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Abstract
In chip-based quantum key distribution (QKD) systems, the non-ideal quantum state preparation due to the imperfect electro-optic phase modulators (EOPM) decreases the secret key rate and introduces potential vulnerabilities. We propose and implement an on-chip transmittance-invariant phase modulator (TIPM) to solve this problem. Simulated and experimental results show that TIPM can eliminate the correlation between phase, intensity, and polarization of quantum states caused by phase-dependent loss. The design can tolerate a significant fabrication mismatch and is universal to multi-material platforms. Furthermore, TIPM increases the modulation depth achievable by EOPMs in standard process design kit (PDK). The proposal of TIPM can improve the practical security and performance of the chip-based QKD systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) provides a feasible approach to realize unconditionally security key sharing between legitimate correspondents (usually called Alice and Bob) based on the fundamental principles of quantum mechanics [1,2]. To promote its practicality, integrated QKD, as a burgeoning technology, has been demonstrated to reduce the cost and size of QKD systems in recent years [3–8]. In integrated QKD systems, high-speed phase modulation is a crucial function because it is used not only as the phase modulator (PM) but also as the core component of the intensity modulator and the polarization modulator. It involves almost all the on-chip high-speed operations of quantum state preparation, including pulse generation, phase encoding or randomization, decoy-state preparation, polarization encoding, and basis selection [3–6]. As an important device to achieve high-speed modulation, electro-optic phase modulator (EOPM) has been implemented on many different material platforms such as the silicon on insulator (SOI) platform [3–7,9–11] and InP platform [4,12–14] in integrated QKD systems.
However there are always some potential safety vulnerabilities in QKD systems due to the imperfection of realistic devices [15,16] and the same is true for integrated systems. Since the different working mechanisms between the on-chip and fiber devices, the practical security of integrated QKD systems needs additional study [7,17]. For on-chip EOPMs. Both free-carrier plasma dispersion effect-based EOPM in the SOI platform and the quantum-confined Stark effect based or Franz–Keldysh effect (FKE) based EOPM in the InP platform suffer the phase-dependent loss despite their different modulation mechanisms [18,19]. In some chip-based polarization encoders, the polarization-dependent loss (PDL) is essentially original from the phase-dependent loss of EOPM [3,6]. Therefore, the presence of phase-dependent loss causes a correlation between the phase, intensity, and polarization of the modulated quantum states.
To deal with this problem, a polarization-loss-tolerant protocol was proposed by Li et al. [17,20] and has been empirically implemented [21]. In this protocol, Alice and Bob negotiate and discard the signal states with higher intensity with a postselection probability ${\rm 1}-P$. Since discarding signal states implies the loss of the secret key, more solutions are proposed from the device design level. Sibson et al. reduced phase-dependent loss by utilizing thermo-optic phase modulators (TOPMs) to provide a suitable bias [3]. But there is still a transmittance variation of about 20% for EOPMs. Dupuis et al. designed a structure called shift-and-dump phase shifter (SDPS) to eliminate phase-dependent loss [22] and similar solutions are also used on the InP platform to reduce the residual amplitude modulation [23]. Although this method can achieve perfect transmittance-invariant phase modulation, it modulates only two certain phase points and its structure must be redesigned when the demanded phases change. Semenenko et al. modulated a $\pi$ phase difference between adjacent pulses without transmittance variant through a Mach–Zehnder Interferometer (MZI) based on InP platform [8]. The limitation of this scheme is that it only modulates the $\pi$ phase. Additionally, a pass-block architecture was proposed to eliminate the variation originating from the phase-dependent loss in intensity modulation but cannot be used to phase modulation [24,25].
In QKD systems, except for the non-correlation modulation of signal states, more phase points (more than two points) are required for encoding or phase randomization. Therefore, none of these solutions can meet all requirements of quantum state preparation in chip-based QKD systems while eliminating phase-dependent loss.
Here, we propose a transmittance-invariant phase modulator (TIPM) to solve this problem. TIPM can eliminate the correlation between phase, intensity, and polarization of quantum states and thus can be used to solve the practical security problems of on-chip QKD systems due to phase-dependent loss. Meanwhile, TIPM breaks the restriction that only two-phase points can be reached in previous schemes and can achieve even $2 \pi$ continuous modulations, which enables more phase state preparation and perfect phase randomization for on-chip QKD systems and can be applied to more QKD protocols such as the continuous variable QKD [26,27]. TIPM has a considerable fabrication tolerance, and its design method of TIPM can be easily extended to other EOPMs and material platforms. The proposal of TIPM improves the practical security and performance of chip-based QKD systems.
