Optical Fiber Telecommunications V A || Fiber-optic quantum information technologies

21Fiber-optic quantum informationtechnologies

Prem Kumar*, Jun Chen†, Paul L. Voss†, Xiaoying Li†,

Kim Fook Lee†, and Jay E. Sharping‡

*Technological Institute, Northwestern University, Evanston,IL, USA†Center for Photonic Communication and Computing,EECS Department, Northwestern University, Evanston, IL, USA‡University of California, Merced, CA

21.1 INTRODUCTION

Quantum mechanics (QM), born almost a century ago, is one of the most aston-

ishing pieces of knowledge that human beings have ever discovered about

Nature. Its rules are surprisingly simple: linear algebra and first-order partial

differential equations, and yet its predictions are so unimaginably precise and

unbelievably accurate when compared with experimental data. Such a successful

theory, however, is not without its own imperfections (or mysteries). For

example, the orthodox interpretation of QM—the Copenhagen interpretation—

does not give a satisfactory explanation about how and why the wavefunction

of a particle (e.g., an electron) would suddenly collapse once some measure-

ment has been made on it (e.g., an electron has been registered by a particle

detector). This so-called measurement problem has not been properly understood

since the very early days of QM, until recently when the process of quantum

measurement is thoroughly studied, and the concept of “decoherence” is pro-

posed [1].

Optical Fiber Telecommunications V A: Components and Subsystems

Copyright � 2008, Elsevier Inc. All rights reserved.

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Another “mysterious” feature of QM, which we will explain in further detail, is

the superposition principle and the ensuing quantum entanglement. The former

allows a quantum mechanical system to be in any state that is spanned by the basis

vectors of its Hilbert space. For example, if the system can be in two orthogonal

states j0i and j1i (i.e., h0j1i = 0), it can also be in a linear combination of these

two states—�j0i þ �j1i, where � and � are complex numbers satisfying j�j2 þj�j2 = 1. This innocuous-looking principle is, in fact, the origin of a lot of

“quantum weirdness” not observed in our classical everyday experiences.

Entanglement is such a counterintuitive example. Consider the following bipartite

state for particles A and B:

j�i= 1ffiffiffi2p ðj0iAj0iB þ j1iAj1iBÞ; (21.1)

which states that whenever particle A is detected to be in state j0i, particle B must

also be found in j0i (in its own subspace), and vice versa. The same rule applies if

particle A is found in state j1i. That is, we are assured to find particle B in its own

j1i state with unit probability. This may not sound so surprising at first sight; after

all, classical objects sometimes exhibit this kind of correlation too. For example,

we can take a coin and split it into half. Then we put the two half-coins into two

separate envelopes, which are sealed afterward. Suppose we do it in such a way

that nobody, not even we, know exactly which half of the coin ends up in which

envelope. It is obvious that if we open one of the envelopes and find that we get the

“head” portion of the coin, we can infer with 100% certainty that the other

envelope contains the “tail” portion of the coin, and vice versa. The two halves

of the same coin, just like the two particles in Eqn (21.1), can be spatially

separated. The deterministic correlation between the two parties in both cases

remain the same, no matter how far they are from each other. An obvious question

naturally arises: as the coin game is something we can play everyday, what is so

shocking about the correlation that we find in the entanglement example?

The fundamental difference between quantum entanglement and classical corre-

lation lies in the fact that particles are quantum-mechanical objects which can exist

not only in states j0i and j1i but also in states described by �j0i þ �j1i (allowed by

the superposition principle!), while half-coins, being classical objects, can only live

in one of the two deterministic states (“head” or “tail”), and not something in

between. To put it more bluntly, in the case of coins, even though we do not

know a priori which portion of the coin ends up in which envelope, we are confident

that one of the envelopes must contain the “head” portion of the coin, the other

envelope the “tail” portion of the coin. This “confident ignorance” about the results

of classical correlation cannot be safely extended into the regime of quantum

entanglement. In fact, before we decide which basis to use for measuring the states

of the particles in Eqn (21.1), we do not know anything about what results we will

eventually get. We cannot even say, before measurement, each particle is in some

deterministic state, we just do not know which is which. It is not that we do not have

830 Prem Kumar et al.

the knowledge about the states of individual particles; they, in fact, do not come into

being until we make the measurement. In other words, the individual particle in

quantum entanglement does not have a well-defined pure state before measurement.

Each of them are in a mixed state; the joint state of both particles constitutes the pure

state in Eqn (21.1), which we call “entanglement.”

It was Schrodinger who first realized the strangeness of entanglement, or

“Verschrankung” as it was originally coined in German. He pondered the quantum-

to-classical-transition problem at the same time, and extended the concept of

entanglement to the ill-defined boundary between quantum and classical worlds,

where the contrast is most extreme. He imagined a macroscopic object, e.g., a cat

(which later becomes the notorious Schrodinger’s cat), is somehow entangled with a

microscopic object, e.g., an atom. The poor cat”s fate depends solely on the decay-

ing property of the atom. If the atom decays, the cat dies; if the atom does not decay,

the cat lives. As discussed before, the atom can live in a superposition state “decay–

not decay,” and as the two are entangled, the cat is then forced to be living in a state

“dead–alive.” This is very counterintuitive, since normally we do not observe a half-

dead, half-alive cat in our daily lives.

This problem was brought into focus by Einstein, Podolsky, and Rosen (EPR) [2]

in a famous paper in 1935, in which they pointed out the incompatibility between

QM and local realism. The latter notion consists of two parts: locality is a very

reasonable assumption that directly follows our everyday physical intuition, which

postulates the nonexistence of “action-at-a-distance”; realism demands the existence

of “elements of physical reality” in every physical system, which should take

definite values prior to any conceivable measurements. In their example, EPR

considered a quantum system composed of two particles such that neither one of

them has well-defined position or momentum, but the sum of their positions (their

center-of-mass) and the difference of their momenta (their individual momenta in

the center-of-mass system) are precisely defined. It then follows that measurement

of either particle’s momentum (position) would immediately determine the measure-

ment outcome for the other particle’s momentum (position), without even interact-

ing with that particle. Since the two particles can be separated by arbitrary distances,

and properties like position and momentum of a particle are “elements of reality”

according to EPR that must assume definite values before any measurement, EPR

then suggest this “spooky action-at-a-distance” must imply that QM is at least

incomplete, if not incorrect; and that there should be a deeper theory, possibly

with some hidden degrees of freedom (later known as “hidden variables”), which

can faithfully reproduce every result that QM has achieved, and hopefully retain our

familiar deterministic classical world view—local realism. Niels Bohr [3] replied by

arguing that the two particles in the EPR case are always parts of one quantum

system, and thus measurement on one particle changes the possible predictions that

can be made for the entire system and consequently on the other particle; QM is

indeed complete and there is no need for a “more complete” theory.

While for a long time, the famous Einstein–Bohr debate has been widely regarded

as merely philosophical, David Bohm [4] in 1951 introduced spin-entangled

21. Fiber-Optic Quantum Information Technologies 831

systems, as a discrete version of the original continuous EPR-entangled systems. In

1964, John Bell [5] pointed out that for such spin-entangled systems, classical

hidden-variable theories would make different predictions from QM on measure-

ments of correlated quantities. The theorem he published, later known as Bell’s

theorem, quantified just how strongly quantum particles were correlated than would

be classically allowed. This effectively opened up the possibility of experimentally

testing quantum mechanical predictions against those of classical hidden variable

theory. By now a number of experiments have been performed, and the results are

almost universally accepted to be fully in favor of QM [6–10]. However, from a

strictly logical point of view the problem is not completely closed yet, because some

loopholes in these existing experiments still make it at least logically possible to

uphold a local realist world view [11, 12].

More recently, since the beginning of 1990s, the field of quantum information

and quantum communication has opened up and expanded rapidly [13, 14].

Quantum entanglement, once the core concept and the sole mystery of the decade-

long Einstein–Bohr debate, has begun to take on a new look. It is still an unresolved

mystery, philosophically speaking, as it forces us to abandon either one of the two

familiar notions that we hold dear since the beginning of modern-day science:

locality and realism. But this sacrifice we have to make, much in the spirit of

Niels Bohr’s comment that we have no right to tell God what to do; we are only

entitled to discover what God’s plans are and accept them, or better yet, utilize them

to our full advantage. After taking this more humble and more practical point of

view, a whole new world of “quantum ideas” have been ignited and are actively

being pursued. Quantum teleportation [15] and quantum cryptography [16] are two

prominent examples. Quantum entanglement plays a central role in the former and

can lead to many advantages in the latter.

21.2 FIBER NONLINEARITY AS A SOURCEFOR CORRELATED PHOTONS

Efficient generation and transmission of quantum-correlated photon pairs, espe-

cially in the 1550-nm fiber-optic communication band, is of paramount impor-

tance for practical realization of the quantum communication and cryptography

protocols [17]. The workhorse source employed in all implementations thus far

[18] has been based on the process of spontaneous parametric down-conversion

(SPDC) in second-order (�(2)) nonlinear crystals. Such a source, however, is not

compatible with optical fibers as large coupling losses occur when the pairs are

launched into the fiber. This severely degrades the correlated photon-pair rate

coupled into the fiber, because the rate depends quadratically on the coupling

efficiency. From a practical standpoint it would be advantageous if a photon-

pair source could be developed that not only produces photons in the commu-

nication band but also can be spliced to standard telecommunication fibers with

high efficiency. Over the past few years various attempts have been made to


develop more efficient photon-pair sources, but all have relied on the �(2)

down-conversion process [19–25]. Of particular note is Ref. [26], in which the

effective �(2) of periodically poled silica fibers was used. In this chapter, we report

the first, to the best of our knowledge, photon-pair source that is based on the Kerr

nonlinearity (�(3)) of standard fiber. Quantum-correlated photon pairs are observed

and characterized in the parametric fluorescence of four-wave mixing (FWM) in

dispersion-shifted fiber (DSF).

The FWM process takes place in a nonlinear-fiber Sagnac interferometer

(NFSI), shown schematically in Figure 21.1. Previously, we have used this NFSI

to generate quantum-correlated twin beams in the fiber [27]. The NFSI consists of

a fused-silica 50/50 fiber coupler spliced to 300 m of DSF having zero-dispersion

wavelength �0 = 1537 nm. It can be set as a reflector with proper adjustment of

the intraloop fiber polarization controller (FPC) to yield a transmission coefficient

<�30 dB. When the injected pump wavelength is slightly greater than �0, FWM in

the DSF is phase-matched [19]. Two-pump photons of frequency !p scatter into a

signal photon and an idler photon of frequencies !s and !i, respectively, where

!s þ !i = 2!p. Signal/Idler separations of ’20 nm can be easily obtained with

use of commercial DSF [27]. The pump is a mode-locked train of ’3 ps long

pulses that arrive at a 75.3-MHz repetition rate. The pulsed operation serves two

important purposes: (i) the NFSI amplifier can be operated at low-average powers

(typical values are � 2 mW, corresponding to � 9-W peak powers) and (ii) the

production of the fluorescence photons is confined in well-defined temporal

windows, allowing a gated detection scheme to be used to increase the signal-to-

noise ratio. A 10% (90/10) coupler is employed to inject a weak signal, which is

parametrically amplified, and the output signal and the generated idler are used for

alignment purposes. For the photon-counting measurements described in this

chapter, the input signal is blocked.

Pumpin

FPC

300 mDSF loop

Signalin

Primarydiffraction

grating

Secondarydiffractiongratings

APD2

APD1

Coincidencecounter

Signal andidler out

FPC

50/50coupler

90/10

Rejectedpump

Figure 21.1 Diagram of the experimental setup; FPC, fiber polarization controller (this figure may be

seen in color on the included CD-ROM).


After passing through the 90/10 coupler, the fluorescence photons are directed

toward free-space filters that separate the signal and the idler photons from each

other and from the pump photons. To measure the nonclassical (i.e., quantum)

correlations between the signal and the idler photons, one must effectively sup-

press the pump photons from reaching the detectors. Because a typical pump pulse

contains ’108 photons and we are interested in detecting ’0.01 photons/pulse, a

pump-to-signal (idler) rejection ratio in excess of 100 dB is required. To meet this

specification, we constructed a dual-band spectral filter based on a double-grating

spectrometer. A primary grating (holographic, 1200 lines/mm) is first employed

to spatially separate the signal, the pump, and the idler photons. Two secondary

gratings (ruled, 600 lines/mm) are then used to prevent the pump photons that are

randomly scattered by the primary grating (owing to its nonideal nature) from

going toward the signal and idler directions. The doubly diffracted signal and idler

photons are then recoupled into fibers, which function as the output slits of the

spectrometer.

Transmission spectrum of the dual-band filter, measured with a tunable source

and an optical spectrum analyzer (OSA), is shown in Figure 21.2. The shape is

Gaussian in the regions near the maxima of the two transmission bands, which are

centered at 1546 nm (signal) and 1528 nm (idler), respectively, and the full-width

at half-maximum (FWHM) is ’0.46 nm. For pulse trains separated by 9 nm, which

is the wavelength difference between the pump and the signal (or idler), this filter

is able to provide an isolation ‡75 dB; the measurement being limited by the

intrinsic noise of the OSA. The combined effect of the Sagnac loop and the

double-grating filter thus provides an isolation ‡105 dB from the pump photons

in the signal and idler channels. The maximum transmission efficiency in the

–90

–80

–70

–60

–50

–40

–30

–20

–10

0

1523 1528 1533 1538 1543 1548

Wavelength (nm)

Filt

er tr

ansm

issi

on (

dB)

Instrumentnoise

Idler Signal

Figure 21.2 Transmission curves of the signal and idler channels in the dual-band filter (this figure

may be seen in color on the included CD-ROM).


signal channel is 45% and that in the idler channel is 47%. The total collection

efficiency for the signal (idler) photons is thus 33% (35%), with inclusion of the

losses in the Sagnac loop (18%) and at the 90/10 coupler (10%).

