2, so one can use the classical nonlinear
Schrödinger equation, Eq. (30), to predict the evolution of
∆t(z), C(z), and ∆ω(z) accurately. The evolution of
T̂ 2(z)
can subsequently be calculated analytically or numerically us-
ing Eq. (43) and the classical evolution of ∆t(z), C(z), and
∆ω(z), analogous to the linearized approach [11, 15].
It is worth noting that the chirp-induced jitter, Eq. (45), de-
pends on the cross-correlation between the position and mo-
mentum noise in Eq. (40), so it can be positive as well as neg-
ative, but the sum of the three sources of jitter must obviously
remain positive.
IV. EFFECT OF LOSS ON ADIABATIC SOLITON
CONTROL
A. Review of the ideal case
Consider the scheme proposed in Ref. [3] and depicted in
Fig. 1. Assume that the dispersion coefficient of the first fiber
β (z) is negative and its magnitude increases along the fiber
slowly compared with the soliton period. The classical soliton
solution of Eq. (30), assuming adiabatic change in parameters
FIG. 1: (Color online) Schematic (not-to-scale) of the adiabatic soli-
ton control scheme. An optical pulse is coupled into a dispersion-
increasing fiber of length L with a negative dispersion coefficient β ,
followed by a much shorter dispersion-compensating fiber of length
L′ with a positive dispersion coefficient β ′.
β (z) and N(z), is [18]
A(z, t) = A0(z)sech
dz′|A0(z′)|2
, (47)
A0(z) =
2τ(z)
, τ(z) =
2|β (z)|
κN(z)
. (48)
The adiabatic approximation is satisfied when
β (z)
dβ (z)/dz
dN(z)/dz
≪ Λ, (49)
where Λ is the soliton period,
Λ(z)≡
τ2(z)
|β (z)|
. (50)
The root-mean-square pulse width ∆t(z) and bandwidth
∆ω(z) then become
∆t(z) =
τ(z) =
|β (z)|
κN(z)
, (51)
∆ω(z) =
3τ(z)
κN(z)
|β (z)| . (52)
The bandwidth ∆ω(z) is thus reduced in the first fiber. If the
second fiber has a positive dispersion coefficient β ′ so that
the net dispersion is zero (
0 dzβ (z)+β
′L′ = 0), the quantum
dispersion effect given by the second and third term on the
right-hand side of Eq. (43) can be eliminated. Furthermore, if
β ′ has a very large magnitude compared with β (z) so that the
second fiber can be very short compared with the first fiber, the
effective nonlinearity experienced by the pulse in the second
fiber can be neglected, and ∆ω(z) remains essentially constant
in the second fiber. In the lossless case, the final timing jitter
T̂ 2(L+L′)
is therefore the same as the input
T̂ 2(0)
, but
∆ω(L+L′) has been reduced and the standard quantum limit
T̂ 2(L+L′)
, Eq. (21), is raised. Provided that the initial
timing jitter of a laser pulse obeys the coherent-field statistics
given by Eq. (22), the final timing jitter is
T̂ 2(L+L′)
T̂ 2(0)
∆t2(0)
β 2(0)
, (53)
while the final standard quantum limit is
T̂ 2(L+L′)
4N∆ω2(L+L′)
3β 2(L)
. (54)
A timing jitter squeezing ratio, analagous to the squeezing ra-
tio defined by Haus and Lai [11], can be defined as
T̂ 2(L+L′)
T̂ 2(L+L′)
β 2(0)
β 2(L)
. (55)
The factor of π2/9 arises because the initial jitter for a sech
pulse shape is slightly higher than the standard quantum limit
given by Eq. (21) in terms of the bandwidth. As long as β (L)
at the end of the first fiber is significantly larger than the initial
value, the timing jitter becomes lower than the raised standard
quantum limit, R becomes smaller than 1, and quantum en-
hancement of position accuracy is accomplished. This semi-
classical analysis is valid in all practical cases, where N ≫ 1,
R ≫ 1/N,
≪ ∆t2,
≪ ∆ω2, and is consistent with
the analysis of exact quantum soliton theory in Ref. [3]. R
is related to the quantum enhancement factor γ defined in
Ref. [3] by R = 1/γ2. The semiclassical analysis is no longer
valid when R is close to the Heisenberg limit 1/N, but as the
next sections will show, owing to decoherence effects, it is ex-
tremely difficult for the enhancement to get anywhere close to
the Heisenberg limit.
