1. Introduction

Interferometry is one of the most important measurement techniques in physics. Its goal is to estimate the quantity of interest, normally a relative phase gathered by one “arm” of the interferometer. The enhanced phase estimation benefits multiple areas of scientific research, such as quantum metrology, imaging, and sensing. Non-classical optical states of light could beat the classical diffraction limit and the shot noise limit [1–4]. In the absence of photon loss and phase noise, the phase measurement precision of the entangled states, such as N00N states, can reach to the Heisenberg limit [5, 6] and achieve N times enhanced resolution. The advantage of entanglement results in the development of the quantum remote sensor, the quantum radar and lidar [7]. The interferometric quantum lidar is one typical kind of interferometer. Exploiting quantum phenomena to enhance the performance of a classical lidar system is a new paradigm. This kind of lidar is what people commonly referred to as the quantum lidar [8, 9] or quantum enhanced ladar [10, 11]. Most quantum lidars are the coherent (or interferometric) rather than direct-detection lidars. The narrow feature of the interference fringes (super-resolving) of an interferometric quantum lidar would be useful, for example, in LADAR ranging or laser Doppler velocimetry in the small return power regime, where one would lock onto the side of such a feature and then monitor how it changes in time with a feedback loop in the interferometer. Two schemes of these interferometric lidars have been found. One is based on the maximally entangling two photons states to achieve super-resolving and super-sensitivity. Another is to use the coherent state photons coupled with a quantum detection scheme or a quantum strategy to beat the Rayleigh diffraction limit. This kind of lidar has the similar theory to that of the quantum metrology and interferometry. The interested quantity for quantum metrology is the phase resolution ${\delta }_{\phi }$, while the interested one for quantum lidar is the range resolution ${\delta }_R$. The relationship between them is about in the form of ${\delta }_R=\lambda /2\pi \cdot {\delta }_{\phi }$, which is specified in [7].

However, the N00N states mentioned above are extremely sensitive to photon loss [12–24] and difficult to be generated. In a lossy interferometer, it has been shown that a transition of the precision from the Heisenberg limit to the shot noise limit occurred with the increase of the photon number N [16, 17]. All of these reveal that using such quantum states of light as the source of lidar has no advantages than the traditional lidar. So, Gao et al. proposed a super-resolving quantum lidar scheme with coherent states (CS) and photon-number-resolving detectors [25] to beat the classical diffraction limit. Jiang et al. presented a quantum homodyne detection scheme to provide both longitudinal and angular super-resolution [7]. Coherent states of light can mitigate the super-Beer’s law in photon loss and maximize sensitivity [26].

Recently, J. Joo et al. reported a quantum metrology scheme with the entangled coherent state (ECS) which shows a noticeable improved sensitivity for phase estimation compared to that of the N00N states [27]. They found that the ECS outperforms the phase enhancement achieved by the N00N states in the lossless, weak, moderate and high loss regimes. Afterwards, Zhang et al. and Jing et al., almost at the same time, put forward an exact analytical result of the quantum Fisher information (QFI) for the ECS with an arbitrary expectation photon number $N$, and obtained the sub-Heisenberg limit sensitivity [28, 29]. The exact results and the analytical solutions of the quantum metrology with the ECS have already been known in the photon loss situation, however, the determination of the ultimate precision limit in the presence of phase noise is still a challenging issue.

In this paper, we propose a unified description to the scheme of the entangled coherent-state quantum lidar by utilizing the general expression of the conditional probabilities for detecting a binary outcome in photon counting measurement. The experiment and theory of the binary-outcome data processing scheme have been widely discussed [30–32]. We first have derived out the conditional probabilities and the average value of detecting an outcome with the parity photon counting measurement. By the numerical calculation, in the case of identical mean photon numbers, the resolution of our scheme with parity detection is $\sqrt{N}$ ($N$) times enhanced relative to that of the coherent-state strategy with the same (intensity) detection. In other words, our scheme beats the Rayleigh diffraction limit in the absence of loss and noise. Simultaneously, we investigate the resolution and the best sensitivity in the presence of photon loss. Numerical results show that the ECS with parity photon counting detection gives better sensitivity and resolution than those of the CS in the whole loss regimes. Secondly, we focus on the roles of the phase noise on the resolution and sensitivity according to the master equation adopted in [33–41]. Taking the same average photon number into account, the ECS outperforms the resolution enhancement achieved by the CS and produces better resolution and sensitivity than that of the N00N states in the whole diffusion regimes. However, the sensitivity of the ECS precedes the CS only in the low ($\kappa <{10}^{-2}$) and high ($\kappa >0.35$) diffusion regions. Finally, the zero-nonzero counting (i.e., the so called Z-detection or on-off detection) shows worse sensitivity than those of the parity detection, which is just opposite from the case as demonstrated in the recent coherent-light Mach-Zehnder experiment [31] and theory [32].