The separated and filtered signal and idler photons are directed toward fiber-

pigtailed InGaAS/InP avalanche photodiodes (APDs, Epitaxx EPM239BA). In

recent years, the performance of InGaAs APDs as single-photon detectors for

use in the fiber communication window around 1550 nm has been extensively

studied by several groups [28–31]. The pulsed nature of the photon pairs allows

us to use the APDs in a gated Geiger mode. In addition, the quality of our

APDs permits room-temperature operation with results comparable to those

obtained by other groups at cryogenic temperatures. A schematic of the electro-

nic circuit used with the APDs is shown in the inset in Figure 21.3. A bias

voltage VB (’�60 V), slightly below the avalanche breakdown voltage, is

applied to each diode and a short-gate pulse (�8 V, 1 ns FWHM) brings the

diodes into the breakdown region. The gate pulse is synchronized with the

arrival of the signal and idler photons on the photodiodes. Due to limitations

of our gate-pulse generator, the detectors are gated once every 128 pump pulses,

giving a photon-pair detection rate of 75.3 MHz/128 = 588 kHz. We expect this

rate to increase by more than an order of magnitude with use of a better pulse

generator. The electrical signals produced by the APDs in response to the

incoming photons (and dark events) are reshaped into 500-ns-wide transistor-

transistor logic (TTL) pulses that can be individually counted or sent to a TTL

AND gate for coincidence counting.

In Figure 21.3 we show a plot of the quantum efficiency vs the dark-count

probability for the two APDs used in our experiments. A figure of merit for the

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7

Dark-count probability (×10–3 counts/pulse)

Qua

ntum

effi

cien

cy

APD 1

APD 2

VBTo counter

or AND gate

Invertingamplifier

Pulseshaper

Figure 21.3 Quantum efficiency vs dark-count probability for the two APDs used in the experiments.

The inset shows a schematic of the electronic circuit used with the APDs.


APDs can be introduced through the noise-equivalent power NEP = (h�/�)(2RD)1/2

[20], where h is the Planck constant, � is the frequency of light, � is the detector

quantum efficiency, and RD is the dark-count rate measured during the gate time.

The best values of NEP obtained by optimizing VB are 1.0 � 10�15 W/Hz�1/2 for

APD1 and 1.6 � 10�15 W/Hz�1/2 for APD2. These values are comparable to those

reported in Refs [28–31] for cryogenically cooled APDs. Under the optimized

conditions, the efficiency of APD1 (APD2) is 25% (20%) and the corresponding

dark-count probability is 2.2 � 10�3/pulse (2.7 � 10�3/pulse).

As a first test of our photon-pair source, and of the filtering process, we measure

the number of scattered photons detected in the signal (idler) channel, NS (NI), as a

function of the number of pump photons, NP, injected into the NFSI. The results

for the idler channel are shown in the inset in Figure 21.4. We fit the experimental

data with NS =ND þ s1NP þ s2N2P, where ND is the number of dark counts during

the gate interval, and s1 and s2 are the linear and quadratic scattering coefficients,

respectively. The fit clearly shows that the quadratic scattering owing to FWM in

the fiber can dominate over the residual linear scattering of the pump due to

imperfect filtering.

In Figure 21.4 we present the coincidence counting results. The diamonds

represent the rate of coincidence counts as a function of the rate of the signal

and idler photons generated during the same pump pulse. For convenience, we

0

0.0004

0.0008

0.0012

0.0016

0.002

0 0.005 0.01 0.015 0.02 0.025 0.03

Signal/idler photon rate (counts/pulse)

Coi

ncid

ence

rat

e (c

ount

s/pu

lse)

0.10.1 1 10 100 1000

1

10

100

ND +

s1NP

+ s2NP

2

ND

s2NP2

s1NP

Pump photons (×106/pulse)

Sig

nal p

hoto

ns (

×10– 3

/pul

se)

Figure 21.4 Coincidence rates as a function in the single-photon rates in two different cases: signal/

idler fluorescence produced by a pump pulse (diamonds) and signal/idler fluorescence produced by two

consecutive pump pulses (triangles). The line represents the calculated “accidental” counts. The inset

shows a plot of the detected idler photons as a function of the injected pump photons (hollow circles).

A second-order polynomial is shown to fit the experimental data. The contributions of the dark counts,

linear scattering, and quadratic scattering are plotted separately as well (this figure may be seen in color

on the included CD-ROM).


have plotted the coincidence rate as a function of the geometric mean of the signal

and idler count rates; in fact, because the efficiency of the two detectors is

different, we measure different single-photon count rates in the two channels.

Dark counts have been subtracted from the plotted count rates. For the coincidence

rates, both dark–dark and photon–dark coincidences have to be taken in account,

but for the counting rates in our experiment the former are negligible. Thus far,

we have achieved a maximum coincidence rate of 103 counts/s (= coincidence

rate/pulse� gate-pulse rate), which is expected to go up by at least a factor of

10 with use of a higher repetition-rate gate-pulse generator.

We have performed two independent experiments to demonstrate the nonclassi-

cal nature of the coincidences. Results of the first experiment are shown by the

triangles in Figure 21.4, which represent the measured coincidence rate as a

function of the signal-photon count rate when the signal is delayed with respect to

the idler by one pulse period. The delay was achieved by inserting a fiber patch-cord

of appropriate length in the signal path from the output of the filter to APD1. For

two independent photon sources, each with a count rate RS� 1, the “accidental”

coincidence rate RC is given by RC =R2S, regardless of the photon statistics of the

sources. This quadratic relation is plotted as the solid line in Figure 21.4, which fits

the delayed-coincidence data (triangles) very well. These measurements then show

that while the fluorescence photons produced by the adjacent pump pulses are

independent, those coming from the same pump pulse show a strong correlation,

which is a signature of their nonclassical behavior.

In the second experiment, measurements were performed to demonstrate the

nonclassicality test described in Ref. [24]. It can be shown that the inequality

RC � RðaÞC � 2 RS=2 � R

ðaÞS=2þ RI=2 � R

ðaÞI=2

� �� 0 (21.2)

is valid for two classical light sources, where RC is the coincidence-count rate for

the two sources, RðaÞC is the calculated “accidental” coincidence-count rate corre-

sponding to the same photon-count rate for the two sources, RS/2 and RI/2 are the

coincidence-count rates measured by passing the light from each of the two

sources through a 50/50 splitter and detecting the two halves independently, and

RðaÞS=2 and R

ðaÞI=2 are the calculated “accidental” coincidence-count rates in the 50/50

splitting measurements. When we substitute the experimental data, Eqn (21.2)

yields (64 – 9) 10�6� 0, where the error is statistical. The inequality for classical

sources is thus violated by over seven standard deviations.

In conclusion, we have demonstrated, for the first time to our knowledge, a

source of quantum-correlated photon pairs that is based on FWM in a fiber near

1550 nm. We have also developed and tested a room-temperature coincidence

detector for the photons in that window. The photon-pair detection rate (’103

coincidence counts/s) at present is limited by the electronics employed in our

setup. In addition, we believe that the spectral filter used for rejecting the pump

photons can be implemented with fiber Bragg gratings, making this source integr-

able with the existing fiber-optic infrastructure.


21.3 QUANTUM THEORY OF FOUR-WAVE MIXINGIN OPTICAL FIBER

Four-wave mixing has long been studied, especially in the context of isotropic

materials, e.g., optical fibers [25, 32]. Generally speaking, it is a photon–photon

scattering process, during which two photons from a relatively high-intensity beam,

called pump, scatter through the third-order nonlinearity (�(3)) of the material (silica

glass in the case of optical fibers) to generate two daughter photons, called signal

and idler photons, respectively. The frequencies of the daughter photons are

symmetrically displaced from the pump frequency, satisfying the energy conserva-

tion relation !s þ !i = 2!p, where !j ( j = p, s, i) denotes the pump/signal/idler

frequency, respectively. They are predominantly copolarized with the pump beam,

owing to the isotropic nature of the optical Kerr nonlinearity:

�ð3Þxxxx =�

ð3Þxxyy þ �ð3Þxyxy þ �ð3Þxyyx = 3�

ð3Þxxyy. The daughter photons also form a time–

energy-entangled state, in the sense that the two-particle wavefunction cannot be

factorized into products of single-particle wavefunctions: �ð!s; !iÞ 6¼ �ð!sÞ� ’ð!iÞ.This four-photon scattering (FPS) process is intrinsically interesting and particularly

useful when applied to the field of quantum information processing (QIP), in which

generation of entangled states and test of Bell’s inequalities play an important role.

A great amount of original work, both theoretical and experimental, has been done

in the rapidly expanding field of QIP (see, e.g., Ref. [33] for a general review). The

workhorse process for generating entangled states is the process of SPDC in second-

order (�(2)) nonlinear crystals, which has been studied exhaustively during the past

decades. However, unlike its �(2) counterpart, the �(3) process of FWM has received

relatively lesser theoretical attention in the quantum mechanical framework, despite

its apparent benefits in the applications of QIP. To name a few, the ubiquitous readily

available fiber plant serves as a perfect transmission channel for the FWM-generated

entangled qubits, whereas it remains a technical challenge to efficiently couple �(2)-

generated entangled photons into optical fibers due to mode mismatch. Besides, the

excellent single-mode purity of the former makes it suitable for applications that

require multiple quantum interactions. Furthermore, it is also possible to wavelength

multiplex several different entangled channels from the broadband parametric spec-

trum of FWM by utilizing the advanced multiplexing/demultimplexing devices

developed in connection with the modern fiber-optic communications infrastructure.

The only drawback of this scheme that has been identified is the process of sponta-

neous Raman scattering (SRS), which inevitably occurs in any �(3) medium and

generates uncorrelated photons into the detection bands, leading to a degradation in

the quality of the generated entanglement [28]. Various efforts have been made to

minimize the negative effect that SRS imposes [29, 30].

In this section, we present a quantum theory that models the FWM process in an

optical fiber, without inclusion of the Raman effect. The pump is treated as a classical

narrow (picosecond-duration) pulse due to its experimental relevance. The signal and

idler fields form a quantum mechanical two-photon (or “biphoton” [31, 34]) state at


the output of the fiber. From the experimental point of view, what we are mostly

interested in is the nonclassicality that the two-photon state exhibits. It is this

unique quantum feature that makes the two-photon state a valid candidate for various

quantum-entanglement-related experiments, including quantum cryptography [35],

quantum teleportation [15], etc. Coincidence-photon counting, or second-order coher-

ence measurement of the optical field [36], serves as a measurement technique that

distinguishes a quantum mechanically entangled state from a classically correlated

state, which will form a central part of our investigation.

A sample coincidence-counting result from Ref. [37] is shown in Figure 21.5.

The top (bottom) series of data points represents the total (accidental) coincidence-

count rate as a function of the single-channel count rate. SRS and dark counts from

the detectors account for the major part of the accidental coincidence counts. Our to-

be-developed theory, however, only takes into account the photon counts generated

by the FWM process. To reconcile the theory with experiments, the contributions

from SRS and dark counts from the detectors are independently measured [29, 30],

and subsequently subtracted from both the single counts and the total coincidence

counts. Overall quantum efficiencies of detection in both the signal and idler

channels are also separately measured. The single-count rates are divided by the

respective quantum efficiencies and the coincidence-count rate by the product of the

efficiencies in the signal and idler channels to arrive at rates at the output of the fiber

for comparison with the prediction of our theory. The dependence of the photon-

counting results on various system parameters, for instance, the pump power, pump

bandwidth, filter bandwidth, etc., can be studied.

Polarization entanglement has also been generated by time and polarization

multiplexing two such FWM processes [38, 39]. However, the theory for that

0

0.0004

0.0008

0.0012

0.0016

0.002

0 0.005 0.01 0.015 0.02 0.025 0.03

Signal/idler photon rate (counts/pulse)

Coi

ncid

ence

rat

es (

coun

ts/p

ulse

)

Figure 21.5 (Color online) Experimental results. Diamonds, total coincidences; triangles, accidental

coincidences; curve, theoretical fit y = x2 for statistically independent photon sources (this figure may

be seen in color on the included CD-ROM).


particular experiment is a straightforward extension of our current theory, and

therefore will not be included in the analysis to follow.

Having described the experiment in the previous section, we are ready to start

building up the theoretical model for that experiment. We take the standard

approach of modern quantum optics, i.e., finding out the interaction Hamiltonian

and calculating the evolution of the state vector using the Schrodinger picture. We

accomplish the first task by seeking connections with the well-known classical

FWM theory in optical fibers [32]. The coupled classical-wave equations for the

pump, signal, and idler fields are

qAp

qz¼ ijApj2Ap;

qAs

qz¼ i 2jApj2As þ A2

pA�i e�iDkz

h i;

qAi

qz¼ i 2jApj2Ai þ A2

pA�s e�iDkz

h i;

(21.3)

where the usual undepleted-pump approximation has been made, and we only keep

terms that are significant, i.e., to OðA2pÞ. Fiber loss is neglected from the above

equations. The Aj ( j = p, s, i) denote electric-field amplitudes for the pump,

signal, and idler, respectively, and all of them have been normalized such that

their unit isffiffiffiffiffiWp

Dk= ks þ ki � 2kp is the magnitude of the wave-vector mis-

match. = 2n2/�Aeff is the nonlinear parameter of interaction, wherein

n2 = ð3=4n2�0cÞReð�ð3ÞxxxxÞ is the nonlinear-index coefficient, �0 is the vacuum

permittivity, Aeff is the effective mode area of the optical fiber, and ��p,s,i is

the wavelength involved in the FWM interaction.

Due to the highly nonresonant nature of FWM in optical fibers, we expect the

quantum equations of motion, which describe the interplay between and evolution

of the fields at the photon level, to fully correspond with their classical counterparts.