B. Numerical analysis of a realistic case
To investigate the impact of noise and the validity of the
adiabatic approximation in practice, a numerical evaluation of
∆t(z), C(z), ∆ω(z), and
T̂ 2(z)
, using Eqs. (30) and (43) and
realistic parameters, is necessary. β (z) is assumed to have the
following profile used in Ref. [19],
β (z) =
−12.75 ps2/km
1+(L− z)/Lβ
. (56)
Lβ = 1 km is used here instead of the Lβ = 1/12 km used
in Ref. [19], in order to satisfy the adiabatic approximation
for a longer pulse in this example. Other fiber parameters are
α = 0.4 dB/km, n2 = 2.6×10−16 cm2/W, Aeff = 30 µm2 [19],
λ0 = 1550 nm, ω0 = 2πc/λ0, so that κ = h̄ω0(ω0n2/cAeff).
L is assumed to be 2 km. A dispersion-compensating fiber
with β ′ = 127.5 ps2/km, α = 0.4 dB/km, n2 = 2.7× 10−16
cm2/W, Aeff = 15 µm2 [20], and L′ = 110 m is used in the
numerical analysis as the second fiber. The classical nonlin-
ear Schrödinger equation, Eq. (30), is numerically solved us-
ing the Fourier split-step method [14]. An initial sech soliton
pulse with τ(0) = 1 ps, N(0) = 1.9×107, and an initial energy
of 2.4 pJ is assumed.
Figure 2 plots the numerical evolution of pulse intensity and
spectrum in the two fibers. As expected, the bandwidth is nar-
rowed in the first fiber and remains approximately constant
in the second (z > 2000 m), owing to the latter’s relative short
length. Figure 3 plots the evolution of pulse width ∆t(z), chirp
−8 −6
−4 −2 0 2 4 6 8
z (m)
|A(z, t)|2
t (ps)
1.5 0
z (m)
|a(z, ω)|2
ω (THz)
FIG. 2: (Color online) Numerical evolution of pulse intensity (top)
and spectrum (bottom). The denser plots for z > 2000 m indicate
pulse propagation in the second fiber. The color codes are in the
same arbitrary units as the heights of the plots.
C(z), and bandwidth ∆ω(z), compared with the adiabatic ap-
proximation, Eqs. (51) and (52). The adiabatic approximation
is evidently not exact, and the pulse acquires a chirp due to
excess dispersion in the first fiber, leading to slight refocusing
in the second fiber. The bandwidth is reduced by a factor of
2.2, as opposed to the ideal factor of 3.6.
Figure 4 plots the evolution of the diffusive jitter given by
Eq. (44), the chirp-induced jitter given by Eq. (45), and the
Gordon-Haus jitter given by Eq. (46). It can be seen that al-
though the Gordon-Haus jitter increases much more quickly
than the other jitter components in the first fiber, the for-
mer drops abruptly in the second fiber (z > 2000 m) due
to the opposite dispersion. This kind of Gordon-Haus jit-
ter reduction by dispersion management is well known [21].
The chirp-induced jitter component drops below zero in the
second fiber, but as noted before, the total noise jitter re-
mains positive. The final jitter values are numerically deter-
mined to be
T̂ 2(L+L′)
= 0.71
T̂ 2(0)
T̂ 2(L+L′)
−0.93
T̂ 2(0)
, and
T̂ 2(L+L′)
= 1.42
T̂ 2(0)
, result-
ing in a total jitter of
T̂ 2(L+L′)
T̂ 2(0)
T̂ 2(L+L′)
T̂ 2(L+L′)
T̂ 2(L+L′)
= 2.19
T̂ 2(0)
. (57)
The final squeezing ratio is hence
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Pulse Width ∆t(z)
Numerical
Adiabatic Approximation
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Chirp C(z)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
z (m)
Bandwidth ∆ω(z)
FIG. 3: (Color online) Evolution of pulse width ∆t(z) (top), chirp
C(z) (center), and bandwidth ∆ω (bottom), compared with the adia-
batic approximation (dash). Plots of ∆t and ∆ω are normalized with
respect to their initial values, respectively.