2. Design
The basic design idea of the TIPM can be described as follows. For EOPM with the presence of phase-dependent loss, the phase and intensity of the modulated results are correlated. Another typical structure that also connects phase with intensity is the MZI. If we construct a proper structure so that the intensity of EOPM and MZI varying with phase could cancel each other, we can achieve a transmittance-invariant phase modulation.
Here we take carrier depletion phase modulator (CDPM) as an example to design our TIPM. The schematic diagram of TIPM is shown in Fig. 1(a). The input light field is assumed to be $\begin {bmatrix} 1 & 0 \end {bmatrix}^{\top }$ and then the output of TIPM can be written as
Fig. 1. Schematic diagram of TIPM. (a) The design structure and (b) the die picture of the TIPM. There is a TOPM and a CDPM at both two arms. The light field can be well balanced using the same devices on two arms. TOPM can be seen as an ideal PM that modulates phase without phase-dependent loss while CDPM leads to transmittance variation. The grating couplers (GC) are used for coupling input and output optical signals.
According to Eq. (2), the synchronization of electrical signals for the two CDPMs is important and we should ensure that the signals are loaded correctly during the duration of the optical pulse. On this basis, to achieve invariant transmittance, it must satisfy the following equation
3. Results
To simulate the output of TIPM, we first need to characterize the actual performance of the prepared CDPM. The modulated phase and transmittance of CDPM can be expressed as [3]
Fig. 2. The normalized transmission of MZI with two 50:50 BSs is measured by modulating the voltage on CDPM at one arm. (a) These data are measured by initializing the phase bias of TOPM at $\pi$. The circle and the corresponding color line refer to the experimental data and fitted line respectively. (b) and (c) shown the normalized transmittance $T(V)$ and the modulated phase $\varphi (V)$ of CDPM, respectively.
Based on these performance parameters of CDPM, the output of TIPM is defined. After optimizing the device parameters $\{\phi, \kappa \}$ and the working parameters $\{\varphi _a[\theta ], \varphi _b[\theta ]\}$, Fig. 3(a) presents the simulation results on a phase constellation diagram, which gives the normalized amplitude $A$ (its modular square is the transmittance) and phase $\theta$ through radial and angular distributions, respectively. The result shows that in addition to achieving the transmittance-invariant phase modulation under the device parameters $\{\phi = 0^{\circ }, \kappa = 0.5\}$, TIPM enables a $2\pi$ continuous phase modulation, which already exceeds the phase modulation depth of the CDPM (which is $\alpha$). We call this the phase-enhancement-characteristic (PEC) of TIPM. PEC can significantly broaden the use scenarios of existing EOPMs, such as the phase randomization and continuous variable QKD [26,29–31].
Fig. 3. The simulation results of TIPM. (a) The red line is the optimal continuous modulation output of TIPM with invariant amplitude $A=0.49$. The gray region corresponds to all output that TIPM can reach by traversing all $\varphi _a$ and $\varphi _b$ under the device parameters $\{\phi = 0^{\circ }, \kappa = 0.5\}$. (b) The lower part shows the maximum amplitude of TIPM to enable continuous phase modulation of $[0, 2\pi ]$ for $\phi \in [0,2\pi ]$ and $\kappa \in [0.01,0.99]$. The upper part shows the maximum amplitude of TIPM at each $\kappa$. The red and black markers correspond to each other and indicate the maximum amplitude 0.49. (c) the red points are the optimal discrete modulation output of TIPM with invariant amplitude $A=1.04$. (d) The lower part shows the maximum amplitude of TIPM to enable discrete phase modulation of {0, $\pi /2$, $\pi$, $3\pi /2$} for $\phi \in [0,2\pi ]$ and $\kappa \in [0.01,0.99]$. The upper part shows the maximum amplitude of TIPM at each $\kappa$ and indicate the maximum amplitude 1.04. The yellow circles in (b) and (d) correspond to the device parameters in (a) and (c) respectively. Both (a) and (c) are normalized to the 0-phase point (about 1.29 rad) to show the intrinsic relative phases modulated by TIPM.
