1. Introduction

Continuous-variable quantum key distribution (CV- QKD), dramatically developed over the past few years, is known as the impressive alternative to traditional discrete variable (DV) QKD [1]. Using usual telecom technologies instead of expensive and low-efficiency photon-counting-based technologies while producing a higher key rate has attracted considerable attention. Among the renowned CV-QKD protocols, Gaussian-modulated coherent state (GMCS) QKD, proposed by Grosshan and Grangier [2], has been the subject of much investigation. In this protocol, the emitter, represented as Alice, selects two random numbers $x$ and $p$, drawn from a Gaussian distribution:

Accordingly, the quantum signal and LO are constructed from the same laser at Alice’s side and jointly sent to Bob. Unfortunately, transmitting LO (TLO), introduces vulnerabilities that could compromise security. For example, it has been demonstrated that Eve can simulate the intensity fluctuations of the TLO concealing her collective attack [4]. Additionally, Diamanti et al. [5] have indicated Eve’s ability to manipulate the clock recovery process by tampering with the LO, enabling a successful calibration attack. They have emphasized that, for security, real-time measurement of the shot-noise is imperative. In addition to these security concerns, the practical implementation of the detection process in such schemes, relies on the highly intense LO, leading to significant challenges in preparation, particularly in long-distance CV-QKD, and imposing limitations on the protocol’s speed [6]. Moreover, the huge difference in optical power between the quantum signal and LO exacerbates the risk of cross-talk, introducing practical complexities in multiplexing and demultiplexing arrangements.

These drawbacks have led to renewed attempts to locally construct a genuine LO [7–9]. In the locally local oscillator (LLO)-based schemes, quantum signal and LO are generated from distinct laser sources. In these protocols, a paramount challenge arises in establishing a reliable phase reference between the input signal and the LLO. Owing to the inherent uncorrelated noise sources associated with the two lasers, the adoption of phase and frequency synchronization methods seems unavoidable.

In these schemes, to achieve a stable and well-known phase and frequency between signal and LO, weak coherent states, instead of employing a bright LO, are sent to Bob. These weak pulses, termed ‘the reference signals’ play a central role in accurately recovering the phase and frequency of Alice’s LO at Bob’s side. The relatively low intensity of these pulses simplifies the multiplexing and demultiplexing procedures compared to schemes using TLO. Time-division multiplexing or sequential LLO [7–9] is the first attempt in this field, but the saturation of the homodyne detection and the time difference between quantum and reference signals degraded the scheme’s performance. Subsequently, novel approaches like delay-line and displacement self-coherent LLOs were introduced [6]. In these schemes, both quantum and reference signals are simultaneously generated from the same optical source wavefront, eliminating concerns about drift excess noise experienced in the sequential LLO scheme. Nevertheless, using complex algorithms and digital signal processing (DSP) to compensate for the phase estimation error is inevitable [10]. Further refinements include time and polarization multiplexing [11] as well as frequency and polarization multiplexing pilot-assisted feed-forward, have been proposed in the most advanced versions [12]. The combination of both multiplexing methods, polarization, and frequency [13], has demonstrated significant improvements in mitigating cross-talk arising from the finite extinction ratios of multiplexer and demultiplexer devices. Then, many efforts have been directed toward enhancing and optimizing the performance of these LLO schemes, including the incorporation of amplifiers [14,15], the utilization of non-Gaussian operations [16], and the development of more efficient phase-tracking algorithms [17].

This paper sought to assess the feasibility of employing the optical injection phase-locked loop (OIPLL) for the phase and frequency recovery of the reference signal on Bob’s side. It is attempted to demonstrate that the noise imposed by OIPLL is low in comparison with the noise accumulated during the phase estimation process in the current LLO schemes. Benefitting from the attributes of OIPLL, notably in amplification and noise suppression, the OIPLL-based LLO scheme exhibits the potential for achieving higher key rates over extended communication distances, surpassing certain commonly employed LLO schemes. The specific focus here pertains to the key rate calculations tailored to the no-switching variant of GMSC CV-QKD under collective attack [3].

The remaining sections of the current paper are organized as follows: Section 2 explains the schematic set-up of the proposed scheme. Section 3 offers a succinct overview of the fundamental principles underlying OIPLL, accompanied by the presentation of an analytical model describing the steady-state behavior of the laser under OIPLL. This section further investigates the noise characteristics associated with this method as well as the phase offset which may persist during the carrier recovery process. In Section 4, we review significant noise sources that impact conventional LLO schemes, subsequently comparing these with the noise profile observed in OIPLL-based LLO. Moreover, we calculate the key rate of this protocol under the collective attack and then compare it with previously established LLO schemes. All results are discussed in Section 5. To provide additional clarity, Section 6 is devoted to the examination of security concerns. Finally, conclusions are summarized in Section 7.