T̂ 2(L+L′)
T̂ 2(L+L′)
T̂ 2(L+L′)
T̂ 2(0)
N(L+L′)
∆ω2(L+L′)
∆ω2(0)
= 0.42 =−3.8 dB. (58)
Despite taking into account the increased timing jitter and the
non-ideal bandwidth narrowing, a squeezing ratio of −3.8 dB
is predicted by the numerical analysis, suggesting that one
should be able to observe the quantum enhancement experi-
mentally using current technology.
C. Potential improvements
As shown in the previous section, the Gordon-Haus effect
contributes the largest amount of noise in the soliton control
scheme, despite its partial reduction by dispersion manage-
ment. Its magnitude at the end of the first fiber can be esti-
mated roughly as
T̂ 2(L)
T̂ 2(L)
(αL). (59)
As the length of the first fiber must be at least a few times
longer than the soliton period Λ for the adiabatic approxima-
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Diffusive Jitter
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Chirp-Induced Jitter
0 200 400 600 800 1000 1200 1400 1600 1800 2000
z (m)
Gordon-Haus Jitter
FIG. 4: (Color online) Evolution of diffusive jitter (top), chirp-
induced jitter (center), and Gordon-Haus jitter (bottom). All plots
are normalized with respect to the initial jitter
T̂ 2(0)
tion to hold and for the bandwidth to be significantly reduced,
L/Λ is approximately fixed, and the Gordon-Haus jitter can
be reduced only if a figure of merit,
FOM ≡ 1
, (60)
is enhanced. Since this is a rough order-of-magnitude esti-
mate, a representative value of Λ, say at z = L, can be used.
The figure of merit suggests that the performance of the soli-
ton control scheme can be improved by reducing the pulse
width, increasing the overall dispersion coefficient, or reduc-
ing the loss coefficient.
Reducing the pulse width is the most convenient way of
obtaining better enhancement, as the adiabatic bandwidth re-
duction can be achieved over a shorter distance with less loss
of photons. For example, using τ(0) = 500 fs, L = 1 km,
Lβ = 0.3 km, L
′ = 44 m, and otherwise the same parame-
ters as in Sec. IV B, the squeezing ratio becomes −6.0 dB,
while using τ(0) = 200 fs, L = 500 m, Lβ = 1/12 km, and
L′ = 16.2 m gives a squeezing ratio of −7.3 dB. The shorter
pulse width, however, significantly enhances higher-order dis-
persive and nonlinear effects. Raman scattering, in particular,
contributes additional quantum noise because of coupling to
optical phonons [22]. It is beyond the scope of this paper to
investigate these higher-order effects, so a more conservative
pulse width of 1 ps is used in the preceding section. A larger
overall dispersion coefficient, on the other hand, means that
more photons or a higher nonlinearity are required for a soli-
ton to form, so the Raman effect may also become more sig-
nificant with a larger dispersion coefficient. The Raman effect
can be reduced by cooling the fiber and reducing the number
of thermal phonons [22], if it becomes a significant problem.
Further advance in optical fiber technology should be able
to increase the figure of merit by reducing loss, since the spe-
cialty fibers assumed in Sec. IV B have a higher loss than usual
transmission fibers by a factor of two. Using α = 0.2 dB/km
instead of 0.4 dB/km in Sec. IV B, for instance, reduces the
squeezing ratio to −4.7 dB. Spectral filtering or frequency-
dependent gain [23] provides another way of controlling the
Gordon-Haus effect, although it adds another level of com-
plexity to the experimental setup, and it is beyond the scope
of this paper to investigate how the frequency-dependentdissi-
pation or amplification might help the quantum enhancement
scheme. Finally, the design of the setup assumed in Sec. IV B
is not fully optimized, and further optimization of parame-
ters, fiber dispersion profiles, and bandwidth narrowing strat-
egy should be able to improve the enhancement.
V. EFFECT OF LOSS ON THE TRANSMISSION OF
QUANTUM-ENHANCED TIMING INFORMATION
Provided that quantum enhancement of pulse position ac-
curacy can be achieved, the information still needs to be
transmitted through unavoidably lossy channels. It is hence
an important question to ask how loss affects the quantum-
enhanced information in optical information transmission sys-
tems. Equation (43) governs the general evolution of the tim-
ing jitter under the effects of loss, dispersion, and nonlinearity,
but in order to estimate the relative magnitude of the decoher-
ence effects and gain more insight into the decoherence pro-
cesses, in this section Eq. (43) is explicitly solved for various
systems and compared with the standard quantum limit.