In light of this correspondence principle, we write the quantum equations of motion

by replacing the classical amplitudes in Eqn (21.3) with electric-field operators:

qEðþÞp

qz¼ i� Eð�Þp EðþÞp EðþÞp ;

qEðþÞs

qz¼ i� ½2Eð�Þp EðþÞp EðþÞs þ E

ð�Þi EðþÞp EðþÞp ;

qEðþÞi

qz¼ i� ½2Eð�Þp EðþÞp E

ðþÞi þ Eð�Þs EðþÞp EðþÞp ;

(21.4)

where EðþÞj =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�h!j=2�0VQÞ

paj ð j= p; s; iÞ are the positive-frequency electric-

field operators, corresponding to photon annihilation operators, and VQ is the


quantization volume. Here we omit the Hermitian-conjugate equations corre-

sponding to Eqn (21.4) for simplicity. We have assumed that the photon fields

phase match, i.e., Dk = 0. In Eqn (21.4), �= � ð�ð3ÞAeff nL!=2VQcÞ is a const-

ant similar to in the classical Eqn (21.3); the exact form of this constant differs

from its classical cousin to compensate for the unit discrepancy between the two

sets of equations (note that the operator EðþÞj is of unit V/m, and the amplitude Aj is

of unitffiffiffiffiffiWp

). The correct form of the interaction Hamiltonian that we are

seeking should lead to Eqn (21.4) through the Heisenberg equation of motion

for the field operators, namely, i�hðqE=qtÞ= ½E;HI, where E stands for any

electric-field operator. Utilizing the mathematical facts q/qt (c/n)(q/qz) and

½EðþÞj ðzÞ;Eð�Þk ðz0Þ ¼ ð�h!=2�0VQÞ � ðz� z0Þ �jk, we arrive at the following form

for our interaction Hamiltonian:

HI ¼ � �0 �ð3ÞZ

V

dV Eð�Þp Eð�Þp EðþÞp EðþÞp

hþ 2Eð�Þs E

ð�Þi EðþÞp EðþÞp

þ 2Eð�Þp Eð�Þp EðþÞs EðþÞi þ 4Eð�Þp EðþÞp Eð�Þs EðþÞs þ 4Eð�Þp EðþÞp E

ð�Þi E

ðþÞi

i; (21.5)

where � is an overall unknown constant related to the specific experimental

details, which will be determined later when we compare our theory with the

experiment; �(3) is the nonlinear electric susceptibility whose tensorial nature is

ignored as all the optical fields are assumed to be linearly copolarized. The integral

is taken over the entire volume of interaction, namely, the effective volume of the

optical fiber. We label the first term in the integrand of Eqn (21.5) as the self-phase

modulation (SPM) of the pump field, the next two terms as the FPS among the

optical fields, and the last two terms as the cross-phase modulation (XPM)

between pump and signal (idler) fields.

After obtaining the Hamiltonian responsible for the quantum FWM process,

we are ready to tackle our next task: calculate the state vector evolution. It is

worthwhile, at this point, to define the various electric field operators appearing

in the Hamiltonian, in accordance with the experiment we are trying to model.

The pump field is taken to be a classical narrow pulse, which is linearly

polarized, propagating in the z direction (parallel with the fiber axis), with a

central frequency �p and an envelope of arbitrary shape eEp. Mathematically, it

can be written as

EðþÞp ¼ e�i�pteEpðz; tÞ

¼ e�i�pt

Zd�p Epð�pÞ eikpz�i�pt; (21.6)

wherein the bandwidth of the pump field is much smaller than �p, satisfying the

quasi-monochromatic approximation. The signal and idler fields are quantized


electromagnetic fields, copolarized and copropagating with the pump, as given by

the following multimode expansion:

Eð�Þs =X!s

ffiffiffiffiffiffiffiffiffiffiffiffi�h!s

2�0VQ

sayks

nð!sÞe�i½ksð!sÞz�!st; (21.7)

Eð�Þi =

X!i

ffiffiffiffiffiffiffiffiffiffiffiffi�h!i

2�0VQ

sayki

nð!iÞe�i½kið!iÞz�!it; (21.8)

where ayks

is the creation operator for the signal mode with frequency !s, ks (!s) =n(!s)!s/c is its wave-vector magnitude. The idler field is defined in an analogous

fashion. The central frequencies of the signal and idler fields are individually

denoted by �s and �i, which are symmetrically distanced from the central fre-

quency of the pump field �p, satisfying the energy conservation relation

�s þ �i = 2�p.

To simplify our calculation and to compare our results with the experiments,

two assumptions are further made about the pump field: it has a Gaussian spectral

envelope and its SPM is included in a straightforward manner, i.e.,

EðþÞp = e�i�pte�iPpzEp0

Zd�p e

�ð�2p=2 2

pÞ eikpz�i�pt; (21.9)

where Pp 2ffiffiffip

Aeff�0 c n 2p E

2p0 is the peak power of the pump pulse, which is

treated as a constant under the undepleted pump approximation, and p is the

optical bandwidth of the pump. The first assumption is justified by the fact that

our experimental optical filter for the pump can be well approximated by a

Gaussian function in the frequency domain. The validity of the second assumption

can be seen when we solve the classical equation of motion for the pump

field, namely, the complex conjugate form of the first equation in Eqn (21.3),

which reads

qA�pqz

= � ijApj2A�p: (21.10)

We choose to study the complex conjugate form of the equation because it is A�pthat corresponds to E

ðþÞp . Straightforward calculations show that the solution to

Eqn (21.10) is A�pðzÞ=A�pð0Þ e�iPpz, where Pp = jApj2 is the same undepleted

peak power of the pump pulse. The SPM term of the pump, e�iPpz, which is the

nonlinear phase factor in the classical FWM theory that determines the phase-

matching condition [32], now manifests itself in our quantum mechanical calcula-

tion as a “phase tag” for the pump field through its propagation along the optical

fiber. Finally, the undepleted-pump approximation holds because the loss in the

fiber is negligible and only a few photons are scattered (�1 out of 108) through the

nonlinear interaction.


The two-photon state at the output of the fiber is calculated by means of first-

order perturbation theory, i.e.,

j�i= j0i þ 1

i�h

Z 1�1

HIðtÞ dtj0i: (21.11)

Retaining of higher order terms in the perturbation series involves generation of

multiphoton states, which will be ignored in our calculation owing to their

smallness. We can see that only the FPS terms in the interaction Hamiltonian

contribute to the formation of the signal/idler two-photon state. This is because all

terms vanish when acting on the vacuum state j0i with the exception of

Eð�Þs E

ð�Þi E

ðþÞp E

ðþÞp þ h:c:, which we denote as

HFPS ��0�ð3ÞZ

V

dVðEð�Þs Eð�Þi EðþÞp EðþÞp þ h:c:Þ; (21.12)

where � = 2�, and h.c. stands for Hermitian conjugate.

The state vector is then given by

j�i= j0i þ 1

i�h

Z 1�1

HFPS dtj0i; (21.13)

which is a superposition of the vacuum and the two-photon state. Substituting

Eqns (21.7–21.9) and (21.12), into Eqn (21.13), after some algebra, leads to the

following form of the state vector:

j�i= j0i þXks;ki

Fðks; kiÞ ayks aykij0i; (21.14)

Fðks; kiÞ ¼g

Z 0

�L

dz1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� ik00ð�pÞ 2pz

q exp � ik00ð�pÞz4

ð�s � �i þ DÞ2�

�2iPpz� ð�s þ �iÞ2

4 2p

); (21.15)

g =�2�ð3Þ Pp

i �0VQ n3�p p: (21.16)

The function F(ks, ki) is called the two-photon spectral function [34]. Here

k00ð�pÞ= d2k=d!2��!¼�p

is the second-order dispersion at the pump central

frequency [also known as the group-velocity dispersion (GVD)], which can be

obtained from k00ð�pÞ¼ � ð�2p=2 cÞDslopeð�p � �0Þ, where �0 is the zero-

dispersion wavelength of the DSF, Dslope = 0.06 ps/nm2 km is the experi-

mental value of the dispersion slope in the vicinity of �0. D�s��i is the


central-frequency difference between signal and idler fields. �s and �i are related to

!s and !i through the following relation: �s = !s��s, �i = !i��i.

In lieu of giving the detailed derivation of the two-photon state (which is

lengthy), we highlight several noteworthy mathematical maneuvers along the

way. The following identification of the Dirac �-function is useful in handling

the time integral:

Z 1�1

eið!þ!0�2�p��p��0pÞtdt = 2�ð!þ !0 � 2�p � �p � �0pÞ; (21.17)

which reinforces the energy conservation requirement in the FPS process. The

volume integralRdV is reduced to a length integral

Rdz by usingR R

dxdy! Aeff, which is a valid approximation for single spatial-mode

propagation and interaction in optical fibers. Taylor expansion of the various

wave-vector magnitudes kp, ks, ki around the pump central frequency �p has

been used to simplify their relationship. In terms of the mathematical structure

of the two-photon spectral function, we note that the GVD term k00(�p) as well

as the pump SPM term Ppz play important roles in shaping the two-photon

state, in contrast with the observation that the pump SPM term is virtually

nonexistent in the �(2)-generated two-photon states. The appearance of the

pump SPM is therefore a unique signature of the �(3) two-photon state, when

comparing with its �(2) counterparts.

In this and the next section, we will make use of the previously derived

formulas for the two-photon state [Eqns (21.14)–(21.16)] to obtain the photon-

counting formulas for the single channels as well as for the coincidences. The

mathematics involved for the two cases are similar to each other, so it suffices to

present a detailed version for the former. The signal-band single-photon counting

rate can be calculated using the following formula [36]:

Sc =

Z 10

h�jEð�Þs EðþÞs j�i dT: (21.18)

It is obvious that an analogous approach can be applied to the idler band as well.

As Sc denotes single-photon counting probability for one pump pulse, it is by

definition a dimensionless quantity. It is customary, in this case, to use the photon-

number unit for the electric field operator [40]. In this unit, the electric field

operator has dimensionality 1=ffiffiffiffiffiffiffisecp

, as shown below:

EðþÞs =X

ks

ffiffiffiffiffiffiffiffiffiffic Aeff

4VQ

saks e

�i!s te�½ð!s��sÞ2=2 20; (21.19)


where the Gaussian filter in front of the detector has been included. The integrand

in Eqn (21.18) can be written as

h�jEð�Þs EðþÞs j�i ¼c Aeff

4VQ

Xki;k

0i

h0jakiayk0ij0i X

k1;k2;ks;k0s

h0jaksayk1

ak2ayk0sj0iei!s t

� e�½ð!s��sÞ2=2 20e�i!

0s te�½ð!

0s��sÞ2=2 2

0 F�ðks; kiÞFðk0s; k0iÞ:

(21.20)

Nonvanishing results emerge only when the wave vectors observe the following

restrictions:

ki = k0i; ks = k1; k0s = k2: (21.21)

The integrand may be further simplified into

h�jEð�Þs EðþÞs j�i ¼c Aeff

4VQ

Xki

Xks

ei!s te�½ð!s��sÞ2=2 20F�ðks; kiÞ

" #

�½Xk0s

e�i!0s te�½ð!

0s��sÞ2=2 2

0Fðk0s; kiÞ

=c Aeff

323uð!iÞu2ð!sÞ

Zd!i

Zd!s e

�i!ste�½ð!s��sÞ2=2 20Fð!s; !iÞ

�� 2, (21.22)

where in the last step we have invoked the following identity to transform wave-

vector summations into angular frequency integrals:

Xkj

!V

1=3Q

2

Zdkj =

V1=3Q

2

Zd!j

uð!jÞ: (21.23)

Here u(!j) = d!j/dkj, j = s, i, is the group velocity of the jth mode, and is to be

taken as a constant c/n in our simplified calculation.

Equation (21.18) can be written in the following form after all the above steps

have been absorbed:

Sc ¼2�2½�ð3Þ2 Aeff P2

p

16 �20 V2

Q n3 c2 �2p

2p

Z 0

�L

dz1

Z 0

�L

dz2

e�2iPpðz1�z2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� ik00 2

pz1Þð1þ ik00 2pz2Þ

q Zd�s

Zd�i

� exp �ð�s þ �iÞ2

2 2p

� �2s

20

� ik00

4ðz1 � z2Þð�s � �i þ DÞ2

( ): (21.24)


The frequency double integral can be analytically integrated through a change of

variables and completion of squares, namely, let

�þ=�s þ �i

2; (21.25)

��= �s � �i: (21.26)

The frequency double integral can be rewritten in terms of the new variables asZd�s

Zd�i e

�½ð�sþ�iÞ2=2 2p � �

2s

20

� ik00

4ðz1 � z2Þð�s � �i þ DÞ2

¼Zd�þ exp �

2 20 þ 2

p

20

2p

�þ þ 2p��

2ð2 20 þ 2

pÞ

" #28<:

9=;Zd�� exp �

�2�

2ð2 20 þ 2

pÞ� ik00ðz1 � z2Þð�� þ DÞ2

4

( ): (21.27)

The first part of the integral, concerning only Gaussian functions with real variable

as arguments, is easily integrated as

Zd� þ exp �

2 20 þ 2

p

20

2p

�þ þ 2p��

2ð2 20 þ 2

pÞ

" #28<:

9=; ¼ffiffiffip

p 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2

0 þ 2p

q : (21.28)

The second part of the integral, having a Gaussian function with complex argu-

ment as integrand, has a closed analytical form by using the integral formula from

Ref. [41], i.e.,Zd� � exp � u2

�2ð2 2

0 þ 2pÞ� ik00ðz1 � z2Þð�� þ DÞ2

4

( )

¼ffiffiffipffiffiffi

ap ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ b24p exp � cb2

1þ b2þ i

2arctanðbÞ þ ir

1þ b2

� �; (21.29)

where

a =1

2ð2 20 þ 2

pÞ; b ¼ �

k00ðz1 � z2Þð2 20 þ 2

pÞ2

;

c =D2

2ð2 20 þ 2

pÞ; and r = � k00ðz1 � z2ÞD2

4:


We therefore obtain the following final form of the single-photon counting

formula:

Sc = A1ðPpLÞ2 0

pIsc; (21.30)

A1 =�2 n A3

eff

18ffiffiffi2p

V2Q

; (21.31)

Isc¼1

L2

Z 0

�L

dz1

Z 0

�L

dz2

exp �2iPpðz1�z2Þ�cb2

1þb2þ i

2arctanðbÞþ ir

1þb2

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�ik00 2

pz1Þð1þik00 2pz2Þ

q ffiffiffiffiffiffiffiffiffiffiffiffi1þb24p ; (21.32)

where A1 is an unknown constant with � and VQ as fitting parameters, and Isc is a

double-length integral that has to be investigated numerically. Despite the seemingly

complicated form of the single-counts formula, the physics behind it is quite clear.