2. Proposed scheme

The proposed setup for polarization-multiplexing and OIPLL CV-QKD is plotted in Fig. 1. In this setup, the continuous wave (CW) laser beam with narrow linewidth ($<100$ Hz linewidth NKT photonics [18]), is used as the transmitter source, and the master laser (ML) for optical injection locking (OIL). The ML output is split into two paths: one for encoding quantum information (red path) and the other for sharing the phase reference (blue path). An unbalanced beamsplitter (BS1) with an intensity ratio of 99:1 is used for this purpose. In the quantum path, an amplitude modulator (AM1) generates pulses which are fed into another AM2 and the phase modulator (PM1) to encode Gaussian information on both quadratures. The encoded information depends on the voltage injected into the AM2 and PM1 by an arbitrary wave generator (AWG ). To operate at the quantum level, the intensity of the quantum signal undergoes attenuation through two variable optical attenuators (VOA1 and VOA3). In the classical path, the intensity of the reference signal is adjusted using VOA2 and VOA3 such that at Bob’s LLO input, the injection optical power is -57 dBm (with an injection ratio of -60 dB). The reference signal is crucial for regenerating Alice’s phase reference frame in Bob’s detection system.

 

Fig. 1. Schematic configuration of the CV-QKD based on OIPLL and polarization multiplexing techniques. CV-QKD: Continuous-variable quantum key distribution; OIPLL: Optical injection phase-locked loop; ML: master laser; SL: slave laser; AM: amplitude modulator; PM: Phase modulator; AWG: Arbitrary wave generator; VOA: variable optical attenuator; PBS: polarizing beamsplitter; PBC: polarizing beam combiner; EPC: electronic polarization controller; BS: beamsplitter; LPF: low pass filter; PID: proportional–integral–derivative controller; BPD: balanced photodetector; PC: polarization controller HD: heterodyne detection system.

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Quantum and reference signals are orthogonally multiplexed and jointly transmitted to Bob through a polarizing beam combiner (PBC). In Bob’s system, the polarization of the input quantum and reference signals is controlled by an electric polarization controller (EPC). Subsequently, a polarizing beamsplitter (PBS) demultiplexes the quantum and reference signals. The reference signal is split using a unbalanced BS2. A significant portion of this signal, after polarization correction, is directed into the slave laser (SL) via an optical circulator for optical injection locking (OIL), while the remaining portion is used for optical phase-locked loop (OPLL). The phase difference between SL and reference signal is significantly reduced by OIL. The output of SL and the other portion of the reference signal are both directed to a balanced photodetector (BPD) to generate an error signal. This signal includes information about the phase difference between the incoming reference signal and SL. After eliminating the high-frequency component through a low-pass filter (LPF), the BPD output is directed into a proportional-integral-derivative (PID) controller. The output of this controller serves as the error signal and is fed back into SL to synchronize its phase and frequency with the reference signal. The major part of SL’s output is separated by an unbalanced beamsplitter (BS3). After undergoing polarization correction, it is jointly transmitted with the quantum signal to the heterodyne detector (HD).

3. Theoretical model

OIL is a technique used to synchronize the phase and frequency of a semiconductor laser with the help of a stable laser known as the ’master.’ During this process, a laser with low coherency and stability characteristics commonly called the ‘slave’, will be forced to lase at the ML frequency provided that the frequency difference between lasers ($\Delta \omega _{inj}$) falls within a specific range known as the ’locking range’ ($\Delta \omega _{LR}$). This implies that the SL has the capability of tracking the phase and frequency drifts of the ML, albeit with a constant phase offset. The locking range is obtained from the steady-state solutions of the governing equations. Various studies have experimentally assessed the efficacy of OIL in laser linewidth reduction [19,20]. In contrast with other phase and frequency synchronization methods, OIL is more cost-effective and has more power consumption efficiency; however, it has some operational problems such as narrow locking-range in the low injection ratio regime (e.g., around MHz at an injection ratio of -40 dB) [21–23]. In other words, by a sudden drift in ML frequency due to the noise, $\Delta \omega _{inj}$ drifts outside the locking-range, leading to restrictions in the application of this method [21,24–26]. To improve the capability of phase and frequency tracking in OIL, using an OPLL seems unavoidable.

OPLL is an optoelectronic feedback loop that can synchronize the local signal (SL) to the phase and frequency of the incoming signal (ML). The feedback loop consists of three pivotal elements: a BPD, a LPF and a PID controller. The phase and frequency difference between two lasers are detected at the BPD. The output signal of the BPD includes the beat term at a frequency corresponding to the frequency difference between ML and SL. The LPF and PID controller collaboratively undertake the elimination of high-frequency component of the error signal, and process the error signal, respectively. The refined error signal is then injected into the SL to force it to track the phase and frequency of the ML. The effectiveness of the OPLL is significantly influenced by the linewidth of the ML involved.

Seed et al. [21] introduced a noteworthy approach that combines two methods to reduce the drawbacks of both OIL and OPLL, resulting in what they termed the optical injection phase locked loop (OIPLL). In the OIPLL technique, OIL reduces the wide bandwidth noise, while OPLL suppresses the residual low-phase noise, enhancing the tracking capacity of OIL. It’s worth noting that OPLL demands rapid electronic components and strongly depends on the bandwidth of the loop, aspects that OIPLL notably alleviates [21,27]. The significant role of OIL in reducing relative intensity noise (RIN), nonlinearities in the dynamics of the lasers, and chirp has been investigated in almost high injection ratios [28]. Recently, some studies have utilized this method at a low injection ratio of -55 dBm in the deep-space communication [18,29].