A. Linear non-dispersive systems
Without dispersion, the timing jitter increases only due to
the diffusive component
T̂ 2(z)
. An analytic expression for
T̂ 2(z)
can then be derived from Eq. (43), as ∆t(z) and ∆ω(z)
remain constant,
T̂ 2(z)
T̂ 2(0)
∆t2(0)
(1− e−αz). (61)
If the initial variance obeys coherent-field statistics, that is,
T̂ 2(0)
= ∆t2(0)/N(0) according to Eq. (22), the subsequent
jitter is
T̂ 2(z)
∆t2(0)
, (62)
and obeys the same coherent-field statistics but for the re-
duced photon number N(z). This is consistent with intu-
ition. On the other hand, in the high loss limit (αz ≫ 1), the
term ∆t2(0)/N(z) is likely to dominate over the initial jitter
T̂ 2(0)
, so in most cases the position of a significantly atten-
uated pulse relaxes to coherent-field statistics independent of
its initial fluctuation. This justifies the assumption in Sec. IV
that a laser pulse exiting a laser cavity has such statistics, re-
gardless of the quantum properties of the pulse inside the cav-
Equation (61) can be renormalized as
R(z)≡
T̂ 2(z)
T̂ 2(z)
= R(0)e−αz + 4∆t2(0)∆ω2(0)(1− e−αz). (63)
≪ ∆t2 and
≪ ∆ω2, classical theory pre-
dicts that 4∆t2∆ω2 ≈ 1. Equation (63) then suggests that the
relative increase in timing jitter is independent of the initial
squeezing ratio R(0). This is nevertheless not true in general,
as ∆t may depend on both ∆ω and R when the classical theory
fails. In Appendix C, the exact dependence of ∆t on ∆ω and R
is calculated for a specific multiphoton state with a Gaussian
pulse shape called the jointly Guassian state. The expression
4∆t2∆ω2 is given by
4∆t2∆ω2 =
(1− 1/N)2
1− 1/(NR)
, (64)
which results in the following exact expression for an initial
jointly Gaussian state,
R(z) = R(0)e−αz +
(1− 1/N)2
1− 1/(NR)
(1− e−αz). (65)
For a large photon number (N ≫ 1) and moderate enhance-
ment (1 ≥ R ≫ 1/N), 4∆t2∆ω2 ≈ 1, as classical theory would
predict for a Gaussian pulse. In this regime, the quantum-
enhanced information is just as sensitive to loss as standard-
quantum-limited information. When R gets close to the
Heisenberg limit 1/N, however, ∆t∆ω approaches infinity.
This is because maximal coincident-frequency correlations
are required to achieve the Heisenberg limit [1], but heuris-
tically speaking, if the photons have exactly the same mo-
mentum, they must have infinite uncertainties in their rela-
tive positions, leading to an infinite pulse width ∆t. Owing to
the abrupt increase in 4∆t2∆ω2 when R approaches 1/N, the
quantum enhancement becomes much more sensitive to loss.
In the Heisenberg limit of R → 1/N, ∆t → ∞, any loss com-
pletely detroys the timing accuracy and leads to an infinite
jitter, according to Eq. (65).
B. Linear dispersive systems
If the system is lossy, dispersive, but linear, it is not difficult
to show that
∆t2(z) = ∆t2(0)+C(0)
dz′β (z′)
+∆ω2(0)
dz′β (z′)
, (66)
C(z) =C(0)+ 2∆ω2(0)
dz′β (z′), (67)
∆ω2(z) = ∆ω2(0). (68)
The following result can then be obtained from Eq. (43) after
some algebra,
T̂ 2(z)
T̂ 2(0)
T̂ (0)Ω̂(0)+ Ω̂(0)T̂ (0)
dz′β (z′)
Ω̂2(0)
dz′β (z′)
∆t2(z)
(1− e−αz).
This result is similar to that in the previous section, except for
the presence of quantum dispersion and the dispersive spread
of the pulse width ∆t(z) that leads to increased jitter. With ini-
tially coherent-field statistics,
T̂ 2(0)
Ω̂2(0)
are given
by Eqs. (22) and (20), respectively, while by similar argu-
ments, the coherent-field statistics for
T̂ Ω̂+ Ω̂T̂
T̂ Ω̂+ Ω̂T̂
. (70)
This leads to the following position variance for a pulse with
initially coherent-field statistics,
T̂ 2(z)
∆t2(z)
, (71)
which still maintains the coherent-field statistics for the dis-
persed pulse width and the reduced photon number. In the
high loss limit (αz ≫ 1), the coherent-field statistics is again
approached regardless of the initial conditions.
