Sub-shot-noise-limited phase estimation via single-mode inputs

Jian-Dong Zhang; Chenglong You; Shuai Wang

doi:10.1364/OE.474643

1. Introduction

Parameter estimation was and is a major issue in the field of high precision measurements. In this regard, optical interferometers come across as a kind of indispensable platform, which can be used to measure many physical quantities, temperature [1–3], concentration [4,5], and angular displacement [6–8], to name a few. They function by converting the measured physical quantity to an unknown phase and by estimating the phase. As an important performance metric in optical interferometry, phase sensitivity is tied to the minimal variance on phase that can be distinguished by the measurement scheme. Related to this, $1/\sqrt {\cal N}$ and $1/{\cal N}$ are referred to as the shot-noise limit and the Heisenberg limit, where $\cal N$ is the average photon number inside the interferometer. With the development of precision measurements, comes the need for further improvement on phase sensitivity. In this context, quantum phase estimation, which utilizes quantum resources to achieve the phase sensitivity beyond classical approaches, emerges as the times require.

In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer cannot surpass the shot-noise limit with using single-mode inputs [9]. Since then, there has been an enormous interest in enhancing phase sensitivity with exotic two-mode inputs, such as two-mode squeezed vacuum states [10], N00N states [11] and entangled coherent states [12]. However, Takeoka et al. have demonstrated that this statement holds if and only if two antisymmetric phase shifts are embedded in two arms [13]. In other words, when the estimated phase only occurs in one of two arms, it is possible for the phase sensitivity to break through the shot-noise limit.

In recent years, with specific measurement schemes, using single-mode inputs to achieve sub-shot-noise-limited phase sensitivity has received a lot of attention. Spagnolo et al. presented a coherent-state-based measurement scheme, of which the phase sensitivity outperforms the shot-noise limit by a factor of $\sqrt 2$ [14]. However, it is worth noting that an optical parametric amplifier was used for the measurement, which could potentially uses more resources. Takeoka et al. reported a measurement scheme in which the phase sensitivity can exceed the Heisenberg limit through the use of a squeezed vacuum state [13,15]. A measurement scheme with sub-shot-noise-limited phase sensitivity is demonstrated by Liu et al., where the input is one of the coherent-light-boosted beams [16]. More recently, with one of two-mode squeezed vacuum states as input, Huang et al. showed that the phase sensitivity can surpass the shot-noise limit [17]. Zhang et al. proposed a measurement scheme with the phase sensitivity twice the shot-noise limit when the input is a displaced squeezed state [18].

The above studies show attractive prospects since phase estimation with single-mode inputs is convenient in some scenarios. The relevant schemes can circumvent phase matching and photon number matching that may be encountered by schemes using two-mode inputs. This advantage motivates us to investigate phase estimation with single-mode inputs. In this paper, we further address this subject in terms of the quantum Fisher information (QFI) analysis. The optimal input is determined by comparing the QFIs of various states. We provide the specific measurement scheme for the optimal input, which can give phase sensitivity saturated with the sensitivity limit. On this foundation, we consider the effects of several realistic factors on the phase sensitivity and analyze the robustness of measurement scheme against these factors.

The remainder of this paper is organized as follows. Section 2. introduces the phase estimation protocol with single-mode inputs and gives the general expression of the QFI. In Sec. 3, we calculate the QFIs of common states and discuss the optimal input. In Sec. 4, we show the optimal measurement scheme and analyze the effects of several realistic factors on the phase sensitivity. Finally, we summarize our work in Sec. 5.

2. Single-phase estimation protocol with single-mode inputs

Figure 1 shows the phase estimation protocol using single-mode inputs, where solid line and dashes line represent mode $A$ and mode $B$, respectively. A single-mode state along with a vacuum state are incident on a 50:50 beam splitter and then an estimated phase $\theta$ is encoded onto the state in mode $A$. All operations after the estimated phase are regarded as measurement.

Fig. 1. Schematic of phase estimation protocol based on single-mode inputs. A single-mode state is injected into a 50:50 beam splitter along with a vacuum state. Then the state in mode $A$ passes through the estimated phase. Finally, a specific measurement strategy is performed at the output.

Download Full Size | PDF

For the phase estimation protocol given by Fig. 1, the ultimate phase sensitivity limit can be calculated from the single-parameter QFI, which is found to be

(1)$${\cal F} = \langle {{{\hat a}^{\dagger} }\hat a{{\hat a}^{\dagger} }\hat a} \rangle - {\langle {{{\hat a}^{\dagger} }\hat a} \rangle ^2} + \langle {{{\hat a}^{\dagger} }\hat a} \rangle,$$

where ${\hat a}^{\dagger}$ and $\hat a$ are creation and annihilation operators for mode $A$, and expectation values are taken over by state $\left | \psi \right \rangle$. Regarding the same number of photons on average, the state which can maximize the QFI is the optimal input.

In terms of the Mandel Q-parameter defined as

(2)$${\cal Q} = \frac{\langle {{{\hat a}^{\dagger} }\hat a{{\hat a}^{\dagger} }\hat a} \rangle - {\langle {{{\hat a}^{\dagger} }\hat a} \rangle ^2}}{{\langle {{{\hat a}^{\dagger} }\hat a} \rangle }} - 1,$$

the expression of QFI can be rewritten as

(3)$${\cal F} = \left( {{\cal Q} + 2} \right)\langle {{{\hat a}^{\dagger} }\hat a} \rangle.$$

Equation (3) suggests that the maximal QFI can be achieved when Mandel Q-parameter is positive, i.e., the optimal input must obey super-Poissonian distribution. As we shall show, this property is useful in analyzing the QFIs of some cases.

