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Abstract
In comparison with the rapidly developing progress on the optical C-band (1525-1565 nm) master-to-slave injection-locked transmitter to perform the coherent single-photon quantum key distribution (QKD), the development on the O-band (1250-1350 nm) QKD transmitter is somewhat delayed as the commercially available wavelength-matched narrow-linewidth distributed feedback laser diode (DFBLD) pair is hardly accessible up to now. By using the DFBLDs with only sub-MHz linewidth and relatively deviated wavelength for the first time, this work demonstrates the optically differential-phase-shift-keying (DPS) QKD by an O-band master-to-slave injection-locked DFBLD pair. The master and slave DFBLDs with wavelength fluctuations of ±0.05% and ±0.2 pm are controlled by a thermo-electric cooler with a feedback gain of 100. The 1-bit delay interferometer (DI) under thermo-insulation maintains its visibility at >96% with dPo/Po and dPo/dt measured below ±0.1% and ±1 × 10−3 mW/s. By RZ-OOK modulating the master DFBLD with step-like power coding at 150 µW to induce π phase shift in the injection-locked slave DFBLD, the rising-/falling-edge DPS envelope distortion of the slave DFBLD diminishes by decreasing the bias current of the master DFBLD from 7Ith (35 mA) to 2Ith (10 mA). This phenomenon enables the 128-bit DPS-QKD transmission with a quantum bit-error rate (QBER) of 3.57% and a secure key rate of 3.524 kbit/s in the 6-km SMF link. The O-band injection-locked single-photon DPS-QKD bit-stream with a mean photon number of 0.2 #/bit minimizes its decoding QBER to 3.88% and 4.84% for 512-bit and 1024-bit, respectively.
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1. Introduction
Single-photon quantum cryptography is an emerging technology for enabling highly secure networking without attacking and cracking risks. The early approach development for data encryption/decryption is mainly based on the Diffie-Hellman proposal with secure key exchange in 1976. This technology relied on transmitting a set of shared keys with unsure security over a common channel. With the rapid development of quantum computers, the new-era algorithms caused the attacking and cracking threats. Even the advanced secure keys established by discrete logarithms or factorization were extensively used in traditional communication networks to hardly guarantee encryption safety [1,2]. Quantum key cryptography has become the emerging mainstream technology to employ versatile protocols with anti-theft and anti-attack functionalities. For example, Bennett and Brassard created the earliest cryptography (called as the BB84) to transmit the encrypted keys by adjusting the polarization of photons in 1984 [2]. In 2002, Inoue et al. proposed the differential phase shift protocol to manipulate the phase of photon for quantum key encryption [3]. This method can retrieve the differential shift-keying phase with a delay interferometer (DI) for decryption [3]. At the transmitting end, the secure keys are used as the phase difference of adjacent optical pulses for the uploading transmission [4]. The received and decoded timing sequence at the receiving end is sent back to perform the identification at the transmitting end by the ideal authentication classic channel [4]. With these progressive achievements, a compact and efficient optical quantum key distribution (QKD) apparatus with the differential phase shift (DPS) keying protocol has been proposed [5]. This protocol can repulse the photon number splitting (PNS) attacks and provide high secure key rate (SKR) communication among potential optical quantum key cryptography methods [5]. In addition, this system also involves the direct amplitude modulation of a master laser diode to induce the indirect phase modulation of a pulse-modulated slave laser diode in the injection-locking scheme [5]. This proposal unveils highly secure quantum communication with distributing secret keys between remote parties in optical fiber networks. Up to now, various quantum key cryptography experiments have confirmed the feasibility of exchanging secure quantum keys over optical fiber links with transmission distances from tens to hundreds of kilometers [6–10]. In addition, these architectures further extend the heterogeneous coverage to the free space and submarine networks [11–13]. However, the transmission distance is still limited as compared to the classical communication scheme when the average photon number of pulsed transmitted bits in a quantum communication channel is usually less than one. Typically, the quantum key cryptography carrier mainly utilizes C-band lasers (1530-1565 nm) owing to the low-loss feature of the standard single-mode fiber during the long-reach transmission. Although the O-band coherent carriers were less used for quantum key transmission than the C-band carriers, the C-band light source often meets significant Raman scattering noise [14,15] and severe chromatic dispersion to decrease signal-to-noise ratio (SNR) [16,17] as compared to the O-band light source. Therefore, the O-band light source is comprehensively used for data-center applications nowadays.