For an initial jointly Gaussian quantum state, on the other
hand, the normalized version of Eq. (69) is
R(z) =
R(0)+
4∆t2(0)∆ω2(0)+ 4ζ 2
(1− e−αz), (72)
where ζ is the normalized effective propagation distance,
ζ ≡ ∆ω2(0)
dz′β (z′), (73)
and 4∆t2(0)∆ω2(0) is given by Eq. (64) evaluated at z= 0. As
long as the loss is moderate so that e−αz ≫ 1−e−αz, quantum
dispersion, given by the term proportional ζ 2/R(0), becomes
the dominant effect and overwhelms the initial enhancement
when ζ exceeds R(0)/2.
If the net dispersion
′β (z′) is zero, both quantum and
classical dispersion are eliminated, and the jitter growth be-
comes identical to that in a non-dispersive and linear system
given by Eq. (61).
C. Soliton-like systems
The previous sections show that coherent-field statistics is
maintained in a linear system, but as Sec. IV clearly shows,
non-trivial statistics can arise from the quantum dynamics of
a nonlinear system. The complex evolution of ∆t(z), C(z), and
∆ω(z) in general prevents one from solving Eq. (43) explic-
itly, except for special cases such as solitons.
If the dispersion is constant and the pulse propagates in the
fiber as a soliton, C(z) is zero, while ∆t(z) and ∆ω(z) can be
regarded as constant if
≪∆t2 and
≪∆ω2 through-
out propagation. Equation (43) can then be solved explicitly,
T̂ 2(z)
T̂ 2(0)
Ω̂2(0)
β 2z2 +
∆t2(0)
(eαz − 1)
2β 2∆ω2(0)
eαz − 1
, (74)
where
T̂ (0)Ω̂(0)+ Ω̂(0)T̂ (0)
is asssumed to be zero for
simplicity. If 4
T̂ 2(0)
Ω̂2(0)
= (π2/9)[1/N2(0)] is also
assumed for a soliton pulse for simplicity, Eq. (74) can be
normalized to give
R(z) = R(0)e−αz +
e−αz +
(eαz − 1)
eαz − 1
. (75)
In the low loss regime with αΛ ≪ 1 and αz ≪ 1, Eq. (75) can
be further simplified,
R(z)≈ R(0)+
(αz). (76)
Quantum dispersion is again the dominant effect in this
regime, while decoherence effects are much smaller, by a fac-
tor of αz approximately.
Even if the net dispersion is zero and quantum dispersion is
compensated, the Gordon-Haus effect cannot be completely
eliminated by dispersion management in the presence of non-
linearity and may become significant, as the numerical anal-
ysis in Sec. IV B shows. An order-of-magnitude estimate of
Gordon-Haus jitter can be performed by considering soliton
propagation in a constant negative dispersion fiber, just as
in the previous case, followed by a dispersion-compensating
fiber of length L′ with positive dispersion coefficient β ′. If
L′ is short, the effective nonlinearity experienced by the pulse
in the second fiber can be neglected, and ∆ω(z) can be re-
garded as constant. Assuming that β L+β ′L′ = 0, the integral
in Eq. (46) can be solved to give the Gordon-Haus jitter,
T̂ 2(L+L′)
≈ α∆ω
6N(0)
β 2L2(L+L′)
≈ α∆ω
6N(0)
β 2L3. (77)
The normalized contribution to the squeezing ratio is therefore
T̂ 2(L+L′)
T̂ 2(L+L′)
(αL). (78)
Compared with the Gordon-Haus jitter at the end of the first
fiber given by the last term of Eq. (76), dispersion manage-
ment cuts the jitter by half, but the expression maintains its
functional dependence on the parameters of the first fiber.
This estimate also justifies the use of Eq. (59) to estimate the
Gordon-Haus jitter at the end of the two fibers in Sec. IV C. To
minimize the impact of Gordon-Haus jitter on the quantum-
enhanced timing accuracy in a dispersion-managed soliton
system, the condition
L3 ≪ 54
R (79)
is required.
