Improvement of phase sensitivity in SU(1,1) interferometer via a Kerr nonlinear

Quantum metrology has a close relation to various-important information areas, such as Bose-Einstein condensate 1 ; 2 ; 3 , gravitational wave detection 4 ; 5 , and quantum imaging 6 ; 7 ; 8 . It has been widely concerned and highly developed in recent years. To meet the high precision demand, all kinds of optical interferometers have been proposed. For instance, as a general model, the Mach–Zehnder interferometer (MZI) has been used to be an essential tool to provide insight into tiny variations on phase shift 9 ; 10 ; 11 .

In order to improve the precise measurement, generally speaking, we can focus on the following three stages 12 : probe generation 13 ; 14 ; 15 ; 16 , probe modification 17 ; 18 ; 19 ; 20 ; 21 and probe readout 22 ; 23 , as illustrated in Fig. 1(a). In particular, for the probe generation, the phase sensitivity is always confined to the standard quantum limit (SQL) when the classical resources are injected into the input ports of the MZI. To surpass the SQL, non-classical quantum states have widely used as the input of the MZI, such as entangled states 13 , twin Fock states 24 , and NOON states 25 , by which the Heisenberg limit (HL) even can be reached 26 ; 27 . Although the usage of nonclassical states can greatly improve the phase sensitivity of optical interferometers, these states with large average photon numbers are not only more difficult to prepare, but also very fragile especially in the presence of environmental interferences 28 ; 29 ; 30 . Thus, from the viewpoint of resource theory, it will be a challenging task with simple input state (such as coherent state (CS)) to further improve the precision of measurement, especially in the realistic case.

On the other hand, many efforts have been paid to the stage of probe modification, especially when Yurke et al. first proposed an SU(1,1) interferometer with a linear phase shift 20 . In this system, the active nonlinear optical devices, such as four-wave mixers (FWMs) and optical parametric amplifiers (OPAs), are used instead of the passive linear beam splitters (BSs) used in the conventional MZI 31 ; 32 ; 33 ; 34 ; 35 ; 36 ; 37 . To beat the SQL, Plick et al. applied strong CS as the inputs into the SU(1,1) interferometer 33 . Subsequently, Li et al. proposed a scheme of reaching HL sensitivity via a squeezed vacuum state (SVS) plus a CS with the homodyne detection 31 . It is interesting that, Hudelist et al. pointed out that the signal-to-noise ratio (SNR) of the SU(1,1) interferometer is about 4.1 dB higher than that of the MZI under the same phase-sensing intensity 38 . This point may be one of reasons focusing on this interferometer. Actually, this implies the role of nonlinear process for improving the precision.

Expect the SU(1,1) nonlinear process, the nonlinear phase shifts have also been proposed for enhancing the phase estimation, which can be viewed as another way of the probe modification 9 ; 39 ; 40 ; 41 ; 42 . For instance, in the traditional MZI, Zhang et al. investigated the phase estimation by replacing linear phase shift by nonlinear one 9 using a CS and parity measurement. Jiao et al. proposed an improved protocol of nonlinear phase estimation by inserting a nonlinear phase shift into the traditional MZI, with active correlation output readout and homodyne detection 39 . More recently, Chang et al. suggested a scheme for enhancing phase sensitivity by introducing nonlinear phase shifter to the modified interferometer consisting of a balanced BS and an optical parameter amplifier (OPA) 41 . It is shown that the OPA potential can be stimulated by nonlinear phase shifter, which is absent for the case of linear phase shifter. In addition, the estimation of nonlinear phase has also lots of applications, such as the third-order susceptibility of the Kerr medium 39 , phase-sensitive amplifiers 43 ; 44 and nonclassical quantum state preparations 45 ; 46 . These results show that the nonlinear optical devices can be considered as powerful tools to effectively achieve both high accuracy and sensitivity. However, on one hand, the researches on nonlinear phase estimation are not as systematic as those on linear one. On the other, most works on the phase precision are based on either specific measurement, especially in the presence of photon loss, or direct calculation of quantum Fisher information for ideal case.