2. Two-mode interferometric lidar scheme with entangled coherent states

A two-mode interferometric quantum lidar scheme is illustrated in Fig. 1, which is consisted of a standard Mach-Zehnder Interferometer (MZI), two fictitious beam splitters $\widehat{B}(T)$ (FBS) and a phase shifter $\Delta \phi$. The anti-clockwise path a and the clockwise path b represent the “arms” of the probe signal and the local signal, respectively. A coherent superposition state (CSS) $|CSS〉=N_{\alpha }\left(|\alpha /\sqrt{2}〉+|-\alpha /\sqrt{2}〉\right)$, with $N_{\alpha }={\left[2\left(1+e^{-{\left|\alpha \right|}^2}\right)\right]}^{-1/2}$and $\alpha =\sqrt{N}$, is injected into one input port of the MZI and the other port is left in a coherent state $|\alpha /\sqrt{2}〉$. After the first 50-50 beam splitter (BS), the entangled coherent states $|ECS〉=N_{\alpha }\left({|\alpha 〉}_a{|0〉}_b+{|0〉}_a{|\alpha 〉}_b\right)$ can be obtained. We consider the roles of photon loss by introducing the FBS in path a and path b. The FBS couples the interferometric mode and the environment mode E with the transmissivity $T$. The reflectivity (or loss rate) corresponding to T is R.

 

Fig. 1 Scheme of a two-mode interferometric quantum lidar.

Download Full Size | PDF

The phase shift $\phi \left(t\right)$ accumulated in path a can be expressed by an unitary evolution operator $\widehat{U}\left[\phi \left(t\right)\right]$ and its value is given by $\phi =2kr$, where $r$ is the target distance and $k=2\pi /\lambda$ is the wave number ($\lambda$ is the wavelength). The 50-50 BS is denoted by an unitary operator ${\widehat{B}}_{1/2}$. The state after the first BS is expressed as $|{\psi }_{in}〉=|ECS〉\otimes {|0〉}_{Ea}\otimes {|0〉}_{Eb}$. The density matrix of this state is given by ${\widehat{\rho }}_{in}=|{\psi }_{in}〉〈{\psi }_{in}|$. The evolution of the second BS, the FBS and the phase shifter $\Delta \phi$ on ${\widehat{\rho }}_{in}$ are described by an unitary transformation $\widehat{M}={\widehat{B}}_{1/2}\widehat{B}\left(T\right)\widehat{B}\left(T\right)\widehat{U}\left(\phi \right)$, thus the output states of the MZI after tracing out the environment mode take the following form

3. Binary-outcomes parity photon counting measurement

Photon counting measurement is a common detection method in the fields of quantum metrology and interferometry. A general photon counting is described by a set of projection operators $\left\{|n,m〉〈n,m|\right\}$ with the two-mode Fock states $|n,m〉={|n〉}_a{|m〉}_b$. According to Eq. (1), the probability for detecting $n$ photons at the output port a and $m$ photons at the output port b, i.e., coincidence rate $P\left(n,m|T,\phi \right)=〈n,m|{\widehat{\rho }}_{out}\left(T,\phi \right)|n,m〉$, is given by

3.1. Parity detection

Parity detection was originally proposed in the context of trapped ions [42] by Bollinger et al. in 1996, and later adopted for phase estimation by Gerry [43]. Gao et al proposed the quantum lidar scheme which exploits parity detection resulting in super-resolution at shot-noise limit [25]. The parity detection described by a parity operator ${\widehat{\Pi }}_a={\left(-1\right)}^{{\widehat{n}}_a}=e^{i\pi {\widehat{a}}^{†}\widehat{a}}$ at the output port a divides the photon counting data $\left\{n,m\right\}$ into binary outcomes $\pm$, according to even or odd number of photons $n$ at that output port. In other words, if $n$ is an even number, ${\Pi }_a=+1$, otherwise ${\Pi }_a=-1$. And if a detection method groups the counting data $\left\{n,m\right\}$ into binary outcomes 0 or Ø, according to the zero or nonzero number of photons n at that port, this detection is the so called zero-nonzero detection which will be detailed in Sec. 4. Now, we can obtain the conditional probabilities $P\left(\pm |T,\phi \right)$ for parity photon counting detection through a sum of $P\left(n|T,\phi \right)$ over the odd or the even number of $n$, with its explicit form [cf. Equation (3)]