The maximum amplitude of TIPM for more device parameters is shown in the lower part of Fig. 3(b). The blank part of this figure corresponds to the parameters that cannot enable the continuous phase modulation of $[0, 2\pi ]$. In previous schemes, the fabrication error of $\kappa$ (the BSR of BS) is the main factor why the device can not reach the designed performance [22]. Although the BS with an adjustable BSR can be utilized to alleviate this problem, it increases the complexity of the structure and needs additional control. Here, in these two device parameters, $\kappa$ is defined during manufacture and cannot be modified once the fabrication is finished, whereas $\phi$ can be adjusted flexibly during device running. As a result, we can modulate $\phi$ so that TIPM can still achieve the desired performance no matter what the actual value of $\kappa$ (from 0.01 to 0.99) is. The maximum normalized amplitude at the output of TIPM that can achieve by modifying $\phi$ under each $\kappa$ is displayed in the upper part of Fig. 3(b). This indicates a significant increase in tolerance for fabrication error. Furthermore, $\kappa = 0.5$ in Fig. 3(a) means that we do not need to design the BS with a special BSR, just the commonly used 50:50 BS, which is a standard device in PDK, is enough to construct TIPM.
We point out that TIPM enables transmittance-invariant modulation and PEC with the cost of transmittance. But the depletion of transmittance (about −6.2 dB in Fig. 3(a) is almost negligible for the transmitter due to the security assumption of QKD that the intensity of pulses must be attenuated to the single-photon level [31] (about −70 dBm for GHz systems). This depletion originates from the insufficient modulation depth of the EOPM. Therefore, if we substitute the CDPM here with other EOPMs with greater modulation depth such as CDPM based on other structures [32,33] or carrier injection phase modulator (CIPM) [5,22], the depletion can be reduced.
On the other hand, this depletion also comes from the excessive demands for PEC, which is the continuous phase modulation of $[0, 2\pi ]$ here. The depletion will be further reduced if there are not such high demands. In fact, for the encoding of discrete variable QKD, it is not necessary to modulate the phase continuously, just some discrete phase points are sufficient. For instance, $\left \{ 0,\pi \right \}$ for Differential Phase Shift (DPS) protocol [4], $\left \{ 0,\pi /2 ,\pi,3 \pi /2 \right \}$ for polarization and phase-encoded QKD [11,34–36]. Among these phase points, $3\pi /2$ exceeds the modulation depth of CDPM, so taking advantage of the PEC is still necessary. Aiming the discrete phase modulation of {0, $\pi /2$, $\pi$, $3\pi /2$}, the optimized results are shown in Fig. 3(c) and Fig. 3(d). As we speculated, the depletion of transmittance is greatly reduced while maintaining the tolerance of $\kappa$. Since the modulation principle of CDPM we mentioned above, the amplitude here is even greater than 1.
A fascinating case in our method is when TIPM is used for $\left \{ 0,\pi \right \}$, the output of TIPM is $b_{out}(\varphi _{a}, \varphi _{b})=\left [ \sqrt {T(\varphi _{a})} e^{i \varphi _{a}} - \sqrt {T(\varphi _{b})} e^{i \varphi _{b}}\right ]\cdot i/2$. Obviously, TIPM can always achieve $\left \{ 0,\pi \right \}$ modulation whatever $\varphi _{a}$ and $\varphi _{b}$ is. $0$ and $\pi$ corresponds to $b_{out}(\varphi _{a}, \varphi _{b})$ and $b_{out}(\varphi _{b}, \varphi _{a})$, respectively, and, more particularly, to $b_{out}(0, \varphi )$ and $b_{out}(\varphi,0 )$. That is
Another feature worth mentioning is the capability to modulate the intensity as well as the phase at the same time. This is because TIPM is essentially a MZI, a typical intensity modulation structure. Based on this feature, a TIPM can be used directly as the transmitter of some QKD protocols such as DPS and phase-encoding BB84. Considering the same device parameters in continuous and discrete modulation, TIPM can be used as an all-in-one device that allows different modulations by simply changing the working parameters without re-fabrication [22].
