February 1, 2007 / Vol. 32, No. 3 / OPTICS LETTERS 301

Frequency-coded quantum key distribution

Matthieu Bloch and Steven W. McLaughlinGeorgia Tech—CNRS, UMI 2958, 2-3 rue Marconi 57070 Metz, France

Jean-Marc MerollaDépartement d’optique P. M. Duffieux, FEMTO-ST UMR 6174, Université de Franche-Comté, 16 route de Gray

25000 Besançon, France

Frédéric PatoisSmart Quantum, 4 rue Ampère, 22300 Lannion, France

Received September 11, 2006; accepted October 26, 2006;posted November 3, 2006 (Doc. ID 74963); published January 12, 2007

We report an intrinsically stable quantum key distribution scheme based on genuine frequency-coded quan-tum states. The qubits are efficiently processed without fiber interferometers by fully exploiting the nonlin-ear interaction occurring in electro-optic phase modulators. The system requires only integrated off-the-shelf devices and could be used with a true single-photon source. Preliminary experiments have beenperformed with weak laser pulses and have demonstrated the feasibility of this new setup. © 2007 OpticalSociety of America

OCIS codes: 270.0270, 060.0060.

Quantum key distribution (QKD) is the only knownmethod providing unconditionally secure key genera-tion between a sender and a legitimate receiver. Ap-propriate protocols make it possible to assess the in-formation leaked to an all-powerful eavesdropper(Eve) during a transmission and subsequently to gen-erate a provably secure key by public discussion.1

Typical QKD systems operating over optical fibersencode the information in the relative phase betweentwo time bins2 and therefore require the use of twounbalanced fiber interferometers prone to unstabilitydue to environmentally dependent temperature andmechanical fluctuations. Several solutions for stabi-lizing the interferometer path lengths within thewavelength of the photons have already been pro-posed, using either an active compensation3 or an in-genious round-trip plug and play architecture.4 Inthis Letter we propose an alternative approach basedon frequency-coded optical qubits. The encoding ofqubits in different frequency modes has known an in-creasing interest,5,6 since frequency modes separatedby microwave frequencies are easily resolvable andcontrollable by using optical and optoelectronic de-vices. We show here that the nonlinear interactionoccurring in off-the-shelf electro-optic phase modula-tors can be exploited to design an efficient and intrin-sically stable QKD system.

The quantum nonlinear interaction between thelight and microwave fields in an electro-optic phasemodulator is the key operation used in the proposedfrequency-coded scheme. It involves an infinite num-ber of discrete light modes and is fundamentally dif-ferent from the interaction occuring in acousto-opticmodulators.7 For brevity we will state here only themain result required for analyzing the QKD setup.We consider a lossless phase modulator with half-wave voltage V�, driven by a voltage at angular fre-quency � with amplitude V and phase �. The photon

number states in mode � are denoted �n��. It can then

0146-9592/07/030301-3/$15.00 ©

be shown that a state �1�� sent through the device istransformed according to

�1�� → �p=−�

�

�1��+p�Jp�a�ejp��−�/2�, �1�

where Jp is the pth-order Bessel function of the firstkind and a=�V /V� is the normalized amplitude.

The QKD system proposed here is depicted in Fig.1. At the transceiver (Alice) a monochromatic single-photon source SPS emits photons with angular fre-quency �0, which are then sent through a phasemodulator, PM1. The phase modulator is driven byvoltage-controlled oscillator, VCO1, operating at an-gular frequency ���0, and its output is filtered byfiber Bragg grating FBG1 and a circulator, which se-lect only the frequency bands �0 and �0±�. If alllosses are neglected, one can see from Eq. (1) thattuning the normalized amplitude a and phase � ofthe driving voltage transforms the state �1��0

emittedby the source into any normalized state of the form

J0�a��1��0+ J1�a�ej��1��0+� + J−1�a�e−j��1��0−�

�J0�a�2 + 2J1�a�2, �2�

with efficiency ��a�=J0�a�2+2J1�a�2�1, where a=�V /V�0. Hence Alice can generate the followingfour states:

