# Experimental Demonstration of an Efficient Mach–Zehnder Modulator Bias Control for Quantum Key Distribution Systems

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## Abstract

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## 1. Introduction

## 2. Method

- During the QKD protocol, Alice prepares N (≥2) types of diagnostic pulses whose MZM phases are modulated by ${\theta}_{mod}^{i}=\frac{2\pi}{N}\left(i-1\right)$ with uniformly distributed probabilities for $i=\left\{1,\dots ,N\right\}$. Thereafter, Alice sometimes transmits them to Bob as substitutes for the decoy pulses. Similar to the signal and decoy pulses, the diagnostic pulses are attenuated to single-photon levels. This method does not weaken the security of the QKD significantly because nobody except Alice can distinguish between the decoy and diagnostic pulses.Conventionally, the MZM output intensity can be described as [34,35,39,51,52]:$${I}_{out}={L}_{in}{I}_{in}{\mathrm{cos}}^{2}\left(\frac{{\theta}_{mod}+{\theta}_{drift}}{2}\right)$$
- Bob receives and measures the incoming pulses using single-photon detectors (SPDs). After measuring, he publicly announces the time indexes where the signals are detected. Thereafter, Alice and Bob perform the remaining protocols, such as key sifting, error correction and privacy amplification.
- Simultaneously, Alice accumulates the detection results of the diagnostic pulses unless there are no significant phase drifts. The optimal accumulation time strongly depends on the ambient environment, channel loss, detection efficiencies and pulse intensities.
- After accumulation, Alice calculates the normalized detection probabilities ${p}_{i}$ as [53,54]$${p}_{i}=\left(\frac{N}{2}\right)\times {C}_{i}/\mathsf{\Sigma}{C}_{i},$$$$\mathrm{Err}\left({\theta}_{drift}^{T}\right)=\mathsf{\Sigma}{\left[{p}_{i}-{p}_{Ti}\left({\theta}_{drift}^{T}\right)\right]}^{2},$$
- Alice finds ${\theta}_{drift}^{T}$ minimizing $\mathrm{Err}\left({\theta}_{drift}^{T}\right)$ by adjusting ${\theta}_{drift}^{T}$ from $0$–${360}^{\xb0}$. Subsequently, the found value is estimated as ${\theta}_{drift}$ because the error is minimized when ${\theta}_{drift}^{T}$ maximally matches the practical value ${\theta}_{drift}$. For example, Alice can assume ${\theta}_{drift}={70}^{\xb0}$ with the smallest $\mathrm{Err}\left({\theta}_{drift}^{T}\right)$ at ${\theta}_{drift}^{T}={70}^{\xb0}$, as shown in Figure 1c. The estimation accuracy depends on the adjustment interval of ${\theta}_{drift}^{T}$. As the interval becomes smaller, the prediction accuracy becomes better. However, more computational power and time are required.
- Finally, Alice compensates for the estimated phase drift by applying ${\theta}_{mod}={\theta}_{mod}-{\theta}_{drift}$; therefore, the MZM bias point is maintained at the desired point. Accordingly, as the ${\theta}_{drift}$ term of the MZM output intensity is erased, Equation (1) becomes$${I}_{out}={L}_{in}{I}_{in}{\mathrm{cos}}^{2}\left(\frac{{\theta}_{mod}-{\theta}_{drift}+{\theta}_{drift}}{2}\right),\phantom{\rule{0ex}{0ex}}={L}_{in}{I}_{in}{\mathrm{cos}}^{2}\left(\frac{{\theta}_{mod}}{2}\right).$$

## 3. Experimental Results

_{3}crystal optimized for a wavelength of 1550 nm and has a bandwidth of up to 10 GHz. A refrigerant spray was used to abruptly reduce the temperature of the MZM. The control method was implemented on a STM32 Nucleo-144 board with a STM32F413ZH microcontroller unit (MCU). We set N = 4, considering the accumulation time and random bit consumption for the diagnostic signals. The desired bias point was set to the null point, and an adjustment interval of 1° was used to calculate ${p}_{{T}_{i}}\left({\theta}_{drift}^{T}\right)$ in Equation (3). A 16-bit digital-to-analog converter and a 12-bit analog-to-digital converter built on the STM32 board were used to generate the diagnostic signals and to measure the output of the photodiode (PD).

_{2π}of approximately 7 V.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Transfer functions of the Mach–Zehnder modulator (MZM) with and without a transmission distortion owing to the bias drift ${\theta}_{drift}$; (

**b**) Block diagram of the proposed method applied to a decoy-state BB84 quantum key distribution (QKD) system; (

**c**) Calculated $\mathrm{Err}\left({\theta}_{drift}^{T}\right)$ according to the theoretical phase drift ${\theta}_{drift}^{T}$. The error is minimized when the ${\theta}_{drift}^{T}$ maximally matches the practical value ${\theta}_{drift}$. Thus, Alice can estimate ${\theta}_{drift}={70}^{\xb0}$ from the minimum error at ${\theta}_{drift}^{T}={70}^{\xb0}$.

