Waveguide Mach-Zehnder interferometer to enhance the sensitivity of quantum parameter estimation

X. N. Feng; H. Y. Liu; L. F. Wei

doi:10.1364/OE.487793

1. Introduction

Optical interferometers are the typical devices to implement the high-precision measurements of such as electric field [1], magnetic field [2], angular velocity [3,4] and gravitational wave [5,6] etc. Basically, the measurement sensitivity of an interferometer, using only the relevant classical resource, is limited by the so-called standard quantum limit (SQL): $1/\sqrt {N}$ [7–9] (with $N$ being the measurement number). To beat such a limit, interferometers using various quantum sources with nonclassical correlations, typically including the quantum squeezings and entanglements [10–16], have been demonstrated with experimental quantum optical and atomic systems [17–21].

Recently, with the rapid development of micro-nano fabrication technologies, the quantum interferometers on chip, instead of those in the free space, have been fast developed by making use of the waveguide quantum networks with the nonlinear light-matter interaction [22–25]. Specifically, the waveguide FPIs, formed by using a series of atomic mirrors to control the transporting probability of the photons transporting along a one-dimensional waveguide, have been successfully demonstrated [26–31]. Compared with the conventional FPI in free space, here the reflectivity of the waveguide FPI can be dynamically adjusted by using the atomic mirrors [24], which can be generated by the superconducting qubits [22,32], trapped ions [33,34], semiconducting quantum dots, cavity, and the color centers diamond [35,36], etc.. Benefited from the transport properties of the waveguide photons scattered by the atomic mirrors can be exactly solved [37–40], the quantum routings [41–43] and switches [44] can be effectively realized. As a consequence, the relevant physical parameters can be estimated precisely by probing either the frequency shifts or the spectrum broadenings in the measured transmitted/reflected spectra of the transporting waveguide photons [45]. Indeed, the precise measurements of the temperature [46], photon absorption [47], optical pressure [48], mechanical displacements [49,50] and rotation [51] have been achieved by using the waveguide interferometer configurations of the one-dimensional waveguide coupled to a ring resonator. Therefore, developing various waveguide interferometers to implement the quantum metrologies are particularly desirable.

Inspired by the recent progress in the precision measurement by using the waveguide FPIs [26–31], in this paper we propose a waveguide MZ-type interferometer to further enhance the sensitivity of the phase estimation. Different from the usual MZIs in free space, here a pair of two-level atoms (i.e., the atomic mirrors) served as the beam splitters to control the routings of the transporting photons between the waveguides [52,53]. Basically, if a single photon injected into the network from one waveguide, it would be routed by the atomic mirror into another waveguide. The phase of the photon acquired when it passes through a phase shifter in a waveguide arm can be detected by measuring the transmitted or reflected spectrum of the transporting photon at the outputs. As the transport properties of the photon are related to both the relative positions of the atomic mirrors and the acquired phase $\theta$ of the photon passing through the phase shifter, the quantum interferences of the photon transporting along the interferometer can be utilized to implement the desired phase estimated precisely.

The paper is organized as the following. In Sec. 2 we introduce the basic configuration of the proposed waveguide MZI, and discuss how to implement the desired quantum metrology by using the proposed interferometer. Specifically, in Sec. 3 we show how to implement the sensitive phase estimation with the proposed MZI and discuss the realizable sensitivities for different cases. It is shown that, by optimizing the relative positions of the atomic mirrors the sensitivity of the phase estimated can be further optimized. Compared with the FPI with only one waveguide, we argue that the sensitivity of the phase estimation could be further improved, even the system parameters are set as the same. Finally, we summarize our results and discuss the potential applications of the proposed MZI in Sec. 4.

2. Waveguide Mach-Zehnder interferometer

The proposed waveguide Mach-Zehnder interferometer is schematically shown in Fig. 1, wherein two one-dimensional waveguides (denoted by waveguide-a and -b) are side-coupled by a pair of two-level atomic mirrors [22,23] (denoted by the atom-A and -B, respectively) acting as the beam splitters of the photons. While the atom-A and -B are coupled to a single waveguide, either waveguide-a or waveguide-b, the configuration is then reduced to the waveguide FPI proposed previously in Refs. [23–25]. In the recent works, similar interferometer configurations [52,53] were proposed to implement the phase measurements of the photons by using the nonlinear photon-atom interactions, in which the atomic mirrors are introduced to replace the usual classical-beamsplitters scenario and the transmitted and reflected probabilities of the photons are probed. In order to show how the phase estimation can be sensitively implemented, a phase shifter is introduced in one of the paths of the interferometer to get the relative phase $\theta$. For simplicity, we assume that the photon is input from the left of the waveguide-a, scattered sequentially by the atomic mirror-A (located at $x_0$) and waveguide-b (located at $x_1$), and then is detected by one of the four single-photon detectors at the out ports. Physically, the probability of the photon propagating along each of the paths could be controlled by the system parameters typically e.g., the photon-atom detuning and the dissipation rates [39,40]. Interestingly, the quantum interference of the single photon when it transports through the interferometer is useful to improve the sensitivity for the estimation of the phase $\theta$.