3. Analysis of the optimal single-mode state

In this section, we analyze common single-mode states and identify the optimal input. Let us consider a displaced squeezed number state, which has the following form

(4)$$\left| \psi \right\rangle = \hat D( \alpha )\hat S( \xi )\left| m \right\rangle,$$

where $\hat D( \alpha )$ and $\hat S( \xi )$ are displacement and squeezing operators with $\alpha = |\alpha |e^{i\varphi }$ and $\xi = r e^{i\delta }$.

Further, the average photon number of a displaced squeezed number state is found to be

(5)$${\cal N}_{\rm {DSNS}} = m\cosh ( {2r} ) + {\sinh ^2}r + {\left| \alpha \right|^2}.$$

The first term suggests that the photon number is amplified due to squeezing operation, while the third term suggests that displacement operation has no amplification effect.

Using Eqs. (1) and (5), one can calculate the QFI with a displaced squeezed number state as input,

(6)$${\cal F} = {\left| \alpha \right|^2}\left[ {( {2m + 1} ){e^{2r}} + 1} \right] + ( {4m + 1} ){\sinh ^2}r+ m + ( {2{m^2} + 2} ){\cosh ^2}r{\sinh ^2}r + 2m{\sinh ^4}r,$$

where the phase-matching condition $2\varphi - \delta = \pi$ is used in order to maximize the QFI.

3.1 Displaced squeezed vacuum states

We first direct our attention to the case where $m$ equals 0. At this point, the input is a displaced squeezed vacuum state, also known as a pure Gaussian state. In the light of Eq. (5), the average photon number of a displaced squeezed vacuum state is given by

(7)$${\cal N}_{\rm {DSVS}} = {\sinh ^2}r + {\left| \alpha \right|^2}.$$

Further, we can obtain the QFI according to Eq. (6),

(8)$${\cal F}_{\rm {DSVS}} = ( {{e^{2r}} + 1} ){\left| \alpha \right|^2} + 2{\cosh ^2}r{\sinh ^2}r + {\sinh ^2}r.$$

There are two parameters, $|\alpha |$ and $r$, in Eq. (8); however, there is only one free parameter when the total average photon number is fixed. Based on Eq. (7), we can rewrite the QFI as a function of $r$. By taking the derivative of the QFI with respect to $r$, we have

(9)$$\frac{{\partial {\cal F}_{\rm{DSVS}}}}{{\partial r}} = \frac{{{e^{ - 4r}}}}{2}\left[ {\left( {4{{\cal N}_{\rm{DSVS}}} + 2} \right){e^{6r}} - {e^{8r}} - 1} \right] \ge 0.$$

Equation (9) means that the QFI is a monotonic increasing function of $r$, and the maximal QFI occurs when $|\alpha |$ is equal to 0. Hence, a squeezed vacuum state gives large QFI compared with a displaced squeezed vacuum state or a coherent state. Related to this, the QFI with a squeezed vacuum state as input is given by

(10)$${\cal F}_{\rm {SVS}} = 2{{\cal N}_{\rm{DSVS}}^2} + 3{{\cal N}_{\rm{DSVS}}}.$$

The result reveals that the corresponding phase sensitivity limit is slightly superior to the Heisenberg limit. For high photon numbers, the phase sensitivity surpasses the Heisenberg limit by a factor of $\sqrt 2$.

3.2 Displaced number states and squeezed number states

Here we turn our attention to two kinds of non-Gaussian states, displaced number states and squeezed number states. Both of them can be regarded as the special cases of displaced squeezed number states; the former corresponds to $r = 0$ and the latter corresponds to $|\alpha | = 0$.

For a displaced number state, the average photon number is given by

(11)$${\cal N}_{\rm {DNS}} = m + {\left| \alpha \right|^2},$$

and the corresponding QFI is found to be

(12)$${\cal F}_{\rm {DNS}} = \left( {2m + 2} \right){\left| \alpha \right|^2} + m.$$

This QFI can be rewritten as a function of $m$ using Eq. (11). Further, the derivative of the QFI with respect to $m$ is calculated to be

(13)$$\frac{{\partial {\cal F}_{\rm{DNS}}}}{{\partial m}} = 2{\cal N}_{\rm{DNS}} - 4m - 1.$$

By setting the derivative to 0, we get the condition for achieving the maximal QFI,

(14)$$m = \frac{2{\cal N}_{\rm{DNS}} - 1}{4}.$$

The expression for the maximal QFI is

(15)$$\max[{\cal F}_{\rm {DNS}}] = \frac{4{\cal N}_{\rm{DNS}}^2 + 12{\cal N}_{\rm{DNS}} + 1}{8}.$$

It should be noted that the QFI given by Eq. (15) is unattainable in most cases since $m$ must be a positive integer, which is not satisfied for any ${\cal N}_{\rm {DNS}}$. Fortunately, this QFI is approximately attainable for high photon numbers. At this point, we have $m \simeq |\alpha |^2 \simeq 1/2{\cal N}_{\rm {DNS}}$, thereby resulting in $\max [{\cal F}_{\rm {DNS}}] \simeq {\cal N}_{\rm {DNS}}^2/2$.

