Bit error rates in a frequency coded quantum key distribution system

Optics Communications 282 (2009) 3827–3833

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Bit error rates in a frequency coded quantum key distribution system Pradeep Kumar, A. Prabhakar * Dept. of Electrical Engineering, Indian Institute of Technology-Madras, Chennai 600 036, India

a r t i c l e

i n f o

Article history: Received 26 November 2008 Received in revised form 29 May 2009 Accepted 2 June 2009

Keywords: Bit error rate Quantum key distribution Gated avalanche photo-detection

a b s t r a c t The quantum bit error rate (QBER) is a measure of the performance of a frequency coded quantum key distribution system. We present a detailed analysis of the system, taking into account the statistics of light at different points in the link. We show that the statistics depends on the choice of phase of both the transmitter and receiver. We also evaluate the effects of crosstalk, out of band noise, and dark count of the gated avalanche photodetector on the QBER. Finally we use QBER to evaluate our implementation of the B92 protocol. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The quality of signal transmission in quantum key distribution (QKD) systems is measured by the quantum bit error rate (QBER). The QBER depends on several factors such as type of the protocol used, transmission impairments to the quantum bits, noise and imperfections of the components in the link [1]. QKD systems differ in many aspects and we focus on one implementation of the B92 protocol known as frequency coded quantum key distribution (FC-QKD) [2–4]. A recent proposal also suggests using acousto-optic modulators to implement the BB84 protocol [5]. FC-QKD encodes a classical bit of information as the relative phase between the carrier and sideband. If the transmitter (Alice) and the receiver (Bob) choose an identical phase, constructive interference occurs and the photon is detected. We transmit secret key information on the sidebands and not in the carrier and hence, a fiber Bragg grating (FBG) filter is used to separate the sidebands from the carrier [4,6]. QBER is defined as the ratio of the number of bits in error to the total detected bits. Most QBER analysis focus on the effect of a specific component in the link. Errors due to loss, dispersion, imperfect sources and detectors are lumped together as dark count. Recently, comprehensive analysis of QBER have appeared in literature for polarization coded QKD implementing the BB84 protocol [1,7]. However, to the best of our knowledge, there exists no similar analysis of QBER for a FC-QKD implementation of the B92 protocol. Our objective is to present a sufficiently detailed analysis of an implementation of the FC-QKD protocol over an optical fiber.

* Corresponding author. E-mail addresses: [email protected] (P. Kumar), [email protected] (A. Prabhakar). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.030

In Section 2, we describe our implementation of the B92 protocol, using our FC-QKD system. In Section 3, we present a mathematical model of the scheme and derive the photon statistics at various points in the link which determine the probability of false detection and the error rate. In addition to the detector and source noise, an additional noise source exists due to the wavelength division multiplexing (WDM) of several optical signals on the same fiber. A portion of these signals leak into the quantum channel and causes detection errors [8]. This is called the crosstalk noise. Also the optical carrier, if not extinguished completely, leaks into the sideband and causes detection errors. We term this as the out of band (OOB) noise. In a DWDM system this severely impacts the error rate in the quantum communication system. Section 4, discusses how we arrive at the QBER for a FC-QKD system, and is followed by a summary of our results.

2. B92 protocol using FC-QKD FC-QKD is a practical realization of the B92 protocol [2,9] which allows two users, Alice and Bob, to exchange a secure key by executing the following steps.  To transmit a bit b 2 f0; 1g, Alice phase modulates an optical pulse with the corresponding phase /A 2 f0; pg on the sideband relative to the carrier. She transmits the optical pulse to Bob.  Bob modulates the incoming optical pulse with a phase /B 2 f0; pg, chosen independently of Alice. The probability of Bob detecting a photon in the sideband depends on the relative phase difference, D/ ¼ /A  /B , as

P / cos2

  D/ : 2

ð1Þ

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 Bob reveals over a public channel the time slots during which he detected a photon but not his choice of phase.  Alice then retains the bits during those time slots as the private key, thus generating a sifted key. The loss of bits reduces the effective key rate between Alice and Bob, but does not compromise the security.  Key reconciliation and privacy amplification are used to amplify the security of bits and to correct errors in transmission.