VI. CONCLUSION
In conclusion, the decoherence effect by optical loss on
adiabatic soliton control and on the transmission of quantum-
enhanced timing information has been extensively studied. It
is found that an appreciable enhancement can still be achieved
by the soliton scheme using current technology, despite an in-
crease of timing jitter due to the presence of loss. It is also
found that the quantum-enhanced timing accuracy should be
much lower than the Heisenberg-limited accuracy to avoid in-
creased sensitivity to photon loss during transmission, and the
net dispersion in the transmission system should be minimized
in order to reduce quantum dispersion and the Gordon-Haus
effect.
Although the most important pulse propagation effects have
been considered in this analysis, higher-order effects, such as
third-order dispersion, self-steepening, and Raman scattering
[14] might provide further adverse impact on the quantum en-
hancement if the optical pulse is ultrashort. In particular, the
inelastic Raman scattering process is expected to be a signifi-
cant source of decoherence for ultrashort pulses [22]. It is be-
yond the scope of this paper to investigate these higher-order
effects, but they should be of minor importance for picosec-
ond pulses and the propagation distances considered in this
paper.
Finally, it is worth noting that while this paper focuses on
optical pulses, the developed formalism is equally valid for
describing the transverse position and momentum of optical
beams [24] and the center-of-mass variables of Bose-Einstein
condensates [9]. Decoherence by loss of particles in those
systems can be studied using the formalism developed in this
paper and parameters specific to those systems.
VII. ACKNOWLEDGMENTS
This work is financially supported by the DARPA Center
for Optofluidic Integration and the National Science Founda-
tion through the Center for the Science and Engineering of
Materials (DMR-0520565).
APPENDIX A: DERIVATION OF EXACT HEISENBERG
LIMITS
An exact Heisenberg limit can be derived if the inverse
photon-number operator N̂−1 is used instead of 1/N in the
definitions of T̂ , Ω̂, ∆t, and ∆ω in Eqs. (9), (11), (14), and
(15), just as in Refs. [9] and [10],
T̂ ′ ≡ N̂−1
dt tÂ†Â, (A1)
Ω̂′ ≡ N̂−1
dω ω â†â, (A2)
∆t ′ ≡
dt t2Â†Â
, (A3)
∆ω ′ ≡
dω ω2â†â
. (A4)
These operators are well defined as long as the quantum state
has zero vacuum-state component (〈0|ρ̂|0〉 = 0). Starting
from the Heisenberg uncertainty principle for T̂ ′ and Ω̂′,
T̂ ′2
, (A5)
and the inequality
dω ′ (ω −ω ′)2â†â′†ââ′
≥ 0, (A6)
one can obtain the exact inequality
≤ ∆ω ′2, (A7)
and the exact Heisenberg limit for the new position operator,
T̂ ′2
4∆ω ′2
. (A8)
The difference between Eqs. (28) and (A8) is negligible for
small photon-number fluctuations. The exact Heisenberg limit
is similar.
APPENDIX B: NOISE STATISTICS
In this section the expression
ŜT (z)ŜT (z
in Eq. (38) is
calculated. The derivations of
ŜΩ(z)ŜΩ(z
in Eq. (39) and
ŜT (z)ŜΩ(z
′)+ ŜΩ(z)ŜT (z
in Eq. (40) are similar. Substi-
tuting Eq. (35) into Eq. (38) gives
ŜT (z)ŜT (z
dt ′ tt ′
ŝ†Âŝ′†Â′
ŝ†ÂÂ′†ŝ′
Â†ŝÂ′†ŝ′
Â†ŝŝ′†Â′
, (B1)
where N = N(z), N′ = N(z′), ŝ = ŝ(z, t), Â = Â(z, t), ŝ′ =
ŝ(z′, t ′), and Â′ = Â(z′, t ′). If the noise reservoir is in the vac-
uum state, ŝ|0reservoir〉= 〈0reservoir|ŝ† = 0, so only the last term
in Eq. (B1) is non-zero,
ŜT (z)ŜT (z