In this paper, we mainly focus on the nonlinear phase estimation of a Kerr SU(1,1) (KSU(1,1)) interferometer by replacing the linear phase shift with the Kerr nonlinear one, together with CS plus vacuum state (VS) (denoted as $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$) as inputs and homodyne detection. Both the phase sensitivity and the quantum Fisher information (QFI) are analytically investigated with and without photon losses. The phase sensitivity can beat the SQL and approach the HL. Compared to the traditional SU(1,1) interferometer with a linear phase and other inputs resources including CS+CS and SVS+CS, our scheme presents much QFI and higher phase sensitivity closer to the quantum Cramér-Rao (QCRB) 34 ; 47 ; 48 ; 49 ; 50 . From the viewpoint of resource theory, CS+VS can be seen as the most simple and easily available input, thus our scheme has an obvious advantage of low-cost input by inserting nonlinear phase shift into the SU(1,1) interferometer.

The remainder of this paper is arranged as follows. Section II introduces the KSU(1,1) interferometer in our scheme. In Section III, we investigate the phase sensitivity of the output signal with homodyne detection, and compare them with the conventional SU(1,1) interferometer. In Section IV, we get the QFI of Kerr nonlinear phase shift for the KSU(1,1) interferometer by invoking the characteristic function (CF) approach. In Section V, we discuss the effects of photon losses on both phase sensitivity and QFI of Kerr nonlinear phase shift, respectively. Conclusions are made in the last section.

Let us begin with the description of nonlinear phase estimation model, as shown in Fig. 1(b). The nonlinear interferometer consists of two OPAs (or FWMs) and a Kerr-type medium. Here we consider a CS $\left|\alpha\right\rangle_{a}$ with $\alpha=\left|\alpha\right|e^{i\theta_{\alpha}}$ and a VS $\left|0\right\rangle_{b}$ as the inputs in mode $a$ and mode $b$, respectively. Thus the probing state can be shown as $\left|\psi_{in}\right\rangle=$ $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}.$ After going through the first OPA, the resulting state is given by $\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$ i.e., a two-mode squeezed CS, where the operator $\hat{S}\left(\xi_{1}\right)=\exp(\xi_{1}^{\ast}\hat{a}\hat{b}-\xi_{1}\hat{a}^{\dagger}\hat{b}^{\dagger})$ represents the OPA process, $\xi_{1}=g_{1}e^{i\theta_{1}}$, with a gain factor $g_{1}$ and a phase shift $\theta_{1}$ and $\hat{a}(\hat{a}^{\dagger}),\hat{b}(\hat{b}^{\dagger})$ are the annihilation (creation) operators, for modes $a$ and $b$, respectively. For simplicity, we assume that the Kerr-type medium is inset into the path $b$ between the first and second OPAs to generate a nonlinear phase shift $\phi$ to be estimated.

After the interaction between the state $\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$ with the Kerr-type medium, the corresponding modified state becomes

depending on the phase parameter $\phi$, where

is the nonlinear phase shift operator and the exponent $k$ is the order of the nonlinearity. In particular, for the case of $k=1$, $\hat{S}\left(\phi,1\right)=e^{i\phi\hat{b}^{\dagger}\hat{b}}$ just reduces to the linear phase shift, while for the case of $k=2$, $\hat{S}\left(\phi,2\right)=e^{i\phi(\hat{b}^{\dagger}\hat{b})^{2}}$ corresponds to Kerr nonlinear case. Throughout this paper, we only consider both Kerr-type nonlinear medium and linear one by taking $k=1,2$, respectively.

After the second OPA, the final output state is given by

where $\hat{S}\left(\xi_{2}\right)=\exp(\xi_{2}^{\ast}\hat{a}\hat{b}-\xi_{2}\hat{a}^{\dagger}\hat{b}^{\dagger}),$ with $\xi_{2}=g_{2}e^{i\theta_{2}},$ is a two-mode squeezing operator, corresponding to the second OPA process. The Kerr nonlinear phase shifter $\hat{S}\left(\phi,2\right)$ satisfying the transformation relation:

which is useful for the calculation of phase sensitivity. One can refer to Appendix A about more details of derivation for this relation. Finally, the homdyne detection is performed on mode $a$, so that one can read information about the value of $\phi$.