As O-band lasers have never been employed for implementing quantum key cryptography by the master-to-slave optical injection-locking (OIL) with purely optical phase synchronizing and shift-keying, this study aims to investigate the use of O-band lasers in quantum key transmission within OIL systems. In addition, this work also addresses the invention of O-band master-to-slave OIL quantum key cryptography with the specific design for adiabatic packaging, the suppression of the phase drift by over-injection, and the feedback thermal control for wavelength stabilization. The compact TO-can package equipped with a thermal-electric cooler mount is employed for two (master and slave) distributed feedback laser diodes (DFBLDs) at O-band. The differential phase shift keying (PSK) algorithm for quantum key cryptography is demonstrated by executing the master-to-slave (M-to-S) OIL with emphasized heat dissipation and phase calibration via precise feedback control. In addition, the accurate biasing and encoding levels, periods, delays, and durations are optimized for quantum key generation to perform the differential phase modulation by initially analyzing the driving stability, drifting efficiency, polarizing adjustment, and locking range of both lasers. The adiabatic packaging of the delayed interferometer scales down the influence of parametric disturbance by several orders of magnitudes. This phenomenon ensures the bit error ratio (BER) and secure key rate (SKR) of the single-photon quantum key decoding to satisfy the criterion of error-free correction. The presented quantum key cryptography algorithm also highlights its unique features to suppress the encoded phase drift caused by the over-injection photon heating within the master laser cavity. This algorithm also compensates the gradually accumulated phase error originating from the residual phase accumulation in the pulsated slave cavity. Such a stabilized quantum key transmitter and decoder implemented by readily cost-effective components demonstrate its potential to realize quantum key cryptography with high efficiency and adaptability in harsh environments.
2. Experimental setup
Figure 1 shows the O-band DPS-QKD system based on the master-to-slave injection-locked DFBLD pair. Both the upper (used as the master) and the lower (used as the slave) DFBLDs were sealed, wired, and packed with the same style. The master and slave DFBLDs were individually packed within the copper mounts adhered upon a thermo-electric cooler (TE-cooler) to control the laser temperature by a thermistor. The thermistor was inserted near the TO-can package to monitor and feedback the laser temperature to the driving circuit. A self-designed SMA adapter was used to wire with two metallic pins of the TO-can for laser driving. In addition, a bias-tee was utilized to combine the DC bias current (for power level control) with the AC encoding signal (for quantum bit/key generation). The copper-based heat sink with an electric fan was placed under the TE-cooler plate to achieve better thermal stabilization of both master and slave lasers. The master DFBLD shows a central wavelength of 1308.396 nm with a full width at half maximum (FWHM) of 0.04 nm when the current is biased at 30 mA. The slave DFBLD exhibits its central wavelength at 1308.372 nm with an FWHM of 0.07 nm under the bias current of 12 mA. The FWHMs of both master and slave lasers were over-estimated by the optical spectrum analyzer with limited resolution bandwidth.
At the transmitter (TX) end, the current source supplies bias currents of 10 mA and 1.2 mA to individually drive the master and slave DFBLDs. The temperatures of the master and slave DFBLDs were respectively set as 24.5°C and 27°C to align their wavelength each other to execute the perfect OIL. The feedback temperature control was performed by setting the TEC modules to support the thermistor sensor with a monitoring feedback of 100 µA. In addition, the C1, C2, and C3 coefficients are detuned at 1.1291 × 10−3, 2.3411 × 10−4, and 9.07 × 10−8, respectively, to convert the feedback of the thermistor sensor into the temperature. A dual-channel arbitrary waveform generator (AWG) separately generates the offset return-to-zero on-off-keying data for modulating the master DFBLD and the non-offset return-to-zero on-off-keying data for pulsating the slave DFBLD. A high-gain power amplifier was further added after the AWG output to amplify the signal for gain-switching the slave DFBLD. Subsequently, the master DFBLD output injection-locked the slave DFBLD from Port 1 to Port 2 of the circulator. The signal of the slave DFBLD was sent from Port 2 to Port 3 of the circulator to deliver the optical quantum key toward the receiver end.
The quantum key carrier with the single photon per bit was implemented by a tunable attenuator to reduce the pulsed slave DFBLD output by more than 70 dB. To enable the single-photon carrier transmission with an average photon number of 0.28, an optical power meter with pico-watt sensitivity was used to determine whether the average output power level meets the single-photon standard or not. A high-precision wavemeter with the wavelength resolution of sub-picometer was employed to monitor the wavelength perturbations. After delivering the single-photon quantum key by a standard single-mode fiber (SSMF), a DI was used to resume the differential phase-shift-keying code back to the amplitude-shift-keying code through one-bit-delay interference. The outputs from dual-channel DI were sent into two single photon avalanche detectors (denoted as SPAD 1 and SPAD 2). Then, the detected TTL bits from the SPAD output were injected into a real-time oscilloscope (RTO) for grabbing the received quantum key stream and for timing sequencing all quantum keys under the identification. To achieve single-photon quantum key carrier transmission, the required attenuation for the pulsed slave DFBLD can be estimated with the formula of ηdB = -10·log(Pouttrep/hν) with Pout, trep, hν respectively denoting the average optical power, the repetition period, and the photon energy of the pulsed slave DFBLD output. For example, the output power of the slave DFBLD, the repetition period, and the energy of a single photon at 1310 nm are respectively assumed as 0.32 mW, 5.1 ns (equivalent to the frequency of 196 MHz), and 1.52 × 10−19 J. The pulsed quantum key carrier with 1.07 × 107 photons per bit needs to be attenuated by nearly 70.03 dB to retrieve single-photon quantum key transmission. To verify the stability of the driver, the time-dependent disturbance of the current source is measured as 0.0065% during 10 min. In addition, the operating temperature without any fluctuation (under the displayed resolution of two decimal places) is observed for 10 min.