This section presents a physical approach for describing the behavior of the laser under OIPLL, considering the essential parameters of the SL, including phase, photon number, and carrier number. Although the behavior of the laser under OIL is frequently investigated [22,24–26], to the best of our knowledge, no study has so far described its behavior with OIPLL through the framework of laser rate equations. This section, therefore, incorporates the impact of OPLL within rate equations and proceeds to analyze the noise characteristics of OIPLL utilizing the Langevin noise model.

We chose the nonlinear ordinary differential laser rate equations as the governing equations [30,31] adapted for OIL [22,23]. To demonstrate that these equations can provide appropriate information about laser behavior, in Supplement 1 Section 1, we showed that these equations were resulted from the quantum Langevin-Heisenberg approach, and in Section 2, we investigated the capability of OIPLL laser rate equations in distinguishing injected quantum state. In this study, the adapted rate equations for OIL were changed for OIPLL as follows:

In Eqs. (2), (3) and (4), $S_{inj}$ and $S$ are the photon numbers of the ML (before injection), and SL. In addition, $\phi$ and $N$ represents the phase difference between ML and SL, and the carrier number of the SL, respectively. Further, $\Delta \omega _{inj}=\omega _{ML}-\omega _{SL}$ is the difference between two free-running state frequencies of ML and SL. Moreover, $\gamma _N$ and $\gamma _P$ are the carrier recombination and photon decay rates. Likewise, $g$, $J$, $N_{th}$, and $\alpha$ denote the linear gain coefficient, electric current density, carrier number at threshold, and linewidth enhancement factor, respectively [30]. Additionally, $K_{inj}= 1/\tau _{in}\big ((1-r_0^2)r_{inj}/r_0\big )$ represents the injection strength, where $\tau _{in}$, $r_0$, and $r_{inj}$ are the round-trip time, reflection coefficient, and rate of the injected electric field, respectively [22]; In these equations, the SL internal photon number is related to the electric field (A) as $S=A^2$. The parameters used in this paper were listed in Supplement 1 Table S1.

$f(S_{inj},S,\phi )$, is the error signal injected into the SL to synchronize it to the phase and frequency of the incoming ML signal. The dynamic of the feedback ($f(S_{inj},S,\phi )$) is determined by the LPF, PID controller, sensitivity of the BPD, and gain of the amplifier. In Supplement 1 Section 3, the steady-state solutions of Eqs. (2), (3) and (4) were calculated.

Random recombination and generation of the photons and carriers in the semiconductor lasers lead to fluctuations in the output intensity, phase, and frequency. The noise in OIL semiconductor lasers was investigated using different approaches such as quantum approach [32], or introducing Langevin sources [33] into rate equations, while considering the effect of partition noise [23,31]. For the OIPLL system, it is possible to linearize Eqs. (2)–(4) to determine the spectral density of physical quantities. In this study, we adopted the second approach and calculated RIN, phase noise, and frequency noise in semiconductor OIPLL lasers. The Langevin noise sources are based on the shot-noise model [31,33,34]. From the Langevin model perspective, noise is resulted from the discrete nature of carriers and photons and is directly proportional to the average rate of their flows. To determine the RIN, phase noise , and frequency noise, the spectral density of the carrier number, photon number, and phase must be calculated. Section 4 from Supplement 1 was devoted to the details of related calculations.

4. Performance analysis of CV-QKD with LLO generated by OIPLL

This section focuses on investigation of the secure key rate in CV-QKD employing a LLO generated by OIPLL. Our primary objective is to evaluate the secure key rate while considering various sources of noise and subsequently compare it with existing LLO-based CV-QKD schemes. In CV-QKD, providing a reliable phase reference between Alice and Bob is one of the most important challenges. In TLO schemes, this happens by constructing LO and quantum signal from Alice’s optical source. Then, this LO is sent to Bob for coherent detection [2]. However, the transmission of a potent optical signal alongside a delicate quantum signal introduces the risk of contaminating the quantum signal. Although sending such a bright coherent state seems unavoidable for coherent detection in the shot-noise limit, it can produce nonlinear effects such as spontaneous anti-Stokes Raman scattering (SASRS) [35,36], Brillouin scattering, and four-wave mixing (FWM) [35]. Further, owing to the finite extinction ratios of multiplexing components, some classical signal photons can inadvertently leak into the quantum signal during the wavelength and frequency multiplexing. These nonlinear effects and leakage are proportional to the power of the LO, consequently complicating the multiplexing procedure. By proposing LLO schemes, most of the reported problems were solved at the expense of new challenges. In the following sections, we initiate by providing an overview of conventional LLO schemes, highlighting crucial noise sources that have constrained their performance. Subsequently, our attention turns to the OIPLL-based LLO CV-QKD scheme, where we theoretically examine the effective noise sources and conduct a comparative assessment against established LLO-based counterparts.

4.1 Noise analysis in conventional CV-QKD with LLO

The most important challenge in the LLO CV-QKD schemes is the excess noise derived from the phase estimation process [6,7,9]. In these schemes, Alice frequently sends some weaker reference pulses relative to the LO but brighter than the quantum signal. Bob measures the quadratures of the reference signal by coherent detection utilizing his own LO. The mean amplitude value of the reference signal is publicly announced, enabling Bob to estimate the phase difference between the two reference frames. Subsequently, during the reverse reconciliation process, this phase difference is rectified. However, imperfections during the phase estimation process lead to the error, which can be expressed as follows [6]:

The total noise imposed on the quantum signal in the LLO scheme is as follows:

In Supplement 1 Section 6, a relatively complete description about the effective conventional noise sources in LLO CV-QKD schemes has been provided.