Now we move on to squeezed number states. After a simple calculation, the average photon number is given by

(16)$${\cal N}_{\rm {SNS}} = (2m+1) {\sinh ^2}r + m,$$

and the QFI is given by

(17)$${\cal F}_{\rm {SNS}} = ( {4m + 1} ){\sinh ^2}r + m + 2m{\sinh ^4}r + ( {2{m^2} + 2} ){\cosh ^2}r{\sinh ^2}r.$$

Similarly, we rewrite the QFI as a function of $m$. By taking the derivative of the QFI with respect to $m$, we get

(18)$$\frac{{\partial {\cal F}_{\rm{SNS}}}}{{\partial m}} ={-} \frac{{2(1 + 2m + 6{m^2} + 8{m^3} + 4{m^4} + 3{\cal N}_{\rm{SNS}}^{} + 3{\cal N}_{\rm{SNS}}^2)}}{{{{( {1 + 2m} )}^3}}} < 0.$$

This result indicates that the QFI is a monotonic decreasing function of $m$ and the maximal QFI occurs when $m$ equals 0. In other words, the best choice is to input a squeezed vacuum state.

3.3 Schrödinger cat states

Finally, we discuss an important kind of single-mode states–Schrödinger cat states, which have strong nonclassical properties.

In general, a Schrödinger cat state is a superposition of two coherent states with equal amplitude and opposite phase, i.e.,

(19)$$\left| \psi \right\rangle = \frac{1}{{\sqrt {2[ {1 + \exp ( { - 2{{\left| \alpha \right|}^2}} )\cos \phi } ]} }}\left( {\left| \alpha \right\rangle + {e^{i\phi }}\left| { - \alpha } \right\rangle } \right).$$

In this section, we merely consider three important cat states: even coherent states ($\phi = 0$), odd coherent sates ($\phi = \pi$), and Yurke-Stoler coherent states ($\phi = \pi /2$).

As has been previously pointed out, the Mandel Q-parameter is an effective tool in analyzing the QFIs with some inputs. Here we take $\alpha$ real and the Mandel Q-parameter for even coherent states is found to be

(20)$${{\cal Q}_{\rm{ECS}}} = {\left| \alpha \right|^2}\frac{{4\exp ( { - 2{{\left| \alpha \right|}^2}} )}}{{1 - \exp ( { - 4{{\left| \alpha \right|}^2}} )}} > 0.$$

Further, the Mandel Q-parameter for odd coherent states is

(21)$${{\cal Q}_{\rm{OCS}}} ={-}{\left| \alpha \right|^2}\frac{{4\exp ( { - 2{{\left| \alpha \right|}^2}} )}}{{1 - \exp ( { - 4{{\left| \alpha \right|}^2}} )}} < 0$$

and that for Yurke-Stoler coherent states is given by

(22)$${{\cal Q}_{\rm{YSCS}}} = 0.$$

It can be seen that the Mandel Q-parameters for odd and even coherent states approach 0 with the increase of $|\alpha |^2$. For high photon numbers, the QFIs of three cat states tend to be the same,

(23)$${{\cal F}_{\rm{ECS}}} \simeq {{\cal F}_{\rm{OCS}}} \simeq {{\cal F}_{\rm{YSCS}}} = 2\langle {{{\hat a}^{\dagger} }\hat a} \rangle.$$

That is, the corresponding phase sensitivity limit surpasses the shot-noise limit by a factor of $\sqrt {2}$.

3.4 Completely mixed states

All of the previous discussions focus on pure states, here we briefly analyze the QFIs of completely mixed states. For a completely mixed state without off-diagonal terms, $\rho = \sum \nolimits _n {{p_n}\left | n \right \rangle \left \langle n \right |}$, the QFI is calculated to be [19]

(24)$${{\cal F}_{\rm{MS}}} = \sum\nolimits_n {p_n}{\cal F}(\left| n \right\rangle) = \langle {{{\hat a}^{\dagger} }\hat a} \rangle,$$

suggesting that the corresponding phase sensitivity limit is the shot-noise limit. Regarding a more general mixed state, it can be expressed as the superposition of completely pure states and completely mixed states. In terms of the convexity of QFI, the corresponding QFI is between the QFI given by the optimal pure state and that given by completely mixed states (shot-noise limit).

3.5 Discussion

In the previous sections, we showed the phase sensitivity limit given by single-mode inputs. The results indicate that the limit of completely mixed states is bounded by the shot-noise limit whereas the limit of pure states can break the shot-noise limit and even the Heisenberg limit. The optimal choice is to inject a squeezed vacuum state into the interferometer. Indeed, the optimal input is determined through comparison of various states instead of a rigorous proof. We have reason to believe that one can design an exotic state that makes it possible to give larger Mandel Q-parameter than a squeezed vacuum state. However, this state is generally engineered by performing a post-selection on the states we discuss in this paper [20,21]. In other words, the QFI has an amplification effect at the expense of less successful probability of post-selection [22,23]. Therefore, a squeezed vacuum state is the optimal input for deterministic measurements.

4. Specific scheme saturated with the sensitivity limit and analysis of realistic factors

In Sec. 3, we identified that the optimal input is a squeezed vacuum state. Here we show the optimal measurement scheme and analyze the effects of several realistic factors on the phase sensitivity.