� ± ;1� =1

�2�1��0

±1

2�1��0+�

1

2�1��0−�,

� + ;2� = �1��0,

�− ;2� =1

�2�1��0+� −

1

�2�1��0−�, �3�

with respective efficiencies

2007 Optical Society of America

302 OPTICS LETTERS / Vol. 32, No. 3 / February 1, 2007

��± ;1� = J0�a0�2 + 2J1�a0�2 � 0.953,

��+ ;2� = J0�0�2 + 2J1�0�2 = 1,

��− ;2� = 2J1�a1�2 � 0.539, �4�

where the parameters a0�1.161 and a1�2.405 arechosen such that J0�a0�=�2J1�a0� and J0�a1�=0. Al-though these four states belong to a three-dimensional Hilbert space, the sets ��+ ;1� , �−;1� and��+ ;2� , �−;2� form two incompatible bases of thesame two-dimensional subspace and can therefore beused to implement the standard BB84 protocol. Asshown by Eqs. (4), the encoding efficiency differs fromone state to another; however, an appropriate biasingof Alice’s statistics compensates this unbalance. Byselecting the four previous states with respectiveprobabilities 0.212, 0.212, 0.202, and 0.374, all fourstates are then produced with uniform probability0.202 at the output of the transceiver. Note that thetotal qubit encoding efficiency is then above 80%.

At the receiver (Bob) the incoming state is sentthrough a second phase modulator, PM2, driven by anoscillator VCO2 synchronized with VCO1. The fiberBragg grating FBG2 reflects the frequency band �0 tosingle-photon detector D2, while all other frequenciesare transmitted to single-photon detector D1. Thissimple detection setup allows Bob to choose his detec-tion basis by either driving PM2 at angular frequency� or not modulating at all. In fact, when no modula-tion is performed the states �+ ;2� and �−;2� are al-ways perfectly discriminated by the passive filtering.On the other hand, when Bob drives PM2 at angularfrequency � with normalized amplitude a� and phase��, one can see from Eq. (1) that, upon reception ofthe states �± ;1�, D2 and D1 click with respectiveprobabilities

PD2�� ± ;1�� =

J0�a��2

2+ J1�a��2 cos2 ��

Fig. 1. (Color online) Frequency-coded QKD scheme. PC,polarization controller; other abbreviations defined in text.

�2J0�a��J1�a��cos ��,

PD1�� ± ;1�� = 1 − PD2

�� ± ;1��. �5�

Setting a�=a0 and ��=0 yields PD2��+ ;1��=0 and

PD2��−;1��=0.953�1. Erroneous detections may hap-

pen upon reception of �−;1�; however, the overall in-crease in the quantum bit-error rate (QBER) is below1.2%. If we reasonably assume that Eve does not con-trol Bob’s receiver, Bob’s nonperfect measurementsdo not create a security loophole. Still, the increase inQBER has to be accounted for when distilling the se-cret key, which slightly reduces the key rate andtransmission distance.

Let us emphasize that many so-called frequency-coded schemes have already been proposed,8,9 butthey are merely the counterparts in the frequency do-main of the traditional phase-coded setups, as infor-mation is encoded only in the relative phase of twofrequency bands. We believe that our coding methodis the first practical attempt to fully exploitfrequency-domain coding. Since interferences occurthrough nonlinear interactions driven by preciselycontrolled microwave fields and passive filtering, oursetup also benefits from a high intrinsic stability. Infact phase modulators do not suffer from tempera-ture drifts and can be driven at higher frequenciesthan acousto-optic modulators (several gigahertz),which allows the use of low-loss wide-bandwidth op-tical filters. Hence little thermal or mechanical isola-tion is necessary to guarantee the stability of thesetup over long periods of time. Moreover, the fre-quency jitter of the light source has almost no effectas long as the fluctuations do not alter the filtering.Finally, the polarization sensitivity of the receivercan be removed by using standard polarization diver-sity techniques.10 Losses in real devices will of coursereduce the practical sifted key rate; however, achiev-ing high efficiency with our setup is not unrealistic,since the total losses introduced when low-loss phasemodulators and fiber Bragg gratings are used couldbe as low as 3 dB.