3.1. Faint laser pulse source (FLPS) A FLPS is obtained by attenuating a laser pulse such that the average number of photons/pulse is approximately unity. The photon emission of an ideal laser is approximated by the coherent state representation,

jwi ¼

n¼1 X n¼0

Thus, after the final step of the protocol, Alice and Bob are left with a K-bit secure key. Our objective is to determine the QBER of the sifted key. The errors in FC-QKD can be due to the imperfections in the transmission link or be introduced by an eavesdropper, Eve. Eve can employ the intercept/resend strategy, where she intercepts Alice’s qubits and transmits fabricated pulses on a better channel to Bob. In such a scenario, she introduces an error in 25% of the bits detected by Bob [1]. Hence successful statistical monitoring of the channel, to detect the presence of Eve, requires a link error less than 25%. 3. Model of the system Fig. 1 shows the essential components of our implementation of FC-QKD. The DFB laser is pulsed and attenuated to form a faint laser pulse source (FLPS) with the average photon number l. It is then phase modulated by an RF signal to generate the side bands whose phase relative to the carrier is /A and forms the quantum (Q) channel of the FC-QKD system. This is combined with the synchronization (S) channel by a multiplexer (MUX) and launched in the optical fiber. At the receiving end, the two channels are separated using a demultiplexer (DEMUX). The combination of MUX, fiber and DEMUX forms a lossy element. Our experiments suggest that the phase relationship between carrier and the sidebands is maintained over long distances in an optical fiber [10]. The quantum signal is phase modulated by Bob with phase /B and the carrier is filtered out by a FBG filter. The residual photons in the carrier act as the OOB noise of the detector and contributes to the detection error. The detector is a gated avalanche photodiode (GAPD) detector with a noise equivalent power (NEP) less than 90 dBm, and a sensitivity of approximately 0.1 photons/pulse.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi elA

lA n!

ej/A jni:

ð2Þ

The phase factor ej/A does not affect the photon statistics [7]. The probability, PðnÞ, that an outgoing pulse contains n photons follows a Poissonian distribution. For an average of lA photons/pulse,

PðnÞ ¼ jhnjwij2 ¼ elA

lnA n!

:

ð3Þ

It follows that Pð1Þ is greater than both Pð0Þ and Pð2Þ for 1:1 < lA < 1:25. Hence we choose lA ¼ 1:25 at the input end. We will show, in Section 4, that lA ¼ 1:25 also minimizes the bit error rate, though such a claim relies on an assumption that multiphoton pulses do not necessarily constitute a security threat1 but will reduce the key creation rate [1]. Other studies have reported a value of lA ¼ 1:1 as the optimum for a wide range of fiber-optic based QKD implementations [11]. To calculate the QBER, we need to know the effective average photon number lA at the receiver. In contrast to the polarization based or single photon-QKD schemes, where the phase of the photon can be neglected, in a FC-QKD system information about the relative phase of the sideband photon is required to calculate the effective lA and consequently the QBER. 3.2. Phase modulator As described in Section 2, Alice encodes her classical bit as the relative phase between the optical carrier and the sideband. This is achieved by modulating the incoming light pulses by an RF signal of frequency X rad/s using an optical phase modulator. The action of a phase modulator on the quantum states of light is described in [12,13]. If a coherent state, jwi, of light of angular frequency x0 rad=s and average photon number lA is input to the optical phase modulator, the output state is a tensor product of sidebands of frequency xp ¼ x0 þ pX:

  pffiffiffiffiffiffi jwi ! p J p ðcÞ lA ejp/A ;

ð4Þ

where c ¼ mT= h is the modulation index, m / jvð2Þ j the second order nonlinear coefficient, T is the interaction time of the coherent state of light in the modulator cavity with the RF signal,  h is the rationalized Planck’s constant and /A is Alice’s phase corresponding to the classical bit. The optical phase modulator creates an infinite number of sidebands around the optical carrier with the average photon number of the pth band being lp ¼ lJ2p ðcÞ. Note that the statistics of the output of the phase modulator are not affected by the phase factor /A . However this is no longer true when the states undergo an additional phase modulation at the receiver. The receiver phase /B , corresponding to Bob’s choice, determines the average number of photons in the carrier and sidebands due to the interference effect as described in Section 3.4. 3.3. Fiber and excess loss

Fig. 1. Schematic diagram of the FC-QKD system. An RF signal of frequency X rad/s and phase /A (and /B ) is used to generate sidebands by optical phase modulation. The fiber is a standard single mode optical fiber of length L km and attenuation constant af ¼ 0:2 dB=km. V th is the GAPD threshold voltage that determines the dark count of the device. FLPS: faint laser pulse source, Q, S: quantum and synchronization channels, PM: phase modulator, FP-FBG: Fabry–Perot fiber Bragg grating, GAPD: gated avalanche photodiode detector.