Apart from a small contribution from the double integral, the single counts scale

quadratically with pump power, which coincides with the intuitive FPS picture that

requires two pump photons to scatter into the signal/idler modes. It also scales

linearly with the ratio of the filter bandwidth to pump bandwidth. This makes

sense in that if one broadens the filter bandwidth, more photons will be collected;

and conversely if the filter bandwidth is narrowed, one would expect to count less

photons. The dependence on pump bandwidth is more clearly seen in the time

domain. As the pulse width becomes wider (thus the pump bandwidth narrower)

while maintaining the peak power to be the same, the probability of FPS increases

linearly with pulse width (thus decreases linearly with pump bandwidth) simply

because there is more time for the pump photons to interact; the reverse is also true.

The more intricate dependence on pump power, pump bandwidth, and filter band-

width is described by the double integral Isc, which takes into account phase match-

ing, SPM of the pump, and the Gaussian shapes of pump and filter spectrum.

Calculations of the coincidence-counting rate with Gaussian filters can be

performed in a similar way to those of the single counting rate. We start with

the probability of getting a coincidence count for each pulse [36]:

Cc =

Z 10

dT1

Z 10

dT2h�jEð�Þ1 Eð�Þ2 E

ðþÞ2 E

ðþÞ1 j�i: (21.33)

The electric fields are free fields propagating through Gaussian filters evaluated at

detectors 1 and 2, defined in the photon-number unit:

EðþÞ1 ¼

Xk1


4 VQ

sak1

e�i!st1 e�½ð!s��sÞ2=2 20; (21.34)


EðþÞ2 ¼

Xk2


4 VQ

sak2

e�i!it2e�½ð!i��iÞ2=2 20; (21.35)

where the Gaussian filters take the form fð!j � �jÞ= fje�½ð!j��jÞ2=2 2j , fj = 1 and

j = 0 for j = s, i are assumed to simplify the calculation. ti = Ti� li/c is the

time at which the biphoton wavepacket leaves the output tip of the fiber, which in

our case is almost the same for the signal and idler as there is negligible group-

velocity difference between the two closely spaced (in wavelength), copolarized

fields. li denotes the optical path length from the output tip of the fiber to the

detector i, i = 1, 2, and can be carefully path matched to be the same.

The integrand in Eqn (21.33) can be written in the following form:

h�jEð�Þ1 Eð�Þ2 E

ðþÞ2 E

ðþÞ1 j�i ¼ jh0jE

ðþÞ2 E

ðþÞ1 j�ij

2

¼ jAðt1; t2Þj2; (21.36)

where A(t1, t2) is the biphoton amplitude introduced in Refs [31, 34]. While the

concept of a biphoton amplitude plays an important role in the study of frequency

and wave-number entanglement inherent in the two-photon state, it serves merely

as a calculational shorthand for our purpose in determining the coincidence

counting rate. It is straightforward to show that

Aðt1; t2Þ=cAeff

4 VQ

Xks;ki

Fðks; kiÞe�ið!st1þ!it2Þe�½�2sþ�2

i=2 2

0: (21.37)

The fact that the biphoton cannot be written as a function of t1 times a function of

t2 may be readily observed from the form that Eqn (21.37) takes. It is also

nonfactorable in the wave numbers ks and ki, displaying its entangled nature in

those degrees of freedom. However, it is not entangled in polarization, due to the

fact that all the fields involved are collinear with respect to one another and the

polarization states can be factored out.

When everything is taken into account, after some similar steps shown in

Section 21.4, we arrive at the following form for the coincidence counting

formula:

Cc = A2ð PpLÞ2 20

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p þ 2

0

q Icc; (21.38)

A2 =�2 n2 A4

eff

144 V8=3Q

; (21.39)


Icc¼1

L2

Z 0

�L

dz1

Z 0

�L

dz2

exp �2iPpðz1� z2Þ�ðc0b02=1þb02Þþði=2Þarctanðb0Þþðir0=1þb02Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1� ik00 2pz1Þð1þ ik00 2

pz2Þq ffiffiffiffiffiffiffiffiffiffiffiffi

1þb24p ; (21.40)

where

b0= � ½k00ðz1 � z2Þ 20=2; c0= ðD2=2 2

0Þ; and r0= � ½k00ðz1 � z2ÞD2=4:

From Eqns (21.30) and (21.38), we can see that the single counts and the

coincidence counts both scale quadratically with the pump peak power. This is a

distinct feature of the �(3) interaction, in contrast to the linear dependence on pump

power in �(2) SPDC. Whereas one might expect to see an exact linear relation

between the single counts and the coincidence counts under ideal detection con-

ditions (unity quantum efficiency of the detectors, no loss, no dark counts), the

linearity is absent due to the broadband nature of the pump field and the presence

of the filters. Some of the correlated twin photons are lost during the filtering

process, and some uncorrelated photons are detected instead. The explicit depen-

dence of Cc on the quantity 20= p


qcan be understood from its limiting

cases. When the pump bandwidth is wide compared with the filter bandwidth, i.e.,

p= 0, every individual frequency component of the pump spectrum will

generate its own energy-conserving signal/idler pairs. The filters, being narrow,

are only effective at collecting a small portion of the correlated photons. There-

fore, the coincidence counts should be proportional to 20= 2p. However, if the

pump bandwidth is sufficiently narrow, i.e., p� 0, the photons being filtered

(and subsequently collected by the detectors) are more likely to be correlated with

each other, in which case the coincidence counts should scale with 0= p. Both

cases are verified when we look at the asymptotic limits:

lim p� 0

20

p


0

q = 2

0

2p

; (21.41)

lim p� 0

20

p


0

q = 0

p: (21.42)

To pin down the unknown parameters � and VQ in Eqns (21.31) and (21.39), we

fit our theory to two sets of experimental data, where the ratio of pump bandwidth

to filter bandwidth is varied. The commonly used least-squares fitting technique

has been employed. The results are shown in Figure 21.6, where the central solid

curve corresponds to the optimum fitting parameters, which are determined to be


� = 0.237 and VQ = 1.6 � 10�16 m3. k00 has also been found to be �0.116 ps2/km,

corresponding to the wavelength difference �p��0 = 1.52 nm, which agrees well

with the measured experimental value. We also show the robustness of the fit by

perturbing either one of the fitting parameters around its optimum value by as

small as 5%. For example, the dotted curve corresponds to the case where we

set VQ = 1.5 � 10�16 m3 while keeping � optimum, and the dot-dashed curve

corresponds to the case where we set VQ = 1.7 � 10�16 m3 while keeping �optimum. The remaining two curves are generated when we keep VQ optimum

and set � = 0.250 (short-dashed curve), or � = 0.220 (long-dashed curve), respec-

tively. The large discrepancies between the experiment and the theory induced by

this operation are shown in the same figure, which boosts our confidence in the

correctness of the theory.

We have provided a detailed discussion of the two-photon state originating

from the third-order nonlinearity in optical fibers. This �(3) two-photon state shares

Pump bandwidth = 0.8 nm

Filter bandwidth = 0.8 nm


Filter bandwidth = 0.8 nm Pump bandwidth = 0.45 nm




00 0.5 1 1.5

Pp (W)2 2.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Sc

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Sc

Cc

00.0050.01

0.0150.02

0.0250.03

0.0350.04

0.045

Cc

(a)

0 0.5 1

Pp (W)1.5

(c)

0 0.5 1

Pp (W)1.5

(d)

0 0.5 1 1.5

Pp (W)2 2.5

(b)

Figure 21.6 Experiment vs theory: squares correspond to experimental data and the curves correspond to

theoretical predictions. (a) Single counts with p = 0.8 nm and 0 = 0.8 nm; (b) Coincidence counts with

p = 0.8 nm and 0 = 0.8 nm; (c) Single counts with p = 0.45 nm and 0 = 0.8 nm; (d) Coincidence

counts with p = 0.45 nm and 0 = 0.8 nm. The central solid curve represents the theoretical fit with the

optimum fitting parameters (� = 0.237, VQ = 1.6 � 10�16 m3), whereas the other curves correspond to

fits with nonoptimum fitting parameters: dotted, � = 0.237, VQ = 1.5 � 10�16 m3; dot-dashed,

� = 0.237, VQ = 1.7 � 10�16 m3; short-dashed, � = 0.250, VQ = 1.6 � 10�16 m3; long-dashed,

� = 0.220, VQ = 1.6 � 10�16 m3 (this figure may be seen in color on the included CD-ROM).


some similar features with the �(2) two-photon state generated from SPDC, yet it

also has some distinct characteristics. Coincidence photon-counting rate, which is

a significant nonclassical figure of merit of the two-photon state, has been shown

to depend heavily upon various experimental parameters. The dependence on the

ratio of the pump bandwidth to filter bandwidth is of practical importance, because

it serves as a guideline for optimizing the measurement of coincidence counts.

Single-photon counting rate has also been studied, and both fit to the experimental

data reasonably well. While in this chapter we are only concerned with parametric

fluorescence from a single pump pulse, the current theory can be readily extended

to include multiphoton-state generation from one pulse [42], and multiple two-

photon-states generation from adjacent pulses [38, 39] to study polarization entan-

glement. The effect of SRS can also be included in our model by taking into

account the noninstantaneous nature of the third-order nonlinearity in optical fiber.

21.4 FIBER NONLINEARITY AS A SOURCE FORENTANGLED PHOTONS

Quantum entanglement refers to the nonclassical interdependency of physically

separable quantum subsystems. In addition to being at the heart of the most

fundamental tests of QM [43–46], it is an essential resource that must be freely

available for implementing many of the novel functions of QIP [47, 48]. In

photonic systems, the ongoing developments in lasers, optical-fiber technology,

single-photon detectors, and nonlinear optical materials have led to enormous

experimental progress in both the fundamental [49–53] and applied domains

[54–56]. A popular approach to generating entangled pairs of photons is based

on the nonlinear process of parametric down-conversion in �(2) crystals [57–59].

Although much progress has been made using this approach, formidable engineer-

ing problems remain in coupling the entangled photons into standard optical fibers

[60] for transmission, storage, and manipulation over long distances.

The coupling problem can be obviated if the entangled photons can be gener-

ated in the fiber itself and, desirably, in the fiber’s low-loss propagation window

near 1.5 mm, since that would minimize losses during transmission as well. Apart

from the inherent compatibility with the transmission medium, a fiber-based

source of entangled photons would have other advantages over its crystal counter-

parts [57, 58, 59, 61–63]. Particularly, the spatial mode of the photon pair would be

the guided transverse mode of the fiber, which is a very pure Gaussian-like single

spatial mode in modern fibers. A well-defined mode is highly desirable for

realizing complex networks involving several entangling operations. In this chap-

ter, we describe the first, to the best of our knowledge, optical fiber source of

polarization-entangled photon pairs in the 1550-nm telecom band. A variety of

biphoton interference experiments are presented that show the nature of the

entanglement generated with this source. All four Bell states can be prepared


with our setup and the CHSH form of Bell’s inequality is violated by up to 10

standard deviations of measurement uncertainty.

Recently, our group has demonstrated that parametric fluorescence accompa-

nying nondegenerate FWM in standard optical fibers is an excellent source of

quantum-correlated photon pairs [64, 65]. The quantum correlation arises from

FPS events, wherein two-pump photons at frequency !p scatter through the Kerr

nonlinearity of the fiber to simultaneously create a signal photon and an idler

photon at frequencies !s and !i, respectively, such that !s þ !i = 2!p. For a

linearly polarized pump with wavelength close to the zero-dispersion wavelength

of the fiber, the FWM process is phase-matched and the accompanying parametric

fluorescence is predominantly copolarized with the pump. Two such parametric

scattering processes can be time and polarization multiplexed to create the desired

polarization entanglement. For example [see Figure 21.7(a)], when the fiber is

pumped with two orthogonally polarized, relatively delayed pulses, the signal/idler

photon pairs scattered from each pulse are copolarized with that pump pulse and

relatively delayed by the same amount. The distinguishing time delay between the

orthogonally polarized photon pairs, however, can be removed by passing the pairs

through a piece of birefringent fiber of appropriate length, wherein the photon-pair

traveling along the fast axis of the fiber catches up with the other pair traveling

along the slow axis. When the emerging signal and idler photons are separated

based on their wavelength, each stream of photons is completely unpolarized

because any polarizer/detector combination is unable to determine which pump

pulse a detected photon originated from. When the signal and idler photons are

detected in coincidence, it is still impossible to determine which pump pulse

created the detected pair. This indistinguishability gives rise to polarization entan-

glement in our experiment.

A schematic of the experimental setup is shown in Figure 21.7(b). Signal and

idler photon pairs at wavelengths of 1547.1 and 1525.1 nm, respectively,

are produced in an NFSI [65, 66]. The NFSI consists of a fused-silica 50/50

fiber coupler spliced to 300 m of DSF that has a zero-dispersion wavelength at

�0 = 1535 – 2 nm. Because the Kerr nonlinearity is weak, for this length of fiber

only about 0.1 photon-pair is produced with a typical 5-ps duration pump pulse

containing �107 photons. Thus, to reliably detect the correlated photon pairs, a

pump-to-signal rejection ratio in excess of 100 dB is required. We achieve this by

first exploiting the mirror-like property of the Sagnac loop, which provides a pump

rejection of >30 dB, and then sending the transmitted fluorescence photons along

with the leaked pump photons through a free-space double-grating spectral filter

(DGSF) that provides a pump rejection ratio in excess of 75 dB [65]. The doubly

diffracted signal and idler photons are then recoupled into fibers, whose numerical

apertures along with the geometrical settings of the gratings determine the pass

bands for the signal and idler channels. The FWHM bandwidth for both the

channels is 0.6 nm.