There are two terms in Eq. (5), the first is a constant term which only affects the amplitude of the phase-sensitive signal. However, the second is the interferometric term which determines the sensitivity and the resolution of the detected signals. The exponential part in the second term is in agreement with Eq. (6) of [25] in the case of $T\to 1$ and $\phi \to \pi /2-\phi$, therefore, the resolution determined by this term gives $\sqrt{N}$ times improvement than that of the CS scheme with the intensity detection. The phase-sensitivity signal will be further modulated by the cosine part, which will results in the final N times super-resolution. The parity signals of Eq. (5) with different photons are plotted in Fig. 2, where all signals of our scheme are shifted $\pi /2$ along $-\phi$ axle direction (all of the following plots in this paper take the same manipulation). The red dot-dashed lines denote the normalized intensity signals for the CS scheme, the green solid lines are the parity signals $〈{\widehat{\prod }}_C〉$ of the CS scheme [7, 25], and the blue thin dotted lines represent the parity signals $〈\widehat{\prod }〉$ for the ECS scheme. According to [7, 25], the resolution of the parity detection for CS is $\sqrt{N}$ times improvement compared with that of the traditional intensity detection. It can be clearly found from Fig. 2 that the resolution of $〈\widehat{\prod }〉$ is $\sqrt{N}$ narrower than that of $〈{\widehat{\prod }}_C〉$ in the sense of multiple narrow peaks and N times resolution enhancement is obvious relative to the traditional intensity signal in the sense of the well-defined narrow feature, which has beaten the Rayleigh diffraction limit.

 

Fig. 2 Output signals for binary-outcome parity photon counting measurement with various number of photons N = 9, 16, 25, 36, 49, 64, 81, 100 (blue thin dotted lines). The green solid lines denote the parity signals for the coherent-state scheme and the red dot-dashed lines represent the traditional intensity signals. In the upper left picture, $\delta \phi$ represents the resolution which is the phase difference between the adjacent wave troughs.

Download Full Size | PDF

3.2. The effect of photon loss on the resolution and sensitivity

The improved resolution is due to the coherence of the ECS, the excellent coherence is an important characteristic of the non-classical state which is different from the classical state and is also the main superiority of the quantum information relative to the classical information. However, for most non-classical states, coherence is excessively sensitive to loss and noise.

We consider the roles of photon loss on the sensitivity and resolution as shown in Fig. 3. The unavoidable losses mainly come from three aspects, i.e., the atmospheric absorption and diffraction, the targets lower-reflection and the imperfect devices of the transmitting and receiving system. All of these losses result in a lower transmissivity which seriously reduces the phase sensitivity and resolution. Figure 3(a) describes the effect of photon loss on the resolution, where the resolution $\delta \phi$ is denoted as the phase difference between the adjacent wave troughs as shown in the upper left picture in Fig. 2. It can be found that the resolution becomes worse and worse with the increasing loss rate $R=1-T$. For comparison, according to [7, 25, 30–32], we plot the resolution of the CS scheme with parity detection as the blue dotted line in Fig. 3(a). It is clear that the resolution of the ECS is always better than that of the CS scheme. The optimal sensitivity (determined by QFI) of the ECS in the loss case has been investigated in detail in [28, 29]. We exploit their results and methods to numerically analyze the roles of photon loss on the optimal sensitivity of the ECS, which is depicted as the red dot-dashed line in Fig. 3(b). For reference, we also plot the sensitivity of the CS scheme as the blue dotted curve. We can see that the ECS outperforms the sensitivity enhancement achieved by the CS (here $\Delta {\phi }_c=1/\sqrt{TN}$, i.e., the shot noise limit with the photon number $TN$) in the whole loss regions. That is to say that the ECS coupled with the binary outcome parity photon counting measurement suffice to beat the shot noise limit in the lossy environment. Apart from photon loss, phase noise is another factor which will also decay the performance of quantum lidar.

 

Fig. 3 (a) The resolution and (b) the optimal sensitivity determined by QFI for ECS as a function of the transmittance T with a fixed number of photons $N=25$.

