Photon nonlinear mixing in subcarrier multiplexed quantum key distribution systems

José Capmany

doi:10.1364/OE.17.006457

1. Introduction

Quantum cryptography features an unique way of sharing a random sequence of bits between users with a certifiable security not attainable with either public or secret-key classical cryptographic systems [1,2]. This is achieved by means of quantum key distribution (QKD) techniques. In essence, QKD relies on exploiting in a positive sense the laws of quantum mechanics, which are often viewed in other contexts of physics as limiting or negative [3,4]. Photonics has proved to be one of the principal enabling technologies for long-distance QKD using optical fiber links. Four main different photonic-based techniques have been reported in the literature for implementing QKD. In 1992 Bennett and co-workers [5] proposed to exploit the polarization of photons to implement the four required states by employing one circular polarization and one linear polarization basis. The main disadvantage of this method rests on the difficulty of preserving the polarization over long lengths of standard telecommunication fibers. The second approach, initially developed by Townsend and co-workers [6-8], relies on the use of optical delays and balanced interferometers at the transmitter and the receiver. The main challenge of this approach resides in keeping the interferometers free from environmental and mechanical variations and preserving the matching of the path difference at the transmitter and receiver. A third approach, based on differential phase shift quantum key distribution [9] has enabled key generation and distribution along distances over 100 Km [10] although with limited security, [11]. The fourth approach, proposed by Merolla and co-workers [12], also known as frequency coding, relies on encoding the information bits on the sidebands of either phase [13] or amplitude [14] radio-frequency (RF) modulated light. The coding principles of subcarrier multiplexing are widely employed in the field of microwave photonics [15] for a variety of applications including remote antenna beam steering and the optical processing of microwave signals [16]. Alice randomly changes the phase of the electrical signal used to drive a light modulator among four phase values, which form a pair of conjugated bases and sends the signal through a fiber link. When it arrives at Bob, he modulates the signal again using the same microwave signal frequency and thus his new sidebands will interfere with those created by Alice [12]. Over the past few years, the work done by Merolla and co-workers has led to considerable dramatic improvements in this kind of system. Originally used for implementing the Bennett 1992 B92 protocol [17],[12], it was subsequently improved by adjusting the modulator characteristics which let them demonstrate the implementation of the Bennett-Brassard 1984 BB84 protocol [2],[18], [19].

2. Theoretical fundamentals

The conceptual approach of frequency coding can be considerably improved in terms of signal speed by incorporating a multiplexing technique widely employed in microwave photonics [15] and known as subcarrier multiplexing (SCM) to provide parallel QKD. In [20] we have proposed the extension of frequency coding to multiple subcarriers, showing that it opens the possibility of parallel quantum key distribution and, therefore, of a potential substantial improvement in the bit rate of such system. The concept behind subcarrier multiplexed quantum key distribution (SCM-QKD) can be explained referring to Fig. 1.

A faint pulse laser source, emitting at frequency w_o is externally modulated by N radiofrequency subcarriers by Alice. Each subcarrier, generated by an independent voltage controlled oscillator (VCO) is randomly phase modulated Φ_1i, among four possible values 0, π and π/2, 3π/2, which, as mentioned above, form a pair of conjugated bases. The compound signal is then sent through an optical fiber link and upon reaching Bob’s location, are externally modulated by N identical subcarriers in a second modulator. These subcarriers are phase modulated Φ_2i among two possible values 0 and π/2 which represent the choice between the two encoding bases. As a consequence, an interference single-photon signal can be generated at each of the sidebands (upper and lower) of each subcarrier. For a given subcarrier Ω_i = 2πf_i if Bob and Alice’s bases match, then the photon will be detected with probability 1 by either the detector placed after the filter centered at w_o + Ω_i or by the detector placed after the filter centered at w_o − Ω_i. If, on the contrary, Bob and Alice’s bases do not match there will be an equal probability of 1/2 of detecting the single photon at any of the two detectors and this detection will be discarded in a subsequent procedure of public discussion.

Fig. 1. SCM_QKD system Layout [20]

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The advantages of SCM-QKD however come at a price. System design is expected to be more demanding and, especially, the impact of nonlinear photon mixing by harmonic distortion and intermodulation needs to be thoroughly addressed. In this letter, we provide, for the first time to our knowledge, this analysis, considering the influence of nonlinear photon mixing on the end-to-end quantum bit error rate (QBER).

