Simultaneously estimating two phase shifts with a power-recycled tri-port interferometer fed by coherent states

Jian-Dong Zhang; Zi-Jing Zhang; Long-Zhu Cen; Jun-Yan Hu; Yuan Zhao

doi:10.1364/OSAC.1.001437

1. Introduction

With the rapid developments in recent decades, quantum communication [1–3] runs to a critical stage of practical industrialization, quantum computation [4–6] and quantum simulation [7,8] are in the ascendant, and quantum precision measurements [9, 10] continue to make headway. We are, as it were, on the cusp of a new era of various quantum technologies. Hitherto, the field of quantum precision measurements is mainly comprised of three kinds of application configurations: cold-atom [11,12], solid-state [13,14], and optical systems. They all have their own innate advantages. Compared with complex cooling equipments of solid-state systems and huge volumes of cold-atom systems, optical systems work at the room temperatures and have the characteristic of miniaturization. Therefore, optics-based measurement systems have got a lot of attention.

In this regard, phase estimation in optical interferometers is of paramount interest to multiple fields, such as sensing [15, 16], imaging [17, 18], and gravitational waves detection [19]. As a consequence, enormous efforts have been devoted to improve the estimation performances. Of the enhanced phase estimation, in general, there are three tried-and-true strategies. The first way of improving performances is to use nonlinear elements in an interferometer, e.g., SU(1,1) interferometers. In addition, the sensitivity and resolution may be enhanced with some novel measurements, parity, on-off, binary homodyne measurements, to name a few. As the last way, one can improve the performances by deploying exotic quantum states: squeezed vacuum states, N00N states, twin Fock states, and entangled coherent states. The first two strategies can be achieved by current technologies. However, preparing a bright non-Gaussian state is still a knotty problem [20–22]. These facts indicate that the protocols based on Gaussian inputs come across as ideal candidate in realistic scenarios. Accordingly, coherent-state-based phase estimation has been thoroughly studied [23, 24]. Expect for the strategies mentioned above, in the last few years, some researches work on exploring other effective approaches, such as weak-value-based post-selections [25, 26], ancilla-assisted technologies [27, 28], and power-recycling technologies [29–31]. Among these approaches, power-recycling is practical since one can significantly economize on the probe state. One the other hand, multi-phase estimation has got more and more attention [32–34], in that it has the potential to provide substantial technological advances as well as deep insights into the fundamental workings of some inherently multi-parameter physical processes, such as microscopy and imaging.

In this paper, we propose a power-recycled bi-phase estimation protocol using two coherent states as inputs. Owing to the partial power is reused, the sensitivities of two estimated phases are both $1 / \sqrt{N}$ when the total mean photon number in the input is 3N/2. The remainder of this paper is organized as follows. Section 2 shows the fundamental principle of estimation protocol and detection strategy. In Sec. 3, we discuss the sensitivity limits of two phase shifts via calculating the quantum Fisher information (QFI). In Sec. 4, the effects of photon loss on the sensitivity are studied. Finally, we summarize our work with a conclusion in Sec. 5.

2. Fundamental principle

We start off with the introduction of estimation protocol, as illustrated in Fig. 1. Unlike single phase estimation protocols, here we input two coherent states launched from lasers with the mean photon numbers N and N/2, respectively. The controlled phase is responsible for matching the phase angles of two inputs in the phase space. Consider a tri-port interferometer of which the inputs are |α〉 and $| α / \sqrt{2} 〉$ , two phase shifts φ and θ are the parameters we would like to estimate. After the evolution process showed in Fig. 1, we can simultaneously estimate the two phases by performing balanced homodyne detection onto the outputs of the third and the fourth beam splitters. Two tunable phases play the self-compensative roles, that is, enabling the interferometer to work at the optimal sensitivity. Through calculation, the two outputs are found to be,

| ψ_{out 1} 〉 = | \frac{(i + 1) e^{i φ} - \sqrt{2}}{2 \sqrt{2}} α 〉,

| ψ_{out 2} 〉 = | \frac{(i - 1) e^{i φ} + \sqrt{2} i + \sqrt{2} (i - 1) e^{i θ}}{4} α 〉 .

