Classically, optical solitons are robust nonlinear waves, capable of
long-distance propagation with little or no distortion. Quantum mechanically,
however, solitons experience phase diffusion and wave packet spreading
during propagation. Progress in quantum optics has led to experimental
confirmation of these effects, with demonstrations of entanglement,
sub-shot noise and non-demolition measurements. These phenomena will
be reviewed, along with recent results describing phase noise reduction
in soliton collisions.
Solitons are localized, shape-preserving waves characterized by ‘elastic’
collisions, in which no change in amplitude or velocity occurs. First
observed as a water wave in 1834 by John Scott Russell in the Union
Canal near Edinburgh, and subsequently recreated in the laboratory ,
solitons arise in a variety of physical systems, as both temporal pulses
which counteract dispersion and spatial beams which counteract diffraction.
For recent reviews of solitons, their interactions and applications,
see Stegeman and Segev  and the book by Kivshar and Agrawal .
Temporal soliton pulses in optical fiber were first predicted in 1973
by Hasegawa and Tappert , originating from a balance between self-phase
modulation and anomalous dispersion. The first experimental observation
occurred in 1980 by a group in Bell Labs . Since then, many theoretical
and experimental advances in soliton science have occurred— among
them, the investigation of the fundamental quantum character of these
nonlinear waves. The inherent robustness of these pulses, low loss of
optical fiber, and low noise of mode-locked laser systems have enabled
experimental measurements of soliton pulses at the quantum level. In
this article, we review the basic principles of quantum soliton theory
and highlight relevant experimental results.
|Figure 1: Schematic of quantum soliton
noise evolution and squeezing process. Initial and propagation-induced
noise distributions are shown on the left and right, respectively.
Amplitude fluctuations DA remain constant,
whereas phase noise Dq increases as a
function of propagation distance z. A sub-shot noise, or squeezed,
measurement in a phase-sensitive interferometric detection scheme
can be made at angle j
Quantum Soliton Propagation,
Squeezing, and Entanglement
A pulse in optical fiber undergoes dispersion, or temporal spreading,
during propagation. This effect arises because the refractive index
of the silica glass is not constant, but is rather a function of frequency.
The pulse can be decomposed into a frequency range—the shorter
the pulse, the broader its spectral width. The frequency dependence
of the refractive index will cause the different frequencies of the
pulse to propagate at different velocities, leading to an overall dispersion,
or temporal spreading, of the pulse. As a result, the pulse develops
a chirp, meaning that the individual frequency components are not evenly
distributed throughout the pulse. There are two types of dispersion:
anomalous and normal. If the longer wavelengths travel slower, the medium
is said to have anomalous dispersion. If the opposite is true, the medium
has normal dispersion.
In a Kerr nonlinear medium such as fiber, the refractive index depends
on the intensity of the propagating field. Because the material responds
almost instantaneously, on the order of femtoseconds, each component
of an intense optical pulse sees a phase shift proportional to its intensity.
The leading edge of the pulse is red-shifted, while the trailing edge
is blue-shifted, an effect known as self-phase modulation (SPM). If
the medium has anomalous dispersion, the pulse is compressed. Under
the proper conditions, this pulse compression can exactly cancel the
anomalous dispersion-induced chirp, resulting in distortionless soliton
propagation. For more details, see the book by Agrawal .
Classically, dispersion and Kerr nonlinearity are described by Maxwell’s
equations, completely deterministic phenomena. However, a full quantum
mechanical description of these effects imposes fundamental uncertainties
in the soliton parameters, including amplitude, phase, position, and
frequency. Frequency fluctuations will cause an overall uncertainty
in the amount of dispersion experienced by the pulse, and hence an increasing
uncertainty in the pulse position as a function of propagation distance.
Similarly, amplitude fluctuations will directly affect the amount of
nonlinearly-induced phase shift, leading to an increasing uncertainty
in the soliton phase as the pulse propagates. For reviews of quantum
soliton theory and experiments, see [7, 8].