Download Full Size | PDF

3.3. The role of phase noise on the resolution and sensitivity

We take the effects of phase noise on the performance of our quantum lidar into account in this section. After the phase accumulation, phase-diffusion process produces a phase noise $\Delta \phi$ in one of the two paths (as depicted in Fig. 1). Generally, the presence of phase noise can be modeled by the following master equation: $\partial \widehat{\rho }/\partial t=\gamma \left(2{\widehat{N}}_a\widehat{\rho }{\widehat{N}}_a-{\widehat{N}}_a^2\widehat{\rho }-\widehat{\rho }{\widehat{N}}_a^2\right)$ with ${\widehat{N}}_a={\widehat{a}}^{†}\widehat{a}$ and the phase-diffusion rate $\gamma$ [33–41]. According to [36, 38], the solution of $\widehat{\rho }$ is given by an integration ${\widehat{\rho }}_{\kappa }\propto {\displaystyle {\iint }_{R}dx{e}^{-x^2/\left(4\kappa \right)}\widehat{U}\left(x\right)\widehat{\rho }\left(\phi \right){\widehat{U}}^{†}\left(x\right)}$, where $\kappa =\gamma t$ is a dimensionless diffusion rate. Note that the phase-encoded state $\widehat{\rho }\left(\phi \right)$ obeys $\widehat{U}\left(x\right)\widehat{\rho }\left(\phi \right){\widehat{U}}^{†}\left(x\right)=\widehat{\rho }\left(x+\phi \right)$ for the noiseless case. Replacing $x\to x-\phi$, we obtain the final state

In the presence of the phase diffusion, all the relevant quantities, such as the output signal, the sensitivity and the conditional probability, can be obtained by integrating the Gaussian with the quantities without diffusion. For example, the binary-outcome parity photon counting measurement gives the output signal

 

Fig. 4 (a) The output signals of binary-outcome parity photon counting detection with $\kappa =0$ (green thick dashed line), $\kappa =5\times {10}^{-4}$ (red solid line), and $\kappa =5\times {10}^{-3}$ (black dot-dashed line) for $N=9$. (b) The resolution as a function of the diffusion rate $\kappa$ for the ECS (red dot-dashed line), the N00N states (black solid line) and the CS (blue dotted line) schemes with the parity detection, and for the ECS scheme (green dashed line) with the Z-detection (which has been specified in Sec. 4).

Download Full Size | PDF

Due to ${\widehat{\prod }}^2=1$, we further obtain the phase sensitivity

According to Eq. (7) and Eq. (8), the effects of phase diffusion on sensitivity are plotted in Fig. 5(a). The phase diffusion has a dramatic influence on the sensitivity in the binary-outcome parity photon counting measurement even though for a very small diffusion rate $\kappa ={10}^{-4}$. Figure 5(b) depicts the best sensitivity $\Delta {\phi }_{\prod }$ for the ECS (red dot-dashed line) as a function of the diffusion rate $\kappa$, from which the best sensitivity becomes worse and worse with the increasing $\kappa$. For comparison, the best sensitivity $\Delta {\phi }_c$ of the CS (blue dotted line) is also depicted in Fig. 5(b). $\Delta {\phi }_{\prod }$ exceeds $\Delta {\phi }_c$ in both high ($\kappa >0.35$) and low ($\kappa <{10}^{-2}$) diffusion rate regimes, however, the opposite results occur in the rest regions.

 

Fig. 5 (a) The sensitivity with $\kappa =0$ (red solid line), $\kappa ={10}^{-4}$ (blue dotted line), $\kappa ={10}^{-3}$ (green dashed line), and $\kappa ={10}^{-2}$ (black dot-dashed line) for $N=9$. (b) The best sensitivity as a function of the diffusion rate $\kappa$ for the ECS (red dot-dashed line), the N00N (black solid line) and the CS (blue dotted line) schemes with the parity photon counting measurement. The green dashed line in (b) denotes the best sensitivity for the ECS with the Z-detection.

Download Full Size | PDF

In order to make performance comparison between the ECS and the N00N states schemes, Fig. 4(b) and in Fig. 5(b) show the numerical results of the resolution and sensitivity for the N00N states in black solid lines. From Fig. 4(b), in the whole phase diffusion windows, the N00N states show worse resolution than that of the ECS scheme. In small diffusion regimes, the resolution of the N00N states is better than that of the CS, however, the opposite cases take place in the rest diffusion regimes. In addition, it can be found from Fig. 5(b) that the sensitivity of the N00N strategy becomes worse and worse with the increasing phase diffusion rates, and the N00N states show no better sensitivity than those of the ECS and the CS schemes at all in the noisy environments. Thus, according to above analysis, the robust ECS will make itself more suitable to be the light source of lidar.