4.1 Optimal measurement scheme

In theory, the measurement operator corresponding to the optimal phase sensitivity can be obtained from symmetric logarithmic derivative. However, in many cases, it is difficult to design the specific measurement operation for the operator. In particular, the optimal measurement scheme for squeezed vacuum was shown in Ref. [13]. In fact, this specific measurement scheme is based on inverse operation, another method which in general is the optimal [24]. Specifically, for our scheme, the state before the estimated phase is $\hat U_{\rm BS}\hat S(r)\left | 0 \right \rangle \left | 0 \right \rangle$; hence, one of the optimal measurement operation is $\left \langle 0 \right | \left \langle 0 \right |\hat S(-r)\hat U^{\dagger} _{\rm BS}$. This way can also be used in Ref. [16] to achieve the optimal phase sensitivity better than the current method. Figure 2 illustrates the scheme, of which phase sensitivity approaches the sensitivity limit corresponding to Eq. (10). The whole setup is similar to a degenerate nonlinear interferometer [25,26]. In mode $A$, an anti-squeezer is placed in front of the detector. The squeezing strength of the anti-squeezer equals that of the input, but there is a $\pi$-phase difference in two squeezing processes. When the estimated phase sits at 0, the anti-squeezer will undo what the squeezer does; as a result, two outputs are both vacua.

Fig. 2. Schematic of phase estimation scheme with a squeezed vacuum state as input. In the measurement stage, the states in two modes are recombined through a beam splitter. Then the state in mode $A$ experiences an anti-squeezer and on-off measurement is performed at the outputs.

Download Full Size | PDF

The detectors used here are Geiger-mode avalanche photodiode (Gm-APD), which can only respond to the presence or absence of photons. An event that there is no photon in two Gm-APDs is recorded as 1, and the remaining events are recorded as 0. This projective strategy is referred to as on-off measurement, and the measurement operator can be written as

(25)$${{\hat \Pi} _{{\rm{off}}}} = \left| 0 \right\rangle \left\langle 0 \right| \otimes \left| 0 \right\rangle \left\langle 0 \right|$$

In view of the fact that the squeezed vacuum state is a Gaussian state with zero mean, one can obtain entire information in the state from its covariance matrix. As a consequence, the phase information can be obtained from the covariance matrix of the outputs. The transformation between the output and input covariance matrices can be expressed as

(26)$${\boldsymbol{\Gamma} _{{\rm{out}}}^{}} = \textbf{U}_{\rm E}^{}{\boldsymbol{\Gamma} _{{\rm{in}}}^{}}\textbf{U}_{\rm E}^{\sf T},$$

where, in phase space, the covariance matrix of the input is given by ${\boldsymbol {\Gamma }_{\rm {in}}} = \boldsymbol {\Gamma _{\rm {SVS}}} \oplus \boldsymbol {\Gamma _{\rm {VS}}}$ with the squeezed vacuum state

(27)$${\boldsymbol{\Gamma}_{\rm{SVS}}} = \left[ {\begin{array}{cc} {{e^{ - 2r}}} & 0\\0 & {{e^{2r}}} \end{array}} \right]$$

and vacuum state

(28)$$\boldsymbol{\Gamma_{\rm{VS}}} = \left[ {\begin{array}{cc} 1 & 0\\0 & 1 \end{array}} \right],$$

the specific form of transformation matrix $\textbf {U}_{\rm E}^{}$ can be seen in Appendix.

According to the covariance matrix of the output, one can get the probability of zero-photon events,

(29)$${P_{{\rm{off}}}} = \langle {{\hat \Pi _{{\rm{off}}}}} \rangle = \frac{4}{{\sqrt {\det \left( {{\boldsymbol{\Gamma} _{{\rm{out}}}} + {\textbf{I}_4}} \right)} }},$$

where ${\textbf {I}_4}$ stands for a four-by-four identity matrix. The expression of ${P_{{\rm {off}}}}$ is quite complicated and thus is showed in Appendix.

Further, phase sensitivity can be calculated in terms of the following formula (see Appendix for details),

(30)$$\Delta \theta = \frac{{\sqrt {P_{{\rm{off}}}^{} - P_{{\rm{off}}}^2} }}{{\left| {{{d P_{{\rm{off}}}^{}} \mathord{\left/ {\vphantom {{d P_{{\rm{off}}}} {d \theta }}} \right.} {d \theta }}} \right|}}.$$

One can find that the sensitivity limit given by Eq. (10) can be obtained when the estimated phase sits at 0. This verifies that our measurement scheme is the optimal. It should be noted that, in the experiment or simulation, $MN$ measurements with large $M$ and $N$ are required for every measurement scheme. $N$ measurements are regarded as single trial to obtain an accurate measurement signal, e.g. Equation (29), and then repeated $M$ trials are performed to asymptotically reach the sensitivity limit scaling of $1/\sqrt {M{\cal F}_{\rm Q}}$ [27–30]. Therefore, the optimal phase sensitivity refers to the sensitivity given by the result of single trail instead of single measurement.

4.2 Effects of anti-squeezing noise

The first realistic factor we consider is anti-squeezing noise. For an actual experimental squeezing processing, there must be extra noise in its anti-squeezing quadrature component [31]. At this point, the covariance matrix of input can be expressed as

(31)$${\boldsymbol{\Gamma} _{\rm{STS}}} = \left[ {\begin{array}{cc} {{e^{ - 2r}}} & 0\\ 0 & {{e^{2r + 2r_0}}} \end{array}} \right],$$

where $r_0$ denotes the extra anti-squeezing parameter describing the noise level of anti-squeezing quadrature component. Throughout this paper, we assume that there is the same noise level in the input and the anti-squeezer. Hence, the covariance matrix of the output in a noisy scenario is given by

(32)$${\boldsymbol{\Gamma} _{{\rm{out1}}}} = \textbf{U}_{\rm E1}^{}( \boldsymbol{\Gamma_{\rm{STS}}} \oplus \boldsymbol{\Gamma_{\rm{VS}}})\textbf{U}_{\rm E1}^{\sf T},$$

where the specific form of evolution matrix $\textbf {U}_{\rm E1}^{}$ and calculation are given in Appendix.