In practical applications, the modulation phase of CDPM also depends on its driving voltage. As shown in Fig. 2, it is about $\pi /2$ at 7.2 V. To better demonstrate TIPM, we cascaded two CDPMs, making it possible to reach the $\pi$ phase for each arm. From our simulation, the maximum transmittance corresponding to a BSR is precisely 0.5 ($\kappa = 0.5$). Hence we build TIPM simply using the 50:50 BS in standard PDK from CUMEC for our experiments.
The experiment setup is shown in Fig. 4. An external fiber interferometer is used to verify the modulated phase of TIPM. PC1 is used to match the polarization between the single-mode fiber and the on-chip GC to maximum the coupling efficiency, which is 4.5 dB here. PC2 and VOA, TDL, PBS1 and PBS2, and PMF BS are employed to ensure the consistency of intensity, delay, and polarization of the interference light in two arms, respectively. PC3 is used here due to the fiber PM device is polarization-dependent. PD1 and PD2 (PD3) measure the transmittance and modulated phase of TIPM, respectively. The electrical signals on CDPMs are supplied by an arbitrary waveform generator (Tektronix AWG5208) and the amplifiers (Conquer KJ-RF-10), and the VOA and fiber PM are controlled by voltage sources (Rigol DP832A). We verify the ability of TIPM for $\left \{ 0,\pi \right \}$ and $\left \{ 0,\pi /2 ,\pi,3 \pi /2 \right \}$ discrete modulation, and [0, $2\pi$] continuous modulation in Fig. 5. Here we test only 8 points in the continuous modulation experiment, which is enough to verify our design. Our results match well with our simulation.
Fig. 4. The setups to verify the modulation of TIPM. LD: laser diode at 1550nm; SMF BS: the single-mode fiber beam splitter with BSR of 50:50; PMF BS: the polarization-maintaining fiber beam splitter with BSR of 50:50; PC: polarization controller; VOA: variable optical attenuator; TDL: tunable optical delay line; PM: fiber phase modulator; PBS: polarization beam splitter; PD: optical power detector. The black and blue lines correspond to the single-mode fiber and polarization-maintaining fiber, respectively.
Fig. 5. The red asterisks (or line) and the blue triangles are the simulation and experiment results of TIPM, respectively, for (a) {0, $\pi$} and (b) {0, $\pi /2$, $\pi$, $3\pi /2$} discrete phase modulation, and (c) [0, $2\pi$] continuous phase modulation. All data are normalized to their respective 0-phase points.
4. Simulation of secret key rate
In the presence of PDL, which causes the inconsistent intensity of different signal states, the security of the polarization encoding system has been studied in Ref. [17,21]. To balance out the average photon number of each signal state, their randomly discard a portion of the signals with higher intensity at postselection probability $1-P$, which depends on the ratio of the intensities of these quantum states. In phase encoding systems, there are some differences from polarization encoding systems. If we denote a single (null) photon in the $i$th pulse ($i = 0,1$ for early and late pulse, respectively) of the time-bin as $\left | 1 \right \rangle _{i}$($\left | 0 \right \rangle _{i}$), the encoded states can be expressed as $\left | \Psi (\theta ) \right \rangle = \left ( \left | 1 \right \rangle _{0} \left | 0 \right \rangle _{1} + e^{i\theta }\sqrt {T(\theta )}\left | 0 \right \rangle _{0} \left | 1 \right \rangle _{1}\right )/\sqrt {2}$ where $\theta$ is the encoded phase and $T(\theta )$ is deduced from Eq. (4). Hence the presence of phase-dependent loss makes $\left | \Psi \right (0) \rangle$ and $\left | \Psi \right (\pi /2) \rangle$ (or $\left | \Psi \right (\pi ) \rangle$ and $\left | \Psi \right (3\pi /2) \rangle$) no longer orthogonal. In this case, the security analysis of the original BB84 protocol is no longer appropriate. To ensure the security of the protocol under such conditions, we analyze the secret key rate of the BB84 protocol under coherent light sources with the vacuum + weak decoy-state method by the scheme proposed by Yin et al. [37], which enables us to generate secure keys without characterizing the error encodes by exploiting the statistics of mismatched-basis cases.