So far, it can be easily understood that the photon loss in this paper is the same as that of a real lidar system, however, the noise terms brought about using the diffusion rates seem to be a slightly different from that of the real world applications. The reason of this phenomenon may be about that the theory of the phase noise caused by the phase diffusion progress in our quantum lidar scheme is different from the one produced by the transmission losses, turbulence and so on in a classical lidar system. There must exist some necessary connections between these two noise theories, unfortunately, it is unable to find out that relations until now. Therefore, this problem will be left as an open question in quantum lidar fields for quite a while.

4. Binary outcomes zero-nonzero photon counting measurement under phase diffusion

Another photon counting measurement scheme is the so called Z-detection. Z-detection gives a slightly better sensitivity than that of the parity detection for the CS scheme [31, 32]. However, the situation in our scheme will be different, we will discuss it in this section.

We consider the zero-nonzero photon counting at the output port a as shown in [31]. In this scheme, the counting data $\left\{n,m\right\}$ are divided into binary outcomes: 0 for $n=0$ and Ø for $n\ne 0$, with the associated probabilities

The numerical results of the output signal and its sensitivity are illustrated in Fig. 6. Even though in the lossless and noiseless cases, the resolution [green dashed line in Fig. 6(a)] and the sensitivity [green dashed line in Fig. 6(b)] of the Z-detection are much lower than those of the parity detection (red dot-dashed line), which shows different results from the CS scheme [31]. In [31], Cohen et al. found that the parity detection gives a slightly worse sensitivity than that of the zero-nonzero detection with unknown reasons. We give a further numerical investigation on the resolution for the CS with Z-detection (whose figure is not shown here) according to the results in [31] and find that the parity detection performs better resolution than the Z-detection. Therefore, the optimal detection strategy for the ECS should be the parity detection, while the best measurement scheme for the CS seems to be the Z-detection. The root causes of these phenomena may be related to the input states. If a classical state, such as the CS, is selected as the input, Z-detection will show better performances, conversely, if the input state is a non-classical state, such as the ECS in this paper, parity detection may be the optimal detection strategy because parity detection guarantees the optimal measurement of phase shift in optical interferometer in a wide range of non-classical input states of light. However, this is just a hypothesis.

 

Fig. 6 The output signal and phase sensitivity of the Z-detection for $N=9$ under the lossless and noiseless cases. (a) The output signal, (b) the phase sensitivity, the red dot-dashed line: the parity detection, the green dashed line: the Z-detection.

Download Full Size | PDF

More surprisingly, we find that the sensitivity of the Z-detection diverges at the phase point of $\phi =\pi /2$ as shown in Fig. 6(b), thus, the best sensitivity for Z-detection occurs at $\phi \ne \pi /2$. Finally, the roles of the phase diffusion on resolution and sensitivity are investigated in this Z-detection. We can see that the phase diffusion degrades the fringe resolution [green dashed line in Fig. 4(b)] and the achievable phase sensitivity [green dashed line in Fig. 5(b)], and we again confirm that the Z-detection gives much worse sensitivity and resolution than those of the parity detection in the present of phase diffusion.

5. Discussions and conclusions

In summary, we have evaluated the performance of super-resolving quantum lidar with the ECS sources in practical environments. Without the photon loss and the phase noise, the ECS with parity detection gives $\sqrt{N}$ ($N$) times resolution enhancement relative to that of the CS strategy with the same (intensity) detection and beats the shot noise limit. In the presence of the photon loss, it is shown that the ECS gives better sensitivity and resolution than that of the CS scheme in the whole loss regimes. Taking the phase noise into account, the ECS outperforms the resolution enhancement limit achieved by the CS and also gives better resolution and sensitivity than those of the N00N states in the whole diffusion regions. However, the sensitivity of the ECS precedes that of the CS only in the low ($\kappa <{10}^{-2}$) and high ($\kappa >0.35$) diffusion regions. In addition, the Z-detection of the ECS shows much worse sensitivity than that of the parity detection, which is just opposite from the case demonstrated in the recent coherent-light Mach-Zehnder experiment [31] and theory [32]. Finally, according to the reports of [44–48], $|ECS〉$ with $\alpha \approx 1.5$ are already feasible in optics with current optical technology. Therefore, the ECS with the characteristics of robust against the photon loss and noise will make it more suitable to be the source of lidar.