In order to observe the effect of anti-squeezing noise on the phase sensitivity intuitively, Fig. 3(a) shows the phase sensitivity as a function of squeezing parameter and extra anti-squeezing parameter. Meanwhile, the shot-noise limit is also plotted for contrast. A clear phenomenon is that the phase sensitivity in the presence of anti-squeezing noise is below than the shot-noise limit for $r_0 < 0.5$, indicating that the scheme is robust against anti-squeezing noise.

Fig. 3. (a) The phase sensitivity (lower surface) and the shot-noise limit (upper surface) versus squeezing parameter and anti-squeezing parameter. (b) Slice across of figure (a) with $r_0 = 0.5$. The dashed line is the shot-noise limit, and the solid line is the phase sensitivity in the presence of anti-squeezing noise.

Download Full Size | PDF

For quantitatively comparing the shot-noise limit and the phase sensitivity, in Fig. 3(b) we give a slice across Fig. 3(a) with $r_0 = 0.5$. From the figure, one can find that the phase sensitivity degenerates into the shot-noise limit for $r = 1$, and the shot-noise limit can be broken with the increase of $r$. This result means that, for a fixed extra anti-squeezing parameter, the robustness can be improved by increasing the squeezing parameter. In other words, sub-shot-noise-limited phase sensitivity is attainable when the ratio of squeezing parameter and extra anti-squeezing parameter is large.

4.3 Effects of photon loss

Now we discuss the effect of photon loss on the phase sensitivity. In an actual measurement process, photon loss is inevitable due to the absorption of devices and environment. For an optical interferometer, this process can be modeled by placing two virtual beam splitters in two arms; photons reflected by virtual beam splitters are regarded as loss. According to the theory of phase space, this amounts to a transformation of the covariance matrix, defined as

(33)$${\boldsymbol{\Gamma} _{\rm L}} = \left( {1 - L} \right){\textbf{I}_4}\boldsymbol{\Gamma} + L{\textbf{I}_4},$$

It should be noted that photon loss occurring at any stage is equivalent, for the loss of photons is linear. As a result, the covariance matrix of the output in a lossy scenario is found to be

(34)$${\boldsymbol{\Gamma} _{\rm {out2}}^{}} = \textbf{U}_{\rm E}^{}[\left( {1 - L} \right){\textbf{I}_4}(\boldsymbol{\Gamma_{\rm{SVS}}} \oplus \boldsymbol{\Gamma_{\rm{VS}}}) + L{\textbf{I}_4}]\textbf{U}_{\rm E}^{\sf T}.$$

To investigate the degradation of phase sensitivity caused by photon loss, we introduce a parameter called the maximal tolerable lossy rate, $L_{\rm {max}}$. This parameter refers to how much loss the scheme can withstand when the phase sensitivity maintains the shot-noise limit. If the actual lossy rate is lower than this parameter, sub-shot-noise-limited phase sensitivity is attainable.

In Fig. 4, we give the dependence of the maximal tolerable lossy rate on the squeezing parameter. One can find that the maximal tolerable lossy rate shows approximately linear decrease with squeezing parameter ranging from 1 to 3. Note that approximately linear decrease refers to the maximal tolerable lossy rate against squeezing parameter instead of average photon number. In the case of $r< 1.5$, the phase sensitivity beyond the shot-noise limit can be obtained for $L\le 10\%$. As to the case of $r = 3$, the lossy rate of less than $\sim$3.8% is required to realize sub-shot-noise-limited phase sensitivity.

Fig. 4. The maximal tolerable lossy rate versus squeezing parameter.

Download Full Size | PDF

These results show that the robustness of the scheme becomes worse with the increase of squeezing parameter. This is mainly due to the fact that two modes after the first beam splitter are of entanglement, of which strength is increased with the increase of the squeezing parameter (see Appendix for details). For the same lossy rate, the state with high entanglement strength is more affected by photon loss, thereby showing low robustness.

4.4 Effects of dark counts

Finally, we analyze the effect of dark counts on the phase sensitivity. As a kind of realistic factors, dark counts widely exist in detectors. For the detector with the rate of dark counts being $d$, the probability of finding $k$ dark counts in single measurement obeys the following Poisson distribution,

(35)$$P\left( k \right) = {e^{ - d}}\frac{{{d^k}}}{{k!}}.$$

We assume that the rates of dark counts of two detectors are the same. Further, the probability of zero-photon events in the presence of dark counts turns to be

(36)$$P_{{\rm{off}}}^{\rm D} = P_1^{}( 0 )P_2^{}( 0 )P_{{\rm{off}}}^{} = {e^{ - 2d}}P_{{\rm{off}}}^{}.$$

That is, only when dark counts in two detectors are zero is the probability of zero-photon events not change. Then the phase sensitivity can be calculated from Eq. (30).

In terms of Ref. [32], where applicable, the rate of dark counts of a detector generally ranges from $10^{-8}$ to $10^{-2}$. In Fig. 5, we plot the phase sensitivity with $d =10^{-2}$ and that with $d =10^{-3}$ against squeezing parameter. It can be seen that the phase sensitivity in the presence of dark counts can break the Heisenberg limit, meaning that the scheme is of strong robustness against dark counts.

Fig. 5. The phase sensitivity and the Heisenberg limit versus squeezing parameter. The dashed line ($10^{-2}$) and dotted line ($10^{-3}$) represent the phase sensitivity in the presence of dark counts, and the solid line is the Heisenberg limit.