It can be seen that the first output is merely associated with φ, whereas the second output is related to φ and θ simultaneously. Therefore, we can estimate φ from Eq. (1), and then estimate θ based on Eq. (2) and φ.

Fig. 1 Schematic diagram of a bi-phase estimation protocol using a tri-port interferometer fed by coherent states. The inset on the upper right shows the details of balanced homodyne detection module, and the inset on the lower right is a sketch of the protocol. |ψ₁〉, |ψ₂〉, and |ψ₄〉 are three inputs. The detailed expressions are as follows: |ψ₁〉 = |α〉, |ψ₂〉 = |0〉, $| ψ_{3} 〉 = | ψ_{4} 〉 = | α / \sqrt{2} 〉$ , $| ψ_{5} 〉 = | ψ_{6} 〉 = | α e^{i π / 4} / \sqrt{2} 〉$ . The abbreviations are defined as follows: L-laser, BS-beam splitter, RM-reflection mirror, CP-controlled phase, TP-tunable phase, EP-estimated phase, HBDM-balanced homodyne detection module.

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Now we turn to our detection strategy––balanced homodyne detection––as manifested in the upper right panel of Fig. 1. It is originally developed by Yuen and Chan [35]. In this strategy, the output to be detected enters one port of a balanced beam splitter while a local oscillator |β〉 (coherent light) with the same frequency is injected via the other port. Subsequently, two detectors measure the two output intensities and subtracts them, the intensity difference is the output signal. Since the detected physical quantity is field intensity rather than photon number, the requirements for detectors in this strategy is considerably lower than those in parity or on-off measurement [36–38]. For Gaussian states, one can substitute balanced homodyne detection for parity or on-off measurement [39,40], in that the Wigner function of output can be reconstituted. In accordance with above description, by selecting an appropriate phase angle for the local oscillator, we can obtain the X quadrature operator in this strategy,

\hat{X} = {\hat{a}}^{†} + \hat{a} .

Using Eqs. (1) and (3), for the first output, the expectation value is given by

〈 {\hat{X}}_{1} 〉 = - | α | [sin (φ - \frac{π}{4}) + 1],

and the expectation value of its square turns out to be,

〈 {\hat{X}}_{1}^{2} 〉 = \frac{N}{2} [1 - sin (2 φ)] + N [1 + 2 sin (φ - \frac{π}{4})] + 1 .

By calculating the visibility defined as [41]

V = \frac{{〈 \hat{X} 〉}_{max} - {〈 \hat{X} 〉}_{min}}{| {〈 \hat{X} 〉}_{max} | + | {〈 \hat{X} 〉}_{min} |},

one can find that the first output has 100% visibility. According to the error propagation, Eqs. (4) and (5), the sensitivity can be expressed as

δ φ = \frac{\sqrt{〈 {\hat{X}}_{1}^{2} 〉 - {〈 {\hat{X}}_{1} 〉}^{2}}}{| \partial 〈 {\hat{X}}_{1} 〉 / \partial φ |} = \frac{1}{| α cos (φ - π / 4) |} .

When φ = π/4, Eq. (7) reaches its minimum (optimal sensitivity)

1 / \sqrt{N}

, the shot-noise limit.

As with the analysis approach of the first output, we can calculate the expectation value of the second output,

〈 {\hat{X}}_{2} 〉 = - | α | [\frac{1}{\sqrt{2}} sin (φ + \frac{π}{4}) + sin (θ + \frac{π}{4})] .

Equation (8) suggests that to such a form of expectation value there corresponds 100% visibility for any values of φ. Then the expectation value of the square of operator goes as

〈 {\hat{X}}_{2}^{2} 〉 = \frac{N}{4} [3 + 2 \sqrt{2} sin (θ + φ) + 2 \sqrt{2} cos (φ - θ) + sin (2 φ) + 2 sin (2 θ)] + 1 .

With regard to the sensitivity, there are some slight changes in the calculation. We can not calculate directly the sensitivity by emulating Eq. (7), since the phase sensitivity of θ may suffer from the perturbation arising out of φ. At this point, the expression of sensitivity calculation is revised as

δ^{2} θ = {| \frac{\partial θ}{\partial 〈 {\hat{X}}_{2} 〉} |}^{2} δ^{2} 〈 {\hat{X}}_{2} 〉 + {| \frac{\partial θ}{\partial φ} |}^{2} δ^{2} φ .