From the preceding discussion, it is clear the inherent quantum noise
in solitons places a fundamental limit on the processes governing their
propagation. Initially, the quantum noise distribution of the soliton
can be viewed as a circle, as shown in the left of Fig. 1. Amplitude
and phase fluctuations DA and Dq are depicted in the vertical and horizontal
directions, respectively. In the absence of additional noise sources,
amplitude fluctuations remain constant, as shown by the dashed lines.
Balanced direct-detection will measure this noise level, which corresponds
to the shot noise limit.
Due to the Kerr nonlinearity, the circle of Fig. 1 will evolve into
an ellipse, as shown on the right-hand side of Fig. 1 after an arbitrary
propagation distance z. Because of the amplitude-dependent phase shift—the
higher the amplitude, the larger the phase shift—the higher amplitude
fluctuations of the noise distribution will acquire a phase lead over
smaller amplitude fluctuations, as shown by the tilted direction of
the ellipse. Overall propagation-induced phase fluctuations, given by
Dq (z), are larger than the initial phase noise level, as expected by
the phase diffusion process. The major consequence of this evolution
is the possibility of observing squeezing, in which the noise dips below
the standard shot noise limit. A squeezed detection is possible using
a phase-sensitive interferometer. In Fig. 1, maximum squeezing is achieved
at a phase angle j to the amplitude quadrature. The first observation
of soliton squeezing using this method measured a noise level 1.1 dB
below the shot noise limit . More recently, directly detectable amplitude
squeezing 2.3 dB below the shot noise limit was observed using spectral
filtering . For a review of fiber-based squeezing experiments, see
In addition to more accurate measurements, squeezed light can be used
to generate entangled soliton pulses. In fact, the linear interference
between two independently squeezed solitons yields entanglement. This
effect was demonstrated by a group in Erlangen, Germany, in which an
entanglement correlation of 4 dB below the shot noise limit was measured
Solution Collisions and Quantum
Solitons experience ‘elastic’ collisions, in which amplitude
and velocity remain unchanged. However, collisions induce phase and
position shifts. The phase shift experienced by each soliton in a collision
is a function of two sets of parameters: the soliton amplitudes and
velocities. Just as the optical intensity self-induces a phase change
through SPM, each soliton ‘feels’ the intensity of the other
in a collision to cause a phase shift. This type of cross-interaction
is called, appropriately, cross-phase modulation (XPM). The magnitude
of the phase shift depends also on the relative velocity; it increases
as the relative velocity decreases, which is intuitively logical given
the increased interaction length in this regime.
Because of quantum fluctuations in the soliton amplitudes and velocities,
an uncertainty in the amount of collision-induced phase and position
shift will result. In addition, a nonclassical correlation arises between
solitons. For example, if two solitons collide—a signal and probe—a
measurement of the probe phase will provide information on the signal
amplitude, without the need for a direct measurement of the signal soliton.
Moreover, the signal amplitude is undisturbed in the collision, with
additional noise introduced only to the signal phase, the conjugate
observable. Quantum theory thus allows a measurement of a quantum observable
in which no noise, or back-action, is introduced to the measured quantity.
No violation of Heisenberg’s uncertainty principle occurs, because
the inevitable back-action is introduced to the conjugate observable
only. Such a measurement is called a quantum non-demolition (QND) measurement
(for reviews of QND measurements, see [12, 13]). Soliton-based QND measurements
demonstrate several advantages over other kinds of QND schemes, including
simplicity of implementation and compatibility with existing optical
An accurate reading of the probe soliton phase is therefore important,
but two sources of noise tend to limit the effectiveness of the QND
measurement. One noise source is guided acoustic-wave Brillouin scattering
(GAWBS) , caused by thermal fluctuations of the refractive index.