Next, we investigate the phase sensitivity of nonlinear phase. For this purpose, we need to choose a special detection method for the readout of phase information at the final output port. Actually, there are many different detection methods, such as homodyne detection 11 ; 31 ; 32 , intensity detection 33 ; 34 ; 47 , and parity detection 27 ; 35 ; 36 . Each way of measurement has its own advantages. For example, the parity detection has been proved to be an optimal detection for linear phase estimation in lots of schemes 27 ; 51 . Compared with both intensity and parity detections, however, homodyne detection can be easily realized with current technologies, thereby playing a key role in the field of continuous-variable quantum key distribution (CV-QKD) 52 ; 53 ; 54 ; 55 ; 56 . Therefore, in our scheme the homodyne detection is employed on mode $a$ at one of output port to estimate the phase parameter $\phi,$ where the detected variable is the amplitude quadrature $\hat{X}$, i.e.,

Based on the error propagation formula, the phase sensitivity can be calculated by

with $\Delta^{2}\hat{X}=\left\langle\hat{X}^{2}\right\rangle-\left\langle\hat{X}\right\rangle^{2}.$ According to Eq. (6), for an arbitrary value of $\phi$, the corresponding phase sensitivity can be analytically derived. For simplicity, the corresponding expression is not shown here. One can refer to Appendix B for more details. In the following discussions, we assume that the KSU(1,1) interferometer is in a balanced situation, i.e., $\theta_{2}-\theta_{1}=\pi$ and $g_{1}=g_{2}=g.$

In Fig. 2(a), we show the phase sensitivity changing with $\phi$ for the linear ($k=1$) and nonlinear ($k=2$) phase shifts when given parameters $g=1,|\alpha|=1,$and $\theta_{\alpha}=\pi/2.$ It is clearly seen that, for both cases above, the minimum value of the phase sensitivity can be achieved at the optimal point $\phi=0.$ In addition, the phase sensitivity for $k=2$ is always significantly superior to that for $k=1$ around the optimal point. This implies that the nonlinear phase shift can be further used to improve the phase sensitivity, compared to the linear one. The reason lies in that the introduction of nonlinear phase shift increases the slope $\partial\left\langle\hat{X}\right\rangle/\partial\phi$ of the output signal $\left\langle\hat{X}\right\rangle$ at $\phi=0,$ which leads to the increase of the denominator in Eq. (6), as shown in Fig. 2(b). In fact, this point will be clear by deriving the phase sensitivity $\Delta\phi_{k}$ for $k=1,2$ at $\phi=0$, which are given by

where $\Delta\phi_{1}$ represents the phase sensitivity of the traditional SU(1,1) interferometer with a linear phase shift $k=1,$ $N_{\alpha}=\left|\alpha\right|^{2}$ is the mean photon number of the input coherent state, and $N_{OPA}=2\sinh^{2}g$ is the total photon number after the first OPA with vacuum inputs. Obviously, $\Delta\phi_{2}>\Delta\phi_{1}.$

Moreover, from Fig. 2(b) it is shown that the peak width for the case of $k=2$ is narrower than that for the case of $k=1.$ In this sense, the nonlinear phase shift can also improve supperresolution compared to the linear one. On the other hand, to display the effects of both the coherence amplitude $\left|\alpha\right|$ and the gain factor $g$ on the phase sensitivity, at the optimal point $\phi=0,$ we also plot the phase sensitivity as a function of $\left|\alpha\right|$ and $g$ with different values of $k=1,2$, in Figs. 3(a) and 3(b), respectively. It is found that, the values of $\Delta\phi$ rapidly decrease with the increase of $\left|\alpha\right|$ and $g$, especially for the case of $k=2.$

To further show the advantage of our scheme, we also make a comparison about phase sensitivities, involving the SQL ($\Delta\phi\thicksim 1/\sqrt{N_{Total}}$), the HL ($\Delta\phi\thicksim 1/N_{Total}$) and the Super HL (SHL) ($\Delta\phi\thicksim 1/N_{Total}^{2}$), as shown in Fig. 4. Note that $N_{Total}=\left\langle\psi_{in}\right|\hat{S}^{\dagger}\left(\xi_{1}\right)(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b})\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$ is the total mean photon number inside the KSU(1,1) interferometer. From Fig. 4, the black dashed line is the sensitivity performance of the linear phase shift, which can only break through SQL and is always surpassed by that of the Kerr nonlinear phase shift. In particular, the latter can break both the SQL and the HL, but cannot beat the SHL. The reason may be that adopting the Kerr nonlinear phase shift to effectively improve the maximum amount of information about the unknown phase shift $\phi$ can result in a transition from HL ($\Delta\phi\thicksim 1/N_{Total}$) to SHL ($\Delta\phi\thicksim 1/N_{Total}^{2}$), as described in Refs. 9 ; 57 ; 58 ; 59 ; 60 ; 61 ; 62 . Nevertheless, the common feature of these two is that the phase sensitivity increases with the increase of $\left|\alpha\right|$ and $g$.