The second term denotes the perturbation which depends on the values of φ and θ. In the light of implicit function theorem, by differentiating on both sides in Eq. (8) with respect to φ, we have

0 = - | α | [cos (φ + π / 4) \partial φ / \partial φ + \sqrt{2} cos (θ + π / 4) \partial θ / \partial φ]

. After felicitously transposing the terms, one can get

\frac{\partial θ}{\partial φ} = \frac{cos (φ + π / 4)}{\sqrt{2} cos (θ + π / 4)} .

It is an interesting fact that when φ = π/4 the value of Eq. (11) is 0 , i.e.,

{\frac{\partial θ}{\partial φ} |}_{φ = π / 4} = 0 .

This means that the phase sensitivity is only dependent of the first term when the condition φ = π/4 is satisfied. The physical interpretation behind this phenomenon is that the path containing phase φ can also be regarded as a reference path with respect to the path containing phase θ, since the two paths on top of Fig. 1 have the same phase for φ = π/4. Note that this condition also makes the estimation sensitivity of phase φ achieve its minimum. In addition, the first term in Eq. (10) equals

{| \frac{\partial θ}{\partial 〈 {\hat{X}}_{2} 〉} |}^{2} δ^{2} 〈 {\hat{X}}_{2} 〉 = \frac{1}{| α cos (θ + π / 4) |} .

Therefore, when φ = π/4 and θ = −π/4, the phase sensitivities of two phases are optimal simultaneously,

δ φ_{min} = δ θ_{min} = 1 / \sqrt{N}

. Two estimated phase shifts of which the sensitivities are shot-noise-limited are reachable by inputting coherent states with 3N/2 photons on average. That is, N/2 photons are reused, equivalently, we save 25% of the probe state.

3. Quantum Fisher information and the sensitivity limit

In this section, we study the sensitivity limit calculated from the QFI [42]. To simplify understanding, our protocol can be divided into two interferometers, as shown in Fig. 2. For the first interferometric part, the input can be written as |ψ₁〉 |ψ₂〉 = |α〉 |0〉, and the state |ψ₃〉 goes as $| α / \sqrt{2} 〉$ . Because of the input of the second coherent state at the second beam splitter in Fig. 1, here the state |ψ₅〉 becomes $| α e^{i π / 4} / \sqrt{2} 〉$ . However, the phase shift π/4 between |ψ₃〉 and |ψ₅〉 has no effect on the QFI of phase φ, since the QFI is independent of the actual value of estimated parameter. Therefore, we can ascribe this shift to φ, i.e., φ → φ + π/4. At this point, we have the density matrix of input,

ρ_{in 1} = | ψ_{1} 〉 | ψ_{2} 〉 〈 ψ_{1} | 〈 ψ_{2} | .

Accordingly, upon undergoing the phase φ, the density matrix can be described as

ρ (φ) = {\hat{U}}_{φ} {\hat{U}}_{BS} ρ_{in 1} {\hat{U}}_{BS}^{†} {\hat{U}}_{φ}^{†},

where Û_φ = exp[iφ(â^†â − b̂^†b̂)/2] and Û_BS = exp[iπ(â^†b̂ + b̂^†â)/4] being the operators for the phase shift and beam splitter. Here â^† (â) and b̂^† (b̂) stand for the creation (annihilation) operators for paths A and B, respectively.

Fig. 2 A simplified separated configuration of the bi-phase estimation protocol. The dotted and solid lines are labeled as paths A and B, respectively.

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Using this density matrix and differential equation,

\frac{\partial ρ (φ)}{\partial φ} = - i [ρ (φ), {\hat{O}}_{φ}],

we can derive the form of estimator Ô_φ. Then, based on the input density matrix and estimator, the QFI of phase φ can be calculated as

ℱ_{φ} = 4 Tr (ρ_{in 1} {\hat{O}}_{φ}^{2}) - 4 {[Tr (ρ_{in 1} {\hat{O}}_{φ})]}^{2} = N .