Figure 2 illustrates a schematic DPS-QKD protocol diagram for the master and slave encoding streams to form binary phase shift keying data between adjacent pulsed bits via the master-to-slave OIL. The master DFBLD is encoded with a negatively biased TTL stream to concurrently induce power/wavelength/phase variations within the duration of a TTL bit. Because the OIL of the adjacent pulsed slave bits occurs at the continuous-wave lasing level of the master DFBLD, the power and wavelength of the master device remain unchanged. Only its phase altered by the TTL bit can differentiate the phase shift between adjacent bits output from the slave DFBLD after OIL. As the power of the master DFBLD remains almost unchanged with extremely small TTL encoding, the differential phase shift of the master DFBLD can be described by the following formula [18],
3. Result and discussion
The power-current-voltage curves of the master and slave DFBLDs shown in Fig. 3 are analyzed to observe the noise performances of the master and slave DFBLDs. The flattened dP/dI slopes of 0.11 and 0.32 mW/mA for master and slave DFBLDs are extracted from the first-order derivative of their P-I curves shown in Figs. 3(a) and 3(b). In addition, the corresponding threshold currents are observed at 5 and 6 mA. In addition, the outputs of the master and slave DFBLDs may exhibit similar power fluctuations of dP/P ≅ 3 × 10−4 when the bias current is driven at 40 mA with a current noise of ±10 µA. To transmit the DPS-QKD bit stream, the master DFBLD must be biased at several times of the threshold current to narrow its spectral linewidth. In addition, the master one is also modulated with a relatively small on-off-keying bit to cause a periodical phase shift. However, the slave DFBLD must be operated below the threshold and pulse-triggered to generate a periodical bit stream by the gain-switching effect [19]. From Figs. 3(c) and 3(d), the differential resistance (dV/dI) extracted from the voltage-to-current (V-I) curves are obtained as 5 Ω for two DFBLDs. The reflection coefficient (Γ) can be represented as (ZLoad-ZLD)/(ZLoad + ZLD) with the ZLoad and ZLD respectively denoting the impedance resistance and the differential resistance of the LD. In this case, the reflection coefficient is as high as 80% to cause the modulation efficiency of only 20% or below for both master and slave DFBLDs [20]. Figures 3(e) and 3(f) exhibit the temperature stabilization of both master and slave DFBLDs under different current operations. From Figs. 3(e) and 3(f), the temperature of master and slave DFBLDs are respectively stabilized at 24.5°C and 27.0°C to align their lasing wavelength coincident with each other. With a powerful heat dissipation control with the TE-cooler and the copper-fin heat sink, the execution of OIL can maintain the highest efficiency under the least power requirement of the master DFBLD.
Fig. 3. P-I-V curve of the master DFBLD and the slave DFBLD. The P-I curves of the (a) master and (b) slave DFBLDs. The V-I curves of the (a) master and (d) slave DFBLDs. The temperature stabilization of (e) master and (f) slave DFBLDs under different current operations.
To declare the impact of intensity noise on the on/off-extinction ratio and SNR of the master and slave DFBLDs [21], Figs. 4(a) and 4(b) illustrate the mode-partition-noise (MPN) and relative-intensity-noise (RIN) spectra for the master DFBLD. In addition, Figs. 4(c) and 4(d) depict the MPN and RIN spectra for the slave DFBLD. These figures compare fluctuations in mean square photon noise within specific frequency ranges (0.1-125 MHz for MPN and 0.5-12.5 GHz for RIN) after the noise is divided by the square of the average optical power. In Fig. 4(a), the master DFBLD exhibits significant noise peaks located around 11.3 and 23-33 MHz with increasing its bias from 3Ith to 7Ith. These noise peaks are caused by the mode partition noise when the high-ordered longitudinal modes become lasing in the DFBLD cavity. It is mandatory to avoid the encoding of DPS codes within the spectral range containing these noise peaks. Otherwise, the signal performance after decoding and receiving could be seriously degraded. Additionally, using the modulation below 40 MHz results in poorer SNR when the master laser has significant low-frequency noise. On the other hand, increasing the biased current of the master DFBLD from 2Ith to 7Ith can effectively up-shift the relaxation-oscillation frequency from 6 to 9 GHz and suppress its spectral peak from -149.42 to -157.58 dBc/Hz. The peak MPN level of the master laser operated at 6Ith is reduced to -131.87 dBc/Hz. As the bias current of the slave DFBLD is operated at 2-3Ith, the MPN noise peaks centered at 20, 43, 65, 80, and 110 MHz with the largest peak below -90.74 dBc/Hz are observed. For the RIN spectra shown in Fig. 4(d), the relaxation oscillating peak of the slave DFBLD up-shifts its frequency from 3.1 GHz to 5 GHz with a corresponding noise level reduced from 121.72 to -123.29 dBc/Hz.
Fig. 4. The noise spectra of the master DFBLD in the frequency ranges of (a) 0.1-125 MHz and (b) 0.5-12.5 GHz. The noise spectra of the slave DFBLD in the frequency ranges of (c) 0.1-125 MHz and (d) 0.5-12.5 GHz. The time-domain noises of the (e) master and (f) slave DFBLDs.