Fig. 1. A simplified waveguide Mach-Zehnder interferometer configuration. Here, the two-level atomic mirrors-A and -B, located at $x_0$ and $x_2$, act as a pair of beam splitters of the photon. A phase shifter located at $x_1$ is introduced to generate a phase shift $\theta$ of the transporting photon. A single photon in Fock state injected from the waveguide-A is firstly scattered by the atom-A (with the eigenfrequency $\omega _0$) and then scattered by the atom-B for interference measurements with the single-photon detectors at the output ports.

Download Full Size | PDF

In real space, the proposed interferometer can be described by the following Hamiltonian [39]: $\hat {H}_s=\hat {H}_{in}+\hat {H}_{\theta }$, with

(1)$$\begin{aligned}\hat{H}_{in}&=\sum_{j=a,b}({-}iv_g)\int dx\left(C_{Rj}^{{\dagger}}(x)\frac{\partial}{\partial x}C_{Rj}(x)-C_{Lj}^{{\dagger}}(x)\frac{\partial}{\partial x}C_{Lj}(x)\right)\\ +& \sum_{j=a,b}\sum_{n=0,2}\int dx\delta(x-x_n)V_{n j}\left(C_{Rj}^{{\dagger}}(x)\sigma_{n}^-{+}C_{Lj}^{{\dagger}}(x)\sigma_{n}^-{+}H.c.\right)+\sum_{n=0,2}(\omega_n-i\gamma_n)\sigma_{n}^+\sigma_{n}^- \end{aligned}$$

describing the interferometer for a single-photon propagating in it. While,

(2)$$\hat{H}_\theta ={-}\int dx V_\theta\delta(x-x_1)\Big(C_{Ra}^{{\dagger}}(x)C_{Ra}(x)+C_{La}^{{\dagger}}(x)C_{La}(x)\Big)$$

describes the interaction between the photon and the phase shifter, which might specifically be related to the difference between the length of the two waveguides. Physically, this length difference (to be estimated) results in the different delays of the propagating photon and thus induces a relative phase. Above, $\gamma _{n}$ is the dissipation rate of the atom-A/B, $V_{0j}$ ($V_{2j}$) is the coupling strength between the atom-A (-B) and the waveguide-j ($j=a,b$), and $\sigma _{0}^{\pm }$ ($\sigma _{2}^{\pm })$ are the raising and lowering ladder operators of the atoms. The generic stationary state of the system can be expressed as:

(3)$$|\psi\rangle=\left\{\sum_{j=a,b}\int dx\Big(\phi_{Rj}(x)C_{Rj}^{{\dagger}}(x)+\phi_{Lj}(x)C_{Lj}^{{\dagger}}(x)\Big)+\sum_{n=0,2}e_n\sigma_{n}^{+}\right\}|\emptyset\rangle,$$

with $|\emptyset \rangle$ being the ground state of the system, wherein the atoms are in the ground states and there is no photon in the waveguides. Above, $e_k (k=0,2)$ are the excited amplitudes of the atoms, $\phi _{Ra}(x)\,(\phi _{Rb}(x))$ and $\phi _{La}(x),(\phi _{Lb}(x))$ denote the probability amplitudes of the photon propagating right and left along the waveguide $a\, (b)$. Specifically, these probability amplitudes can be further expressed as

(4)$$\left\{\begin{array}{l} \phi_{Ra}(x)=e^{iqx}\Big(f(x_0)+f({-}x_{2})t_a+f({-}x_0)f(x_{1})t_{01a}+f({-}x_1)f(x_{2})t_{12a}\Big),\\ \phi_{La}(x)=e^{{-}iqx}\Big(f(x_0)r_a+f({-}x_0)f(x_{1})r_{01a}+f({-}x_1)f(x_{2} )r_{12a}\Big),\\ \phi_{Rb}(x)=e^{iqx}\Big(f({-}x_0)f(x_2)t_{02b}+f({-}x_{2})t_b\Big),\\ \phi_{Lb}(x)=e^{{-}iqx}\Big(f(x_0)r_b+f({-}x_0)f({-}x_{2})r_{02b}\Big). \end{array}\right.$$

Here, $f(x_n)=\theta (x_n-x),f(-x_n)=\theta (x-x_n)$ denotes the step functions, $t_{nka}$ ($t_{nkb}$) and $t_a$ ($t_b$) represent the transmitted amplitudes of the photon in the waveguide-$a$ (-b) in $(x_n,x_k)$ and $x>x_2$, respectively. Analogously, $r_{nka}(r_{nkb})$ and $r_a$ ($r_b$) present the reflected amplitudes of the photon in the waveguide-$a$ (-b) in $(x_{n},x_k)$ and $x< x_0$, respectively.