4.2 Noise analysis in OIPLL-based LLO CV-QKD

In previous LLO schemes, due to the phase estimation process and non-idealities of the devices, a considerable level of noise was dictated on the LLO. For instance, in a recent paper [39], the excess noise attributed to the phase estimation was 0.028 shot-noise unit (SNU) (comprising 0.022 SNU due to error excess noise and 0.006 SNU due to drift excess noise), constituting $34.7{\%}$ of the total excess noise. This paper has used the time multiplexing scheme with a simplified set-up. On the other hand, Wang et al. [40] achieved a mean value of $\xi _{tot}=0.022$ SNU SNU while considering the finite-size key effect, using frequency-polarization multiplexing, along with relatively high-speed DSP equipped with high computational power.This protocol necessitates a relatively intricate phase-tracking algorithm for correcting the phase difference between Alice and Bob’s frames. Accordingly, phase recovery is a challenging problem in LLO CV-QKD, often requiring complex DSP and sample-by-sample phase tracking algorithms. however, OIL can significantly simplify this process.

In the OIPLL-based scheme, phase and frequency adjustments can be made nearly in real-time, and in spite of previous LLO schemes, complex digital phase tracking becomes unnecessary. Additionally, in the OIPLL-based LLO, LLO is commonly locked with a deterministic offset with the ML phase, which can be corrected by Alice. Owing to the low phase noise resulting from the OIL and OPLL, the phase difference between Alice and Bob’s frames [41] can be more accurately compensated, even with a low-intensity reference signal. The residual uncompensated phase resulting from the noise and imperfections in the system can be significantly suppressed by choosing an appropriate electronic feedback loop and employing more stable ML and SL. In this scheme, within the locking range OIL functions similar to a linear amplifier [42] for the injection ML signal [19]. In this perspective, according to the seminal work of Haus et al. [32], the noise introduced by OIL is twice the magnitude dictated by the Heisenberg uncertainty principle, specifically:

On the other hand, some studies investigated the quantum noise performance of the SL under OIPLL, i.e., $\langle (\Delta S_{SL})^2\rangle \langle (\Delta \phi _{SL})^2\rangle \geqslant \frac {1}{4}$ as a laser source. The results of this investigation must align with the quantum noise limit (QNL) set by the Heisenberg uncertainty principle [43,42]. Notably, it has been demonstrated that in an especial conditions characterized by high optical injection levels and substantial injected electric currents, particularly when $\alpha =0$ [43] and internal loss are negligible [32], an ideal laser, when operating at high frequencies [31,42], the product of photon number and phase variances converge toward the noise floor, where the residual noise is fundamentally of a quantum nature. They additionally proved that however the noise of the SL approaches the QNL, never go beyond it [43]. In this research, we adopt the second method; nonetheless, it is easy to show the validity of the first approach.

The real role of OIPLL in the LLO scheme is to change the phase estimation noise into the RIN and phase noise which are negligible in comparison with those resulting from the phase recovery process via DSP. OIL is well-known as a cost-efficient method for reducing the laser linewidth. By this method, a laser with a wide linewidth, g. e., MHz linewidth, can be turned into a laser with a kHz linewidth [19]. Additionally, more reduction is attainable with the appropriate feedback loop.

In brief, the most important difference between an OIPLL-based scheme and the other LLOs is the reduction of the phase recovery noise at the expense of using narrow-band linewidth lasers. In LLO schemes, using a kHz linewidth laser is conventional, while in OIPLL LLO, it is preferable to work with very narrow linewidth lasers. Supplement 1 Section 7 is devoted to the comparison between OIL, OPLL, and OIPLL as well as addressing stability issues.

5. Numerical results and discussion

In this section, we present the results obtained from applying the OIPLL technique in LLO-based CV-QKD. We start with the locking-range for OIL and OIPLL which are illustrated in Fig. 2. Fig. 2(a) shows the locking-range of an OIL laser, while Figs. 2(b) and 2(c) depict the locking-range of an OIPLL laser with different feedback gains. As shown, OIPLL extends the locking-range as reported in experimental studies [21,44]. Based on the data, in the case of low injection ratios (e. g., $-31$ dB), the locking-range in OIL is 0.7 GHz. However, with OIPLL, the locking-range expands and can reach up to 24 GHz, based on the chosen feedback gain and injected electrical current (as depicted in Fig. 2(c)). These results have been verified by experimental research, reporting more than 26 GHz [21]. Fig. 2(d) shows the locking range concerning different injected electric currents ($2I_{th}$ and $3I_{th}$). It is worth noting that in low injection current, the probability of losing locking increases due to temperature drift and noise. As a result, controlling temperature seems necessary [45]. In our simulations, we have omitted the consideration of delay-times associated with the OPLL, primarily due to the large coherence time of the ML, and the capability of OIL in the following fast fluctuations, as experimental studies showed that OIPLL does not need extremely fast electronics [21].