When the coherent state described by (4) propagates through an optical fiber, the average photon number reduces according to

1 Although the statement is true for polarization coded QKD implementations we also expect it to hold for FC-QKD systems.

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lp ðLÞ ¼ lp ð0Þ10af L ;

ð5Þ

where lp ð0Þ is the average photon number of the pth sideband at the input of the fiber, af L is the attenuation over a propagation distance L. Dispersion in an optical fiber induces a phase shift between the carrier and sidebands of the FC-QKD system. We compensate for a dispersion induced phase shift by transmitting a reference phase to the receiver over the S channel [3]. We do not include dispersion in our analysis except to point out that an additional phase term, which depends on dispersion and optical path length, appears in (10a). This reduces the visibility of sideband interference at the receiver and increases the error rate. Other losses in the system such as the MUX, DEMUX and insertion loss of optical components are combined into a single parameter ae , the excess loss. Thus lp ðLÞ at the output of the receiver DEMUX becomes,

lp ðLÞ ¼ lp ð0Þ10ðaf Lþae Þ :

ð6Þ

3.4. Receiver phase modulator At the receiver, a WDM demultiplexer separates the S and Q channel optical pulses. The separated Q channel pulses are remodulated by an RF signal at the receiver phase modulator. Photons in each sideband undergo phase modulation with a phase reference /B corresponding to Bob’s bit. Expanding the first few terms of (4) we obtain,

    jwi !    J 1 ej/A  jJ 0 i  J 1 ej/A     ;

ð7Þ

where we have suppressed the argument c of the Bessel function. We note that the photon in jx0 i remains in the same state with probability amplitude J 0 or makes a transition to jx0  Xi with amplitude J1 ej/A and so on, shown schematically in Fig. 2. The transition amplitudes decay as we move away from the band. The statement is true for c2  1, which implies a low modulation index condition typical of QKD systems. Generalizing (7) for the pth band, the output state at Bob’s end becomes,

 +  J J ejðp1Þ/A ej/B þ J J ejp/A þ p 0  p1 1 ; jwip /   Jpþ1 J 1 ejðpþ1Þ/A ej/B þ Oðc2 Þ



2

 PðnÞ ¼

2

e4lA ðJ0 J1 Þ :

ð11Þ

A Fabry–Perot fiber Bragg grating (FP-FBG) filter was used to extinguish the carrier photons and detect the sideband photons. A tunable filter with a resolution of 50 pm was used to observe the power in the optical spectrum (solid line in Fig. 5). The laser power was then attenuated by another 20 dB before modulation by Alice, such that the power in the sidebands reached single photon levels that were then detected by the GAPD (dashed line in Fig. 5). Note that the efficiency of a QKD system that uses a FLPS

LASER

PM1

PM2

FP−FBG

GAPD

RF AMP SPLITTER RF Fig. 3. Experimental setup to determine probability of sideband photon detection at the output of the second phase modulator. An electrical phase shifter was used to vary the transmitter phase.

ð9Þ 0.4

Experiment Fit

0.35 0.3

l0 ¼ lA J20 þ 2J21 cosðD/Þ ;

ð10aÞ

l1 ¼ 4lA jJ0 J1 j2 cos2

ð10bÞ

  D/ ; 2

n!

3.5. Fiber Bragg filter

Count rate (not normalized)

2

n

Experimental data was collected by varying /A , and separating the sideband using a FBG filter as shown in Fig. 3. The count rate (unnormalized), which is proportional to detection probability, at the output of the second phase modulator is shown in Fig. 4. These results are similar to those reported in an earlier experiment [14]. We observe a dependence on cos2 ðD/=2 þ dp Þ, where dp is the uncorrected phase offset between Alice and Bob. We attribute this phase offset to the different RF path lengths in the circuits used by Alice and Bob. dp – 0 introduces a visibility error, described further in Section 3.6.4.