dt ′ tt ′
Â†ŝŝ′†Â′
dt ′ tt ′
Â†Â′
αδ (z− z′)δ (t − t ′)+
Â†ŝ′† ŝÂ′
δ (z− z′)
dt ′ tt ′
Â†ŝ′†ŝÂ′
. (B2)
The first term on the right-hand side of Eq. (B2) is the desired
result, while the second term can be rewritten as
dt ′ tt ′
Â†ŝ′†ŝÂ′
dt ′ tt ′
Â†, ŝ′†
ŝ, Â′
. (B3)
If the system is linear, the commutator between ŝ and Â is
always zero [13], but because ŝ does not commute with Â†
and Â is coupled to Â† by the nonlinear term in Eq. (31), ŝ
may fail to commute with Â. That said, it can be argued that
the optical field operator must always commute with future
noise operators due to causality and the infinitesimally short
memory of ŝ,
Â†, ŝ′†
= 0 if z < z′, (B4)
ŝ, Â′
= 0 if z > z′, (B5)
so Eq. (B3) can be non-zero only at z = z′. The commutator
between ŝ and Â at z = z′ due to the parametric coupling of Â
and Â† can be estimated by a perturbative technique. Consider
an integral form of Eq. (31) with the nonlinear term and the
Langevin noise term only,
Â(z+∆z) = Â(z)+
∫ z+∆z
iκÂ†(z′)Â(z′)Â(z′)+ ŝ(z′)
and Â†(z′) given by the Hermitian conjugate of Eq. (B6),
Â†(z′) = Â†(z)+
−iκÂ†(z′′)Â†(z′′)Â(z′′)+ ŝ†(z′′)
The commutator between ŝ and Â at z+∆z becomes
[ŝ(z+∆z), Â(z+∆z)]
∫ z+∆z
ŝ(z+∆z), Â†(z′)
Â(z′)Â(z′). (B8)
ŝ(z+∆z) commutes with Â(z′) because z+∆z > z′, while it
fails to commute with Â†(z′) because Â†(z′) given by Eq. (B7)
depends explicitly on ŝ†. Thus, in the leading order of ∆z,
[ŝ(z+∆z), Â(z+∆z)]
∫ z+∆z
ŝ(z+∆z), ŝ†(z′′)
Â(z′)Â(z′), (B9)
which approaches 0 in the limit of ∆z → 0. Hence ŝ commutes
with Â at z= z′, and the position noise is given only by the first
term on the right-hand side of Eq. (B2), resulting in Eq. (38).
APPENDIX C: THE JOINTLY GAUSSIAN STATE
A Fock state can be expressed as [13, 25]
dω1 . . .
dωN φ(ω1, . . . ,ωN)|ω1, . . . ,ωN〉,
dt1 . . .
dtN ψ(t1, . . . , tN)|t1, . . . , tN〉, (C1)
where the spectral and temporal eigenstates are given by
|ω1, . . . ,ωN〉 ≡
â†(ω1) . . . â†(ωN)|0〉, (C2)
|t1, . . . , tN〉 ≡
Â†(t1) . . . Â
†(tN)|0〉. (C3)
Theses states are eigenstates of the following operators rele-
vant to our purpose,
Ω̂|ω1, . . . ,ωN〉=
|ω1, . . . ,ωN〉,
T̂ |t1, . . . , tN〉=
|t1, . . . , tN〉, (C5)
dω ω2â†â|ω1, . . . ,ωN〉=
|ω1, . . . ,ωN〉,
dt t2Â†Â|t1, . . . , tN〉=
|t1, . . . , tN〉. (C7)
φ(ω1, . . . ,ωN) is the spectral multiphoton probability ampli-
tude, and it is related to the temporal probability amplitude
ψ(t1, . . . , tN) by the N-dimensional Fourier transform in the
slowly-varying envelope regime. Both amplitudes should also
satisfy normalization and boson symmetry. To study temporal
quantum enhancement, it is convenient to define the probabil-
ity amplitude as a jointly Gaussian function [25],
φ(ω1, . . . ,ωN) =C exp
, (C8)
ψ(t1, . . . , tN) =C′ exp
−N2B2
, (C9)
where B and b are arbitrary and real constants, and C and C′
are normalization constants. Explicit expressions for
, ∆ω2, and ∆t2 can be obtained using Eqs. (C4)-(C7) and
Appendix B of Ref. [25],
= B2, (C10)
4N2B2
, (C11)
∆ω2 = B2 +
b2, (C12)
∆t2 =
4N2B2
. (C13)
In the limit of b → 0,
reaches the Heisenberg limit,
4N2∆ω2
, (C14)
and the quantum state can be written as a state of photons with
maximal coincident-frequency correlations,
|N〉 ∝
dω exp
|ω , . . . ,ω〉. (C15)
On the other hand, when B2 = b2/N,
is at the standard
quantum limit,
4N∆ω2
, (C16)
the quantum state has only one excited Gaussian mode [25],
|N〉 ∝
dω1 . . .