We implemented the setup of Fig. 1 on a benchtopwhere the transmission line was replaced by a vari-able attenuator. Owing to the lack of readily avail-able single-photon sources at telecommunicationswavelengths, we used instead a strongly attenuatedlaser source running in continuous mode at1547.5 nm. The whole system was then operated bydriving two nearly identical phase modulators(EOSPACE with V��5.5 V) at a modulation fre-quency of � /2��8 GHz with a single oscillator. Theamplitudes of the voltages driving the modulatorswere tuned independently with two variable-gainamplifiers, and the relative electric phase was ad-justed with a phase shifter located at the emitter.The filter FBG1 was made of two 80 pm bandwidthtunable fiber Bragg gratings (FBGs) centered aroundthe second and third upper and lower sidebands(�0±2� and �0±3�). Since all higher-order side-bands had negligible amplitude in all cases of inter-est, additional filtering turned out to be unnecessary.The filter FBG2 at the receiver was a single 80 pmbandwidth FBG centered around 1547.5 nm. The iso-

lation of all the filters was above 30 dB at 60 pm, and

February 1, 2007 / Vol. 32, No. 3 / OPTICS LETTERS 303

hence a negligible decrease in visibility was incurred.The losses at the emitter and receiver were botharound 3.2 dB but could be further reduced by slicingall fibers.

We first measured the classical spectra correspond-ing to the detection of the states �± ;1� at the outputof PM2 with an optical spectrum analyzer (APEX,0.07 nm precision, 80 dB dynamic range); see Fig. 2.When modulating the state �+ ;1�, the intensity levelof the central band is 20 dB below those of the firstsidebands, which guarantees a visibility around 98%.Similar results are obtained when modulating thestate �−;1�, but the accuracy of these estimations islimited by the poor amplitude precision of the opticalspectrum analyzer. Since we lacked the switchingelectronics necessary to perform a true key transmis-sion, we recorded only the QBER when each of thefour states was sent independently; however, we ex-pect the complete setup to have similar behavior. Thephotocounts were recorded at a 200 kHz rate by aInGaAs avalanche photodiode (EPITAXX) gated by10 ns pulses. For simplicity we set the source attenu-ation to get a mean photon number of ��0.1 photonper 10 ns gate at the output of the emitter. The quan-tum efficiency of the photodiode was approximately13%, and the dark count was pdark=4.5 10−6. The re-sults obtained for various attenuation levels areshown in Fig. 3. As expected, the lowest QBER is ob-tained when the state �+ ;2� is sent, since encodingand decoding are then purely passive. With the cur-rent results, we believe that the whole system couldeasily operate over distances as large as 60 km.

In summary, we proposed a new implementation ofthe BB84 protocol based on genuine frequency-codedqubits. By combining nonlinear interactions inelectro-optic modulators with filtering to replace tra-ditional interferometers, our system avoids mostthermal and mechanical isolation issues. This stabil-ity is achieved at the cost of a 1.2% increase in theQBER, which has, however, little effect on the finalkey generation rate. Like the single-side band (SSB)scheme,11 our setup could benefit from WDM syn-chronization and therefore be used over large dis-tances. Recent results12 have reported the generation

Fig. 2. Classical spectrum after remodulation ���1�.

of narrow-bandwidth single photons at 850 nm,which could be compatible with our coding scheme. Itmight also be possible to generate narrow-bandwidthsingle photons at 1550 nm by pumping a periodicallypoled lithium niobate waveguide and filtering theoutput.13 Research is now in progress to perform acomplete key distribution under field conditions.

The authors thank Marc Hanna and Anil Prab-hakar for fruitful discussions. M. Bloch’s e-mail ismatthieu@gatech.edu.

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Fig. 3. Evolution of the QBER with the equivalent trans-mission distance (assuming 0.25 dB km−1 loss).