Before calculating the secret key rate, we first compare the phase-dependent loss for PMs from different foundries and platforms, that is, CDPMs from CUMEC and IMEC [3,38], CIPM from IME [5], and the FKE-based PM on the InP platform [23] in Table 1. To make a more intuitive comparison, the phase-dependent loss is represented by the normalized transmittance at the modulated phase $\left \{0, \pi /2, \pi \right \}$. $3\pi /2$ is not included here because the modulation depth of some PMs is not enough and only three phase-encoding states are needed here to calculate the secure key rate [37], that is, $\left \{0,\pi \right \}$ for code basis and $\left \{\pi /2 \right \}$ for test basis. This table shows that the InP-PM has a much higher phase-dependent loss than that in SOI due to the FKE, which gives rise to more electroabsorption while causing electrorefraction [19].
Table 1. The phase-dependent loss for different PMs
The final secure key rate at the asymptotic case is given by
where $q$ is the efficiency of protocol (which we set to $q \approx 1$ at the asymptotic case [39]), $Q_{1}^{L}$ is the lower bound for the single photon response rate at the code basis, $H(x) = -xlog_{2}x-(1-x)log_{2}(1-x)$ is the Shannon’s binary entropy function, $e_{p}^{U}$ is the upper bound for the single photon phase error rate at the code basis, $Q_s$ is the gain of the signal states at the code basis, $f_{EC}$ is the error correction efficiency (we set $f_{EC}=1.16$ here [40]), and $E_{s}$ is the overall quantum bit error rate of the signal states.We calculate the secret key rate of these PMs in Fig. 6 by using the overall transmittance of Bob’s detection apparatus $\eta _{Bob} = 0.2$, the fiber channal loss $\alpha =0.2$ dB/km, and the dark count rate $Y_{0}= 2 \times 10^{-5}$. The particulars of calculation can be found in the literature [37]. Thanks to Yin’s scheme, there is no significant decrease in the key rate of InP-PM even with such large phase-dependent loss. Since TIPM cancels phase-dependent loss completely and enables the $3\pi /2$ phase modulation due to the PEC, the secure key rate of TIPM is calculated by the standard $\left \{0, \pi /2 ,\pi,3\pi /2 \right \}$ phase-encoding BB84 protocol. In contrast, the secure key rate can be largely increased when these PMs in Table 1 are substituted with our TIPM especially in the long distance case.
Fig. 6. The secret key rate as the function of communication distance using different PMs with the presence of phase-dependent loss. The blue line denotes the TIPM that eliminates the phase-dependent loss. The yellow, purple, and red lines indicate three plasma dispersion effect-based PMs in the SOI platform: the CDPM1 from CUMEC, the CDPM2 from IMEC, and the CIPM from IME. The green line denotes the InP platform’s Franz–Keldysh effect PM.
5. Conclusion
In this work, we propose and demonstrate a TIPM to eliminate the correlation between phase, polarization, and intensity of quantum states and it is a benefit to strengthen the practical security of chip-based QKD systems. With this foundation, TIPM increases the modulation depth achievable by the existing EOPMs, thus enabling multi-point discrete (more than two points) or even $[0, 2\pi ]$ continuous phase modulation, which is impossible in previous solutions. This perfectly satisfies the demands of practical QKD systems and dramatically broadens the application scenario of EOPMs with insufficient modulation depth. Additionally, TIPM has a considerable fabrication tolerance and its implementation does not require the design of special components, just that in standard PDK is sufficient. The design method of TIPM can be easily extended to solve the same problem in other types of EOPM such as the CIPM and the FKE-based PM. Furthermore, its capability to simultaneously modulate intensity and phase facilitates TIPM to be an all-in-one device, which can be directly used as a simple transmitter for some QKD protocols. In conclusion, TIPM can significantly benefit a more secure and higher performance chip-based QKD system.
Funding
National Key Research and Development Program of China (2018YFA0306400); National Natural Science Foundation of China (61627820, 61961136004, 62105318, 62171424, 62271463); Anhui Initiative in Quantum Information Technologies (AHY030000); China Postdoctoral Science Foundation (2021M693098); Prospect and Key Core Technology Projects of Jiangsu provincial key R & D Program (BE2022071).
Acknowledgments
We would like to acknowledge Xin Biao Xu for useful discussions. This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
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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