Substituting p ¼ 0 and p ¼ 1 in (9), we get the average number of photons in the carrier and sidebands,



4lA ðJ 0 J 1 Þ2

ð8Þ

where we have neglected the contributions from p þ n bands, for n P 2, on the pth band [13]. Defining the relative phase difference, D/ ¼ /A  /B , and eliminating the common phase term, ejp/A , we obtain the average number of photons in the pth band as

lp ¼ lA Jp1 J1 ejD/ þ Jp J0 þ Jpþ1 J1 ejD/  :

under the approximations J2 J 1  1 and J2 J 1  1. The second term in the bracket of (10a) can be neglected since J1 =J0  1. The average photon number of the sidebands (to the first order) depends on the relative phase between Alice and Bob, similar to that in a classical system [2]. When D/ ¼ p, destructive interference occurs and no photons are detected. The probability of n photons in the sideband (when D/ ¼ 0) is

0.25 0.2 0.15 0.1 0.05 0 −0.05 −1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Phase, φ (rad) A

Fig. 2. Representation of transition amplitudes of the photon in an electro-optic phase modulator.

Fig. 4. Experimental detection probability at the output of second phase modulator. A least squares fit to the data, yields the form 0:3 cos2 ðD/=2 þ 0:38Þ, which follows from (10a). The dotted lines mark the 90% confidence intervals of the fit.

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where m is the optical frequency of S channel. In a DWDM network a number of other classical channels copropagate along with the Q channel. These channels induce crosstalk noise and increase the QBER. Assuming independent channels, the probability that n photons are detected in the ON time of the GAPD is

Pðn; CÞ ¼ elC

ðlC Þn ; n!

lC ¼

PC T on ; hm

ð15Þ

P where PC ¼ Nk¼1 P x is the sum of optical power of N classical channels of power Px and lC is effective average photon number/pulse, respectively. This approximation is valid if we neglect the nonlinear interactions of the optical channels.

Fig. 5. Extinction of optical carrier at 1552 nm using a fiber Bragg grating. The solid and dashed lines represent the optical power and the detection probability, respectively. The detection probability was measured after a 20 dB reduction in laser power.

is limited by the non-zero probability, given by (10b), that a photon remains in the carrier and is observed with a detection probability of 0.1 between the sidebands. Thus, the finite extinction provided by the filter limits the efficiency of our QKD system. The extinction ratio of the filter is defined as

gf ¼

Tðx0  XÞ ; Tðx0 Þ

ð12Þ

where T is the transmission coefficient of the filter and ideally Tðx0 Þ ¼ 0 and Tðx0  XÞ ¼ 1. We model the filter as an attenuator with a probability of transmission p0 for the carrier and p1 for the sidebands. If k photons arrive at the input of the filter with a probability PðkÞ, the probability Pðn; pÞ of n photons of frequency xp at the output is given by

Pðn; pÞ ¼ k C n T np Rkn p PðkÞ;

ð13Þ

where T p is the transmission coefficient of the filter at xp and T p þ Rp ¼ 1. This follows from the classical combinatorial problem of observing n successful events out of k trials with probability of success p and failure 1  p. For the FBG this corresponds to the detection of n photons from k photons with probability of success is T. 3.6. Noise sources We classify as noise any process that triggers a false count in the quantum channel between Alice and Bob. The false counts could have electrical or optical origins. A study of Rayleigh and Raman scattering on a two-way QKD system, which is the result of nonlinear interactions between the classical and Q channel, can be found in [15,16]. Other nonlinear effects that impact the QBER of a QKD system are given in [8]. For the present purpose, we focus on four major contributions to QBER.