In Fig. 4(e), the time-domain noise of the master DFBLD is observed as approximately ±8 mV. However, the slave DFBLD exhibits an extremely large noise level of ±200 mV, as shown in Fig. 4(f). Apparently, both the noise reduction of MPN and RIN would strictly rely on the master-to-slave injection-locking technology. The wavelength fluctuation is another key parameter to maintain the DI at constant visibility during QKD decoding. Figures 5(a) and 5(b) respectively exhibit the wavelength variation of the master and slave DFBLDs under different bias currents. In Figs. 5(a) and 5(b), the respective wavelength drift slopes of 4.65 pm/mA and 5.19 pm/mA for the master and slave DFBLDs are obtained when the bias current enlarges from Ith to 6-7Ith. In addition, the power variation of these two devices can be observed in Figs. 5(c) and 5(d). The dP/dI slopes of the master and slave DFBLDs are respectively evaluated as 0.11 and 0.32 mW/mA.
Fig. 5. The wavelength variation of the (a) master and (b) slave DFBLDs under different bias currents. The power variation of the (c) master and (d) slave DFBLDs under different bias currents.
When selecting the appropriate master DFBLD, the characteristics of two DFBLDs are discussed in Fig. 6. In Figs. 6(a) and 6(b), the DFBLD-1 with a narrower linewidth exhibits less power fluctuation between -0.042% and 0.035% to provide a lower noise in the time domain. In principle, the linewidth of the DFBLD can be measured by the self-heterodyne interferometer. In the beginning, the DFBLD carrier is split into two branches by a 3-dB optical coupler. One of the optical beams adds the additional 25-km delay fiber and a phase modulator (PM) to interfere with another beam. Finally, the interfered light was sent to a photodetector (PD) and an electrical spectrum analyzer (ESA) for spectral linewidth analysis [22]. Figure 6(c) exhibits the measured linewidth of two master DFBLDs. With Lorentzian fitting to extract the full-width-at half-maximum at -3dB and -10 dB decaying points for suppressing the influence of the coherent spike in Fig. 6(d), the master DFBLD-1 reveals a linewidth of 489.24 kHz at -3dB decay as compared to the master DFBLD-2 with a linewidth of 797 kHz. In principle, the laser linewidth affects the probability of the photon leaking into the wrong detector at another output port of the DI to cause the quantum bit-error rate (QBER) degradation, as expressed by [23]
Fig. 6. (a) The power stability, (b) the time-domain noise, (c) the measured linewidth, and (d) the Lorenz-fitting linewidth of the master DFBLD-1 and DFBLD-2.
To perform the optimized injection-locking of the slave DFBLD with a sufficiently wide lockable wavelength range for DPS-QKD transmission, the allowable injection-locking range of the slave DFBLD is analyzed by altering the master power as well as the master-to-slave injection ratio. The relationship can be described as [24]
Fig. 7. The optical spectra of (a) the master laser (tunable single-mode DFBLD), (b) the slave DFBLD biased at Ith, and (c) the master-to-slave injection-locked slave DFBLD. (d) The allowable injection-locking range of the slave DFBLD as a function of the master-to-slave injection ratio.
Traditionally, the optical QKD is often operated at the C band because of the well-established network architecture to demonstrate low-loss and long-haul telecommunication. In addition, the larger wavelength tolerance for the DI device under the C-band operation benefits the QKD transmission. Figures 8(a)–8(d) exhibit the wavelength-dependent power variation of the DI device at the O band and C band to compare the output stability. In principle, the relationship between the laser wavelength and the transmittance of the DI device can be described as T%=cos2(πnSMFΔLDI/λ’DFBLD) with the nSMF, ΔLDI, and λ’DFBLD respectively denoting the refractive index of the SMF, the length difference between two arms of the DI device, and the first-order differential of the laser wavelength. The λ’DFBLD can be expressed as λ+Δλ with λ and Δλ indicating the wavelength and wavelength change of the DFBLD. Therefore, the wavelength-dependent power variation of the DI device under the O- and C-band operation can be simulated and shown in Figs. 8(a) and 8(b). As the QKD carrier is centered at 1308.4 nm, the wavelength variation of 0.5712 pm is enough to cause a π shift for detuning the interference from constructive to destructive condition. However, the wavelength variation to increase a π shift is increased to 0.8098 pm for the C-band QKD carrier at 1550 nm. This phenomenon can be verified from experimental results with corresponding values of 0.558 pm (for O-band) and 0.8712 pm (for C-band), as shown in Figs. 8(c) and 8(d). When the current disturbance causes a wavelength variation of 0.0106 pm to cause a phase shift of π/41 for the O-band DFBLD, the theoretical QBER can be increased from 0% to 9 × 10−5%. With the measured steady-state visibility of the DI device as 99.2%, Figs. 8(e)–8(f) exhibit the time-dependent stability of the DI device without and with adiabatic control. In Fig. 8(e), the non-adiabatic-controlled DI device only provides short-term stability in 13 sec. The thermal insulation can lengthen the stability of the DI device to 2100 sec, as shown in Fig. 8(f). The logarithmic power variations of the DI device under 1-hr operation without and with the thermal insulation are shown in Figs. 8(g) and 8(h). With the thermal insulation, the logarithmic power variation is significantly reduced from 5% to 0.2% with the 25-times improvement. In addition, Figs. 8(i) and 8(j) show the monitored instantaneous power fluctuation (dPo/dt) of the DI device under 1-hr operation without and with the thermal insulation. From Figs. 8(i) and 8(j), the thermal insulation can effectively suppress the instantaneous power fluctuation from ±0.15 mW/s to ±6 × 10−3 mW/s to benefit the long-term DPS-QKD operation.