We assume that the output signal from the laser, which is well above threshold can be described by a coherent state |ψ〉 and that the probability of observing a photo-count using a detector with efficiency ρ at time t is proportional to [21]:

P = ρ 〈 ψ ∣ E^{-} (t) E^{+} (t) ∣ ψ 〉

Where:

E^{+} (t) = j \sum_{w} ξ (w) a_{w} e^{- jwt}

ε_o is the dielectric permittivity, V_o the mode volume, a_w, a_w ⁺ the annihilation and creation operators respectively. The coherent state describing the field at the output of Alice’s modulator can be expressed as:

∣ ψ_{A} 〉 = ∣ (1 + e^{j Ψ_{1}}) α_{w_{o}} 〉 ∣ j e^{- j Φ_{1 N}} e^{j Ψ_{1}} α_{w_{o} - Ω_{N}} 〉 ∣ j e^{- j Φ_{1 N - 1}} e^{j Ψ_{1}} α_{w_{o} - Ω_{N - 1}} 〉 \dots

The coherent state at the output of Bob’s modulator is given by:

∣ ψ_{B} 〉 = ∣ (1 + e^{j Ψ_{1}}) (1 + e^{j Ψ_{2}}) α_{w_{o}} 〉 \dots \dots ∣ j {(1 + e^{j Ψ_{1}}) e^{j Ψ_{2}} e^{- j Φ_{2 i}} + (1 + e^{j Ψ_{2}}) e^{j Ψ_{1}} e^{- j Φ_{1 i}} e^{- jD (- Ω_{i})}} α_{w_{o} - Ω_{i}} 〉 \dots

In Eq. (4) the subindices k and l run across all their possible values and:

D (Ω_{i}) = β_{1} L Ω_{i} + \frac{β_{2} L}{2} Ω_{i}^{2}

Represents the fiber link chromatic dispersion operator for subcarrier Ω_i. Now, assuming that an optical filter centred at w_o + Ω_i is placed after Bob’s modulator to select the content of that sideband we can obtain the coherent state at its output by adding the contribution of the nonlinear terms |α_{w_o+Ω_l-Ω_k}〉, |α_{w_o+Ω_l+Ω_k}〉 such that Ω_l ± Ω_k = Ω_i. To include them properly we must bear in mind that for these nonlinear photon beating terms:

∣ α_{w_{o} + Ω_{l} \mp Ω_{k}} 〉 = \frac{m}{2} ∣ α_{w_{o} + Ω_{i}} 〉

Where m represents the subcarrier index of modulation, which is assumed to be the same for all of them.

Therefore from Eq. (3)-Eq. (6) we get:

∣ ψ_{+ Ω_{i}} 〉 = ∣ 0 〉 ∣ 0 〉 ∣ 0 〉 \dots \dots ∣ 0 〉 \dots ∣ 0 〉 ∣ c_{w_{o} + Ω_{i}} α_{w_{o} + Ω_{i}} 〉

Where:

c_{w_{o} + Ω_{i}} = [j {\begin{matrix} (1 + e^{j Ψ_{1}}) e^{j Ψ_{2}} e^{j Φ_{2 i}} + \\ + (1 + e^{j Ψ_{2}}) e^{j Ψ_{1}} e^{j Φ_{1 i}} e^{- jD (Ω_{i})} \end{matrix}} - \frac{m}{2} e^{j (Ψ_{1} + Ψ_{2})} {\begin{matrix} \sum e^{- jD (Ω_{l})} e^{j (Φ_{1 l} - Φ_{2 k})} + \\ Ω_{k}^{k, l} - Ω_{l} = Ω_{i} \\ + \sum e^{- jD (Ω_{r})} e^{j (Φ_{1 r} + Φ_{2 s})} \\ Ω_{r}^{r, s} + Ω_{s} = Ω_{i} \end{matrix}}]

A similar procedure for the corresponding lower sideband yields:

∣ ψ_{- Ω_{i}} 〉 = ∣ 0 〉 ∣ 0 〉 ∣ 0 〉 \dots \dots ∣ 0 〉 \dots ∣ 0 〉 ∣ c_{w_{o} - Ω_{i}} α_{w_{o} - Ω_{i}} 〉

c_{w_{o} - Ω_{i}} = [j {\begin{matrix} (1 + e^{j Ψ_{1}}) e^{j Ψ_{2}} e^{- j Φ_{2 i}} + \\ + (1 + e^{j Ψ_{2}}) e^{j Ψ_{1}} e^{- j Φ_{1 i}} e^{- jD (- Ω_{i})} \end{matrix}} - \frac{m}{2} e^{j (Ψ_{1} + Ψ_{2})} {\begin{matrix} \sum e^{- jD (- Ω_{l})} e^{- j (Φ_{1 l} - Φ_{2 k})} + \\ {- Ω}_{k}^{k, l} + Ω_{l} = - Ω_{i} \\ + \sum e^{- jD (- Ω_{r})} e^{- j (Φ_{1 r} + Φ_{2 s})} \\ {- Ω}_{r}^{r, s} - Ω_{s} = - Ω_{i} \end{matrix}}]

Direct application of Eq. (1) to Eq.(7)-Eq.(10) yields the photo-count when an upper or lower sideband subcarrier is selected:

P_{\pm Ω_{i}} = {∣ c_{w_{o} \pm Ω_{i}} ∣}^{2} ρ ξ^{2} (w_{0} \pm Ω_{i}) 〈 n_{w_{o} \pm Ω_{i}} 〉

Where [21]:

〈 n_{w_{o} \pm Ω_{i}} 〉 = 〈 α_{w_{o} \pm Ω_{i}} ∣ a^{+} a ∣ α_{w_{o} \pm Ω_{i}} 〉

Is the average number of photons in the mode of frequency w_o ± Ω_i.