 

Fig. 2. Locking-range with respect to different injection ratios: (a) optical injection locking without feedback loop (OIL). The dashed line presents the unstable locking zone. (b) and (c) Optical injection phase locked loop (OIPLL) with different feedback gains. As shown, the locking-range is extended in terms of the feedback gain. In these figures, the output power and the injection current are 30 mW and $3I_{th}$, respectively. In (c), locking range is 24 GHz at $-31$ dB. Experimental results adapted from [21] reported $>26$ GHz. (d) Locking range for two different injected currents ($I=\{2I_{th},3I_{th}\}$).

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Next, we calculate the noise of the OIPLL laser. The results of the RIN of the SL for different injection ratios are depicted in Fig. 3. As shown, the noise suppression depends on the injection ratio. For the low values of injection ratio, the effect of injected coherent photons of the ML, in comparison with amplified spontaneous emmision in salve, is insignificant and SL behaves similarly to the free running state. Having a peak in RIN has been reported in previous theoretical and experimental research (e.g., [23]). This peak happens in the resonance frequency of the photon-carrier number of the SL system function.

 

Fig. 3. RIN and phase offset in OIPLL laser: (a) RIN with respect to the different injection ratios, when the phase offset is zero. (b) RIN for two injected electric currents, when the injection ratio and the phase offset are -60 dB and zero, respectively. (c) RIN for various injection ratios when the phase offset is $-\frac {\textstyle \pi }{\textstyle 4}$. (d) Phase offset of the SL in the steady-state during the OIPLL for different injection ratios versus detuning frequency. SL: salve laser RIN: relative intensity noise; OIPLL: Optical injection phase locked loop, and $R_{inj}\equiv P_{inj}/P$.

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Another parameter that affects the RIN is the offset phase between ML and SL. The results of the phase offsets for values of zero and $-\pi /4$ are plotted in Figs. 3(a) and 3(c), respectively. The results highlight that RIN can surpass that of the free-running state for larger offset angles, especially at the extremes of the locking range. Fig. 3(d) illustrates the accumulated phase offset for different injection ratios versus the detuning frequency. The error phase in OIL is conventional but can be reduced by OPLL [19]. As shown, the zero offset can be provided by tuning the feedback gain, injection level, and detuning frequency. For having zero phase offset, nonzero detuning is unavoidable due to the influence of the linewidth enhancement factor ($\alpha$) inherent to semiconductor lasers [30].

Despite the RIN that does not change considerably in the low injection regime, the phase and frequency noises in the OIPLL state are significantly reduced. The results related to the optimum feedback gain with different injection ratios, along with a summary of some experimental results [20,46] are depicted in Figs. 4(a) and 4(b). However, the difference between experimental results relies on the applied set-up. For example, the linewidth of the ML is $<100$ Hz in one study [46], while it is about 5 kHz in another [20].

 

Fig. 4. Phase and frequency noises of an OIPLL laser versus frequency for different injection ratios; Dash lines are related to the experimental studies [20,46]. (a) Phase noise at low (a1) and high (a2) frequency offsets, respectively. As [20] reported the noise bump (a1) is due to the OIPLL electronic feedback. Also, noise peak happens at the relaxation oscillation frequency of the SL, and depends strongly on the laser parameters. Additionally, with increasing the injection ratio, noise peak decreases. (b) Frequency noise of an OIPLL laser with respect to different injection ratios; OIPLL: optical injection phase locked loop; RIN: relative intensity noise; OIPLL: Optical injection phase locked loop, and $R_{inj}\equiv P_{inj}/P$.

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Jumping in the phase noise (Fig. 4(a2)) was theoretically and experimentally shown in OIL near the resonance frequency [47]. Noise suppression exhibits a profound dependence on the injection ratio and phase offset, similar to the behavior observed in RIN, and is improved in the high injection ratios [48]. Far away from the resonance frequency of the transfer function (the poles of the transfer function), both phase noise and RIN are extremely low [49]. The performance of the OIPLL can be optimized by adjusting the poles and zeros of the transfer function of the feedback loop.

Although OIPLL can reduce the noise of the SL, it cannot surpass the inherent quantum noise limit predicted by the Heisenberg uncertainty principle on the laser noise as previously demonstrated by studies [32,43]. In Fig. 5, the product of SL’s output photon number and phase within the bandwidth of the of the SL in high frequency regime where it is anticipated that the laser approaches the QNL [31] is depicted, while considering different injection ratios. As depicted in the figure, the variances of SL noise approach the QNL, but they never surpass it. Even if both the SL and ML are QNL lasers, the noise of the ML will be amplified through OIPLL. This is because, typically, the product of the SL’s noise variances and that of the ML is greater than the quantum noise limit, as observed in previous studies [42].

 

Fig. 5. The product of photon number and phase variances of the SL.

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5.1 Noise in OIPLL based LLO

This subsection evaluates the secure key rate for OIPLL CV-QKD in the asymptotic regime under collective attack which is given by the Devetak–Winter formula [50,51]. Then the results are compared with pre-existing LLO schemes. More details about the secure key rate calculations can be found in Supplement 1 Section 8. The parameters adopted for our simulations are outlined in Supplement 1 Table S2. In this section, four LLO schemes, including time multiplexing (sequential and delay-line) [6,9], time (delay-line)-polarization multiplexing [52], and polarization multiplexing-OIPLL have been compared. The results of these evaluations are summarized in Supplement 1 Table S3. All the noise sources subject to investigation have been detailed in Section 4.1, with further elucidation provided in Supplement 1 Sections 5 and 6. For a fair comparison, we used the same parameters and formulas which may be different from the original paper.