|ω1, . . . ,ωN〉
dω exp
â†(ω)
|0〉, (C17)
and therefore also satisfies the coherent-field statistics [12,
13]. These limits and the corresponding quantum states are
consistent with those suggested in Ref. [1]. With Eqs. (C12)
and (C13), the pulse width ∆t can be determined explicitly in
terms of ∆ω and the squeezing ratio R = ∆ω2/(NB2),
∆t2 =
(1− 1/N)2
1− 1/(NR)
. (C18)
[1] V. Giovannetti, S. Lloyd, and L. Maccone, Nature (London)
412, 417 (2001); V. Giovannetti, S. Lloyd, and L. Maccone,
Phys. Rev. A 65, 022309 (2002).
[2] V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong,
Phys. Rev. Lett. 88, 183602 (2002); O. Kuzucu, M. Fiorentino,
M. A. Albota, F. N. C. Wong, and F. X. Kärtner, Phys. Rev. Lett.
94, 083601 (2005).
[3] M. Tsang, Phys. Rev. Lett. 97, 023902 (2006).
[4] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96,
010401 (2006); V. Giovannetti, S. Lloyd, and L. Maccone, Sci-
ence 306, 1330 (2004).
[5] S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby,
Phys. Rev. Lett. 58, 1841 (1987); P. D. Drummond and S. J.
Carter, J. Opt. Soc. Am. B 4, 1565 (1987); M. J. Potasek and
B. Yurke, Phys. Rev. A 35, 3974 (1987); M. J. Potasek and
B. Yurke, ibid. 38, 1335 (1988); H. A. Haus, Electromagnetic
Noise and Quantum Optical Measurements (Springer, Berlin,
2000).
[6] J. M. Fini and P. L. Hagelstein, Phys. Rev. A 66, 033818 (2002).
[7] P. L. Hagelstein, Phys. Rev. A 54, 2426 (1996); J. M. Fini, P. L.
Hagelstein, and H. A. Haus, ibid. 60, 2442 (1999).
[8] B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992);
R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, ibid. 52,
4823 (1995).
[9] T. Vaughan, P. Drummond, and G. Leuchs, Phys. Rev. A 75,
033617 (2007).
[10] Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989).
[11] H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[12] U. M. Titulaer and R. J. Glauber, Phys. Rev. 140, 676 (1965);
U. M. Titulaer and R. J. Glauber, ibid. 145 1041 (1966).
[13] L. Mandel and E. Wolf, Optical Coherence and Quantum Op-
tics (Cambridge University Press, Cambridge, 1995).
[14] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San
Diego, 2001).
[15] H. A. Haus, J. Opt. Soc. Am. B 8, 1122 (1991).
[16] Y. Lai and H. A. Haus, Phys. Rev. A 40, 854 (1989).
[17] J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
[18] H. H. Kuehl, Opt. Lett. 5, 709 (1988).
[19] V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov,
P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A.
Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A.
N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, J.
Lightwave Technol. 9, 561 (1991).
[20] L. Grüner-Nielsen, M. Wandel, P. Kristensen, C. Jørgensen,
L. V. Jørgensen, B. Edvold, B. Pálsdóttir, and D. Jakobsen, J.
Lightwave Technol. 23, 3566 (2005).
[21] N. J. Smith, W. Forysiak, and N. J. Doran, Electron. Lett. 32,
2085 (1996).
[22] F. X. Kärtner, D. J. Dougherty, H. A. Haus, and E. P. Ippen,
J. Opt. Soc. Am. B 11, 1267 (1994); J. F. Corney and P. D.
Drummond, ibid. 18, 153 (2001).
[23] A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett.
16, 1841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17,
31 (1992).
[24] S. M. Barnett, C. Fabre, and A. Maı̂tre, Eur. Phys. J. D 22,
513 (2003); N. Treps, U. Andersen, B. Buchler, P. K. Lam, A.
Maı̂tre, H.-A. Bachor, and C. Fabre, Phys. Rev. Lett. 88, 203601
(2002); N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Ba-
chor, P. K. Lam, Science 301, 940 (2003).
[25] M. Tsang, “Relationship between resolution enhancement and
multiphoton absorption rate in quantum lithography,” e-print
quant-ph/0607114 (to appear in Phys. Rev. A).
http://arxiv.org/abs/quant-ph/0607114