Such QFI implies that the fundamental sensitivity limit of phase φ is the shot-noise limit, which is achieved by our detection strategy.

In what follows, we direct our attention to the QFI calculation of phase θ. Unlike the first interferometric part, the second part is a bi-phase estimation process, as shown in the right of Fig. 2. As with above calculation approach, we have the density matrix of input

ρ_{in 2} = | ψ_{5} 〉 | ψ_{6} 〉 〈 ψ_{5} | 〈 ψ_{6} |

with

| ψ_{5} 〉 = | ψ_{6} 〉 = | α e^{i π / 4} / \sqrt{2} 〉

, and the density matrix after two phase shifts is expressed as

ρ (φ, θ) = {\hat{U}}_{θ} {\hat{U}}_{φ} {\hat{U}}_{BS} ρ_{in 2} {\hat{U}}_{BS}^{†} {\hat{U}}_{φ}^{†} {\hat{U}}_{θ}^{†},

here the forms of the operators are Û_φ = exp(ib̂^†b̂φ) and Û_θ = exp(iâ^†âθ).

Of multi-parameter estimation, the sensitivity limits are given by the QFI matrix [43–45]. In particular, for a bi-phase estimation, the sensitivity limits are related to a two-by-two QFI matrix

ℱ = (\begin{matrix} ℱ_{φ φ} & ℱ_{θ φ} \\ ℱ_{φ θ} & ℱ_{θ θ} \end{matrix})

with

ℱ_{p q} = 4 Tr (ρ_{in 2} {\hat{K}}_{p} {\hat{K}}_{q}) - 4 Tr (ρ_{in 2} {\hat{K}}_{p}) Tr (ρ_{in 2} {\hat{K}}_{q}) .

Where the subscripts p and q stand for the phases φ and θ, and the estimators K̂_φ and K̂_θ can be calculated from the density matrix and differential equations,

\frac{\partial ρ (φ, θ)}{\partial φ} = - i [ρ (φ, θ), {\hat{K}}_{φ}],

\frac{\partial ρ (φ, θ)}{\partial φ} = - i [ρ (φ, θ), {\hat{K}}_{θ}] .

For such a bi-phase estimation, the sensitivity limit of phase θ is explicitly given by the second diagonal element of the QFI inverse matrix, i.e.,

δ θ_{min} = ℱ_{22}^{- 1} = \sqrt{\frac{ℱ_{φ φ}}{ℱ_{φ φ} ℱ_{θ θ} - ℱ_{φ θ} ℱ_{θ φ}}} = \frac{1}{\sqrt{2 N}},

Where the details of QFI matrix elements are found to be

ℱ_{φ φ} = ℱ_{θ θ} = 2 N,

ℱ_{φ θ} = ℱ_{θ φ} = 0 .

The QFI in Eq. (24) indicates that the sensitivity limit is a sub-shot-noise limit, which is greater than the shot-noise limit with a factor of $\sqrt{2}$ . Overall, from the results calculated by the QFI and QFI matrix, balanced homodyne detection is an optimal strategy for phase φ, also, a nearly optimal strategy for phase θ.

4. Effects of photon loss

In this section, we study the effects of photon loss on the sensitivity. This phenomenon exists widely in the field of precision measurements. As a consequence, it has been received lots of attention [46–50]. In generally, photon loss is caused by environmental absorption and non-ideal devices, like imperfect detectors and beam splitters. For modeling linear photon loss, one can insert two fictitious beam splitters, transmissivity T₁ and T₂, into the two paths of interferometer, L₁ = 1−T₁ and L₂ = 1−T₂ being the lossy ratios of two paths, respectively. Here, for simplifying, we merely consider the effect on phase φ since all conclusions are applicable to phase θ. At this point, the scenario can be boiled down to the configuration in Fig. 3.

Fig. 3 A simplified configuration of photon loss towards a two-port interferometer, v̂_a and v̂_b are two operators for surrounding vacuum environment.