The secure key rate values for these schemes are plotted in Fig. 6(a). Based on the data, the achievable distance by OIPLL is more noticeable. This is predictable because, in OIPLL-based LLO, we do not confront the estimation phase excess noise in spite of other LLO schemes. In the OIPLL scheme, the key point is that the noise associated with the estimation process is replaced by the RIN and the phase noise of the OIPLL which are negligible in comparison with other noise sources in the set-up. Additionally, owing to the little residual phase noise resulting from the OIPLL process, this scheme is less affected by the noise, thus the available distance is relatively large compared with previous LLO schemes. Our proposed scheme simultaneously implemented by an independent group [46]. The approach of this group is completely experimental. For more clarification, a comparison of our results with those of this empirical study is plotted in Fig. 6(b). For a fair comparison, we use their parameters (i.e., $V_a=5.25$ SNU, $f_{sym}=10$ Mbaud, $R_{inj}=-60$ dB, $\beta ={\%}98$, and $\eta =0.6799$). As shown, our proposed theory and their results are in good agreement.

 

Fig. 6. (a) The secure key rate analysis for four different LLO schemes: time-multiplexing (sequential) [9], time-multiplexing (delay-line) [6], time-polarization multiplexing [52] and polarization-multiplexing and OIPLL [53]. All the noise sources considered for this simulation were explained in Section 4.1 and Supplement 1 Sections 3 and 4. This simulation is based on the untrusted noise model. (b) The secure key rate in CV-QKD with the LLO generated by the OIPLL and polarization-multiplexing methods. Experimental study [46] reported the average value of 0.83 Mbps at 22-km. The estimated key rate by this group are between 0.62 Mbps and 1.07 Mbps at 22 km. It is assumed that $f_{sym}=10$ MBaud, $\beta ={\%}98$, $V_A=5.25$ SNU, and the injection power is 1 nW. Our simulation demonstrates 0.836 Mbps secure key rate at this distance. LLO: locally local oscillator; OIPLL: optical injection phase locked loop.

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The contribution of the phase noise and RIN to the total excess noise is illustrated in Fig. 7. The phase fluctuations are considered in the output photo current fluctuations of the BPD ($\Delta n$). As shown, this noise is negligible in comparison with other important noises listed in Table S3 in Supplement 1.

 

Fig. 7. Excess noise due to the RIN: (a) related to the quantum signal at $f=200$ MHz. The bandwidth of the laser and RIN are assumed 100 Hz and $-135~ \text {dBHz}^{-1}$, respectively. (b) Related to the LLO, when $P_{out}\approx 2$ mW at $f= 10$ MHz. The bandwidth of the laser is 120 kHz.

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Although employing more narrow linewidth lasers in prior LLOs can yield better results, some difficulties remain, including the finite extinction of amplitude modulator (AM), the saturation of the balanced photodetector (BPD) when detecting consecutive quantum and reference signals in time multiplexing (sequential), addressing concerns related to photon leakage, and establishing stable interferometric setups in delay-line time multiplexing [6]. Moreover, these schemes often necessitate the implementation of relatively complex algorithms for tracking the residual phase through sample-by-sample pilot tone signals.

The remainder of this section focuses on investigating the effect of two key parameters, namely, Alice’s modulation variance and the intensity of the reference signal on the secure key rate. By reducing the reference signal intensity, the error excess noise is considerably increased in the previous LLOs, while the leakage and ADC noises represent a decrease (for more details see Supplement 1 Section 6.). Consequently, there exists a constraint on significantly reducing the intensity of the reference signal in these schemes. While, in the OIPLL scheme, the RIN and the phase noise reduction do not significantly affect the total excess noise (the contribution of these noises is extremely low compared with the other excess noises in LLO scheme [38]). These noises are further reduced due to the OIPLL. Fig. 8(a) illustrates the secure key rate at a distance of 22 km, regarding different reference signal intensities for time (delay-line)-polarization multiplexing [52], and OIPLL-polarization multiplexing LLO. Further, Fig. 8(b) depicts the secret key rate versus distance concerning two different reference signal photon numbers for two LLO CV-QKD schemes. As shown, OIPLL can have a secure key rate, even in low reference signal intensity (approximately 60 photons). In fact, the reference signal intensity can be reduced to the extent that the injection intensity at Bob’s system reaches -60 dB before the input of the SL. In such a situation, the output power of the SL is $\approx 2$ mW, and the injection power is $\approx 2$ nW. The intensity per pulse can be further reduced by increasing the symbol rate. For fewer injection ratios, experimental studies proposed using the EDFA pre-amplifier [18] at the expense of more noise.

 

Fig. 8. The effect of the reference signal intensity on the secure key rate in time-polarization multiplexing LLO [52], and polarization-multiplexing and OIPLL LLO. All parameters used in this simulation were listed in Table S2 in Supplement 1. (a) The secure key rate with respect to different reference signal intensities over 22 km distance for two LLO schemes. Time-polarization multiplexing scheme cannot have a secure key rate in low reference signal intensity. (b) The secure key rate versus distance for two different values of reference signal intensities (500 and 1000 photons) in two LLO schemes. LLO: Locally local oscillator.