3.6.2. Out of band noise We also consider the photons outside of the sidebands x0  X as an additional source of noise. The APD treats these photons on par with the signal photons. The main source of this out of band (OOB) noise is the carrier photons in the Q channel which are not completely extinguished by the FBG filter. 3.6.3. GAPD dark current An important noise source in the GAPD detector circuit is the dark current of the device itself. Noise statistics of an APD are empirical at best. In our experiments we gate the APD and minimize the after-pulsing effects. In [17,18], we discussed our implementation of the GAPD detector. In Fig. 6, we plot the dark count as a function of threshold voltage for various values of bias voltage. By properly selecting the operating point of the GAPD we can reduce the dark count below 106 and the corresponding NEP is 0.1 photons/pulse. Thus the chosen lower bound for reliable photon detection by Bob, in either sideband, is l1 ¼ 0:1. We also know from (5) that l decreases with the propagation distance. Thus, given the sensitivity of GAPD to be 0.1 photons/pulse, a fiber loss coefficient of af ¼ 0:2 dB=km, and assuming Alice transmits pulses with an average photon number of lA ¼ 1:25 we estimate a maximum link length of 50 km. However the link length reduces when reconciliation and privacy amplification are applied to raw bits. Our estimate of 50 km neglects cross talk and assumes that P oob can be decreased by adding in series another FP-FBG filter. 3.6.4. Interference visibility Finally, yet another source of error is due to the reduced visibility (or contrast) of the sideband interference, V defined as,

V¼

Imax  Imin ; Imax þ Imin

ð16Þ

3.6.1. Crosstalk due to classical channels The weak optical signal from the S channel affects the detection on the Q channel due to the crosstalk between the two. The photons from the S channel act as an additional source of noise in the Q channel and are characterized by the Poisson statistics over the ON period of the GAPD detector. If the leakage power in the S channel at the input of the receiver phase modulator is P x , the number of photons in the time T on of gated APD is Poisson distributed:

Pðn; SÞ ¼ elS

ð lS Þ n ; n!

lS ¼

Px T on ; hm

ð14Þ

Fig. 6. GAPD dark count probability for different values of reverse bias voltage. The operating point of detector is set by fixing both the bias and threshold voltage.

P. Kumar, A. Prabhakar / Optics Communications 282 (2009) 3827–3833

3831

where Imax and Imin are maximum and minimum sideband intensities, respectively. Ideally V ¼ 1 and no error is incurred in the detected bits. However, when V < 1, the error bits are non-zero and the error rate, due only to V, QBERV ¼ ð1  VÞ=2 [1]. Using the fit to the experimental data in Fig. 4, we estimate that V  96% in our experiment. 4. QBER QBER is defined as the ratio of number of bits in error over the number of bits received. Following [1], we write

Rerror QBER ¼ ; Rerror þ Rsift

ð17Þ

Fig. 7. Experimental setup to determine the crosstalk probability P x . The S channel optical signal is coupled to the Q channel through a 3 dB coupler. A tunable Fabry– Perot filter (FFP) is used to monitor the FP-FBG filter output by tapping the optical signal from a 99:1 coupler. VOA: Variable Optical Attenuator.

where

1  frep  Pnoise ; 2

1 ¼  frep  Psig þ Pnoise : 2

Rerror ¼ Rsift

ð18Þ

The factor of 1/2 appearing in (18) indicates that, on an average, Alice and Bob choose identical phases half the instants. We denote the probability of the GAPD registering a detection due to signal and noise photons by P sig and Pnoise respectively. We recall that, in the FC-QKD scheme, Alice and Bob choose their phases, /A and /B as 0 and p to represent bits 0 and 1, independently of each other. Thus, the different scenarios that occur during a given time slot are summarized in Table 1. More concisely we have,  D/ ¼ p with probability 2 Pnoise .  D/ ¼ 0 with probability 2 Psig þ 4P noise . Assuming Bob detects both the upper and lower sidebands, the signal probability Psig is given by,

Psig ¼ ð1  el Þ;

ð19Þ

where l ¼ l1 as defined by (10b). The total noise probability, Pnoise at the receiver is the sum of GAPD dark current, crosstalk and OOB noise:

Pnoise ¼ Pdark þ P x þ Poob :

ð20Þ

The probability of detecting one or more photons, Poob ðn > 0Þ, when k photons of carrier frequency is incident on the FP-FBG filter is obtained by setting p ¼ 0, T 0 ¼ gf , and R ¼ 1  gf :

h i Poob ðn > 0Þ ¼ ½1  Pðn ¼ 0Þ PðkÞ ¼ 1  ð1  gf Þk PðkÞ;