Substituting Eqs. (4) and (3) into the Schrödinger equation: $\hat {H}_{S}|\psi \rangle =\omega |\psi \rangle$, we obtain

(5)$$\left\{\begin{array}{l} -iv_g\partial_x\phi_{Ra}(x)+V_{0a}g(x_0)e_0+V_{2a}g(x_2)e_{2}-V_\theta g(x_1)\phi_{Ra}(x)=\omega\phi_{Ra}(x), \\ iv_g\partial_x\phi_{La}(x)+V_{0b}g(x_0)e_0+V_{2b}g(x_{2})e_{2}-V_\theta g(x_1)\phi_{La}(x)=\omega\phi_{La}(x), \\ -iv_g\partial_x\phi_{Rb}(x)+V_{0a}g(x_0)e_0+V_{2a}g(x_2)e_{2}=\omega\phi_{Rb}(x), \\ iv_g\partial_x\phi_{Lb}(x)+V_{0b}g(x_0)e_0+V_{2b}g(x_{2})e_{2}=\omega\phi_{Lb}(x), \\ V_{0a}\phi_a(x_0)+V_{0b}\phi_b(x_0)+(\omega_0-i\gamma_0)e_0=\omega e_0, \\ V_{2a}\phi_a(x_{2})+V_{2b}\phi_b(x_{2})+(\omega_2-i\gamma_2)e_{2}=\omega e_{2}, \end{array}\right.$$

where $g(x_n)=\delta (x_n-x)$, $\phi _a(x)=\phi _{Ra}(x)+\phi _{La}(x)$, and $\phi _b(x)=\phi _{Rb}(x)+\phi _{Lb}(x)$. Next, by substituting $\partial _x\phi _{Ra}(x)$, $\partial _x\phi _{Rb}(x)$, $\partial _x\phi _{La}(x)$ and $\partial _x\phi _{Lb}(x)$ into the Eqs. (5), we can obtain a set of algebraic equations. By solving these algebraic equations set, the transmitted and reflected amplitudes of the photons at the output ports can be expressed as

(6)$$\left\{\begin{array}{l} t_a=\frac{-1}{B}\left[\alpha_0\alpha_2\beta_0\beta_2+(\beta_0^2\beta_2^2+i\beta_0^2\Delta_2v_g+ i\beta_2^2\Delta_0v_g)e^{i\theta} -(\alpha_0\alpha_2e^{i\theta}+\beta_0\beta_2)e^{i(2\theta+\theta_2)}\right],\\ r_a=\frac{1}{B}\left[\alpha_0^2(\alpha_2^2\!+\!\beta_2^2\!+\!i\Delta_2v_g)\!-\!i\alpha_2^2\Delta_0v_ge^{i(\theta+\theta_2)}\!-\!2\alpha_0\alpha_2\beta_0\beta_2e^{i(\theta+2\theta_2)} \!+\!\alpha_2^2(\beta_0^2-\alpha_0^2)e^{2i(\theta+\theta_2)}\right],\\ t_b=\frac{1}{B}\left[\alpha_0\beta_0(\alpha_2^2-i\Delta_2v_g)-i\alpha_2 \beta_2 e^{{-}i\theta_2}\Delta_0v_g+\beta_0^2\alpha_2\beta_2(1-e^{2\theta_2})e^{i\theta} -\alpha_0\beta_0\alpha_2\beta_2e^{2i(\theta+\theta_2)}\right],\\ r_b=\frac{1}{B}\left[\alpha_0\beta_0(\alpha_2^2+\beta_2^2+i\Delta_2v_g-2\beta_2^2e^{2i\theta_2})+i\alpha_0\beta_2\Delta_0v_ge^{i(\theta+2\theta_2)}-\alpha_0\beta_0 \alpha_2^2e^{2i(\theta+\theta_2)}\right], \end{array}\right.$$

where

(7)$$B={-}(\alpha_0^2+\beta_0^2+i\Delta_0v_g)(\alpha_2^2+\beta_2^2+i\Delta_2v_g) +(\alpha_0\alpha_2e^{i(\phi+\theta_2)}+\beta_0\beta_2e^{i\theta_2})^2.$$

Above we have let $\alpha _{j}=V_{ja}$, $\beta _{j}=V_{jb}$ and $\Delta _j= (\omega _j-\omega -i\gamma _j)$ for brevity, and $\theta =\arctan (V_\theta /v_g)\approx V_\theta /v_g$ for $\theta \ll 1$. Simply, if the atomic mirrors are identically coupled to both waveguides and resonant with the waveguide photon, i.e., $V_{jx}=V$ and $\Delta _j=i\gamma _j=i\gamma$ ($j=0,2; x=a,b$), then the transmitted and reflected amplitudes of the photons of Eq. (6) can be simplified as

(8)$$\left\{\begin{array}{l} t_a = \lambda_0\Big(e^{2 i \theta_{2}}\left(e^{2 i \theta}+e^{i\theta}\right)-(1+\kappa)^{2} e^{i \theta}-1\Big),\\ t_b =\lambda_0\left(e^{i \theta}+1\right)\left(1+\kappa-e^{2 i \theta_{2}}e^{i \theta}\right),\\ r_a =\lambda_0\Big(2+\kappa-e^{2 i \theta_{2}}\left(2 e^{i \theta}-\kappa e^{2 i \theta}\right)\Big),\\ r_b =\lambda_0\Big({2+\kappa-e^{2 i \theta_{2}}\left(1-\kappa e^{i \theta}+e^{2 i \theta}\right)}\Big), \end{array}\right.$$

with $\lambda _0=1/\left ((\mathrm {e}^{i\theta }+1)^2e^{2i\theta _2}-4(1+\kappa /2)^2\right )$, and $\kappa =v_g\gamma /V^2$. Consequently, the transmitted and reflected probabilities can be calculated by Eqs. (12) and (14) which depend on the value of phase $\theta$ and $\theta _2$. Although similar results were also obtained in Refs. [52,53], we investigate alternatively how the sensitivity of the phase estimation, i.e. $\Delta \theta$, can be achieved by using the MZI presented above, and then discuss how the dissipations of the atomic mirrors influence on the sensitivity of their relative position measurement.