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As depicted in Fig. 8(b), the error excess noise in the time-polarization multiplexing scheme highly depends on the reference signal intensity. Hence, it plays an important role in the total excess noise especially in the low intensities in contrast with the OIPLL scheme. Additionally, it can open security threats against the protocol. Section 6 is devoted to the security analysis of the OIPLL LLO versus conventional LLOs.

Another important parameter impacting the secure key rate is the modulation variance. Fig. 9(a), shows the effect of Alice’s modulation on the secret key rate over 16 km distance for two LLO schemes. Based on the data, there is an optimum value for $V_A$, and with increasing the modulation variance, the secure key rate in the time-polarization multiplexing LLO decreases more rapidly. In this scheme, the phase estimation excess noise is directly proportional to the modulation variance (Eq. (5)), causing the secure key rate rapidly to decrease to zero, and there is no secure key rate for $V_A > 10$ SNU.

 

Fig. 9. The effect of Alice’s modulation on the secure key rate for time-polarization multiplexing LLO [52], and polarization-multiplexing and OIPLL LLO. The error excess noise in the time-polarization multiplexing LLO directly depends on the modulation variance (Eq. (5)); hence, with increasing the modulation variance, the secure key rate reduces significantly. (a) The secure key rate at 16 km for two schemes with respect to different modulation variances. (b) The secure key rate of two LLO schemes versus distance when $V_A=\{10,20\}$ SNU. LLO: locally local oscillator; OIPLL: optical injection phase locked loop; SNU: shot-noise unit.

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Although the contribution of the RIN excess noise to increasing the $V_A$ increased as well, in the OIPLL scheme, as mentioned earlier, the magnitude of this noise is negligible compared to the other noises, thus it does not considerably affect the secure key rate. Fig. 9(b) illustrates the secure key rate with respect to the distance for two LLO schemes when $V_A=\{10,20\}$. According to the results, the secure key rate for both schemes reduces with increasing the $V_A$. More relevant figures can be found in Supplement 1 Section 8.

6. Security analysis

In the CV-QKD, LO plays two crucial and almost interdependent roles. The initial role pertains to quadrature measurement, wherein coherent detection employing a robust LO can be viewed as a quantum measurement of electric field quadratures, as expounded upon by Yuen and Shapiro [54]. Further, to eliminate the quantum and excess noises of the LO, they proposed both output ports of the BS must be coherently subtracted with a balanced photodetector. Under these conditions, the fundamental quantum noise inherent in two-port coherent detection is related to the signal quantum noise rather than LO noise [55,56]. As a consequence, even an amplitude-squeezed state LO cannot improve the performance of a coherent detection [55–57]. Indeed, LO carries information about the Alice’s phase reference, and in the primary versions of CV-QKD protocols was even co-transferred with the quantum signal. Thus, the intensity of reference pulses or TLOs were conventionally announced in public, and Alice and Bob agreed on the intensity and the expected noise of these signals.

To date, there have been no reported attacks that demonstrate how knowledge of reference signal properties could compromise the security of CV-QKD protocols despite DV-QKD protocols. However, Eve can intercept the reference signal and perform measurements on both quadratures. Subsequently, she might make new states based on her measurements and transmit them to Bob. But, in this case, Alice and Bob can identify her, since her measurement introduces noise into the system. In CV-QKD, the zero-error attacks [58] are of crucial importance which means despite the attack, Alice and Bob cannot normally detect the adversary unless they use more tools to check or fix the loopholes. These attacks correspond to the second role of the LO in CV-QKD protocols which will be explained in the following.

The next important role of the LO in CV-QKD protocols is related to the estimation of excess noise [59]. Based on the results of the noise estimation process, one may recognize the influence of the adversary during the reconciliation process. For this, the shot-noise must be precisely measured, since all the excess noises imposed on the protocol are conventionally normalized to the shot-noise value. Typically, shot-noise is quantified when the LO illuminates the BS, while the signal port is vacuum [59]. Eve, by manipulating the estimation of shot-noise, can mount an attack on the protocol. She can open the security loopholes provided that she deceives communication parties during the noise estimation process. In other words, she attacks such that the total noise estimated by Alice and Bob does not change. This means that the excess noise is reduced while the quantum signal noise is increased due to the Eve’s measurement. This leads to Eve giving access to the information, without imposing further noise. In these circumstances, Alice and Bob typically overestimate the noise, while the real noise is slightly suppressed by Eve. To compensate this reduction, Eve attempts to compromise the quantum signal [5,60]. For instance, Haseler et al. [60], introduced an intercept-resend attack in which Eve transmits equal coherent amplitudes for signal and LO instead of the main signals. This attack can yield lower variances in the Bob’s detection system than obtained from the actual input states, effectively granting Eve access to the information shared between Alice and Bob. Hence, they recommended measuring the intensity of the LO as the countermeasure to prevent such attacks. Here, we ignore some kind of attacks where LO indirectly plays a role [61,62], as well as some experimental complexities that a high intensity LO imposes on the system [6].