During the experiment, for alignment and phase-control purposes, input signal

and reference pulses are also needed that are temporally synchronized with the


Signal

0 20 40 60 80 100 120 140 160 180 200

Time (s)

Ref.

(a)

(b)

Delay

PM fiber

Hs Hi

Vi

Vs

f

sPH

PV

Temporally delayedorthogonally polarized

pump pulsesTemporally delayed

orthogonally polarizedsignal/idler pairs Polarization-entangled

signal/idler pairs

is

iφis

HHeHHψ

χ(3) FWM

OP

O

APD1

G1

M4M3

Delay

QWP1

PZT

QWP2

Pump

300mDSF Loop

Out

FPC1

EDFA 90/1050/50

FPC2

4% BS

HWP2 &QWP3

F

APD2

Countingsystem

Sig. Det.

Ref. Det.

PM fiber

Signal in

Ref. light

50/50

FPC4

FPC3

P1

P3

P4 &HWP3

G2

P5 &HWP4

M5

M2

M1 G3

G4 P2

HWP1

(c)

Figure 21.7 (a) Conceptual representation of the multiplexing scheme used to create polarization

entanglement. (b) Schematic of the experimental setup. P1–P5, polarization beam splitters; G1–G4,

diffraction gratings; M1–M5, mirrors; FPC1–FPC4, fiber polarization controllers; QWP, quarter-wave plate;

HWP, half-wave plate; F, flipper mirror. (c) Sinusoidal variations (or constancy at the peaks and troughs) of

the photocurrents obtained from the signal (top traces) and the reference detectors (bottom traces) upon

linearly sweeping the voltage (or maintaining a fixed voltage) on the PZT, piezoelectric transducer. The

clarity of the traces demonstrates minutes-long stability of the polarization interferometer formed between P1

and P3 (P2) for signal (reference) light (this figure may be seen in color on the included CD-ROM).


pump pulses. The main purpose of the signal pulses is to ensure that the time

distinguishability between the orthogonally polarized photon pairs is effectively

removed. By spectrally carving [66] the �150-fs pulse train from an optical

parametric oscillator (OPO) [Coherent Inc., model Mira-OPO], we obtain trains

of 5-ps pump pulses, 2.8-ps signal pulses, and 4-ps reference pulses at central

wavelength of 1536, 1547, and 1539 nm, respectively. The pump pulses are then

amplified by an erbium-doped fiber amplifier (EDFA) to achieve the required

average pump power. Light at the signal and idler wavelengths from the OPO that

leaks through the spectral-carving optics and the amplified spontaneous emission

(ASE) from the EDFA are suppressed by passing the pump pulses through a 1-nm

bandwidth tunable optical filter (Newport, TBF-1550-1.0).

A 30-ps relative delay between the two orthogonally polarized pump pulses is

introduced by adding separate free-space propagation paths for the two pulses with

use of a polarization beam splitter (PBS) P1, quarter-wave plates (QWP) QWP1

and QWP2, and mirrors M1 and M2. Mirror M2 is mounted on a piezoelectric

transducer (PZT)-driven translation stage, which allows precise adjustment of the

relative delay and phase difference between the orthogonally polarized pump-

pulse pairs. After the NFSI, the delay is compensated by propagating the scattered

photon pairs along the fast- and slow-polarization axes of a 20-m-long polariza-

tion-maintaining (PM) fiber. A careful alignment procedure is implemented to

properly orient the axes of the PM fiber, taking into consideration the change of

polarization state incurred by an input signal-pulse pair upon maximally amplified

reflection from the NFSI [67]. Alignment is performed prior to the actual experi-

ment by injecting weak path-matched signal-pulse pairs, having identical temporal

and polarization structure as the pump pulses, into the NFSI through the 50/50 and

90/10 couplers. First the signal amplification is maximized by adjusting FPC2,

while monitoring the signal gain on a detector (ETX500) placed after P3. Then the

fringe visibility of the polarization interferometer formed between P1 and P3 is

maximized by adjusting FPC3, HW2, and QWP3, while observing the fringes in

real time upon periodic scanning of M2. Once the alignment is completed, the

injected signal is blocked and further measurements are made only on the para-

metric fluorescence.

After compensation of the time delay, the following polarization-entangled

state is generated at the output of the PM fiber: j�i = jHisjHii þ e2i�pjVisjVii,where �p is the relative phase difference between the two delayed, orthogonally

polarized pump pulses. This source can produce all four polarization-entangled

Bell states. When �p = 0, /2, the states j�–i = jHisjHii – jVisjVii are created.

The other two Bell states j�–i = jHisjVii – jVisjHii can be prepared by insert-

ing a properly oriented HWP in the idler channel. Nonmaximally entangled

pure states with an arbitrary degree of polarization entanglement can also be

created with our setup by choosing the two-pump pulses to have unequal

powers.

To actively monitor and control the relative phase �p during the course of data-

taking, weak reference-pulse pairs of about 50 mW average power are injected into


the NFSI through the 50/50 and 90/10 couplers. The reference-pulse pairs have

identical temporal and polarization structure as the pump pulses, except the

temporal location of the reference-pulse pairs is mismatched with respect to the

pump-pulse pairs and their wavelength is slightly detuned, so that they neither

interact with the pump pulses nor are seen by the single-photon detectors used in

the signal and idler channels. During the course of measurements on the polariza-

tion-entangled states, the relative phase between the reference-pulse pairs, �ref, is

monitored by measuring the photocurrent from a low-bandwidth reference detector

placed after P2 to make observations on one output port of the polarization

interferometer [see Figure 21.7(b)]. The voltage created by this photocurrent is

compared to a reference voltage and the difference is used to stabilize �ref by

feeding back on the PZT through an electronic circuit. The excellent overall

stability of the system is shown by the near-perfect classical interference fringes

displayed in the inset in Figure 21.7(c), which were simultaneously obtained with

injected signal light and with reference light while scanning �ref by ramping the

voltage on the PZT. The relative phase between the reference-pulse pairs, �ref, is

related to the relative phase between the pump-pulse pairs through �p = �ref þ �,where � results from dispersion in the DSF owing to slightly different wavelengths

of the pulse pairs.

The photon-counting modules used for detecting the signal and idler photons

consist of InGaAs/InP APDs (Epitaxx, EPM 239BA) operated in a gated Geiger

mode [65]. The measured quantum efficiencies for the two detectors are 25% and

20%, respectively. The overall detection efficiencies for the signal and idler

photons are about 9% and 7%, respectively, when the transmittance of the

Sagnac loop (82%), 90/10 coupler, DGSF (57%), and other optical components

(90%) are included. Given a parametric scattering probability of ’0.1 pairs/pulse

in the DSF, corresponding to 0.39 mW of average pump power in each direction

around the Sagnac loop, and the gate rate of 588 kHz, we typically observe

’4000 counts/s in the signal and idler channels when detecting the parametric

fluorescence.

The polarization-entanglement generation scheme described here uses the fact

that the FPS efficiency does not depend on the pump-polarization direction. We

verify this by monitoring the parametric fluorescence while varying the polariza-

tion direction of the injected pump pulses with use of a half-wave plate (HWP1).

The individual counts for the signal and idler photons, and their coincidence

counts, vs the HWP1 angle are shown in Figure 21.8(a). The slight variation

observed in the count rates is due to polarization-dependent transmission of the

DGSF. Note that for the measurements shown in Figure 21.8(a) the input pump

delay, the PM-fiber delay compensation, and the detection analyzers were

removed.

Polarization correlations are measured by inserting adjustable analyzers in the

paths of signal and idler photons, each consisting of a PBS (P4, P5) preceded by an

adjustable HWP (HWP3, HWP4). For the state j�i = jHisjVii þ ei2�pjVisjHii,when the polarization analyzers in the signal and idler channels are set to �1 and


–4.71 –3.14 –1.57 0 1.57 3.14

Relative phase φ ref (rad)

Ref

. det

. out

put (

a.u.

)

0

4

8

12

–4.71 3.14

0

40

80

120

0

1

2

3

4

5

0 50 100 150 200HWP1 orientation (°)

Sin

gle

coun

ts(×

103 /

s)

Single counts(×

103/20

s)

Single counts(×

103/30

s)

0

50

100

150 Coincidences (s

–1)

–0.2

1.8

3.8

5.8

7.8

–50 150 350

Relative angle θ1 – θ2 (°)

Coi

ncid

ence

s(×

102 /2

0 s)

Coi

ncid

ence

s(×

102 /

30 s

)

0

20

40

60

80

(b)

δ

(c)

(a)

(d)

Figure 21.8 (a) Observed polarization (in)dependence of parametric fluorescence in the DSF. (b) Coincidence counts and single counts detected over 30 s when the

relative phase fref is varied. The solid curve is a fit to Eqn (21.43). (c) Output from the reference detector vs fref showing the ordinary one-photon interference with

twice the fringe spacing as in (b). (d) Measurement of polarization entanglement: Coincidence counts and single counts detected over 20 s as the analyzer angle in

the idler channel is varied while keeping the signal-channel analyzer fixed at 45 relative to vertical (this figure may be seen in color on the included CD-ROM).

85

6P

remK

um

aret

al.

�2, respectively, the single-count probability for the signal and idler photons is

Ri = �i/2 (i = 1, 2) and the coincidence-count probability R12 is given by

R12 ¼ 2�1�1�2½sin2 �1 cos2 �2 þ cos2 �1 sin

2 �2

þ 2 cosð2�pÞ sin �1 cos �1 sin �2 cos �2; (21.43)

where �i is the total detection efficiency in each channel.

We performed three sets of experiments to evaluate the degree of polarization

entanglement of our source. The first measurement consisted of setting both

analyzers at 45 and slowly scanning �ref by applying a voltage ramp on the

PZT. As shown in Figure 21.8(b), the coincidence counts reveal sinusoidal varia-

tion with a fringe visibility of 93% (dark counts and accidental-coincidence counts

have been subtracted), while the single counts remain unchanged. The output

from the reference detector is also recorded simultaneously, which is shown in

Figure 21.8(c). The relative shift of the sinusoidal variation of two-photon inter-

ference in Figure 21.8(b) from that of reference-light interference in Figure 21.8(c)

is a direct measure of the phase shift �, which is used below to properly set �p for

measurements of the violation of Bell’s inequality.

In the second set of measurements on polarization entanglement, we locked the

generated state to j��i = jHisjVii� jVisjHii by applying an appropriate feedback

on the PZT, fixed the angle of the polarization analyzer in the signal channel to

45 , and varied the analyzer angle in the idler channel by rotating HWP4. The

result is shown in Figure 21.8(d). As expected, the coincidence-count rate displays

sinusoidal interference fringes with a visibility of 92%, whereas the variation in

the single-count rate is only 4% (once again, dark counts and accidental coin-

cidences have been subtracted).

In the third set of experiments, we characterized the quality of polarization

entanglement produced with our source through measurements of Bell’s

inequality violation. By recording coincidence counts for 16 different combina-

tions of analyzer settings with �1 = 0 , 90 , �45 , 45 and �2 = � 22.5 , 67.5 ,22.5 , 112.5 , we measured the quantity S in the CHSH form of Bell’s inequal-

ity [46], which satisfies jSj � 2 for any local realistic description of our experi-

ment. The results, which are presented in Table 21.1, show that (a) the CHSH

Table 21.1

Measured values of S for the four Bell states.

Bell state S Violation ( )

jHisjHii þ jVisjVii 2.75 – 0.077 10

jHisjHii� jVisjVii 2.55 – 0.070 8

jHisjVii þ jVisjHii 2.48 – 0.078 6

jHisjVii� jVisjHii 2.64 – 0.076 8


inequality is violated, i.e., jSj > 2, for all four Bell states produced with our

setup and (b) the violation occurs by up to 10 standard deviations ( ) of

measurement uncertainty.

To ascertain the degree of entanglement produced by the true FPS events in our

setup, the accidental coincidences resulting from the uncorrelated background

photons and the dark counts in the detectors were measured for each set of data

acquired in the three polarization-entanglement experiments described above. The

rate of accidental coincidences was as large as the rate of “true” coincidences,

plotted in Figure 21.8 by subtracting the accidental coincidences, and the raw

visibility of TPI was only ’30%. We believe, the majority of background photons

in our setup arise from SRS as verified by our recent measurement of the noise

figure of fiber-optic parametric amplifiers [68–70]. Our recent measurement with a

modified DGSF has shown that the contribution of accidental coincidences can be

made <10% of the total measured coincidences [64]. With these improvements, a

raw TPI visibility of >85% would be obtained, i.e., without any postmeasurement

corrections.

In conclusion, we have developed and characterized a fiber-based source of

polarization-entangled photon pairs. The pair-production rate can be dramati-

cally increased by using state-of-the-art pulsed lasers that have been developed

for fiber-optic communications. These lasers operate at 10–40-GHz repetition

rates and can have the requisite peak-pulse powers with use of medium-power

EDFAs. Bulk-optic implementations of the pump delay apparatus and the

detection filters were used in these proof-of-principle experiments for purposes

of tunability and control. All-fiber versions of these subsystems can be readily

realized with use of PM fibers, wavelength-division-multiplexing filters, and

fiber polarizers. Finally, we have understood the origin of the large number of

accidental coincidences in the experiment and subsequent system improvements

are expected to significantly improve the degree of entanglement produced with

our system. Therefore, we believe that such fiber-based entangled-photon pairs

will prove to be an efficient source for developing quantum communication

technologies.

21.5 HIGH-FIDELITY ENTANGLEMENT WITHCOOLED FIBER

For many QIP applications it is desirable to produce entangled photon-pairs at

telecom wavelengths directly in the fiber by use of the fiber’s Kerr nonlinearity.