3. Sensitive phase estimations with the waveguide MZI

In the present waveguide MZI, the measurements are implemented by using the single-photon detectors, which have been well developed in recent years (see, e.g., [54,55]), at the outports. Different from the optical MZI in free space, here the single photon inputs from one port of the interferometer can output at one of the four ports. Therefore, the measurement signals are the desired photon counts, which can be generally expressed as:

(9)$$\langle\hat{M}_d\rangle=\langle\psi|C_{S}^{(j)\dagger}(x_d)C_{S}^{(j)}(x_d) |\psi\rangle=|\psi^{j}_{S}(x_d)|^2,\quad j=a,b$$

where $\psi ^{a}_{S}(x)=\phi _{S}(x)$, $\psi ^{b}_{S}(x)=\tilde {\phi }_{S}(x)$, with $S=R$ for $x_d>x_2$ and $S=L$ for $x_d<x_0$, respectively. The measured uncertainty of the photon accounts can be expressed as $\Delta M_d=\sqrt {\langle {\hat {M}_d^2}\rangle -\langle \hat {M}_{d}\rangle ^2}$, due to the statistical nature of the photon detections. For the present single-photon inputs such an uncertainty reads: $\Delta M_d=\sqrt {|\psi ^{j}_S(x)|^2-|\psi ^{j}_S(x)|^4}$. Therefore, the sensitivity of the phase estimation can be derived through the error-propagating formula

(10)$$\Delta\theta=\frac{\sqrt{|\psi^{j}_{S}(x)|^2-|\psi^{j}_{S}(x)|^4}}{d|\psi^{j}_{S}(x)|^2/d\theta},$$

From Eq. (4), we have $\psi ^a_{R}(x_d)=t_ae^{iqx_d}$, $\psi ^b_{R}(x_d)=t_be^{iqx_d}$ and $\psi ^a_{L}(x_d)=r_ae^{-iqx_d}$ and $\psi ^b_{L}(x_d)=r_be^{-iqx_d}$, and thus $|\psi ^{a}_{R}(x_d)|^2=|t_a|^2$, $|\psi ^{b}_{R}(x_d)|^2=|t_b|^2$, $|\psi ^{a}_{L}(x_d)|^2=|r_a|^2$ and $|\psi ^{b}_{L}(x_d)|^2=|r_b|^2$. Therefore, with the measurements of the transmitted and reflected probabilities of the photon at the output ports, the uncertainty of the phase estimation can be expressed as

(11)$$\Delta\theta=\frac{\sqrt{S_k-S_k^2}}{|dS_k/d\theta|},\quad S=T,R; k=a,b,$$

for the transmitted measurement $T_{a(b)}=|t_a|^2 (|t_b|^2)$ and reflected measurement $R_{a(b)}=|r_a|^2 (|t_b|^2)$, respectively.

3.1 Optimal sensitivity

First, the dissipations of the atomic mirrors are neglected and their eigenfrequencies and the strengths coupled to the waveguides are assumed to be identical, $\omega _0=\omega _{2}=\omega _e$ and $\gamma _0=\gamma _2=0$. With Eq. (8), we have

(12)$$\left\{\begin{array} {l} T_a = \lambda_1\Big (1+\cos(\theta)\Big)\Big(1-\cos(\theta+2\theta_2)\Big),\\ R_a =2\lambda_1\Big(1-\cos(\theta+2\theta_2)\Big),\\ R_b = \dfrac{\lambda_1}{2}\Big(3-2\cos(2\theta_2)-2\cos(2\theta+2\theta_2)+\cos(2\theta)\Big), \end{array} \right.$$

for the resonant photon (i.e., $\omega _e=\omega$). Here, $\lambda _1=1/\Big (\big [3+\cos (\theta )\big ]^2-4\big [\cos (\theta _2)+\cos (\theta +\theta _2)\big ]^2\Big )$, and $T_b=T_a$.

Figure 2 shows how the transmitted and reflected probabilities vary with the relative phase $\theta$ for $\theta _2=qx_2$. It is seen that the value of $\theta _2$, which can be precisely chosen by properly setting the position $x_2$ of the atoms, sensitively influence the spectra of the photon transmissions. A peak of $R_b$ (denoted by peak-A) and dips of $R_{a}$ and $T_{a/b}$ (denoted by dips-A) are observed at $\theta =2n\pi -2\theta _2, n\in Z$. Interestingly, the peak-A and dips-A become sharper and narrower if $\theta _2\rightarrow 0$.

Fig. 2. The transmitted and reflected probabilities of the photon at the output ports of the waveguide MZI for the estimation of the $\theta _2=\pi /2 (a),\,~0.1\pi (b),\,~0.04\pi (c)$, and $~0.01\pi$ (d), respectively. Here, the photon is supposed to be resonant with the atomic mirrors without any dissipation.