Fig. 8. The simulated wavelength-dependent power variation of the DI device under the (a) O-band and (b) C-band operations. The experimental wavelength-dependent power variation of the DI device under the (c) O-band and (d) C-band operations. The time-dependent power variation of the DI device without the thermal insulation under the (e) O-band and (f) C-band operations. The time-dependent power variation of the DI device with the thermal insulation under the (g) O-band and (h) C-band operations. The time-dependent instantaneous power fluctuations of the DI device without the thermal insulation under the (i) O-band and (j) C-band operations.
Before the DPS-QKD transmission, the performance of the binary DPSK transmission at the multi-photon-carrier stage should be evaluated. The electrical DPS code stream with the repetitive pattern of {1, 0, 1, 0, …..} is used to drive the master DFBLD, as shown in Fig. 9(a). The pulsewidth, period time (tp), and peak-to-peak voltage (Vpp) of the electrical DPS code are set as 1.22 ns, 10.2 ns, and 110.5 mV, respectively. From the optical output of the master DFBLD in Fig. 9(b), the optical DPS code exhibits a Vpp of 60 mV as detected by a PD with a responsivity of 0.88 A/W. The salve DFBLD is gain-switched by the reverse electrical pulse train for the pulsed QKD bit generation. The electrical pulsed QKD bit exhibits a pulsewidth of 0.58 ns, a period of 5.1 ns, and a peak-to-peak amplitude of 918.3 mV, as shown in Fig. 9(c). After driving the slave DFBLD, the optical pulsated QKD bit with a pulse width of 0.26 ns is shown in Fig. 9(d). In principle, the gain-switching technology can shorten the optical pulsewidth of the slave DFBLD to make the decoded DPS-QKD bit precisely received within the gate-on window of the SPAD.
Fig. 9. (a) The electrical input and (b) optical output signals of the master DFBLD. (c) The electrical input and (d) optical output signals of the slave DFBLD. The DI output from the inputs of the (e) the free-running pulsed slave DFBLD, (f) the injection-locked pulsed slave DFBLD, (g) the DPS protocol with the sequence of {0, π, 0, π, 0, π, 0, π, 0, π….}, and (h) DPS protocol with the sequence of {π, π, 0, π, 0, π, p, 0, 0, π….}.
In addition, the peak-to-peak power of the optical pulsated QKD bit can be evaluated by the P = V/(GL × RL × ηPD) = 1936mV/(10 × 50Ω×(0.8 A/W)) with V, GL, RL, and ηPD respectively denoting the peak-to-peak voltage, the gain of the scope, the resistance of the scope, and the responsivity of the PD. Therefore, the peak-to-peak power of the optical pulsated QKD bit can be obtained as 4.83 mW. In this work, the optical DPS code transmitted by the master laser will injection-lock the optical pulsated (gain-switched) slave bits for DPS-QKD transmission. For the master-to-slave injection-locking, the direct optical modulation with a step-like power of 150 µW or an equivalent photocurrent of 0.12 mA can induce the π phase shift in the slave QKD bit. In this work, the actual operating voltages for the master and slave DFBLDs may differ from those calculated from the differential rate equations because of the impedance mismatch between the laser and generator. This phenomenon can be confirmed from the reflection coefficient calculated by dV/dI of master and slave DFBLDs, as shown in Fig. 5. To observe the effect of the master-to-slave injection-locking on the decode performance, the decoding results of the DI device for the slave DFBLD under free-running and continuous-wave injection are compared, as shown in Figs. 9(e) and 9(f). From the 9(e), the decoded signals of the DI device for the pulsed slave DFBLD under free-running operation exhibit a disordered output owing to its randomized phase. After the continuous-wave injection by the master laser, the decoded signals of the DI device for the slave DFBLD become flattened with equivalent bit power, as shown in Fig. 9(f). That is because the phase of each decoded bit is identical after injection-locking by the continuous-wave master. When the DPS codes transmitted by the master laser injection lock the slave DFBLD, the biasing and encoding parameters of both master and slave DFBLDs should be properly adjusted to obtain the perfectly 1-bit-delay decoded slave DFBLD pulsed bit. Figures 9(g) and 9(h) show the decoded signals of the DI device for the slave DFBLD when the repeated and random DPS codes with sequences of {0, p, 0, p, 0, p, 0, p, 0, p…} and {π, p, 0, p, 0, p, p, 0, 0, p…}) are used to perform the master-to-slave injection-locking. The amplitude fluctuation of the decoded DPS sequence is obtained as 6.5%-7.6% to achieve the theoretical BER of single-photon-carried DPS in the long-code transmission to 1.153% by the erroneous coding analysis.