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Noticing that, for our inputs, it is equivalent that the two fictitious beam splitters are placed before or after phase φ in Fig. 3. For the second fifty-fifty beam splitter, we have the following standard transformations

{\hat{a}}_{3} = \frac{1}{\sqrt{2}} ({\hat{a}}_{2} + i {\hat{b}}_{2}),

{\hat{b}}_{3} = \frac{1}{\sqrt{2}} ({\hat{b}}_{2} + i {\hat{a}}_{2}),

and for the two fictitious beam splitters, the standard transformations are given by

{\hat{a}}_{2} = \sqrt{T_{1}} {\hat{a}}_{1} + i \sqrt{1 - T_{1}} {\hat{v}}_{a},

{\hat{b}}_{2} = \sqrt{T_{2}} {\hat{b}}_{1} + i \sqrt{1 - T_{2}} {\hat{v}}_{b} .

where v̂_a and v̂_b are the annihilation operators for surrounding vacuum. Combining Eqs. (27)–(30), further, we have

{\hat{a}}_{3} = \frac{1}{\sqrt{2}} (\sqrt{T_{1}} {\hat{a}}_{1} + i \sqrt{1 - T_{1}} {\hat{v}}_{a} + i \sqrt{T_{2}} {\hat{b}}_{1} - \sqrt{1 - T_{2}} {\hat{v}}_{b}),

{\hat{a}}_{3}^{†} = \frac{1}{\sqrt{2}} (\sqrt{T_{1}} {\hat{a}}_{1}^{†} + i \sqrt{1 - T_{1}} {\hat{v}}_{a}^{†} + i \sqrt{T_{2}} {\hat{b}}_{1}^{†} - \sqrt{1 - T_{2}} {\hat{v}}_{b}^{†}) .

Through calculation, we can write the output affected by photon loss as

\begin{array}{l} 〈 {\hat{X}}_{L} 〉 & = 〈 {\hat{a}}_{3}^{†} + {\hat{a}}_{3} 〉 \\ = \frac{1}{\sqrt{2}} (\sqrt{T_{1}} 〈 {\hat{a}}_{1} + {\hat{a}}_{1}^{†} 〉 + i \sqrt{T_{2}} 〈 {\hat{b}}_{1} - {\hat{b}}_{1}^{†} 〉) \\ = - | α | [\sqrt{T_{1}} sin (φ - \frac{π}{4}) + \sqrt{T_{2}}] . \end{array}

Where the expectation value are taken over the direct product state

| ψ 〉 = | \frac{(i + 1) α e^{i φ}}{2} 〉 | \frac{i α}{\sqrt{2}} 〉 | 0 〉 | 0 〉,

and the two vacuum states are the ports introduced by the environment. Meanwhile, it is easy to verify that the visibility of Eq. (33) is also 100%.

One can see that the phase information is only related to the transmissivity T₁. Considering an extreme scenario, T₂ = 0, in fact, there is no interference in the interferometer at this time since only a single beam is incident on the second beam splitter. However, according to Eq. (33), we can also get an output containing phase φ. That is, whatever the value of T₂ is, the output always contains the phase information originating from the interference. At first glance, such a conclusion seems ridiculous and counterintuitive, whereas upon further thinking we can straighten out this phenomenon. This is because the beam including φ can interfere with the local oscillator, a phase reference.

Further, in order to obtain the sensitivity, we need to calculate the expectation value of square of operator

\begin{array}{l} 〈 {\hat{X}}_{L}^{2} 〉 & = 〈 {\hat{a}}_{3}^{†} {\hat{a}}_{3}^{†} + {\hat{a}}_{3} {\hat{a}}_{3} + 2 {\hat{a}}_{3}^{†} {\hat{a}}_{3} + 1 〉 \\ = N [2 \sqrt{T_{1} T_{2}} sin (φ - \frac{π}{4}) + T_{2}] + \frac{T_{1} N}{2} [1 - sin (2 φ)] + 1 \end{array}