Download Full Size | PDF

Certainly, if the peak-A or dips-A is the sharper, the high precision for the estimated phase $\theta$ can be obtained. By substituting Eq. (12) into Eq. (10), we get specifically the sensitivity of the relevant phase estimation. With the estimation of phase $\theta$ (due to such as the length difference of the waveguides), the relative position $(x_2=\theta _2/q)$ between the atomic mirrors in such an interferometer can be estimated effectively. This implies that the proposal could be used to detect certain weak signals, typically e.g., the optomechanical one [56] resulting in the position perturbations of the atomic mirrors. One can see in Fig. 3, that the sensitivity of the estimation is optimal if $\theta =2n\pi -2\theta _2$. Figure 3(a) shows that, the optimal sensitivity by using the peak-A (denoted by $\Delta \theta _{R_b}(\theta _2)$) is a little better than that by using the dips-B (denoted by $\Delta \theta _{tr}(\theta _2)$). In fact, this can be verified by the relevant analytical calculation, i.e., for $\theta _2\ll 1$, we have

(13)$$\Delta\theta_{R_b}(\theta_2)\approx \frac{\theta_2^2}{\sqrt{3}},\, \Delta\theta_{tr}(\theta_2)\approx \theta_2^2.$$

Therefore, the sensitivity of the phase estimation can achieve theoretically any high, as long as $\theta _2$ is sufficiently small as shown in Fig. 3(b-c). The smaller value of $\theta _2$ corresponds to the narrower of the observed dips, which means that the regime of $\theta$ for the optimal sensitivity also become smaller as can be seen clearly on Fig. 3(c). Of course, this is a common problem that is encountered in quantum metrology, which might be resolved by shifting the parameter to approach sufficiently close to the optimal regime via the quantum feedback control [57,58]. Additionally, the presence of decoherence of the system would also limit the realizable sensitivity in the practical interferometer.

Fig. 3. (a) The sensitivity of the phase estimation by probing the $T_{a/b}$- and $R_{a/b}$ parameters for $\theta _2=\pi /10$ ; (b) The sensitivities implemented by the $R_b$ measurements for the different values of $\theta _2$-parameter. Here, the black-dotted line for $\Delta \theta _{R_b}(\theta _2)$ describes the optimal sensitivity for $\theta _2\ll 1$; (c) The contour plot of the sensitivity $\Delta \theta _{R_b}$ changes with the $\theta$ and $\theta _2$ parameters.

Download Full Size | PDF

3.2 Dissipation effect

Next, let us consider the effects of the dissipations of the atomic mirrors [39,40], i.e., $\gamma =\kappa V^2/v_g\neq 0$ ($\kappa$ is a constant). In this case, the transmitted and reflected probabilities of Eq. (12) become

(14)$$\left\{\begin{array}{l} T_a^k = \lambda_\kappa\Big(3+(1+\kappa)^4+2\big[(\kappa+1)^2+1\big]\big[\cos(\theta)-\cos(\theta+2\theta_2)\big]-2\cos(2\theta+2\theta_2) \\ \qquad-2(1+\kappa)^2\cos(2\theta_2)\Big), \\ R_a^k = \lambda_\kappa\Big(2(2+\kappa)^2+4\kappa-4\kappa\cos(\theta)-2(2+\kappa)\big[2\cos(\theta+2\theta_2)+\kappa\cos(2\theta+2\theta_2)\big]\Big), \\ T_b^k = 2\lambda_\kappa\Big(\big[1+(1+\kappa)^2\big]\big[1+\cos(\theta)\big]-(1+\kappa)\big[2\cos(\theta+2\theta_2)+\cos(2\theta+2\theta_2)+\cos(2\theta_2)\big]\Big), \\ R_b^k = \lambda_\kappa \Big((2+\kappa)^2+\kappa^2+2-2(\kappa+2)\big[\cos(2\theta_2)+\cos(2\theta+2\theta_2)+\kappa\cos(\theta+\theta_2)\Big] \\ \qquad-4\kappa\cos(\theta)+2\cos(2\theta)\Big), \end{array}\right.$$

with $\lambda _\kappa =1/4\left (\big [(1+\kappa /2)^2+1+\cos (\theta )\big ]^2-4(1+\kappa /2)^2\big [\cos (\theta _2)+\cos (\theta +\theta _2)\big ]^2\right )$. Correspondingly, the sensitivity of the phase estimation is modified as

(15)$$\Delta\theta=\frac{\sqrt{S_j^\kappa-(S_j^{\kappa})^2}}{|dS_j^\kappa/d\theta|},S=T,R;j=a,b,$$

which is really related to the value of the dissipation rate $\gamma$, with $T_{a(b)}^\kappa =|t_{a(b)}^{\kappa }|^2$ and $R_{a(b)}^\kappa =|r_{a(b)}^\kappa |^2$. It is seen schematically shown in Figs. 4(a) and 4(b) that, in this case the peak-A and dips-A become flatter and thus the phase estimated sensitivity decreases with the increase of the dissipations of the atomic mirrors. Also, the dips of the sensitivities are split at $\theta =2n\pi -2\theta _2$, showing that the sensitivity for the estimated phase $\theta =2n\pi -2\theta _2$ decreases quickly. However, one can see that the peak-A and dips-A are still sufficiently sharp for $\kappa \ll 1$, implying that the optimal sensitivity can still be realizable for $\theta \approx 2n\pi -2\theta _2$, as shown in Figs. 4(c) and 4(d).