By changing the mean photon number of DPS-QKD bit-stream to 0.28#/bit during transmission, Fig. 10 displays the received signals by the paired SPADs operated at a gate-on time of 1 ns and hold-off time at 999 ns. In addition, each normalized TTL bit at the 1-MHz clock rate corresponds to the received single-photon DPS-QKD bit at a specific time slot. Figure 10(a) exhibits the received DPS-QKD data streams with the 2000 DPS-QKD pattern length by two SPADs in the 2-ms timeframe. In general, the probability (P(k)) of photon occurrence within a QKD bit can be determined by the Poisson distribution of e-µ×mk/k! with k and µ denoting the photon count and the mean photon number. Substituting µ=0.28 into the Poisson distribution can obtain the occurrence probability of the zero-photon bit as 75.58%. With selecting arbitrary time slots within 100 µs to observe the received DPS-QKD bit-stream, Figs. 10(b) and 10(c) respectively show the received 12 single-photon DPS-QKD bits by both SPAD1 and SPAD2. The total 24 non-zero-photon counts received by two SPADs contribute to a receiving probability of 24%. Similarly, Figs. 10(d) and 10(e) exhibit the received DPS-QKD bit-stream in another time-slot zone between 1.0 ms and 1.1 ms to monitor the received non-zero-photon DPS-QKD bits. However, only 11 non-zero-photon bits are received by SPAD1 and SPAD2 with a receiving probability of 22% in this timeframe. These two results are all in good agreement with the aforementioned prediction by Poisson distribution as the mean photon number cannot be precisely controlled during attenuation and loss procedures in the attenuator and the DI device.
Fig. 10. (a) The received DPS-QKD data stream with a pattern length of 2000 by the SPAD1 and SPAD2 in a timeframe of 2 ms. The received DPS-QKD bit-stream in the time-slot zone between 0 s and 100 µs by (b) the SPAD1 and (c) SPAD2. The received DPS-QKD bit-stream in the time-slot zone between 1 ms and 1.1 ms by (d) the SPAD1 and (e) SPAD2.
Subsequently, the DPS-QKD transmission over different SMF lengths is demonstrated to realize the receiving penalty on the QBER. Under the fixed mean photon number of 0.28#/bit at the receiving end, the similar transmission condition can be set as the single-photon DPS-QKD bit-stream with the mean photon number at the transmission end of 0.5 #/bit in the 8-km SMF link with corresponding SMF loss of 0.32 dB/km at 1310 nm. In this work, the reason to use the mean photon number of 0.28 #/bit is limited by the input level set to protect the paired SPADs. Figure 11(a) plots the receiving count rate and the dark count rate of the DPS-QKD transmission under different-length SMF transmission. In this work, the dark count rate observed in the absence of incident photon indicates the mean count rate with most of the false detection events caused by thermal noise. In Fig. 11(a), the count rate of the DPS-QKD transmission received by the SPAD1 at the constructively interfered arm of the DI device is decreased from 365.184 to 136.407 kbit/s by extending the SMF distance from 0 to 15.6 km with the corresponding loss enlarging from 0 to 5 dB. However, the DPS-QKD transmission received by the SPAD2 at the destructively interfered arm exhibits its degraded count rate from 8.892 to 1.702 kbit/s. Therefore, the visibility of the DI device operated under the single-DPS-QKD mode is estimated as 97.5%.
Fig. 11. (a) The dark count rate and count rate of the DPS-QKD transmission received by the SPAD1 and SPAD2 under different-length SMF links. (b) The QBER of the DPS-QKD transmission under different-length SMF links.
With the constant visibility of the DI device, the increasing thermal noise by the decreasing mean photon number per bit will affect the QBER of the DPS-QKD transmission. In general, the QBER can be expressed as [0.5·(1-ν)]×{[µ×h×10-(αl + γ)/10 + Pd/(1-ν)]/[µ×h×10-(αl + γ)/10 + Pd]} with ν, α, l, γ, η, and Pd respectively denoting the visibility of the DI device, the fiber loss, the transmission distance, the loss in Bob, the detection probability of the SPAD, and the dark count rate [5]. As a result, the theoretical QBER is evaluated as 1.585% by setting the visibility of the DI device at 0.975. When the transmission distance increases from 0 to 15 km, the QBER of the DPS-QKD transmission is increased from 18.67% to 28.135%. The high QBER is mainly attributed to the overly DC bias current of the slave DFBLD. Therefore, the injection power ratio must be optimized to obtain a complete and perfect master-to-slave injection-locking with a precision phase control of the slave DFBLD. Otherwise, such high QBER cannot be further resumed back to access the error-free codes with the recommended QBER criterion of <4.1% even by executing versatile error-correction procedures.