Previously, our group has developed the use of an NFSI to generate quantum-

correlated twin beams [66], correlated twin-photon pairs [71], and polarization

entanglement [72–74]. We have pointed out that SRS, which gives rise to the

majority of background photons, prevents us from observing TPI with unit visibi-

lity [74–76]. In these experiments, we have reduced the contribution of Raman


photons by lowering pump power, selecting signal/idler photon-pairs at small

detuning from the pump (’5 nm), and using polarizers to remove the cross-

polarized Raman photons. In this chapter, we present our measurements of the

ratio between coincidence counts (coincidences arising from the same pulse) and

accidental-coincidence counts (coincidences arising from adjacent pulses) as a

function of pump power, and with the DSF at three different temperatures: ambient

(’300 K), dry ice (195 K), and liquid nitrogen (77 K). Then, we create polarization

entangled photon-pairs with the fiber at the above three temperatures using a

compact counterpropagating scheme (CPS) [77, 78]. We observe TPI with visibi-

lity >98% and Bell’s inequality violation by >8 standard deviations of measure-

ment uncertainty at 77 K.

As shown in Figure 21.9, the pump is a mode-locked pulse train with pulse

duration ’5 ps and repetition rate of 75.3 MHz, which is obtained by using a

diffraction grating to spatially disperse the output of an optical parametric oscil-

lator (Coherent Inc., model Mira-OPO). The pump central wavelength is set at

1538.7 nm. To achieve the required power, the pump pulses are further amplified

by an EDFA. The ASE from the EDFA is suppressed by passing the pump through

a tunable filter (Newport, model TBF-1550-1.0) with 1-nm FWHM passband.

A fused-silica 90/10 fiber coupler is used to split 10% of the total pump power

to a power meter for monitoring the power and stability of the pump pulses. The

remaining 90% of the pump power goes through " fiber polarization controller

(FPC1) and a linear polarizer (LP), whose purpose is to adjust the input pump field

90/10coupler

FPC2

FP

FP

FPC1

FP

FP

LP

HWP1at 22.5°

PBS1

Signal Idler

PV

PH

Fiber spool

Filter>100 dB

L4

L2 L3

L5

Polarization analyzer

PBS

PBS

HWP

QWP

HWP

QWP

D1

D2

Idler = 1543.5 nm

Signal = 1533.9 nm

Figure 21.9 A schematic of the experimental setup. FP, fiber-port; LP, linear polarizer; L2, L3, L4, L5,

fiber-to-free space collimators; PBS, polarization beam splitter; HWP, QWP, half- and quarter-wave

plates; FPC, fiber polarization controller (this figure may be seen in color on the included CD-ROM).


to be horizontally polarized. A half-wave plate (HWP1) can be inserted in the

pump path to split the pump pulse into horizontally and vertically polarized

components for creating polarization entanglement. Initially, we use a single

pump with horizontal polarization to pump a straight 300 m of DSF whose zero-

dispersion wavelength is at �0 = 1538.7 nm. The fiber is wound on a spool of

9.32 cm diameter. A fiber polarization controller (FPC2) is used to compensate the

bend-birefringence induced changes in the polarization states in the fiber. To

reliably detect the scattered photon-pairs, an isolation between the pump and

signal/idler photons in excess of 100 dB is required. We achieve this by using

two cascaded WDM filters with FWHM of about 1 nm in the signal and idler

channels, which provide total pump isolation greater than 110 dB [74]. The

selected signal and idler wavelengths are 1543.5 and 1533.9 nm, respectively,

corresponding to ’4.8 nm detuning from the pump’s central wavelength. Two

sets of polarization analyzers, each composed of a quarter-wave plate (QWP), a

half-wave plate (HWP), and a PBS are constructed and are individually inserted

into the signal and idler channels. With proper settings of the QWP and HWP in

each channel, the signal/idler photons with horizontal polarization can be made to

arrive at the detectors with negligible loss. Raman photons that are copolarized

with the pump also reach the detectors, while cross-polarized Raman photons are

blocked by the PBSs. The copolarized Raman photons are inevitably detected,

and hence contribute to the background photons in coincidence detection. It is

known that Stokes and anti-Stokes Raman-scattering noise photons are emitted

at a rate proportional to nth þ 1 and nth [79, 86], respectively, where nth =1/[exp(h�/kT)� 1] is the Bose population factor, � is the frequency shift of

Stokes and anti-Stokes from the pump frequency, T is the temperature of the

fiber, h is Planck’s constant, and k is Boltzmann’s constant.

The photon-counting modules consist of InGaAs/InP APDs (Epitaxx, Model

EPM239BA) operated in the gated Geiger mode at room temperature [71].

The 1-ns-wide gate pulses with the FHWM window of about 300 ps arrive at a

rate of 588 kHz, which is downcounted by 1/128 from the original pump

pulses. The gate pulses are adjusted by an electronic delay generator to

coincide with the arrival of the signal and idler photons at the APDs. The

quantum efficiency of APD1 (APD2) is about 25% (20%), with a correspond-

ing dark-count probability of 2.2 � 10�3 (2.7 � 10�3) per pulse. The total

detection efficiencies for the signal and idler photons are about 7% and 9%,

respectively. Because the size of the fiber spool used in our experiment is

smaller than the regular spool provided by Corning, it is convenient to cool

the DSF by dry ice in a small homemade box or by liquid nitrogen in a small

dewar. The loss due to fiber bending and handling, even in the low-tempera-

ture environment, is negligible. The loss due to cooling the fiber is less than

1% (4%) at 195 K (77 K). Cooling the fiber also causes contraction of the

fiber length, which in turn shortens the propagation time for the photon-pairs

to arrive at the detectors. The advancement of the arrival time is 1.6 ns (3.0 ns)

at temperature of 195 K (77 K).


With the fiber at each temperature, we record coincidence and accidental-

coincidence counts with an integration time of 60 s as the pump power. Note

that accidental coincidence is the measured coincidence counts as the signal

channel is delayed with respect to the idler channel by one pulse period. After

subtracting the detector dark counts, we plot the ratio between the coincidence

and accidental coincidence vs signal (or idler) counts per pulse. The plot at each

temperature has a similar shape as those observed by other groups [81, 82]. The

reason for such shape is not yet clearly understood. At high pump powers, the

ratios are similar for the cooled and uncooled fibers. This may be due to

multiphoton effects or leakage of the pump photons through the filters leading

to increasing accidental-coincidence counts. As the pump power is decreased,

we observe a ratio as high as 111:1 at 77 K and 60:1 at 195 K compared to a

ratio of 28:1 at room temperature (300 K). The reason is that as the fiber is

cooled, the Raman scattering is reduced as given by nth, and leads to a decrease

in the accidental-coincidence counts. The relative increase of the ratio is

commensurate with the temperature dependence of nth, which is about a factor

of 4.5 (1.6) for the fiber at 77 K (195 K) compared to its room temperature

value. At 300 and 195 K, the peaks in the ratio occur at an average pump power

of about 50 mW (133 mW peak power). At 77 K, the peak occurs at a higher

average pump power about 75 mW (200 mW peak power) because of cooling-

induced loss in the fiber. At these peaks, the photon-pair production rate

is ’0.01/pulse.

To further test the fidelity of the generated photon pairs, we create polariza-

tion entanglement with the cooled fiber by using a CPS [77, 78]. The single

horizontally polarized pump pulse is now split into two equally powered, ortho-

gonally polarized components PH and PV by inserting and properly setting the

half-wave plate (HWP1) in front of the polarization beam splitter (PBS1). For

low-FPS efficiencies, where the probability for each pump pulse to scatter more

than one pair is low, the clockwise (CW) and counterclockwise (CCW) pump

pulses scatter signal/idler photon pairs with probability amplitudes jHiijHsi and

jViijVsi, respectively. After propagating through the DSF, these two amplitudes

of the photon pair are then coherently superimposed through the same PBS1.

This common-path polarization interferometer has good stability for keeping

zero-relative phase between horizontally and vertically polarized pumps, and

hence is capable of creating polarization entanglement of the form jHiijHsi þei2�pjViijVsi at the output of PBS1, where the relative pump phase �p is set to 0

by setting HWP1 to 22.5 . The polarization analyzers in signal and idler channels

are used to set the detection polarization angles �1 and �2 of the entangled two-

photon state. In our experiment, we set �1 = 45 , vary �2, and record single

counts for both signal and idler channels as well as coincidence counts between

the two channels for each value of �2. We repeat these measurements with the

fiber spool at 300, 195, and 77 K with corresponding integration times of 10, 50,

and 60 s, respectively. At room temperature, we observe TPI with 91% visibility.

As the fiber is cooled to 195 and 77 K, the TPI visibility increases to 95% and 98%,


respectively, as shown in Figure 21.10. All these TPI fringes are obtained when the

pump power is adjusted to match the peak values of the ratio and are recorded

without subtraction of accidental-coincidences. The observed higher visibility at low

temperatures can be attributed to the suppression of Raman photons. The fitting

function used in the above figures is cos2(�1� �2). We believe that the imperfection

in spatial-mode matching of the correlated nondegenerate photon pairs at the PBS1

and the remaining copolarized Raman photons prevent this scheme from achieving

unit-visibility TPI fringes.

0

100

200

300

400

500

0

1 × 104

2 × 104

3 × 104

4 × 104

0

5 × 104

1 × 104

1.5 × 104

2 × 104

2.5 × 104

3 × 104

3.5 × 104

0 50 100 150 200

Coi

ncid

ence

cou

nts

in 5

0 s

0

100

200

300

400

500

600

700

800

Coi

ncid

ence

cou

nts

in 6

0 s

Sig

nal/i

dler

cou

nts

in 6

0 s

Sig

nal/i

dler

cou

nts

in 5

0 s

(a)

Signal

Idler

Two-photoninterference

Signal

Idler

Two-photoninterference

Relative angle θ1 − θ2 (°)

0 50 100 150 200

(b)

Relative angle θ1 − θ2 (°)

Figure 21.10 Two-photon interference with the fiber at (a) 195 K and (b) 77 K. The observed

visibility is about 95% and 98.3%, respectively (this figure may be seen in color on the included

CD-ROM).


We further confirm the nonlocal behavior of the polarization entangled photon

pairs generated from this source by making Bell’s inequality measurements. For

this purpose, we prepare the singlet state jHiijVsi� jViijHsi (where �p = /2) by

inserting a QWP at 0 after HWP1 and by adding a HWP at 45 (HÐV) in the

signal channel. By recording coincidence counts for 16 different combinations of

analyzer settings with �1 = 0 , 90 , �45 , 45 and �2 = � 22.5 , 67.5 , 22.5 ,112.5 , we measure the quantity jSj in the CHSH form of Bell’s inequality

[46], wherein jSj � 2 holds for any local realistic system. The results are shown

in Table 21.2. At 77 K we measure jSj = 2.76 – 0.09, which amounts to Bell’s

inequality violation by over eight standard deviations of measurement uncertainty.

All these measurements are made without subtraction of the background Raman

photons. Given these achievements, we believe we can reliably implement Ekert’s

QKD protocol with our polarization-entangled photon-pairs source.

21.6 DEGENERATE PHOTON PAIRS FOR QUANTUMLOGIC IN THE TELECOM BAND

The seminal paper by Knill et al. [83] has rekindled considerable amount of interest

in the field of linear optical quantum computing (LOQC). For real applications of

LOQC, single-photon as well as entangled-photon sources are indispensable [84].

Quantum interference arising from two indistinguishable photons, such as the well-

known Hong-Ou-Mandel (HOM) interference [85], lies at the heart of LOQC.

Therefore, it is of great significance to develop photon sources that generate

identical photons in well-defined spatiotemporal modes. Additionally, for distributed

LOQC it is desirable to produce such photons in the 1550-nm telecom band.

21.6.1 Polarization-Entangled Degenerate Photon-PairGeneration in Optical Fiber

Entangled photons in well-defined time slots are usually obtained using pulsed

SPDC in �(2) crystals [86], wherein a high-frequency pump photon (!p) fissions

into two identical daughter photons (!1 = !2 = !p/2). Recently, Fan et al. [87]

Table 21.2

Violation of Bell’s inequality for the state jHiijVsi� jViijHsi.s is the standard deviation in the measurement of jSj.

Temperature (K) jSj Violation ( )

300 2.22 – 0.06 4

77 2.76 – 0.09 >8


have also produced identical correlated photons at the mean frequency (!c) of two

pump frequencies (!p1 þ !p2 = 2!c) by using a reverse degenerate FWM process

in a piece of �(3) microstructure fiber (MF). In that experiment, the generated

photons reside in the visible/near-infrared wavelength region (�< 800 nm). The

spatial-mode profile of the identical photons is thus incompatible with standard

single-mode fiber, making them unsuitable for distributed LOQC. Here, we

describe a source useful for distributed LOQC by producing identical photons at

a telecom-band wavelength using standard DSF. As a further endeavor, we use a

novel dual-pump, CPS [78, 88] to make the otherwise identical photons entangled

in polarization. We thus present the first, to the best of our knowledge, telecom-

band, degenerate, correlated/entangled photon source based on a spool of standard

DSF, which constitutes a promising step toward practical implementation of

LOQC using fiber-based devices.

Figure 21.11 depicts our experimental setup. Out of the broadband spectrum of

a femtosecond laser (repetition rate ’50 MHz), we spectrally carve out our desired

pump central wavelengths (�p1 = 1545.95 nm, and �p2 = 1555.92 nm, pulse

width ’ 5 ps) by cascading two free-space double-grating filters (DGF1 and

DGF2, FWHM ’0.8 nm for each passband, see inset in Figure 21.12). An EDFA

is sandwiched in between the two DGFs to provide pump power variability. The

out-of-band ASE photons from the EDFA are suppressed by DGF2. Here we

utilize the same nonlinear process as in Ref. [87], namely, reverse degenerate

(a)

Laser DGF1 EDFA

50/50BS1

DGF2

50/50BS2

Pump

FPC1

FPC2

λp1

λp1

λp2

λp2

(b)

Pump

λp1 λp2

DSF 300 m

OBPF

Coincidencecounter

50/50BS3

λc

FPC3

HWP1

QWP1

LP

HWP2PBS1

HWP3

HWP4

QWP2

QWP3

PBS2

PBS3

APD1APD2

H

V

Figure 21.11 Experimental layout: (a) Pump preparation; and (b) CPS scheme. See text for details (this

figure may be seen in color on the included CD-ROM).