Fig. 4. The transmitted/reflected probability and the sensitivity of the phase estimation versus the atomic dissipation $\kappa =\gamma v_g/V^2$. Here, $T_s=T_a^\kappa +T_b^\kappa +R_a^\kappa +R_b^\kappa$.

Download Full Size | PDF

3.3 Compared with the waveguide FPI

Thirdly, let us compare the realizable sensitivity of the proposed waveguide MZI with that by using the waveguide FPI demonstrated in Refs. [23,25], which can be generated by simplifying the present MZI. In fact, if the atom-A and -B are coupled to only one of the two waveguides, typically such as let $V_{0b}=V_{2b}=0$, the proposed MZI reduces to the waveguide FPI. As a consequence, the transmitted probability of the photon can be simplified as [23]

(16)$$T_a^s=\frac{\kappa^4}{\big[(1-\kappa)^2-1\big]^2+4(1-\kappa)^2\sin^2(\theta_2+\theta)},$$

for $V_{0a}=V_{2a}=V$, $\Delta _0=-i\gamma$ and $\kappa =\gamma v_g/V^2$. Similarly, with the error-propagation formula in Eq. (10), the corresponding sensitivity of the phase estimation can be calculated [23].

From Fig. 5, one can see that, for the same parameter condition the optimal sensitivity realized by the two-waveguide MZI is much higher than that by using the single-waveguide FPI. Also, the regime of the estimated phase $\theta$ by using the MZI, approaching the optimal sensitivity, is much larger than that by using the FPI. Physically, these improvements of the sensitive phase estimation is benefitted from that the MZI provides more quantum interference paths.

Fig. 5. Compared the sensitivities of the phase estimation by using the proposed waveguide MZI (the blue dotted line) with that by using the waveguide FPI (the red line).

Download Full Size | PDF

4. Conclusions and discussions

In summary, we proposed a waveguide MZI to implement the desired quantum metrology for sensitive phase estimation. Different from the optical MZIs in free space, here a pair of atomic mirrors are utilized to implement the photonic beam-splitters and the single-photon detectors are used to implement the measurements of the outputs. Due to the quantum interference, the optimal sensitivity of the phase estimation depends strongly on the relative locations of the two atomic mirrors. As a consequence, by precisely setting the distance between the atomic mirrors, the desired sensitive phase estimation can be realized. We have also discussed how the dissipations of the atomic mirrors influence on the realizable optimal sensitivity. Inversely, by precisely setting the value of the phase estimation $\theta$, the proposed MZI could also be utilized to precisely detect the distance between the atomic mirrors for implementing the sensitive measurements of the distance.

Given the sufficiently small dissipations of the atomic mirrors can be engineered experimentally and also the significantly strong photon-atom couplings can be demonstrated by the current integrated optical technique, it is believed that the proposed waveguide interferometer is feasible and could also be applied to implement various desirable ultra-sensitive sensing of the weak signals. Furthermore, although the waveguide MZI proposed here focuses only on the single-photon Fock state input, the generalization to the waveguide MZI with the other quantum states typically such as the coherent state, squeezed states, and even the entangled photons, etc. is possible and will be discussed elsewhere. Certainly, in addition to the atomic dissipation, in fact, other noises such as the dispersions of the photons, can also affect the measurement sensitivity of the system. For example, the dispersion may lead to the broadening of the pulse packets and limit the visibility of the proposed quantum interference. However, following the Refs. [59,60] the method called the dispersion cancellation can be utilized to suppress the probable nonlinear dispersions. Anyway, benefitted from the strong quantum interference effects, quantum metrology with the waveguide photonic interferometer is particularly desirable.

Funding

Chengdu Science and Technology Program (2021-YF05-02421-GX); National Natural Science Foundation of China (11974290); National Key Research and Development Program of China (2021YFA0718803).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. J. Peng, S. Jia, J. Bian, S. Zhang, J. Liu, and X. Zhou, “Recent Progress on Electromagnetic Field Measurement Based on Optical Sensors,” Sensors 19(13), 2860 (2019). [CrossRef]

2. D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007). [CrossRef]

3. V. D’Ambrosiol, N. Spagnolo, L. D. Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013). [CrossRef]

4. M. Hiekkamäki, F. Bouchard, and R. Fickler, “Photonic Angular Superresolution Using Twisted N00N States,” Phys. Rev. Lett. 127(26), 263601 (2021). [CrossRef]

5. B. P. Abbott, R. Abbott, T. D. Abbott, et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016). [CrossRef]

6. Y. Ma, H. Miao, B. H. Pang, M. Evans, C. Zhao, J. Harms, R. Schnabel, and Y. Chen, “Proposal for gravitational-wave detection beyond the standard quantum limit through EPR entanglement,” Nat. Phys. 13(8), 776–780 (2017). [CrossRef]

7. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007). [CrossRef]

8. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011). [CrossRef]