Finally, the QBER optimization of the DPS-QKD transmission by lengthening the DPS coding patterns and adjusting the bias currents of the slave DFBLD is discussed. From Fig. 12(a), increasing the DPS code pattern from 128 to 1024 bits enlarges the QBER from 3.13% to 4.84% as the injection-locked slave DFBLD is operated at 0.2Ith. Enlarging the bias current of the slave DFBLD to 0.35Ith significantly degrades the QBER to 15.63%, 16.13%, 18.29%, and 19.53% for the 128-bit, 256-bit, 512-bit, and 1024-bit DPS transmission, respectively. The QBER degradation becomes more severe when the bias current of the slave DFBLD is set as 0.5Ith. The QBER of the DPS-QKD transmission is obtained as 35.71% for the 128-bit, 36.73% for the 256-bit, 38.04% for the 512-bit, and 48.83% for the 1024-bit. By marking the correct codes from the decoded DPS-QKD bit-streams by the paired SPADs, the 8 erroneous DPS codes are obtained from 19 receiving DPS codes in the 64-bit DPS-QKD transmission as the bias current of the slave DFBLD is set as 0.5Ith, as shown in Fig. 12(b). In Fig. 12(c), the 2 incorrect DPS codes in 16 receiving DPS codes are observed when the bias current of the slave DFBLD decreases to 0.35Ith. In addition, no erroneous DPS code is detected in the 20 receiving DPS codes carried by the slave DFBLD operated at 0.2Ith, as shown in Fig. 12(d). In this case, the QBER degradation is attributed to the phase disturbances induced by incomplete master-to-slave injection-locking with enlarging the DC bias of the slave DFBLD. Figures 12(e)–12(g) exhibit the laser spectra of the injection-locked slave DFBLD under different bias currents. With the fixed power of the master laser, setting the bias current of the slave DFBLD at 0.5Ith obtains the lowest injection ratio to generate the four-wave mixing observed in the lasing spectrum, as shown in Fig. 12(e). By reducing the bias current of the slave laser to 0.35Ith, the slave DFBLD cannot be perfectly injection-locked to provide single-mode lasing with a precisely controlled phase, as shown in Fig. 12(f). This phenomenon can suppress the QBER of the DPS-QKD transmission. In Fig. 12(g), the perfect master-to-slave injection-locking is achieved by reducing the DC bias of the slave laser to 0.2Ith to demonstrate the narrowest injection-locked single-mode spectrum with the lowest frequency/phase noise. Therefore, the QBER of the 128-bit DPS-QKD transmission can be improved to the lowest value of 3.13%. The secure key rate (Rsecure) can be estimated by a formula of Rsifted{(1-2µ)log2Pc0(e)-f(e)[elog2e + (1-e)log2(1-e))]} with Rsift, Pc0(e), and f(e) respectively denoting the sift key rate, Eve’s total collision probability, and the error correction algorithm [4]. In principle, the Pc0(e) and f(e) can be defined as ∼0.6 and 1. The QBER and sifted key rate are degraded from 3.03% to 6% and from 374.857 kbit/s to 138.604 kbit/s as the SMF transmission distance changes from 0 km to 15 km. In addition, the secure key rate (SKR) is obtained as 23.865 kbit/s at 0 km and 3.524 kbit/s in the 6-km SMF transmission. In this work, the QBER of the DPS-QKD transmission over 6 km can be obtained as 3.57% to achieve the specification demands of QBER below 4.12%.
Fig. 12. (a) The QBER of the DPS-QKD transmission with different DPS code lengths and different bias currents of the slave DFBLD. The received and sifted DPS key data by the paired SPADs when the slave DFBLD is operated at (b) 0.5Ith, (c) 0.35Ith, and (d) 0.2Ith. The lasing mode optical spectra of the injection-locked slave DFBLD biased at (e) 0.5Ith, (f) 0.35Ith, and (g) 0.2Ith.
Table 1 summarizes the performance of the DPS-QKD transmission in five representative works. In 2012, Wang et al. utilized a 1550-nm laser with external IM and PM schemes to achieve the PDS-QKD transmission over 30-km SMF with a QBER of 1.89% and an SKR of 810 kbit/s [10]. A serious SKR degradation as low as 1.85 bit/s is also observed when the SMF transmission extends up to 260 km. In 2019, Paraïso et al. proposed on-chip QKD with a 1550-nm laser under injection-locking to demonstrate the 50-km SMF transmission with a QBER of 3.5% and an SKR of 125 kbit/s [25]. In 2021, Marco et al. presented a QKD transmission based on the C-band injection-locked laser to achieve 70-km SMF transmission with QBER = 2.7% and SKR = 400 kbit/s [26]. From previous works, only the external IM + PM technology was applied to deliver the single-photon DPS-QKD by employing the O-band laser carrier. Takesue et al. preliminarily proposed such a system in 2006 to demonstrate 10-km SMF transmission with a QBER of 8.1% and an SKR of 22.7 kbit/s after [16]. In 2009, Ma et al. completed the O-band DPS-QKD transmission with a QBER of 3.8% and an SKR of 8.5 kbit/s over 50-km SMF [17]. In this work, our experimental architecture is the only work based on the O-band master-to-slave injection-locked DPS-QKD link. In addition, the preliminary demonstration with competitive performances has promoted to become an alternative potential candidate as compared to other works. The current limitation to perform the O-band DPS-QKD in comparison with the C-band system is mainly attributed to the higher propagation loss of the SMF at the same distance. This phenomenon causes an increased QBER to limit the allowable transmission distance. Although a large gap between O-band and C-band DPS-QKD in terms of the transmission distance and the security key rate still exists as shown in Table 1, the necessity and superiority of the O-band carrier instead of C-band carrier to realize the DPS-QKD is addressed in more detail. As the O-band carriers are currently used in the short-reach fiber-optic networks, the commercially available optical channels are provided for data streaming among metropolitan users with the lowest waveform distortion owing to the nearly chromatic-dispersion-free transmission of communication data. Some specific application scenarios also demand secure data communication to fulfill with the requirement for medical, financial, defense, and private data exchange between individuals and parties in the wide-spreading O-band gigabit ethernet links.