FWM. The pump central wavelengths are selected such that their mean wave-

length (�c = 1550.92 nm) is located near the zero-dispersion wavelength of the

300-m-long DSF. The two-pump pulses emerging from the second 50/50 beam

splitter (BS2), before being launched into the CPS, have to satisfy the following

criteria to maximize the FWM efficiency and the resulting entanglement in the

two-photon state: (i) overlapped in time, (ii) parallel polarization, and (iii) equal

power. The first criterion is met by careful path matching, and the resulting

overlapped pulses from the unused port of BS2 can be monitored on a high-

speed oscilloscope. The second criterion is satisfied by adjusting the fiber polar-

ization controllers (FPC1 and FPC2) and the half-wave plate/quarter-wave plate

combination (HWP1 and QWP1) such that the power exiting the LP is maximized.

The last criterion is fulfilled by balancing the transmission efficiencies in the two

arms of the cascaded DGFs. SRS in the DSF is suppressed by cooling the DSF to

77 K [89, 90]. Compared with the cascade-MF approach in Ref. [87], this setup is

straightforward and simpler in design.

We first characterize our degenerate correlated-photon source by measuring

temporal coincidences between the two identical photons. We call a “coincidence”

count when the two detectors fire in the same triggered time slot, and an “accidental-

coincidence” count when they fire in the adjacently triggered time slots. In the

literature [87, 89, 90], a figure of merit for such a source has been established, i.e.,

the coincidence-to-accidental ratio (CAR). We abbreviate it as CAR hereafter.

Intuitively, CAR is a measure of the purity of a correlated photon source, and a

high CAR value indicates a relatively high purity of the source in

coincidence basis, i.e., coincidence events due to uncorrelated noise photons are

0

20

40

60

80

100

120

0 0.001 0.002 0.003 0.004 0.005

77 K

300 K

Single counts/pulse

CA

R

–80

–70

–60

–50

–40

–30

–20

–10

0

–150

–100

–50

0

1540 1545 1550 1555 1560Wavelength (nm)

DGFs

OBPF

Rel

ativ

e tr

ansm

issi

onsp

ectr

um (

dB)

Figure 21.12 CAR plotted as a function of single counts/pulse at two different temperatures [77 K

(dots) and 300 K (squares)]. Inset: spectral shapes of the cascaded DGFs and OBPF (this figure may be

seen in color on the included CD-ROM).


very rare. To measure the CAR, we set HWP2 so that the two-frequency pump field

maintains its horizontal polarization upon entering the CPS. After propagating

through the DSF, the pump and its accompanying degenerate FWM photons are

made to keep the same polarization by proper adjustment of FPC3. We use an optical

bandpass filter (OBPF) composed of two cascaded 100-GHz-spacing wavelength-

division-multiplexing filters (FWHM ’0.3 nm, see Figure 21.12, inset) with a

central wavelength of �c at the output of the CPS to collect the correlated FWM

photon pairs, and to provide the >100 dB pump isolation needed to effectively

detect those photons. The OBPF is followed by a 50/50 beam splitter (BS3) to

probabilistically split the two daughter photons. We are only interested in the cases

where the two daughter photons split up (which happen with 50% probability) to

give coincidence counts; the other equally probable cases where the two photons

bunch together are not recorded in coincidence detection. Nevertheless, the latter

contribute to the single-count measurement results. For this part of the experiment,

the polarization analyzers (HWP3/QWP2/PBS2 and HWP4/QWP3/PBS3) shown in

Figure 21.11 are taken out, as the polarization properties of the detected photons are

not of concern in this part of the experiment. Each photon is directed to a fiber-

coupled APD1 and APD2 (Epitaxx EPM 239BA) operating in the gated Geiger

mode, whose detection results are recorded and analyzed by a “coincidence

counter” software. For the measurement reported in this chapter, the detection

rate is 1/64 of the laser’s repetition rate (’780 kHz), which is limited by the

electronics used to arm and trigger the detectors. The overall efficiencies of the

two detectors, including propagation losses and detector quantum efficiencies, are

7% and 9%, respectively. We subtract detector dark-count contributions from all of

our measurement results.

A sample experimental result is shown in Figure 21.12, in which we plot the

CAR as a function of the single-channel photon-detection rate. A preliminary

version of this result has been reported previously [91]. The CAR is measured at

the ambient temperature (300 K) and when the DSF is immersed in liquid nitrogen

(77 K). A CAR value as high as 116 is obtained at 77 K, whereas the peak of the

CAR at 300 K is around 25. This is consistent with the expectation that SRS is

more severe at a higher temperature, resulting in degraded purity of the source.

The shape of the CAR function is similar to those reported previously [87, 89, 90]

and can be qualitatively explained by a quantum model [92], which correctly

predicts the CAR’s temperature dependence. However, further work is required

for a quantitative comparison between experiment and theory.

After determining the single-count rate for the optimum CAR value, we adjust

HWP2 to split the dual-frequency pump field into two orthogonally polarized dual-

frequency components with equal power (more specifically, P1H = P1V = P2H =P2V). Each pump component probabilistically produces its own degenerate signal/

idler photon pair, and the two probability amplitudes superpose upon each other at

the output of the CPS, as there is no distinguishability between them. As a result of

reverse degenerate FWM and the CPS, polarization entanglement of the form

jHsHii þ jVsVii is generated, wherein the signal/idler photons are of identical


wavelength �c. After passing through the 50/50 beam splitter BS3, the photons are

then detected in coincidence as before, except we add in the polarization analyzers

shown in Figure 21.11(b), because it is the polarization entanglement properties of

the photon pairs that are to be examined here. In the experiment, the polarization

analyzer in one photon’s path is set to a fixed linear polarization angle �0, and we

stepwise rotate the HWP in the other photon’s path to go from an initially parallel

polarization angle �0 to a final polarization angle �0 þ , which corresponds to a

HWP rotation angle of D� ˛ [0, /2]. Both single counts and coincidence counts

are recorded as a function of D�.A sample experimental result is shown in Figure 21.13. We demonstrate the high

purity of our entanglement source by exhibiting TPI with visibility >97% together

with polarization-independent single counts. These results are obtained when we pump

the CPS with relatively low power [P1H = P1V = P2H = P2V ’ 90 mW (peak power

’ 0.36 W)] and the DSF is cooled to 77 K. The high-TPI visibility means that we are

indeed generating the maximally entangled state jHs Hii þ jVsVii. Nonmaximally

entangled states can also be generated if we change the relative pump-power ratio

between the CW and the CCW paths. The relatively large error bars associated with the

TPI data points can be attributed to the low-data-collection rate in our current imple-

mentation, which can be improved by either using an OBPF with a wider passband, or

using faster detection electronics [93]. As is the case with other CPS entanglement

sources [78, 88], the dual-pump CPS source presented here can be easily configured to

produce all four Bell states (jHs Hii – jVs Vii, jHs Vii – jVsHii) and the Bell’s inequality

violation can be inferred from the observed >71% TPI visibility.

We further investigate the scenario when only a single-frequency pump (at

either �p1 or �p2) is injected into the CPS. As shown in Figure 21.14, we

0

50

100

150

200

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

–20 0 20 40 60 80 100

Δθ (°)

Coi

ncid

ence

/120

s

Sin

gle

coun

ts/1

20 s

Signal

Idler

TPI

Figure 21.13 Two-photon interference (TPI) with visibility >97% is shown, while single counts in

both channels exhibit no polarization dependence. Solid curve is a cosine fit to the data (this figure may



observe polarization-independent behavior for both the single counts and the

coincidence counts. In fact, the coincidence counts after subtraction of the dark-

count contributions in both cases are zero within their error-bar ranges. This is

expected because the FWM photons in this case are produced not by the reverse

degenerate FWM process, but by the conventional FWM process involving a

degenerate-frequency pump. As the filters in the experiment are configured

to look at coincidences between photons in a single band (i.e., without its

conjugate), the pairwise nature of the conventional FWM photon production is

not manifested. Hence all the recorded coincidence counts are light–dark

0

–20 0 20 40

Δθ (°)60 80 100

200

400

600

800

1000

0

5,000

10,000

15,000

20,000

signal

idler

CC

(a)

–20 0 20 40

Δθ (°)60 80 100 120

(b)

0

200

400

600

800

1000

0

5,000

10,000

15,000

20,000

25,000

signal

idler

CC

Sin

gle

coun

ts/6

0 s

Sin

gle

coun

ts/6

0 s

Coi

ncid

ence

/60

sC

oinc

iden

ce/6

0 s

Figure 21.14 Single counts and coincidence counts (CC) show no dependence on the HWP angle D�when only a single-frequency pump at �p1 = 1545.95 nm (a), or at �p2 = 1555.92 nm (b), is used (this



coincidences, which after subtraction of the dark counts lead to zero light–light

coincidence counts. This serves as a further evidence that the degenerate-

frequency polarization-entangled photon pairs are indeed produced by the

dual-frequency pumps, and not by the single-frequency pumps. The polariza-

tion-independent single counts in Figure 21.14 are a result of the CPS-induced

polarization indistinguishability.

To conclude this section, we have presented an efficient, high-purity source of

correlated/entangled photon pairs in a single spatiotemporal mode in the telecom

band. It is foreseeable that further development of such fiber-based photon sources

will lead to practical applications of various quantum-communication protocols

and LOQC.

21.6.2 Hong-Ou-Mandel Interference with Fiber-GeneratedIndistinguishable Photons

The nascent field of quantum information science, motivated by the extraordinary

computing power of a full-fledged quantum computer, naturally selects photons—

the fundamental energy packets of the electromagnetic field—as the carrier of

quantum information from one computing node to another, a task typically

associated with quantum communication. The omnipresent telecommunication

fibers constitute the quantum channels for quantum cryptography [16], quantum

gambling [94], and quantum games [95], which utilize remote sharing of quantum

entanglement as a resource. In addition, indistinguishable photons—photons hav-

ing identical wave packets—play a major role in the arena of LOQC [96]. There-

fore, a telecom-band indistinguishable photon-pair source is particularly useful for

the above QIP tasks. The traditional method of producing such photons—SPDC

in second-order (�(2)) nonlinear crystals [97]—faces formidable engineering

challenges when these photons are launched into a single-mode optical fiber for

long-distance transmission or mode cleansing. Large coupling losses inevitably

occur due to mode mismatch [98], limiting the usefulness of such a source.

Recently, we demonstrated a fiber-based source of indistinguishable photon pairs

at a telecom-band wavelength near 1550 nm that utilizes the third-order (�(3))

nonlinear process of FWM [99] in the fiber itself. This approach automatically

takes care of the aforementioned mode-matching issue, because the photonic

spatial mode of the generated photon pair is the same as that of standard optical

fiber.

Four-wave mixing is a third-order process mediated by the Kerr nonlinearity of

optical fiber, wherein two-pump photons annihilate to give birth to a pair of time-

energy entangled daughter photons, usually denoted as signal and idler. Energy

conservation as well as momentum conservation are obeyed during the FWM

process: !p1 þ !p2 = !s þ !i, and ~kp1 þ ~kp2 =~ks þ ~ki, where !j and ~kj stand

for the frequency and wavevector of the jth photon, and subscripts p1, p2, s,


and i denote the two pump photons, the signal photon, and the idler photon,

respectively. Our group has previously demonstrated generation [29, 30], distribu-

tion [100, 101], and storage [38, 39] of nondegenerate photon pairs (s 6¼ i) with a

single-frequency pump pulse (p1 = p2). More recently, we have also achieved

success in generating degenerate photon pairs (s = i) with dual-frequency pump

pulses (p1 6¼ p2) [99]. While it is straightforward to separate the nondegenerate

photon pairs by means of their different wavelengths through a wavelength-

division (de)multiplexer, it remains unclear how to deterministically separate the

fully degenerate photon pairs, as the two photons share the same properties in all

degrees of freedom: spatial, temporal (frequency), and polarization. This situation

is analogous to a type-I down-conversion with collinear output photons [102]. Up

to now, researchers have used a regular 50/50 beam splitter (BS) to probabilisti-

cally split the identical photons [99, 102, 103], which in actuality produces the

following state [schematically shown in Figure 21.15(a)]:

j�iin ¼ j2iaj0ib;

j�iout ¼1ffiffiffi2p j2idj0ic � j0idj2icffiffiffi

2p þ iffiffiffi

2p j1icj1id

1ffiffiffi2p �2002 þ

iffiffiffi2p �11: (21.44)

The presence of the �2002 component limits the usefulness of such a probabilistic

identical photon source, which has a maximum attainable HOM dip visibility of

only 50% [102].

However, if the output wavefunction only consists of the �11 component, the

HOM dip visibility can in principle reach 100%.

(a) (b)

d c

FPC

CWCCW

Input

50/50 coupler

P/2 P/2

P

02 ba20 b

iδae

ab

50/50BS

a

b

c

d

ψin ψout

Figure 21.15 (Color online) Quantum interference at a beam splitter. (a) Schematic of input/output

state transformation for a 50/50 beam splitter (BS). (b) Illustration of quantum interference in a Sagnac

loop between photon pairs with different phase shift �. CW, clockwise; CCW, counterclockwise; FPC,

fiber polarization controller (this figure may be seen in color on the included CD-ROM).