This effect can be efficiently suppressed using a closely spaced reference
pulse in such a way that both the reference and probe solitons acquire
the same GAWBS phase fluctuations [15, 16] or by working at liquid-nitrogen
temperatures . Experiments performed at NTT Laboratories in Japan
found that the propagation-induced process of phase diffusion through
SPM (Fig. 1) accounts for another major source of noise [15, 16]. In
recent work , we have shown that phase noise can be reduced in a
soliton collision because of a negative correlation that exists between
phase fluctuations induced by propagation and collision. This effect
can be exploited to improve QND measurement accuracy.
|Figure 2: Contour plots of variance ratio
(ratio of phase noise variances in collision with respect to no
collision) as function of (a) propagation distance z with dp
= 0.6 and (b) relative velocity dp with
z = 10 as a function of signal amplitude As and fixed probe amplitude
Ap = 1. These plots converge at z = 10
and dp = 0.6, and the dashed curves are
contours of minimum variance ratio. All quantities are in dimensionless,
In order to describe the effect of collision-induced phase noise reduction,
we introduce a variance ratio, defined as the ratio between the post-collision
phase noise variance and the variance in the absence of collision. In
Fig. 2, the dependence of the variance ratio on propagation distance
z and relative velocity dp is plotted as a function of signal amplitude
As, keeping the probe amplitude constant at Ap = 1. From Fig. 2, we
can see that by optimizing the relative amplitudes of the solitons,
the propagation distance, and the relative velocities, a soliton collision
can reduce the phase fluctuations in the probe by more than 20%. The
increase in variance ratio for z > 10, shown in Fig. 2(a) with dp
= 0.6, arises due to SPM-induced phase noise, which tends to drown out
the noise reduction at long propagation distances. In addition, the
variance reduction favors a larger probe for z < 16. For fixed z,
phase noise can be minimized for small dp and, consequently, a small
wavelength separation between probe and signal. This trend is shown
in Fig. 2(b), in which z = 10. It should be noted that a collision-induced
phase noise reduction exists also for the case of two-component vector
As discussed above, under the proper conditions, a soliton collision
can reduce the fluctuations in the probe phase by more than 20%. This,
in turn, can provide a more accurate reading of the signal amplitude
in a QND measurement.
To quantify the effectiveness of a QND measurement, we calculate the
normalized error variance S with which a measurement of the probe phase
will infer the signal amplitude . The best possible measurement
corresponds to a value of S = 0, the minimum achievable inference error
of the signal amplitude, and S = 1 indicates no correlation between
the probe phase and signal amplitude. Contours of S are plotted in Fig.
3 with respect to z and dp as a function of As, again keeping Ap = 1.
For short z, approximately equal probe and signal amplitudes are optimal,
as seen in Fig. 3(a). At longer lengths, however, smaller probes favor
a more accurate measurement. When z = 5, the dependence of S is plotted
as a function of dp and As in Fig. 3(b), which demonstrates an optimal
S for small relative velocity. Clearly, the performance of a soliton-based
QND measurement can be optimized by choosing the experimental parameters
properly. The reduction in phase fluctuations provided by the soliton
collision can be leveraged to significantly reduce the error in the
measurement of the signal amplitude.
|Figure 3: Contour plots of normalized QND
error variance S with respect to (a) propagation distance z with
dp = 0.6 and (b) relative velocity dp
with z = 5 as a function of signal amplitude As and fixed probe
amplitude Ap = 1. These plots converge along the horizontal axis,
and the dashed curves are contours of minimum S. All quantities
are in dimensionless, normalized units.
Quantum information science initially developed from the interdisciplinary
work of scientists in the fields of physics, computer science, mathematics,
and information theory. The field has since experienced an overwhelming
surge in interest, with much research focused on finding practical implementations
to diverse applications in communications, cryptography, and computing.
Remarkable features of fiber optics in quantum information include low
noise and strong coherence. In particular, one benefit of using optical
solitons is the fact that the same nonlinearity which allows soliton
propagation is the driving force behind the quantum evolution of the
soliton, enabling squeezing, entanglement, and non-demolition measurements,
all of which have been demonstrated. In this paper, these phenomena
were reviewed, along with recent work describing quantum phase noise
reduction in soliton collisions, which arises due to a negative cross-correlation
between propagation- and collision-induced phase fluctuations .
This effect may lead to improvements in the accuracy of quantum non-demolition
measurements, taking the measurement further into the quantum regime
and furthering the potential of soliton-based quantum information processing.
The authors acknowledge collaboration with Prof. Ken Steiglitz and many
useful discussions with Prof. Jason Fleischer.
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