Table 1. Comparison on DPS-QKD Transmission in Different Research Group
4. Conclusion
This paper preliminarily demonstrates the ultrastable quantum-key encryption and decryption by an O-band master-to-slave injection-locking quantum key distributor with both the single-photon transmitter and the delay interferometric decoder with isothermal packages to improve the BER and SKR. In this work, the O-band master-to-slave injection-locked DFBLD pair at 1308.4 nm is demonstrated for single-photon quantum key distribution with precision wavelength and phase stabilization via the feedback control on both temperature and current of the master laser. In addition, the bias current of the slave DFBLD is also optimized to retrieve the best phase injection-locking condition. This optimization can suppress the QBER of the decoded and sifted DPS-QKD bit-stream. The TO-can packaged DFBLDs at 1308.4 nm with flattened dP/dI slopes of 0.11 and 0.32 mW/mA are respectively employed as master and slave DFBLDs with corresponding threshold currents of 5 and 6 mA.
Their respective wavelength drift slopes are 4.65/5.19 pm/mA under the bias operation below 6-7Ith and 94.3/98.4 pm/°C under the temperature control below 35°C. This optimization can provide the power-to-temperature and current-to-temperature slopes of -0.039/-0.043 mW/°C and 0.11/0.32 mW/mA for master/slave DFBLDs. The differential resistance of 5 Ω induces a modulation reflection coefficient of 80% with a degraded modulation efficiency of only 20%. By optimizing the feedback-controlled TE-cooler gain as 100, the linewidth, power, and wavelength variations of the master DFBLD are respectively 489.24 kHz, ± 0.05%, and ±0.2 pm. In addition, the linewidth, power, and wavelength variations of the slave DFBLD are respectively 11.4 GHz, ± 0.1%, and ±0.8 pm. To induce the π shift in the 1-bit-delay DI with an FSR of 196 MHz, a wavelength shift of 0.558 pm for the O-band laser is relatively smaller than that of 0.8712 pm for the C-band laser. It indicates that the wavelength must be accurately controlled to maintain stability during DPS-QKD transmission. Moreover, the thermal insulation of DI is mandatory to lengthen the decoding stability from 13 to 2100 sec with interferometric visibility above 96%. Setting the TE-cooler gain of the DFBLD at 100 effectively maintains the DI at to preserve its operation time beyond 300 sec at constructive interference with dPo/Po and dPo/dt below ±0.1% and ±1 × 10−3 mW/s. The wavelength injection-locking range is only 25.24 GHz when remaining the master-to-slave injection ratio as small as -20 dB. However, the master wavelength be widely tuned as 108.82 GHz to injection-lock the slave by increasing the master-to-slave injection ratio up to -2 dB.
In the multi-photon DPS experiment, the TTL-like master encoding power of 150 µW equivalent to a photocurrent modulation by 0.12 mA can induce the π phase shift in the slave QKD bit after performing the master-to-slave injection-locking. The free-running gain-switched slave DFBLD delivers the optical pulsated QKD bit with a pulsewidth of 0.26 ns and a peak-to-peak power of 4.83 mW. With master-to-slave injection-locking under DPS encoding, the conjugated DPS code sequences from the dual-port DI outputs reveal identical bit amplitude fluctuation of 6.5%-7.6%. During long-DPS-code transmission, the bias current of the master DFBLD significantly affects the encoded phase of the injection-locked slave QKD bits at the near rising and falling edges of the long-code pattern. Decreasing the bias current from 7Ith (35 mA) to 2Ith (10 mA) essentially suppresses a heating-induced phase instability. After comparing the count rate of the paired SPADs, the visibility of the DI device operated under the single-photon periodical DPS-QKD mode is estimated as 97.5% by reducing the mean photon number (µ) to 0.28 #/bit. By increasing the SMF distance from 0 to 15 km, the analyzed QBER rises from 18.67% to 28.135%. To satisfy with error-correction criterion for decoding the single-photon DPS encrypted QKD, the qualified SMF transmission can only achieve 6 km with a QBER of 3.57% and an SKR of 3.524 kbit/s by the perfect master-to-slave injection-locking DFBLD pair. By reducing the DC bias of the slave DFBLD to 0.2Ith, the lowest frequency/phase noise is retrieved to scale down the single-photon QBER to its respective minimum of 3.88% and 4.84% for 512-bit and 1024-bit DPS-QKD transmission.
Funding
National Science and Technology Council (MOST110-2221-E-002-100-MY3, NSTC 111-2119-M-002-009, NSTC112-3111-E-992-001-, NSTC112-2221-E-002-176-MY3, NSTC112-2119-M-002-013, and NSTC 113-2119-M-002-023).
Acknowledgments
National Science and Technology Council, Taiwan (MOST 110-2221-E-002-100-MY3, NSTC 111-2218-E-A49-011-, NSTC 111-2124-M-A49-004-, NSTC 111-2221-E-002-143-MY3, NSTC 112-3111-E-992-001-, NSTC 112-2221-E-002-176-MY3, and NSTC 112-2119-M-002-013-).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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