with

\begin{array}{l} 〈 {\hat{a}}_{3}^{†} {\hat{a}}_{3}^{†} 〉 & = \frac{1}{2} 〈 T_{1} {\hat{a}}_{1}^{†} {\hat{a}}_{1}^{†} - T_{2} {\hat{b}}_{1}^{†} {\hat{b}}_{1}^{†} - i 2 \sqrt{T_{1} T_{2}} {\hat{a}}_{1}^{†} {\hat{b}}_{1}^{†} 〉 \\ = \frac{{| α |}^{2}}{4} [- i T_{1} e^{- i 2 φ} + T_{2} - \sqrt{2 T_{1} T_{2}} (1 - i) e^{- i φ}], \end{array}

\begin{array}{l} 〈 {\hat{a}}_{3} {\hat{a}}_{3} 〉 & = \frac{1}{2} 〈 T_{1} {\hat{a}}_{1} {\hat{a}}_{1} - T_{2} {\hat{b}}_{1} {\hat{b}}_{1} - i 2 \sqrt{T_{1} T_{2}} {\hat{a}}_{1} {\hat{b}}_{1} 〉 \\ = \frac{{| α |}^{2}}{4} [i T_{1} e^{i 2 φ} + T_{2} - \sqrt{2 T_{1} T_{2}} (1 + i) e^{i φ}], \end{array}

\begin{array}{l} 〈 {\hat{a}}_{3}^{†} {\hat{a}}_{3} 〉 & = \frac{1}{2} 〈 T_{1} {\hat{a}}_{1}^{†} {\hat{a}}_{1} - T_{2} {\hat{b}}_{1}^{†} {\hat{b}}_{1} - i \sqrt{T_{1} T_{2}} ({\hat{a}}_{1}^{†} {\hat{b}}_{1} - {\hat{a}}_{1} {\hat{b}}_{1}^{†}) 〉 \\ = \frac{{| α |}^{2}}{4} [T_{1} + T_{2} + 2 \sqrt{T_{1} T_{2}} sin (φ - \frac{π}{4})] . \end{array}

On the basis of Eqs. (33) and (35), we get the variation in the operator

〈 δ^{2} {\hat{X}}_{L} 〉 = \frac{T_{1} N}{2} [1 - sin (2 φ)] + 1 - T_{1} N {sin}^{2} (φ - \frac{π}{4})

Then using the error propagation, the sensitivity turns out to be,

δ φ_{L} = \frac{\sqrt{1 + T_{1} [1 - sin (2 φ)] / 2}}{\sqrt{T_{1}} cos (φ - π / 4)} .

This equation suggests that the sensitivity merely relies on T₁ regardless of T₂, the physical interpretation of this phenomenon has been shown in the above. As for the optimal sensitivity, only when phase φ takes on the value of π/4 does the sensitivity sit at its minimum

1 / \sqrt{T_{1} N}

. Unlike parity or on-off measurement, even with photon loss, the value of phase corresponding to the optimal sensitivity remains the same [38,51].

Next, we briefly discuss the effect of photon loss introduced by detector imperfection on the sensitivity. Here we utilize D₁ and D₂ to describe the detection efficiencies of the two detectors. Using operator transformation approach, the expectation value of operator is given by

〈 {\hat{X}}_{D} 〉 = \frac{D_{1} - D_{2}}{2} (〈 {\hat{a}}^{†} \hat{a} 〉 + {| β |}^{2}) + \frac{D_{1} + D_{2}}{2} | β | 〈 {\hat{a}}^{†} + \hat{a} 〉 .

This scenario is somewhat complicated since the amplitude parameter of local oscillator is included. One can find that the visibility is a function of three parameters: detection efficiencies D₁, D₂, and intensity ratio N/|β|². We pay our attention to the effect of these parameters on the visibility, the results are illustrated in Fig. 4. It can be seen from the figure, when the ratio N/|β|² is small (strong oscillator), 100% visibility only appears at the region in the vicinity of identical detection efficiencies. As the oscillator becomes weak, the visibility gets substantial improvement. Even when N/|β|² = 10⁴, the visibility can be approximately considered as 100%, regardless of the two detection efficiencies.

Fig. 4 Visibility as a function of detection efficiencies D₁ and D₂ and intensity ratio N/|β|². The values of the parameter N/|β|² in panels are 10⁻⁶, 10⁻⁴, 10⁻², 1, 10², and 10⁴. The color bar on the right of the 2D-planar forms indicates the corresponding value of the visibility.