ð21Þ

We note that (21) is zero for k ¼ 0 as it should. Assuming PðkÞ is zero for k > 2 we can approximate (21) by

  l  0 ; Poob ðn > 0Þ  gf l0 el0 1 þ 2  gf 2

ð22Þ

where l0 is the average photon number of the carrier at the input of FP-FBG filter, as given by (10b). In practice, the dark current probability is determined by the NEP of the GAPD. For our GAPD, NEP was measured to be less than 90 dBm which corresponds to 0.1 photons/pulse at a dark count

Table 1 Different possibilities in a given time slot. Alice’s bit

Bob’s bit

Probability

0 0 1 1

0 1 0 1

P sig þ P noise P noise P noise P sig þ P noise

probability of 106 . Thus we set Pdark ¼ 106 . The probability of emission of crosstalk photons, Px ðn P 1Þ, during the ON time of the GAPD is 1  elS where lS is given in (14). The crosstalk noise depends on the isolation between S and Q channels and the optical power leaking into the Q channel from the S channel. Inserting (18) into (17), and accounting for errors due to fringe visibility, QBER takes the canonical form:

Q ¼ QBER ¼

P noise þ QBERV : 2Pnoise þ Psig

ð23Þ

With a sideband detection probability of about 0.25, from Fig. 5, and a noise detection probability less than 0.01, we obtain a QBER less than 5%. Further system specifications, along with the design of our synchronization channel, are discussed in [17]. We experimentally obtained the crosstalk probability, Px , as a function of S channel optical power by the setup shown in Fig. 7, with the results plotted in Fig. 8. Recall that the detector sensitivity as well as the dark count probability depend on both the GAPD bias and the comparator threshold voltages. In this experiment we used V bias ¼ 67:5 V and V th ¼ 45 mV to obtain Pdark  0:008. We note that Px approaches Pdark when the crosstalk optical power exceeds 94 dBm. The wavelength assignment of S channel determines the isolation between S and Q channels2. If we choose the channels to lie on the ITU-T wavelength grid, commercial WDM multiplexers and demultiplexers provide a minimum isolation of 25 and 40 dB for adjacent and non-adjacent channels respectively. In our FC-QKD setup, the optical power required for clock recovery was experimentally found to be 2:15 lW with a PIN diode receiver [17]. The crosstalk power reaching the Q channel GAPD was 61 dBm and 76 dBm for adjacent and non-adjacent channels, respectively. These reduced to 81 dBm and 96 dBm, respectively, when the PIN receiver was replaced by an APD receiver with a gain of M ¼ 100. From Fig. 8, we find that P x ¼ 0:06 for adjacent channel spacing and less than Pdark for non-adjacent channel spacing. In our final setup we use an APD receiver and adjust the bias and threshold voltages to set Px ’ Pdark ¼ 106 (see Fig. 6). The sifted key rate, due to signal photons, is

Rsig ¼

1 frep  lA 10aloss =10  4ðJ 0 J 1 Þ2  ggapd ; 2

ð24Þ

where aloss is the total optical path loss, ggapd is the GAPD efficiency, frep is the pulse repetition rate and the factor of 1/2 is the probability that D/ ¼ 0. The useful key rate, after reconciliation and privacy amplification, is given by

2 The channel spacing between S and Q channels also determines the effects of dispersion on clock recovery on the S channel [3,17].

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P. Kumar, A. Prabhakar / Optics Communications 282 (2009) 3827–3833 Table 2 System parameters to calculate QBER.

0.1

Phase modulator IL amod Mux/deMux IL amux FBG filter extinction ratio gf GAPD dark current probability IL of FBG filter at k ¼ ks

Detection probability (Px )

0.08

0.06

6 dB 1 dB 40 dB 106 2.5 dB

0.04 300

50km link

Pdark

0.02

QBER = 0.25 250

−92

−90

−88

−86

−84

−82

−80

Optical power (dBm)

Fig. 8. Crosstalk detection probability as a function of optical power.

200

Key rate (bps)

0 −94

150

Multiphoton Regime

100 60 50 50 0 0.2

QBER (%)

40

0.4

0.6

0.8

1

1.2

1.4

1.6

Input avg. photon number ( μA)

30

Fig. 10. Key rate (after reconciliation) as a function of lA . The key rate is maximum for lA  1:25, when using a coherent optical source.