Download Full Size | PDF

Although the scheme we presented was mentioned in Ref. [13], there are three characteristics to be pointed about our work. Firstly, we determine that the squeezed vacuum is the optimal single-mode input for the first time, which shows the upper bound of the performance for relevant schemes using single-mode inputs. Secondly, we explain the reason why measurement scheme can achieve the optimal phase sensitivity from the perspective of parameter estimation, which provides a specific way for other schemes to find the optimal strategy. Finally, we analyze the effects of common realistic factors on phase sensitivity, which makes the measurement scheme more practical.

5. Conclusion

In summary, we studied the phase estimation scheme using single-mode inputs. We considered five kinds of states, including displaced squeezed vacuum states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. In terms of the QFI calculation, the squeezed vacuum state is proved to be the optimal input. The QFI analysis indicates that the corresponding sensitivity limit is below than the Heisenberg limit. We showed a specific measurement scheme which can reach the sensitivity limit. For practical purposes, the effects of anti-squeezing noise, photon loss and dark counts on the phase sensitivity were discussed. We showed that the phase sensitivity under anti-squeezing noise surpasses the shot-noise limit, while the phase sensitivity in the presence of dark counts can break the Heisenberg limit. In addition, we discussed the maximal tolerable lossy rate for the measurement scheme. Our results imply that phase estimation using single-mode inputs is a promising way for practical metrology.

Appendix

A. Expectation value of square of photon number operator

The transformation of displacement operator is given by

(37)$${\hat D^{\dagger} }( \alpha ){\hat a^{\dagger} }\hat D( \alpha ) = {\hat a^{\dagger} } + {\alpha ^ * },$$

(38)$${\hat D^{\dagger} }( \alpha )\hat a\hat D( \alpha ) = \hat a + \alpha,$$

and that of squeezing operator is given by

(39)$${\hat S^{\dagger} }( \xi ){\hat a^{\dagger} }\hat S( \xi ) = {\hat a^{\dagger} }\cosh r - {e^{ - i\delta }}\hat a\sinh r,$$

(40)$${\hat S^{\dagger} }( \xi )\hat a\hat S( \xi ) = \hat a\cosh r - {e^{i\delta }}{\hat a^{\dagger} }\sinh r.$$

For a displaced squeezed number state, the expectation value of square of photon number operator is calculated to be

(41)$$\begin{aligned}\langle {{{\hat a}^{\dagger} }\hat a{{\hat a}^{\dagger} }\hat a} \rangle =&{\left| \alpha \right|^2}\left[ {4m\cosh ( {2r} ) + ( {2m + 1} )\sinh ( {2r} ) + 4{{\sinh }^2}r + 1} \right] + {\left| \alpha \right|^4} + {m^2}{\cosh ^4}r\\ &+ {( {m + 1} )^2}{\sinh ^4}r + ( {4{m^2} + 4m + 2} ){\cosh ^2}r{\sinh ^2}r. \end{aligned}$$

B. Probabilities of zero-photon events in idealized and realistic scenarios

Here we provide the matrix forms of optical processes and give the probabilities of zero-photon events in idealized and realistic scenarios. All calculations are performed by Mathematica.

In an idealized scenario, the transformation matrix of a beam splitter is given by

(42)$${\textbf{U}_{\rm{BS}}} = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{cc} \textbf{I}_2 & \textbf{I}_2\\ -\textbf{I}_2 & \textbf{I}_2 \end{array}} \right].$$

where $\textbf {I}_2$ is two-by-two identity matrix. The transformation matrix of phase shift is given by

(43)$${\textbf{U}_{\rm{PS}}} = \left[ {\begin{array}{cc} {\cos \theta } & {\sin \theta }\\ { - \sin \theta } & {\cos \theta } \end{array}} \right] \oplus \textbf{I}_2,$$

and that of an anti-squeezer is given by

(44)$${\textbf{U}_{\rm{AS}}} = \left[ {\begin{array}{cc} {{e^r}} & 0\\ 0 & {{e^{ - r}}} \end{array}} \right] \oplus \textbf{I}_2.$$

Then the entire transformation matrix can be written as ${\textbf {U}_{\rm {E}}^{}} = {\textbf {U}_{\rm {AS}}^{}}{\textbf {U}_{\rm {BS}}^{\sf T}}{\textbf {U}_{\rm {PS}}^{}}{\textbf {U}_{\rm {BS}}^{}}$, and the probability of zero-photon events is given by

(45)$${P_{{\rm{off}}}} = \frac{8}{{\sqrt{\cal G} }}$$

with

\begin{aligned}\cal{G} = & { 7\cosh (4r) - 4 \left\{ 4\cos \theta [ {3 + \cosh ( {2r} )} ]{{\sinh }^2}r + \cos ( {2\theta } )[ {5 + 3\cosh ( {2r} )} ]\right\}{{\sinh }^2}r}\\ & + 37 + 20\cosh ( {2r} ). \end{aligned}

In the presence of anti-squeezing noise, the transformation matrix of the anti-squeezer is recast as

(46)$${\textbf{U}_{\rm{AS1}}} = \left[ {\begin{array}{cc} {{e^{r+r_0}}} & 0\\ 0 & {{e^{ - r}}} \end{array}} \right] \oplus \textbf{I}_2.$$