Next, we make a comparison about the performance of phase sensitivity. In Fig. 5, the phase sensitivity is plotted as a function of the total average input photon number $N$ with several different input resources, including $\left|\alpha\right\rangle_{a}\otimes\left|\beta\right\rangle_{b}$, $\left|\alpha\right\rangle_{a}\otimes\left|\varsigma,0\right\rangle_{b}$ and $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ used in the traditional SU(1,1) interferometer ($k=1$) and $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ used in the KSU(1,1) interferometer ($k=2$). It is shown that, for the traditional case, the phase sensitivity with the input $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ performs the worst among these three inputs 31 ; 32 . This implies that the introduction of coherence amplitude $\left|\alpha\right|$ or squeezing parameter $\varsigma$ is beneficial for the phase sensitivity improvement. However, for the KSU(1,1) interferometer, it is interesting that the best phase sensitivity can be achieved only using the simplest input $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$, which is significantly superior to those inputs in the traditional SU(1,1) interferometer. That is to say, for a simple input with less energy and less resources, the phase sensitivity can be further improved by introducing Kerr nonlinear phase.

As an elegant approach, the QFI can be used to visually describe the maximum amount of information about the unknown phase shift $\phi$, which is connected with the QCRB. In fact, the QFI is the intrinsic information in quantum state and is independent of any specific detection scheme. In the absence of losses, for a pure state, the corresponding QFI can be calculated as 34 ; 48 ; 49

where $\left|\psi_{\phi}\right\rangle=\hat{S}\left(\phi,k\right)\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$ is the state vector prior to the second OPA and $\left|\psi_{\phi}^{\prime}\right\rangle=\partial\left|\psi_{\phi}\right\rangle/\partial\phi.$ Then, for the linear phase shift $\left(k=1\right)$ and the Kerr nonlinear phase shift $\left(k=2\right),$ the QFI can be, respectively, reformed as 45

where $\hat{n}=\hat{b}^{\dagger}\hat{b}$ is the photon number operator on mode $b$ and $\left\langle\Delta^{2}\hat{n}^{j}\right\rangle=$ $\left\langle\bar{\psi}_{out}\right|\left(\hat{n}^{j}\right)^{2}\left|\bar{\psi}_{out}\right\rangle-\left\langle\bar{\psi}_{out}\right|\hat{n}^{j}\left|\bar{\psi}_{out}\right\rangle^{2},\left(j=1,2\right)$ with the state vector after the first OPA, i.e., $\left|\bar{\psi}_{out}\right\rangle=\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle.$ Using the normal ordering forms of the operators,

and the characteristic function method for calculating average, it is ready to have

with

and $L_{m}(\bullet)$ is the Laguerre polynomials. One can refer to Appendix C for more details about derivations of $\bar{A}_{m}$. Based on Eq. (12), we can give the QCRB $\Delta\phi_{QCRB},$ which represents the lower bound of the phase sensitivity, i.e.,

where $\upsilon$ is the number of trials. For simplicity, here we set $\upsilon=1.$ In general, the smaller the values of $\Delta\phi_{QCRB}$, the higher the phase sensitivity.

To clearly see this point, according to Eqs. (12) and (15), at a fixed $\left|\alpha\right|=1,$ Fig. 6 shows the QFI and the QCRB changing with $g$ for different $k=1,2.$ It is clear that, for the cases of $k=1,2,$ the values of the QFI increase significantly with the increase of $g,$ thereby leading to the more precise phase sensitivity. Moreover, when given the same gain factor $g$, both the QFI and the QCRB of $k=2$ always outperform those of $k=1$, which distinctly shows the superiority of the nonlinear phase shift compared to the linear case. This result originates from the additional item $f$ in Eq. (12), giving rise to the increase of QFI.