20

10 60

0

0.5

1

QBER (%)

0 1.5

Input avg. photon number (μ ) A

Fig. 9. QBER as a function of avg. photon number specifications are given in Table 2.

μA=0.5 μA=1.25

40 20

lA for a 50 km. The system 0 0

10

20

30

40

50

60

70

80

90

100

Link distance (km) 4

ð25Þ

where Hr and Hpa are the entropy functions of QBER. The exact form of Hr and Hpa depend on the type of reconciliation protocol and we specify a rough upper bound [1,19],

Hr ðQ Þ ¼



10

Key rate (bps)



Ruseful ¼ Rsig ð1  Hr ðQ ÞÞ 1  Hpa ðQ Þ ;

μ =0.5 A

μA=1.25

3

10

2

10

1

Q log2 ðQ Þ  ð1  QÞlog2 ðQ Þ; Q 6 1=4 1;

Q P 1=4:

10

ð26Þ

Thus, there is no key transmission when Q > 1=4 which agrees with our earlier specification of a maximum error rate in a FC-QKD system. The formula in (26) does not take into account the multiphoton pulse probability. For a detailed discussion of distilling key bits from raw bits, we refer the reader to [19,20]. Since we are using an FLPS and relying on coherent optical states, we treat the average photon number, lA , used by Alice as a variable parameter. Fig. 9 shows the estimated QBER as a function of lA for a 50 km link with an APD receiver for clock recovery on the S channel3. Fig. 10 shows the key rate as a function of lA . We see that there is an optimal region where we achieve secure key transmission. Low values of lA result in loss of data, making QBER unacceptably high and the link susceptible to eavesdropping. Meanwhile lA > 1:25 results in Pðn P 2Þ > Pðn ¼ 1Þ, exposing the link to multiphoton attacks. We find that the rate is a maximum at lA  1:25, and choose this value as our input average photon number. 3 There has been a recent demonstration of QKD over 200 km [21]. Our choice of 50 km is motivated by metro-area applications.

0

10

20

30

40

50

60

70

80

Link distance (km)

Fig. 11. QBER and Key rate as a function of link distance for different values of

lA .

The top half of Fig. 11 shows the estimated QBER as a function of link distance for l = 0.5 and 1.25. We see that QBER increases with the increasing link distance. The bottom half shows the key rate after reconciliation (not including privacy amplification) is applied. As expected, the key rate drops with increasing distance. Note that the distilled key rate will be less than the rate in (25) once multiphoton pulse effects and privacy amplification are included. 5. Summary Recent articles have demonstrated the unconditional security of the B92 protocol over lossy and noisy channels [20]. However, polarization dependent losses can significantly reduce the key rate. FC-QKD is polarization insensitive and hence not limited to short distances. When compared with a phase coded scheme, frequency coding is less sensitive to optical path length variations. In addi-

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tion, the use of a separate co-propagating channel to send the clock information to the receiver, makes it easy to electronically synchronize transmitter and receiver clocks. However, the main difficulty of the FC-QKD setup is that it requires a narrow band, high extinction, and stable optical filter to separate the carrier from the sidebands. The use of electro-absorption modulators, with modulation frequencies in the tens of GHz, will make carrier extinction easier. As with other fiber-optic based QKD schemes, the quantum channel must be isolated from the other classical channels to minimize crosstalk noise. Practical implementations of FC-QKD systems require that we bridge the gap between quantum bit error rates and classical metrics such as signal to noise, insertion loss and modulation depth. We have used our implementation of the B92 protocol as a basis to understanding the system design methodology. We have derived expressions for crosstalk and out of band noise while the dark count of gated APD receivers represents the lower bound on the total system noise. For a wavelength division multiplexed (WDM) optical network, the crosstalk noise is the most dominant source of noise which severely impacts the QBER of system. Although the optical carrier does not (to lowest order) carry any information about the key, the photons in the carrier give rise to false counts. This can be exploited by Eve and she can masquerade as Alice. Thus, suppression of the optical carrier before gated photo-detection is a key requirement in our FC-QKD system. This is likely to require thermally stabilized FP-FBG filters with a high extinction ratio. In any commercial implementation, the QBER would also be affected by crosstalk from the adjacent channels on a WDM network and an isolation as high as 40dB between channels may be required. Thus, a careful design of the entire optical transport system would be necessary to maintain QBER below acceptable levels. The maximum key rate of a QKD system depends on the number of bits lost and the details of key reconciliation and privacy amplification. Furthermore, a certain fraction of bits are discarded by Alice and Bob to minimize Eve’s information about the key bits. The precise nature of key rate reduction will depend on the assumed strengths of Eve’s equipment as well as her possible attacks on the protocol. Such a study is too involved for the scope of the present paper. However, we believe that the QBER can still yield sufficient information to quantify the physical characteristics of a FC-QKD link.