While the experimental challenges posed by high-intensity TLOs have been solved in the LLO schemes, phase reference estimation noise was introduced. Unfortunately, this added noise is highly dependent on the amplitude of the phase reference signal, as depicted in Fig. 8(b). As reported by previous studies [63,64], this introduces security vulnerabilities. For example, if Alice and Bob agree on an expected level of noise, Eve could potentially diminish the excess noise by either employing a lossless channel [63] or amplifying the intensity of the reference signal [64]. Then, she might compensate for this noise reduction by attacking the quantum signal. Meanwhile, if Bob continues to estimate the noise based on the previously measured power of the reference signal, he would overestimate the noise, allowing Eve to elude detection by Alice and Bob. Therefore, monitoring the intensity of the reference signal is of paramount importance [63].

In OIPLL based LLO, there are two important points that should be noted, firstly, as mentioned before, the performance of SL highly relies on the characteristics of the input injection signal, such as injection rate, detuning frequency, and the type of the injected quantum state. Secondly, the excess noise ($\xi _{RIN}$) added to the protocol due to the LO regeneration is negligible compared to the phase estimation noise ($\xi _{error}$) in other LLO schemes. Here, the phase estimation noise is replaced with the RIN of the LLO and quantum signal, which is negligible (about $10^{-6}$ and $10^{-10}$ shot-noise-unit) concerning other noise sources in the CV-QKD system (See Table S3 from Supplement 1 and Figs. 7). These properties help Alice and Bob to recognize any potential eavesdropper in the middle, making the protocol more secure against attacks than the previous schemes. In the following, further details about how these properties can be helpful will be provided.

Suppose Eve decides to attack. She should adopt a strategy that reduces the excess noise. For this, she can change three quantities, the injection ratio, the detuning frequency, and the choice of the injected quantum state. However, it is important to note that only increasing the injection ratio yields a reduction in excess noise, as depicted in Figs. 3(a) and (4). While Eve can indeed diminish excess noise by raising the injection rate, this reduction might not be particularly advantageous, since the excess noise attributable to RIN ($\xi _{RIN}$) usually plays an insignificant role in comparison to other effective noise sources. Additionally, increasing the injection ratio is easily detectable by Bob. Also, other interventions such as manipulating the detuning frequency or injected quantum state would manifest itself with more noise in the output of the SL. In our analysis, we assume that Alice and Bob choose the optimum detuning frequency to minimize the phase offset and noise. Details regarding the injection of different quantum states were provided in Supplement 1, Section 2. It’s worth noting that such alterations may potentially lead to the unlocking or instability of locking [31], which imposes constraints on Eve’s choice of specific values for frequency detuning or injection ratio. In addition, it is worth noting that any noise related to the ML would be amplified by OIPLL technique, as a consequence, it can be detected by Alice and Bob during the reconciliation process [Supplement 1 Section 2].

Additionally, in this study, we assume that the minimum output power of the OIPLL-based LLO is approximately greater than 2 dBm, which is in line with experimental schemes like those presented in [12,46]. In Ref. [12], the output power of the LLO is indeed 12 dBm. However, after passing through an optical hybrid and experiencing a 10 dBm loss, the power that reaches the PIN detector is reduced to 2 dBm. The purpose is to demonstrate the feasibility of OIPLL even in scenarios with low injection ratios. For better clarity, refer to Fig. (10(a)), which depicts the output power of the SL as a function of the injected optical power. This figure illustrates that in the low injection regime, the output power remains relatively independent of the injection ratio and remains similar to the free-running state output power. In contrast, in the high injection regime, the effect of the optical pump becomes dominant. Additionally, Fig. (10(b)) displays the output power of the SL as a function of different injected electric currents. In summary, we can control and adjust the output power of the SL using both the electric current and optical injection. In this respect, the phase noise variance of the OIPLL SL output, given a pulse length of 1 ns at a 100 Hz bandwidth with -90 dBc/Hz phase noise (Fig. 4(a)) is $3\times 10^{-4}$ rad. This is comparable to the phase noise of an ideal coherent state LO with approximately $10^{8}$ photons which is around $2\times 10^{-4}$ rad. Hence, the OIPLL technique can help generate an ideal coherent classical source for the coherent detection.

 

Fig. 10. SL output power with respect: (a) different injection ratios, (b) different injection electric currents.

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Briefly, any intervention by Eve can potentially impact the SL output. Consequently, it is sufficient to monitor these parameters both before and after injection. These attacks are categorized as the non-zero error side-channel attacks [51] and do not compromise the unconditional security, since can be monitored and elimitated by Bob and Alice during the post-processing. Nevertheless, it seems necessary to throughly examine the ML signal parameters, such as power, frequency, and polarization, before OIPLL, especially for having stable locking [23].

7. Conclusion

This study investigated the feasibility of using OIPLL in the generation of an LLO in CV-QKD. It was revealed that the noise associated with the OIPLL technique is extremely low compared to pre-existing LLO-based CV-QKD schemes which suffer from excess noise due to the phase estimation process. Rate equations with Langevin noise sources were adapted to explain the behavior of the laser when a combination of both techniques, OIL and OPLL, are used. Based on the findings, using the OPLL in addition to OIL with a proper gain could enhance the locking-range which is the main drawback of OIL. In addition, the secure key rate under collective attack was calculated for this scheme, and compared with previous LLO schemes. The results demonstrated that due to the role of OIPLL as an amplifier with low noise, the secure key rate is achievable over longer distances. Our results corroborate with those of experimental research.