As a comparison, we also consider the $\Delta\phi_{QCRB}$ as a function of $N$ for several different input resources in Fig. 7, similar to the phase sensitivity (in Fig. 5). Some similar results are obtained. For instance, although the QCRB with $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ is the worst than that with other resources in the traditional SU(1,1) interferometer 63 , the smallest $\Delta\phi_{QCRB}$ can be realized by the simplest input $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ in the KSU(1,1) interferometer. In addition, comparing Fig. 5 with Fig. 7, it is found that compared with several different input resources in the traditional SU(1,1) interferometer, the input $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ in the KSU(1,1) interferometer is closer to the ultimate phase precision $\Delta\phi_{QCRB}.$

In the realistic scenario, the decoherence process is unavoidable because there always exist interaction between the interferometer system and its surrounding environment, thereby leading to the information leakage from the system to the environment. For this season, the system performance would drop severely. In general, there are many types of decoherence process 64 ; 65 , such as photon loss, imperfect detector, phase diffusion, and so on. Here, for simplicity, we only study the effects of photon losses on both the phase sensitivity and the QFI in our scheme.

In summary, we propose a protocol of second-order nonlinear phase estimation by introducing the Kerr medium into the traditional SU(1,1) interferometer. A kind of simplest inputs, coherent state plus vacuum state, and homodyne detection for one mode are used. Both the phase sensitivity and the QFI or QCRB are analytically derived in both ideal and nonideal scenario by using the CF method and the IWOP technique. It is shown that the increase of both gain factor $g$ and coherent amplitude $\left|\alpha\right|$ is beneficial for improving both the phase sensitivity and the QCRB. Compared to the linear phase estimation, our scheme presents a significantly better performance about both of them, especially around the optimal point $\phi$ $=0$. In particular, for the ideal case, our scheme breaks through both the SQL and the HL, even approaching the SHL, at the large $g$ and $\left|\alpha\right|$ levels. In addition, we consider the effects of the decoherence on the phase sensitivity, from the internal or external photon losses. It is found that the former has a more obvious decoherence than the latter. Even so, our scheme still provides a significant improvement for the performance of phase estimation by homodyne detection, compared to the linear case.

To further show the advantages of our scheme, with and without photon losses, we also consider both the phase sensitivity and the QCRB changing with total average input photon number $N$ for several different input resources, including $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b},$ $\left|\alpha\right\rangle_{a}\otimes\left|\beta\right\rangle_{b}$ and $\left|\alpha\right\rangle_{a}\otimes\left|\varsigma,0\right\rangle_{b}$, when given parameters $g=1.$ It is found that the best phase sensitivity can be achieved only using the simplest input $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ in the proposed KSU(1,1) interferometer, which is significantly superior to those inputs in the traditional SU(1,1) interferometer. This means that, for a simple input with less energy and less resources, both the phase sensitivity and the QCRB can be further improved by introducing Kerr nonlinear phase. These results may have important applications in other quantum information processing.

Appendix A: Proof of the transformation relation

For a completeness, here, we give the proof about the transformation relation, i.e., $\hat{S}^{\dagger}\left(\phi,2\right)\hat{b}^{\dagger}\hat{S}\left(\phi,2\right)=e^{-i\phi}\hat{b}^{\dagger}e^{-i2\phi\hat{b}^{\dagger}\hat{b}}$. It is well known that any operator $\hat{\zeta}$ can be expanded in Fock state space,

where $\zeta_{m,n}=\left\langle m\right|\hat{\zeta}\left|n\right\rangle$ is the matrix element $\hat{\zeta}$ in Fock space.

Thus, if we take

then the matrix element $\zeta_{m,n}$ can be calculated as

where we have used the $\hat{b}^{\dagger}\hat{b}\left|n\right\rangle=n\left|n\right\rangle$ and $\hat{b}^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle.$ Substituting Eq. (A3) into Eq.(A1), we can get

Appendix B: Phase estimation based on homodyne measurement

Combining Eqs. (3) and (5), we can derive the standard deviation $\Delta\hat{X}$ as

where we have set

and

with