One way to produce a “clean” �11 output is to utilize a cross-polarized dual-

frequency pump to excite the �xyxy component of the Kerr nonlinear susceptibility

tensor [104, 105]. This is analogous to a type-II collinear down-conversion [106],

where the output orthogonally polarized degenerate-frequency photon pairs are

split by a PBS in a deterministic fashion. However, the intrinsically weaker nature

of �xyxy, which is 1/3 of the collinear tensor component �xxxx in a Kerr nonlinear

medium like fused-silica glass, makes this approach much less efficient than the

copolarized FWM process. Here we introduce a new type (or topology) of copo-

larized identical-photon source, called the “50/50 Sagnac-loop” source, which is

capable of producing a clean �11 state. Due to its intended use to deterministically

split up the identical photons, it is also given the name “quantum splitter,” or QS

for short.

The idea behind the QS source comes from the realization that, not only can a

fiber Sagnac loop be configured as a total reflector [107] (TR) or a total transmitter

(TT), it also can be set in an equally transmissive and reflective state (50/50). The

difference between the above operational modes lies in the different settings of the

intraloop fiber polarization controller (FPC), which results in different relative

phase shifts between the CW and the CCW paths. As shown in Figure 21.15(b), the

pump is injected from port d into the Sagnac loop, which is composed of a 50/50

fiber coupler, a piece of DSF of suitable length, and an FPC. The pump peak power

is assumed to be P, which is split into two equally powered pulses (P/2) by the

coupler. The two-pump pulses traverse the DSF in a counterpropagating manner,

each of which probabilistically scatters copolarized FWM photon pairs. Here we

neglect the case where both pumps undergo FWM scattering, as it corresponds to a

higher order process of multiphoton generation, whose probability is vanishingly

small when the pump power is low. The two identical-photon probability ampli-

tudes, with a differential phase � controlled by the setting of the FPC, are then

recombined at the coupler before coming out of the Sagnac loop. The input state in

this case is written as

j�iin =j2iaj0ib þ ei�j0iaj2ibffiffiffi

2p : (21.45)

The corresponding output state is obtained from the standard BS input/output

relationship, and is given by

j�iout1� ei�

2�2002 þ

ið1þ ei�Þ2

�11: (21.46)

We can readily see, from Eqn (21.46), that a pure �11 state is obtained when we

set the differential phase � to be 0, which corresponds classically to the case of a

50/50 Sagnac loop. This can be physically interpreted as time-reversed HOM

interference. One can also verify that when � is set to be (-), the Sagnac loop

is totally reflective (transmissive), and one obtains the pure state �2002.


We now demonstrate the identical nature of the photons produced by the QS

source. The observation of a HOM dip of high visibility has been established as a

figure of merit for identical-photon sources [108], and has been demonstrated

using a variety of devices, including identical photon pairs from the same SPDC

source [109, 110], independent indistinguishable photons from separate SPDC

sources [111–113], and indistinguishable photons from a quantum-dot single-

photon source [114]. Our scheme bears some resemblance to the first category,

but yet is different enough in that it is fiber-based and thus easily integrable into

fiber-optic networks [115]. The experimental setup is depicted in Figure 21.16

with considerable amount of detail. Figure 21.16(a) shows how the dual-frequency

copolarized pump is prepared [99]. Out of the broadband spectrum of a

femtosecond laser (repetition rate ’50 MHz), we spectrally carve out our desired

pump central wavelengths (�p1 = 1545.95 nm and �p2 = 1555.92 nm, pulse width

’ 5 ps) by cascading two free-space double-grating filters [DGF1 and DGF2,

FWHM ’ 0.8 nm for each passband; see Figure 21.17(a)]. An EDFA is

Pump1

Pump2

50/50 Sagnacinterferometer

OBPF2

OB

PF

1

λ m

λ p1

λ p2

FPC3

Circ

50/50BS3

Coincidence

counter

Translationstage

HWP1

HWP2

QWP2

QWP3

QWP4QWP1 PBS1 PBS2

PBS3

BS

APD1

e

APD2

f

g

h

M1

M2

(b)

Laser DGF1λ p1

λ p2

50/50BS1

EDFADGF2

50/50BS2

FPC1

FPC2

FPBS

Pump1

Pump2

λ p1

λ p2

(a)

λ m

Figure 21.16 (Color online) Schematic experimental setup to observe the Hong-Ou-Mandel

interference between identical photons generated from the QS source. (a) Preparation of the dual-

frequency copolarized pump. BS, beam splitter; DGF, double-grating filter; FPBS, fiber polarization

beam splitter; FPC, fiber polarization controller; EDFA, erbium-doped fiber amplifier. (b) Hong-Ou-

Mandel experiment. PBS, polarization beam splitter; HWP, half-wave plate; QWP, quarter-wave plate;

OBPF, optical band-pass filter; Circ, circulator; APD, avalanche photodiode (this figure may be seen in

color on the included CD-ROM).


sandwiched in between the two DGFs to provide pump power variability. The out-

of-band ASE photons from the EDFA are suppressed by DGF2. The pump central

wavelengths are selected such that their mean wavelength (�m = 1550.92 nm) is

located near the zero-dispersion wavelength of the 300-m-long DSF in the Sagnac

loop to maximize the FWM efficiency [116]. The two-pump pulses emerging from

the second 50/50 beam splitter (BS2) are in turn passed through a fiber polarization

beam splitter (FPBS) to ensure their copolarized property. The other necessary

properties of the two-pump pulses, namely, temporal overlapping and equal power,

are individually addressed by careful path-matching and transmission-efficiency

balancing for the two pulses. Figure 21.16(b) shows the QS source and its intended

use in a HOM experiment. The Sagnac loop is preceded by a circulator (Circ),

which redirects the Sagnac-loop reflected photons to a separate spatial mode. The

output degenerate FWM photons from the Sagnac loop are selected by two optical

bandpass filters (OBPF1 and OBPF2), whose transmission spectrum is shown in

Figure 21.17(a) (center wavelength = 1550.92 nm, passband ’ 0.8 nm). The

OBPFs also provide the necessary >100-dB isolation from the pump to effectively

detect those filtered photons. The alignment of the Sagnac loop to its 50/50 state

consists of using a continuous-wave laser source with its wavelength set to �m as

the input to the Sagnac loop, and adjusting FPC3 so that the transmitted and

reflected powers are equal (i.e., Pe = Pf). In practice, we measure the powers at

the output of the OBPFs (Pg and Ph), and demand that Pg/�eg = Ph/�fh, where �eg

(�fh) is the transmission efficiency from point e (f) to point g (h). The experimental

values of �eg and �fh are 0.749 and 0.768, respectively. SRS in the DSF is

suppressed by cooling the DSF to 77 K using liquid nitrogen [117, 118].

Before the actual HOM experiment, we first characterize the QS source

by measuring its CAR [99], which has been established to be a figure of merit of

Single counts/pulse

CA

R

0

20

40

60

80

100

0

1000

2000

3000

4000

0 0.00125 0.0025 0.00375 0.005

Coi

ncid

ence

/60

s

Single counts/pulse

Coincidence

Accidentalcoincidence

–801540 1550 15551545 1560 0 0.001 0.002 0.003 0.004 0.005

–70

–60

–50

–40

–30

–20

–10

Wavelength (nm)

(a) (b)

Tra

nsm

issi

on s

pect

rum

(dB

) OBPFsDGFs

Figure 21.17 (Color online) Experimental parameters. (a) Transmission spectra of the OBPFs and

DGFs. (b) Experimentally measured CAR function of the QS source with the fiber at 77 K. Inset shows

the measured coincidence and accidental-coincidence counts as a function of single counts/pulse (this



such correlated-photon sources [103, 104, 117, 118]. To do that, the path-matched

outputs of the QS source [points g and h in Figure 21.16(b)] are directly connected to

two APD1 and APD2 (Epitaxx EPM 239BA) for coincidence detection, bypassing all

the intermediate free-space optics. The APDs are operated in a gated Geiger mode,

with detection rate ’780 kHz and overall detection efficiencies of 7% and 9%,

respectively. Single-channel counts as well as coincidence counts are recorded by a

“coincidence counter” software. Detector dark-count contributions are subtracted from

all of our measurement results. We call a “coincidence” count when the two detectors

fire in the same triggered time slot, and an “accidental- coincidence” count when they

fire in the adjacently triggered time slots. We then vary the pump power and record

the corresponding coincidence and accidental- coincidence counts. Their ratio,

CAR, is plotted as a function of the single-channel count rate in Figure 21.17(b).

It has a similar shape to those reported previously [99, 118]. The CAR peak value

of about 100 occurs at a single-count rate of around 4 � 10�4/pulse (photon-pair

production rate ’2.2 � 10�3/pulse), and decreases rather rapidly on both sides.

We then proceed with our HOM experiment. After the identical photons come

out from g, and h, they are collimated into free space through lenses, and two sets

of polarization compensators composed of half-wave plates and quarter-wave

plates (HWP1/QWP1 and HWP2/QWP2) are used to restore each photon’s polar-

ization to horizontal before they enter PBS1. Each photon then traverses one arm

of the Mach–Zehnder-like interferometer before being combined at the 50/50

beamsplitter (BS) cube. Quarter-wave plates QWP3 and QWP4 are each set at

45 , so that when combined with a mirror behind, they function as 45 -oriented

half-wave plates, rotating the horizontal polarization of the incident light by 90 .In principle, the combinations PBS3/QWP3/M1 and PBS2/QWP4/M2 can be

replaced with just two mirrors to direct the photons to the input ports of the BS.

In practice, we choose to implement the more complicated version, because it is

less susceptible to translation-induced misalignment of the photon wavepackets at

the BS. Careful path-matching is done to ensure that the two photons reach the BS

at approximately the same time (i.e., their arrival-time difference is within the

tuning range of the translation stage placed under M1). The identical photons,

before hitting the BS, are both of linear vertical polarization. The output photons

from the BS are each coupled into single-mode fibers, which are connected to the

two APDs for coincidence detection. The HOM experiment is performed by

recording the coincidence counts at each setting of the translation stage, which is

equivalent to recording the coincidence counts as a function of the overlap

between the two identical photon wavepackets. The two scales are related by

�� = 2Ds/c, where �� is the temporal difference between the photon wavepackets,

Ds is the difference in readings of the translation stage, and c = 3 � 108 m/s is the

speed of light in vacuum.

We first pump the QS source with relatively high pump power, and obtain a

sample experimental result as shown in Figure 21.18(a). The single-count rate is

’3.4 � 10�3/pulse, corresponding to a low-CAR value of around 2 according to

Figure 21.17(b). A HOM dip of visibility ’50% is observed. This 50% dip


visibility is the upper bound for the case of two classically random photon sources

as inputs to the BS, and can be explained by using a simple classical electro-

magnetic theory [112]. It is an expected result, because in the high-pump condi-

tion, the photon source emits mainly spontaneously scattered Raman photons,

along with multiple FWM photon pairs, whose random behavior masks the true

correlated nature of a single pair of identical FWM photons. We then lower down

the pump power to observe a much higher HOM dip visibility of 94 – 1%, as

shown in Figure 21.18(b). This result is obtained at a single-count rate of around

4 � 10�4/pulse, which corresponds to the CAR peak in Figure 21.17(b). The near-

unity visibility of the HOM dip is well beyond the classical limit of 50%, and

clearly demonstrates the high indistinguishability of the QS-generated identical

photons. Combined with its telecom-band operation and fiber-optic networkabil-

ity, we expect to be able to achieve more complicated QIP applications with this

fiber-based QS source.

Theoretical modeling of the QS source and its HOM experiment will be

provided elsewhere [119]. Here we show theoretical simulation results of the

HOM dip in Figure 21.18(b), along with the experimental data. The two theory

curves, generated for different OBPF spectral shapes (see Figure 21.18 caption for

details), appear to fit the experimental data remarkably well. Both fits agree on the

ideally attainable HOM dip visibility of 100%, which corresponds to the fact that

the two-photon probability distribution function is symmetric with respect to its

two frequency arguments [106]. The missing 6(�1)% visibility can be explained

by taking into account the following real-life imperfections: (i) The BS used in

the experiment (R = 0.474, T = 0.526, where R and T are the BS intensity

0

100

200

300

400

500

600

700

–15 –10 –5 0 5 10 15 –12 –8 –4 0 4 8 12

Coi

ncid

ence

cou

nts/

30 s

Coi

ncid

ence

cou

nts/

60 s

δπ (ps) δπ (ps)

0

20

40

60

80

100

120

(a) (b)

Figure 21.18 Hong-Ou-Mandel experimental results. (a) Hong-Ou-Mandel dip visibility of 50% is

observed when the pump power is high. The solid curve is a least-squares Gaussian fit to the data.

(b) Hong-Ou-Mandel dip visibility of 94.3% is observed when the pump power is low. The solid curve

is a least-squares Gaussian fit to the data. The dotted curve is a theoretical fitting for Gaussian OBPFs,

while the dot-dashed curve is that for super-Gaussian OBPFs (see Ref. [119] for details) (this figure may



coefficients of reflection and transmission, respectively) deviates from its ideal

performance, but this gives rise to a negligible correction factor [110] 2RT/

(R2 þ T2)’ 0.994. (ii) The spatial-mode mismatch of the two photon wavepackets

at the BS results in some distinguishing information between the two coincidence-

generating probability amplitudes. A simple calculation [119] shows that a small

angular mismatch of around 30 mrad can bring the HOM dip visibility down

to 94%. (iii) Some remaining �2002 component due to nonideal alignment of the

50/50 Sagnac loop may lead to a degradation of the dip visibility [102]. (iv)

Existence of unsuppressed noise photons, such as Raman photons and FWM

photons induced by a single-frequency pump, may also degrade the attainable

HOM dip visibility.

21.7 CONCLUDING REMARKS

Because this volume deals with advances in Optical Fiber Telecommunications,

our focus in this chapter has been on the generation of correlated and entangled

photons in the telecom band with use the Kerr nonlinearity in DSF. Over the past

decade, microstructure or holey fibers (MFs) have come on the scene, which

because of their tailorable dispersion properties allow phase-matching to be

obtained over a wide range of wavelengths. Generation of correlated and entangled

photons in MFs has been demonstrated and rapid progress is taking place.

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