The transformation matrices of other optical processes are the same as those in an idealized scenario. At this point, the entire transformation matrix is described by ${\textbf {U}_{\rm {E1}}^{}} = {\textbf {U}_{\rm {AS1}}^{}}{\textbf {U}_{\rm {BS}}^{\sf T}}{\textbf {U}_{\rm {PS}}^{}}{\textbf {U}_{\rm {BS}}^{}}$. Consequently, the probability of zero-photon events is calculated to be

(47)$$P_{{\rm{off}}}^{\rm N} = \frac{8}{{{e^{r_0}}\sqrt {{\cal A}} }}$$

with

\begin{aligned}{\cal A} = & [ {7 - 4\cos \theta - 3\cos ( {2\theta } )} ]\cosh [ {2( {2r + r_0} )} ] + 2[ {5 - 4\cos \theta - \cos ( {2\theta } )} ]\cosh [ {2( {r + r_0} )} ]\\ & + 2\left\{ ( {3 + \cos \theta } )^2\cosh ( {2r_0} ) {9 + 4\cos \theta + 3\cos ( {2\theta } ) + 4[ {3 + \cos \theta } ]\cosh ( {2r} ){{\sin }^2}\frac{\theta }{2}} \right\}. \end{aligned}

In the presence of photon loss, all optical processes are of the same matrix forms with those in an idealized scenario. The probability of zero-photon events is found to be

(48)$$P_{{\rm{off}}}^{\rm L} = \frac{8}{{\sqrt {\cal B} }}$$

with

\begin{aligned}{\cal B} = & 4( {L - 1} )\left\{ {4\cos \theta [ {3 - L + ( {1 + L} )\cosh ( {2r} )} ] + \cos ( {2\theta } )[ {5 - L + ( {3 + L} )\cosh ( {2r} )} ]} \right\}{\sinh ^2}r\\ & + 37 + L( {17L - 22} ) + 4[ {5 - 3L( {L - 2} )} ]\cosh ( {2r} ) + ( {1 - L} )( {7 + 5L} )\cosh (4r). \end{aligned}

C. Derivation of phase sensitivity

Phase sensitivity with a specific measurement scheme is given by the inverse of the relevant classical Fisher information. On-off measurement is a typical binary projective strategy, the classical Fisher information can be written as

(49)$${{\cal F}_{\rm{c}}} = \frac{1}{{{P_{{\rm{off}}}}}}{\left( {\frac{{d{P_{{\rm{off}}}}}}{{d\theta }}} \right)^2} + \frac{1}{{{P_{{\rm{on}}}}}}{\left( {\frac{{d{P_{{\rm{on}}}}}}{{d\theta }}} \right)^2}.$$

By virtue of the property of probability ${P_{{\rm {on}}}} = 1 - {P_{{\rm {off}}}}$, the classical Fisher information is rewritten as

(50)$${{\cal F}_{\rm{c}}} = {\left( {\frac{{d{P_{{\rm{off}}}}}}{{d\theta }}} \right)^2}\left( {\frac{1}{{{P_{{\rm{off}}}}}} + \frac{1}{{1 - {P_{{\rm{off}}}}}}} \right) = {\left( {\frac{{d{P_{{\rm{off}}}}}}{{d\theta }}} \right)^2}\frac{1}{{{P_{{\rm{off}}}}\left( {1 - {P_{{\rm{off}}}}} \right)}}.$$

Related to this, the phase sensitivity with on-off measurement is found to be

(51)$$\Delta \theta = \frac{1}{{\sqrt {{{\cal F}_{\rm{c}}}} }} = \frac{{\sqrt {P_{{\rm{off}}}^{} - P_{{\rm{off}}}^2} }}{{\left| {{{d{P_{{\rm{off}}}}} \mathord{\left/{\vphantom {{d{P_{{\rm{off}}}}} {d\theta }}} \right.} {d\theta }}} \right|}}.$$

Using the property of measurement operator $\hat \Pi _{{\rm {off}}}^2 = \hat \Pi _{{\rm {off}}}^{}$, one can find that Eq. (51) equals error propagation formula.

D. Proof of entanglement between two modes after the first beam splitter

The covariance matrix of the input can be decomposed into the direct sum of two single-mode states, i.e.,

(52)$${\boldsymbol{\Gamma} _{{\rm{in}}}} = \left[ {\begin{array}{cc} {{\boldsymbol{\Gamma} _{\rm{SVS}}}} & {{\textbf{O}_2}}\\ {{\textbf{O}_2}} & {{\boldsymbol{\Gamma} _{\rm{VS}}}} \end{array}} \right] = {\boldsymbol{\Gamma} _{\rm{SVS}}} \oplus {\boldsymbol{\Gamma} _{\rm{VS}}}$$

with $\textbf {O}_2$ being two-by-two zero matrix. This is due to the fact that two input states are direct product states without entanglement.

After the first beam splitter, the covariance matrix can be calculated to be

(53)$$\boldsymbol{\Gamma} ' = \textbf{U}_{\rm{BS}}^{}{\boldsymbol{\Gamma} _{{\rm{in}}}^{}}\textbf{U}_{\rm{BS}}^{\sf T} = \left[ {\begin{array}{cc} {{\boldsymbol{\Gamma} _{+}}} & {{\boldsymbol{\Gamma} _{-}}}\\ {{\boldsymbol{\Gamma} _{-}}} & {{\boldsymbol{\Gamma} _{+}}} \end{array}} \right]$$

with

(54)$${\boldsymbol{\Gamma} _{{\pm}}} = \frac{1}{2}\left[ {\begin{array}{cc} {1 {\pm} {e^{ - 2r}}} & 0\\ 0 & {1 {\pm} {e^{2r}}} \end{array}} \right].$$

It can be seen that the covariance matrix after the first beam splitter is no longer a direct sum regarding the top-left block and bottom-right block, suggesting that two modes are entangled.