9. L. Pezzè and A. Smerzi, “Quantum metrology with nonclassical states of atomic ensembles,” Rev. Mod. Phys. 90(3), 035005 (2018). [CrossRef]

10. C. Lee, “Adiabatic Mach-Zehnder Interferometry on a Quantized Bose-Josephson Junction,” Phys. Rev. Lett. 97(15), 150402 (2006). [CrossRef]

11. O. Hosten, N. J. Engelsen, R. Krishnakumar, and M. A. Kasevich, “Measurement noise 100 times lower than the quantum-projection limit using entangled atoms,” Nature 529(7587), 505–508 (2016). [CrossRef]

12. P. M. Anisimov, G. M. Raterman, A. C. W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010). [CrossRef]

13. L. Pezzé and A. Smerzi, “Ultrasensitive two-mode interferometry with single-mode number squeezing,” Phys. Rev. Lett. 110(16), 163604 (2013). [CrossRef]

14. B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015). [CrossRef]

15. S. Cialdi, E. Suerra, S. Olivares, S. Capra, and M. G. A. Paris, “Squeezing Phase Diffusion,” Phys. Rev. Lett. 124(16), 163601 (2020). [CrossRef]

16. X. N. Feng, J. Huang, L. F. Wei, and C. Lee, “Squeezing-enhanced parameter estimation with a hybrid spin-oscillator interferometer,” Phys. Rev. A 106(2), 022413 (2022). [CrossRef]

17. Y.-Q. Zou, L.-N. Wu, Q. Liu, X.-Y. Luo, S.-F. Guo, J.-H. Cao, M. K. Tey, and L. You, “Beating the classical precision limit with spin-1 Dicke states of more than 10,000 atoms,” Proc. Natl. Acad. Sci. U. S. A. 115(25), 6381–6385 (2018). [CrossRef]

18. K. Xu, Y.-R. Zhang, Z.-H. Sun, H. Li, P. Song, Z. Xiang, K. Huang, H. Li, Y.-H. Shi, C.-T. Chen, X. Song, D. Zheng, F. Nori, H. Wang, and H. Fan, “Metrological Characterization of Non-Gaussian Entangled States of Superconducting Qubits,” Phys. Rev. Lett. 128(15), 150501 (2022). [CrossRef]

19. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453(7198), 1008–1015 (2008). [CrossRef]

20. J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of Quantum Noise of Motion in a Micromechanical Resonator,” Phys. Rev. Lett. 115(24), 243601 (2015). [CrossRef]

21. S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock, “Quantum amplification of mechanical oscillator motion,” Science 364(6446), 1163–1165 (2019). [CrossRef]

22. G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single Atom as a Mirror of an Optical Cavity,” Phys. Rev. Lett. 107(13), 133002 (2011). [CrossRef]

23. F. Fratini, E. Mascarenhas, L. Safari, J.-P. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot Interferometer with Quantum Mirrors: Nonlinear Light Transport and Rectification,” Phys. Rev. Lett. 113(24), 243601 (2014). [CrossRef]

24. D. E. Chang, L. Jiang, A. V. Gorshkov, and H. J. Kimble, “Cavity QED with atomic mirrors,” New J. Phys. 14(6), 063003 (2012). [CrossRef]

25. M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle, A. J. Keller, A. A.-Garcia, D. E. Chang, and O. Painter, “Cavity quantum electrodynamics with atom-like mirrors,” Nature 569(7758), 692–697 (2019). [CrossRef]

26. J. Belfi and F. Marin, “Sensitivity below the standard quantum limit in gravitational wave detectors with Michelson-Fabry-Perot readout,” Phys. Rev. D 77(12), 122002 (2008). [CrossRef]

27. C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80(4), 043822 (2009). [CrossRef]

28. Y. Tsujiie and Y. Kawamura, “Micro Fabry-Pérot Interferometer at Rayleigh Range,” Sci. Rep. 8(1), 15193 (2018). [CrossRef]

29. C. Valagiannopoulos, “Quantum Fabry-Perot Resonator: Extreme Angular Selectivity in Matter-Wave Tunneling,” Phys. Rev. Appl. 12(5), 054042 (2019). [CrossRef]

30. W. Li, J. Du, R. Wen, G Li, and T. Zhang, “Quantum-enhanced metrology based on Fabry-Perot interferometer by squeezed vacuum and non-Gaussian detection,” J. Appl. Phys. 115(12), 123106 (2014). [CrossRef]

31. N. M. R. Hoque and L. Duan, “A Mach-Zehnder Fabry-Perot hybrid fiber-optic interferometer operating at the thermal noise limit,” Sci. Rep. 12(1), 12130 (2022). [CrossRef]

32. B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. F. Kockum, J. Braumller, D. K. Kim, M. Kjaergaard, P. Krantz, A. Melville, B. M. Niedzielski, A. Vepslinen, R. Winik, J. L. Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, “Waveguide quantum electrodynamics with superconducting artificial giant atoms,” Nature 583(7818), 775–779 (2020). [CrossRef]

33. Y. Zhou, Z. Chen, and J.-T. Shen, “Single-photon superradiant emission rate scaling for atoms trapped in a photonic waveguide,” Phys. Rev. A 95(4), 043832 (2017). [CrossRef]