3. Quantum bit error rate derivation and discussion

From Eq. (11) we can derive a suitable end-to-end QBER expression for the SC-QKD system subject to photon nonlinear mixing following the approach of [21].

QBER (Ω_{i}) = \frac{{(1 - V) + (\frac{1}{{QCNR}_{CSO}^{i}})} ρ T_{L} {\bar{μ}}_{i} + d_{B}}{2 [{1 + (\frac{1}{{QCNR}_{CSO}^{i}})} ρ T_{L} {\bar{μ}}_{i} + d_{B}]}

In Eq. (13) V represents the optical visibility achieved in the filtering process, ρ the detector efficiency, μ¯_i is the average photon number in mode Ω_i, d_B is the dark count rate, T_L the overall fiber link losses T_L = T_B10^-αL/10 where T_B represents the losses in Bob’s modulator and QCNRⁱ_CSO is the Quantum Carrier to Noise Ratio, given by:

{QCNR}_{CSO}^{i} = \frac{16}{m^{2} N_{CSO} (Ω_{i})}

The number of composite second order terms N_CSO(Ω_i) depends on the number of subcarriers employed in the system and also on their frequency spacing. It is composed of the sum of second harmonic distortion terms and frequency sum and difference terms. Figure 2 shows, for example the frequency spectrum, number of second harmonic distortion terms N_CSOHD(Ω_i), number of frequency sum and difference terms N_CSOU(Ω_i)+N_CSOD(Ω_i) and overall number N_CSO(Ω_i) for a low count (i.e 15 channel frequency plan) system.

Fig. 2. Frequency spectrum, number of second harmonic distortion, frequency sum and difference intermodulation terms, and overall CSO number N_CSO(Ω_i) for a frequency plan composed of 15 evenly spaced channels spanning from 2 to 30 GHz.

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We now make use of Eq. (13) to analyze the effect of photon nonlinear mixing on the performance of the SCM-QKD systems. We consider three different frequency plans representative of low, middle and high channel count systems. In the first case N=15 and channels are evenly spaced channels spanning from 2 to 30 GHz, in the second N=30 and channels are evenly spaced channels spanning from 1 to 30 GHz, in the third N=50 and channels are evenly spaced channels spanning from 1 to 50 GHz.

In Fig. 3.a and Fig. 3.b we present the QBER values versus the optical fiber link length in Km, obtained for the three frequency plans (N=15,N=30 and N=50) system previously presented and for the case of a frequency coded (N=1) system.

For the computation of the QBER we have taken standard values for several parameters. For instance the visibility is taken as V=98%, the detector efficiency is ρ = 0.13, α = 0.2dB/Km and T_B = 9.6dB. We have considered the worst case scenario, that is, the QBER has been computed for the channel in the frequency plan exhibiting the highest N_CSO value. The average number of photons per sideband at the laser output is, unless otherwise stated, μ¯ = 0.05.

In Fig. 3.a we represent the QBER values when the modulation index for the SCM channels is 2% (i.e m=0.02). Note that the impact of nonlinear photon mixing in this case is negigible for low, middle and high count channel systems. This situation changes for higher modulation indexes. For instance, in Fig. 3.b we plot the QBER results for the same systems when the modulation index of the SCM channels is a 8%. The impact of the degrading effects of nonlinear photon mixing is now apparent with a greater impact the higher the channel count of the frequency plan. It can be appreciated that, for instance, the QBER can increase by a factor of almost 50% for N=50 as compared to the case where N=1. In practice these results suggest that modulation indexes close to 2% must be employed if the SCM-QKD system is to be considered inmune to photon nolinear mixing effects.

Fig. 3. QBER values versus the optical fiber link length in Km, obtained for the three frequency plan (N=15,N=30 and N=50) systems and for the case of a frequency coded (N=1) system. 3.a) m=2%, 3.b) m=8%

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4. Conclusion

In this paper, we have analysed the influence of nonlinear photon mixing on the end to end quantum bit error rate QBER performance of subcarrier multiplexed quantum key distribution Systems. The results show that negigible impact is to be expected for modulation indexes in the range of 2%. These values are perfectly attainable with available radiofrequency and optoelectronic technologies.

Acknowledgements

The authors wish to acknowledge the financial support of the Spanish Government through Quantum Optical Information Technology (QOIT), a CONSOLIDER-INGENIO 2010 Project and the Generalitat Valenciana through the PROMETEO research excellency award programme GVA PROMETEO 2008/092.

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Photon nonlinear mixing in subcarrier multiplexed quantum key distribution systems

Abstract

1. Introduction

2. Theoretical fundamentals

3. Quantum bit error rate derivation and discussion

4. Conclusion

Acknowledgements

References and links

Cited By

Figures (3)

Equations (21)

Optics Express