Funding

National Natural Science Foundation of China (12104193); Natural Science Research of Jiangsu Higher Education Institutions of China (21KJB140007); Shuangchuang Ph.D Award (JSSCBS20210915); Project for Leading Innovative Talents in Changzhou (CQ20210107).

Acknowledgments

J.-D. Zhang would like to thank Prof. Xiaojun Jia for helpful discussions about anti-squeezing noise.

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. T. Liu, J. Yin, J. Jiang, K. Liu, S. Wang, and S. Zou, “Differential-pressure-based fiber-optic temperature sensor using Fabry-Perot interferometry,” Opt. Lett. 40(6), 1049–1052 (2015). [CrossRef]

2. T. Liu, C. Zhang, S. Wang, J. Jiang, K. Liu, X. Zhang, and X. Wang, “Simultaneous measurement of pressure and temperature based on adjustable line scanning polarized low-coherence interferometry with compensation plate,” IEEE Photonics J. 10(4), 1–9 (2018). [CrossRef]

3. M. Jarzyna and M. Zwierz, “Quantum interferometric measurements of temperature,” Phys. Rev. A 92(3), 032112 (2015). [CrossRef]

4. S.-J. Yoon, J.-S. Lee, C. Rockstuhl, C. Lee, and K.-G. Lee, “Experimental quantum polarimetry using heralded single photons,” Metrologia 57(4), 045008 (2020). [CrossRef]

5. A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012). [CrossRef]

6. O. S. Maga na Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014). [CrossRef]

7. A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011). [CrossRef]

8. V. D’ambrosio, N. Spagnolo, L. Del 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]

9. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981). [CrossRef]

10. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, 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]

11. Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112(10), 103604 (2014). [CrossRef]

12. J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011). [CrossRef]

13. M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017). [CrossRef]

14. N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012). [CrossRef]

15. D. Gatto, P. Facchi, F. A. Narducci, and V. Tamma, “Distributed quantum metrology with a single squeezed-vacuum source,” Phys. Rev. Res. 1(3), 032024 (2019). [CrossRef]

16. J. Liu, Y. Yu, C. Wang, Y. Chen, J. Wang, H. Chen, D. Wei, H. Gao, and F. Li, “Optimal phase sensitivity by quantum squeezing based on a Mach-Zehnder interferometer,” New J. Phys. 22(1), 013031 (2020). [CrossRef]

17. Z. Huang, C. Macchiavello, L. Maccone, and P. Kok, “Ancilla-assisted schemes are beneficial for gaussian state phase estimation,” Phys. Rev. A 101(1), 012124 (2020). [CrossRef]

18. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Classical inputs and measurements enable phase sensitivity beyond the shot-noise limit,” (2020).

19. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009). [CrossRef]

20. G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43(1), 492–497 (1991). [CrossRef]

21. G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46(1), 485–488 (1992). [CrossRef]

22. L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015). [CrossRef]

23. B. Gard, “Advances in quantum metrology: Continuous variables in phase space,” (2016).

24. L. Pezze and A. Smerzi, “Quantum theory of phase estimation,” arXiv preprint arXiv:1411.5164 (2014).

25. M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017). [CrossRef]

26. E. Giese, S. Lemieux, M. Manceau, R. Fickler, and R. W. Boyd, “Phase sensitivity of gain-unbalanced nonlinear interferometers,” Phys. Rev. A 96(5), 053863 (2017). [CrossRef]

27. Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017). [CrossRef]

28. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Super-resolved angular displacement estimation based upon a sagnac interferometer and parity measurement,” Opt. Express 28(3), 4320–4332 (2020). [CrossRef]

29. S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, S. W. Nam, and G. J. Pryde, “Unconditional violation of the shot-noise limit in photonic quantum metrology,” Nat. Photonics 11(11), 700–703 (2017). [CrossRef]

30. C. You, M. Hong, P. Bierhorst, A. E. Lita, S. Glancy, S. Kolthammer, E. Knill, S. W. Nam, R. P. Mirin, O. S. Maga na-Loaiza, and T. Gerrits, “Scalable multiphoton quantum metrology with neither pre- nor post-selected measurements,” Appl. Phys. Rev. 8(4), 041406 (2021). [CrossRef]

31. J. Yu, Y. Qin, J. Qin, H. Wang, Z. Yan, X. Jia, and K. Peng, “Quantum enhanced optical phase estimation with a squeezed thermal state,” Phys. Rev. Appl. 13(2), 024037 (2020). [CrossRef]

32. R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3(12), 696–705 (2009). [CrossRef]

Sub-shot-noise-limited phase estimation via single-mode inputs

Abstract

1. Introduction

2. Single-phase estimation protocol with single-mode inputs

3. Analysis of the optimal single-mode state

3.1 Displaced squeezed vacuum states

3.2 Displaced number states and squeezed number states

3.3 Schrödinger cat states

3.4 Completely mixed states

3.5 Discussion

4. Specific scheme saturated with the sensitivity limit and analysis of realistic factors

4.1 Optimal measurement scheme

4.2 Effects of anti-squeezing noise

4.3 Effects of photon loss

4.4 Effects of dark counts

5. Conclusion

Appendix

A. Expectation value of square of photon number operator

B. Probabilities of zero-photon events in idealized and realistic scenarios

C. Derivation of phase sensitivity

D. Proof of entanglement between two modes after the first beam splitter

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (57)

Optics Express