34. N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019). [CrossRef]

35. T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Experimental Realization of Highly Efficient Broadband Coupling of Single Quantum Dots to a Photonic Crystal Waveguide,” Phys. Rev. Lett. 101(11), 113903 (2008). [CrossRef]

36. M. K. Bhaskar, D. D. Sukachev, A. Sipahigil, R. E. Evans, M. J. Burek, C. T. Nguyen, L. J. Rogers, P. Siyushev, M. H. Metsch, H. Park, F. Jelezko, M. Lonar, and M. D. Lukin, “Quantum Nonlinear Optics with a Germanium-Vacancy Color Center in a Nanoscale Diamond Waveguide,” Phys. Rev. Lett. 118(22), 223603 (2017). [CrossRef]

37. J.-T. Shen and S. Fan, “Strongly Correlated Two-Photon Transport in a One-DimensionalWaveguide Coupled to a Two-Level System,” Phys. Rev. Lett. 98(15), 153003 (2007). [CrossRef]

38. L. Zhou, Z. R. Gong, Y.-X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]

39. J.-T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79(2), 023837 (2009). [CrossRef]

40. C. H. Yan, W. Z. Jia, and L. F. Wei, “Controlling single-photon transport with three-level quantum dots in photonic crystals,” Phys. Rev. A 89(3), 033819 (2014). [CrossRef]

41. X. Li and L. F. Wei, “Designable single-photon quantum routings with atomic mirrors,” Phys. Rev. A 92(6), 063836 (2015). [CrossRef]

42. L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum Routing of Single Photons with a Cyclic Three-Level System,” Phys. Rev. Lett. 111(10), 103604 (2013). [CrossRef]

43. S. He, Q. He, and L. F. Wei, “Atomic-type photonic crystals with adjustable band gaps,” Opt. Express 29(26), 43148–43163 (2021). [CrossRef]

44. S. Itay, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan., “All-optical routing of single photons by a one-atom switch controlled by a single photon,” Science 345(6199), 903–906 (2014). [CrossRef]

45. M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015). [CrossRef]

46. W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112(16), 160801 (2014). [CrossRef]

47. A. Belsley, E. J. Allen, A. Datta, and J. C. F. Matthews, “Advantage of Coherent States in Ring Resonators over Any Quantum Probe Single-Pass Absorption Estimation Strategy,” Phys. Rev. Lett. 128(23), 230501 (2022). [CrossRef]

48. R. Henze, T. Seifert, J. Ward, and O. Benson, “Tuning whispering gallery modes using internal aerostatic pressure,” Opt. Lett. 36(23), 4536–4538 (2011). [CrossRef]

49. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10(9), 095015 (2008). [CrossRef]

50. A. Schliesser, O. Arcizet, R. Rivière, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5(7), 509–514 (2009). [CrossRef]

51. W. Cheng, Z. Wang, and X. Wang, “Quantum interferometry for rotation sensing in an optical microresonator,” Phys. Rev. A 105(2), 023716 (2022). [CrossRef]

52. N. Almeida, T. Werlang, and D. Valente, “Mach-Zehnder interferometer with quantum beamsplitters,” J. Opt. Soc. Am. B 36(12), 3357–3363 (2019). [CrossRef]

53. G. S. Paraoanu, “Fluorescence interferometry,” Phys. Rev. A 82(2), 023802 (2010). [CrossRef]

54. W. H. P. Pernice, C. Schuck, O. Minaeva, M. Li, G. N. Goltsman, A. V. Sergienko, and H. X. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat. Commun. 3(1), 1325 (2012). [CrossRef]

55. K. Inomata, Z. Lin, K. Koshino, W. D. Oliver, J.-S. Tsai, T. Yamamoto, and Y. Nakamura, “Single microwave-photon detector using an artificial Λ-type three-level system,” Nat. Commun. 7(1), 12303 (2016). [CrossRef]

56. M. Aspelmeyer, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]

57. D. W. Berry and H. M. Wiseman, “Optimal States and Almost Optimal Adaptive Measurements for Quantum Interferometry,” Phys. Rev. Lett. 85(24), 5098–5101 (2000). [CrossRef]

58. H. Yuan and C.-H. F. Fung, “Optimal Feedback Scheme and Universal Time Scaling for Hamiltonian Parameter Estimation,” Phys. Rev. Lett. 115(11), 110401 (2015). [CrossRef]

59. D.-G. Im, Y. Kim, and Y.-H. Kim, “Dispersion cancellation in a quantum interferometer with independent single photons,” Opt. Express 29(2), 2348–2363 (2021). [CrossRef]

60. A. Mandilara, C. Valagiannopoulos, and V. M. Akulin, “Classical and quantum dispersion-free coherent propagation by tailoring multimodal coupling,” Phys. Rev. A 99(2), 023849 (2019). [CrossRef]

Waveguide Mach-Zehnder interferometer to enhance the sensitivity of quantum parameter estimation

Abstract

1. Introduction

2. Waveguide Mach-Zehnder interferometer

3. Sensitive phase estimations with the waveguide MZI

3.1 Optimal sensitivity

3.2 Dissipation effect

3.3 Compared with the waveguide FPI

4. Conclusions and discussions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (16)

Optics Express