**Figure 2.**Experimental setup and results of the laboratory tests: (

**a**) Experimental setup of the laboratory test; (

**b**) MZM output intensity of signal modulation. The black and red solid lines represent the output intensities with and without the proposed control method, respectively; (

**c**) Direct-current (DC) bias voltage to compensate for the bias drift; (

**d**) MZM output intensities of the diagnostic modulations. The abbreviations are defined as follows: distributed feedback laser (DFB laser); photodiode (PD); and microcontroller unit (MCU).

**Figure 3.**Field deployment and experimental setup of the 1 × 3 quantum key distribution (QKD) network system: (

**a**) Field deployment in a smart factory in South Korea. Map data: Google, © 2022 Maxar Technologies, TerraMetrics. The insets show the equipment for the transmitter (Alice) and receiver (Bob); (

**b**) Experimental setup. Time- and wavelength-division multiplexing were used to establish the 1 × 3 network. The lengths (in km) and losses (in dB) of the quantum channels were indicated. The abbreviations are defined as follows: tunable laser driver (TLD); circulator (CIR); beam splitter (BS); single-photon detector (SPD); phase modulator (PM); delay line (DL); polarization beam splitter (PBS); dense wavelength division multiplexer (DWDM); p-i-n photodiode (PINPD); variable optical attenuator (VOA); storage line (SL); and Faraday rotator mirror (FM).

**Figure 4.**Experimental results of the field test: (

**a**) Sifted key rates; (

**b**) Quantum bit error rates (QBERs); (

**c**) Extinction ratios (ERs) of the MZMs. The red, black and blue solid lines are the results of Alice 1–3, respectively.

Abbreviation | Description | Symbol | Description |
---|---|---|---|

BS | Beam splitter | ${\theta}_{drift}$ | Practical phase drift |

CIR | Circulator | ${\theta}_{drift}^{T}$ | $\mathrm{Theoretical}\mathrm{phase}\mathrm{drift},\left[{0}^{\xb0},{360}^{\xb0}\right)$ |

CW | Continuous-wave | ${p}_{i}$ | Practical detection probability |

DC | Direct-current | ${p}_{{T}_{i}}\left({\theta}_{drift}^{T}\right)$ | Theoretical detection probability |

DFB | Distributed feedback | $\mathrm{Err}\left({\theta}_{drift}^{T}\right)$ | $\mathrm{Error}\mathrm{between}{p}_{i}$$\mathrm{and}{p}_{{T}_{i}}\left({\theta}_{drift}^{T}\right)$ |

DL | Delay line | ${\theta}_{mod}$ | Phase modulation |

DWDM | Dense wavelength division multiplexer | ${\theta}_{mod}^{i}$ | Phase modulation of the i-th diagnostic pulse |

ER | Extinction ratio | ${I}_{out}$ | Output intensity |

FM | Faraday rotator mirror | ${L}_{in}$ | Insertion loss |

FPGA | Field-programmable gate array | ${I}_{in}$ | Input intensity |

MCU | Microcontroller unit | ${C}_{i}$ | Count for the i-th diagnostic pulse |

MZM | Mach–Zehnder modulator | ||

PBS | Polarization beam splitter | ||

PD | Photodiode | ||

PID | Proportional–integral–derivative | ||

PINPD | P-i-n photodiode | ||

PM | Phase modulator | ||

QBER | Quantum bit error rate | ||

QKD | Quantum key distribution | ||

SL | Storage line | ||

SPD | Single-photon detector | ||

TLD | Tunable laser driver | ||

VOA | Variable optical attenuator |

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**MDPI and ACS Style**

Park, C.-H.; Woo, M.-K.; Park, B.-K.; Jeon, S.-W.; Jung, H.; Kim, S.; Han, S.-W.
Experimental Demonstration of an Efficient Mach–Zehnder Modulator Bias Control for Quantum Key Distribution Systems. *Electronics* **2022**, *11*, 2207.
https://doi.org/10.3390/electronics11142207

**AMA Style**

Park C-H, Woo M-K, Park B-K, Jeon S-W, Jung H, Kim S, Han S-W.
Experimental Demonstration of an Efficient Mach–Zehnder Modulator Bias Control for Quantum Key Distribution Systems. *Electronics*. 2022; 11(14):2207.
https://doi.org/10.3390/electronics11142207

**Chicago/Turabian Style**

Park, Chang-Hoon, Min-Ki Woo, Byung-Kwon Park, Seung-Woo Jeon, Hojoong Jung, Sangin Kim, and Sang-Wook Han.
2022. "Experimental Demonstration of an Efficient Mach–Zehnder Modulator Bias Control for Quantum Key Distribution Systems" *Electronics* 11, no. 14: 2207.
https://doi.org/10.3390/electronics11142207