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Acknowledgements We are grateful for our interaction with the faculty and students at Univ. French-Comte, Besancon, in particular J.M.M. Merolla and M. Bloch. This work was made possible through grants from the Indo-French Center for the Promotion of Advanced Research and the Dept. of Science and Technology, India. We acknowledge the efforts of S. Thiruthakkathevan, for designing the single photon detector, and V. Lakshminarayanan, for the characterization of the FC-QKD system. Finally, we thank the reviewers for suggestions that helped to improve the manuscript considerably. References [1] N. Gisin et al., Rev. Mod. Phys. 74 (2002). [2] J.M. Merolla, Y. Mazurenko, J.P. Goedgebuer, W.T. Rhodes, Opt. Lett. 24 (1999) 104. [3] O.L. Guerreau, J.M. Merolla, A. Soujaeff, F. Patois, J.-P. Goedgebuer, F.-J. Malassenet, IEEE J. Select. Top. Quant. Elect. 9 (2003) 1533. [4] J.M. Merolla, Y. Mazurenko, J.P. Goedgebuer, L. Duraffourg, H. Porte, W.T. Rhodes, Phys. Rev. A 60 (1999) 1899. [5] T. Zhang, Z.-Q. Yin, Z.-F. Han, G.-C. Guo, Opt. Commun. 281 (2008) 4800. [6] O.L. Guerreau, F.J. Malassenet, S.W. McLaughlin, J.M. Merolla, IEEE Photon. Technol. Lett. 17 (2005). [7] G. Gilbert, M. Hamrick, Practical quantum cryptography: a comprehensive analysis, 2003. Available at: http://arXiv:quant-ph/0009027. [8] P. Toliver et al., Demonstration of 1550 nm QKD with ROADM-based DWDM networking and the impact of fiber FWM, Lasers and Electro-Optics, 2007, Conference on CLEO 2007, May 2007, pp. 1–2. [9] C.H. Bennett, Phys. Rev. Lett. 68 (1992) 3121. [10] P. Kumar, A. Prabhakar, Dispersion measurement of single mode optical fibre using intensity modulator, National Conference on Communications, NCC07, 2007. [11] D. Pearson, C. Elliott, On the optimal mean photon number for quantum cryptography, 2004. Available at: http://arxiv.org/abs/quant-ph/0403065v2. [12] M. Bloch, S.W. McLaughlin, J.M. Merolla, F. Patois, Opt. Lett. 32 (2007) 467. [13] P. Kumar, A. Prabhakar, IEEE J. Quant. Electron. 45 (2009). [14] J.M. Merolla et al., Phys. Rev. Lett. 82 (2009) 1656. [15] P. Toliver et al., Impact of spontaneous anti-Stokes Raman scattering on QKD + DWDM networking, in: Proc. LEOS, 2004, pp. 491–492. [16] D. Subacious et al., Appl. Phys. Lett. 86 (2005). [17] P. Kumar, S. Thevan, A. Prabhakar, Designing practical quantum communication systems, in: Proc. National Conf. on Communications NCC08, 2008. [18] P. Kumar, S. Thevan, V. Laxminarayanan, A. Prabhakar, Optimization of gated photodetection for quantum key distribution, in: SPIE Europe Optics and Optoelec. Conf., April 2009. [19] N. Lütkenhaus, Phys. Rev. A 61 (April) (2000) 052304. [20] K. Tamaki, Phys. Rev. A 77 (2008) 032341. [21] H. Takesue, S.W. Nam, Q. Zhang, R.H. Hadfield, T. Honjo, K. Tamaki, Y. Yamamoto, Nat. Photon. (2007) 343.