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Further, through calculation, we can find that the optimal sensitivity $1 / \sqrt{D N}$ appears at D₁ = D₂ = D. It should be noted that when D₁ = D₂ = D the sensitivity is independent of the ratio N/|β|². Overall, the optimal sensitivity of a lossy interferometer with a coherent state as input is equivalent to that of a lossless interferometer fed by a weaker coherent state.

From the above discussions, it can be seen that our protocol is an initial version whose sensitivities are bounded by the shot-noise limit. As the last work, we briefly present some approaches which have the potential for enhancing the performances of our protocol. (i) Squeezed vacuum injection or squeezed-vacuum-assisted measurements may provide a sensitivity breaking the shot-noise limit [45]. (ii) The use of photon addition and subtraction as a probabilistic amplifier is a promising way to improve the sensitivity [52]. (iii) With nonlinear detection schemes one may also obtain a great sensitivity improvement [53].

5. Conclusion

In summary, we propose a novel tri-port interferometric protocol for simultaneously estimating two phase shifts with two coherent states as inputs, and balanced homodyne detection as strategy. Based upon power-recycling technology, two shot-noise-limited sensitivities are reachable with 3N/2 photons on average as inputs. By calculating the QFI, the results state that the optimal sensitivity of the first phase given by our protocol is saturated, and that of the second phase is nearly saturated. Finally, we study the effects of photon loss on the sensitivity. With regard to the loss in transmission process, we show that the sensitivity is only related to the transmissivity of the path containing the phase shift. As to the loss in detection process, a better sensitivity is given when the two detection efficiencies of the detectors are identical.

Funding

National Natural Science Foundation of China (61701139).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017). [CrossRef] [PubMed]

2. S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43 (2017). [CrossRef] [PubMed]

3. J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground entanglement-based quantum key distribution,” Phys. Rev. Lett. 119, 200501 (2017). [CrossRef] [PubMed]

4. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008). [CrossRef]

5. A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008). [CrossRef]

6. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]

7. I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153–185 (2014). [CrossRef]

8. J. Struck, C. Ölschläger, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, “Quantum simulation of frustrated classical magnetism in triangular optical lattices,” Science (2011). [CrossRef] [PubMed]

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

10. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006). [CrossRef] [PubMed]

11. S.-Y. Lan, P.-C. Kuan, B. Estey, P. Haslinger, and H. Müller, “Influence of the coriolis force in atom interferometry,” Phys. Rev. Lett. 108, 090402 (2012). [CrossRef] [PubMed]

12. P. Haslinger, N. Dörre, P. Geyer, J. Rodewald, S. Nimmrichter, and M. Arndt, “A universal matter-wave interferometer with optical ionization gratings in the time domain,” Nat. Phys. 9, 144 (2013). [CrossRef] [PubMed]

13. M. W. Doherty, V. V. Struzhkin, D. A. Simpson, L. P. McGuinness, Y. Meng, A. Stacey, T. J. Karle, R. J. Hemley, N. B. Manson, L. C. L. Hollenberg, and S. Prawer, “Electronic properties and metrology applications of the diamond NV center under pressure,” Phys. Rev. Lett. 112, 047601 (2014). [CrossRef]

14. W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, “Mach-Zehnder interferometry in a strongly driven superconducting qubit,” Science 310, 1653–1657 (2005). [CrossRef] [PubMed]

15. Z.-E. Su, Y. Li, P. P. Rohde, H.-L. Huang, X.-L. Wang, L. Li, N.-L. Liu, J. P. Dowling, C.-Y. Lu, and J.-W. Pan, “Multiphoton interference in quantum fourier transform circuits and applications to quantum metrology,” Phys. Rev. Lett. 119, 080502 (2017). [CrossRef] [PubMed]

16. A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photonics 9, 577 (2015). [CrossRef]

17. T. Ono, R. Okamoto, and S. Takeuchi, “An entanglement-enhanced microscope,” Nat. Commun. 4, ncomms3426 (2013). [CrossRef]

18. C. A. Pérez-Delgado, M. E. Pearce, and P. Kok, “Fundamental limits of classical and quantum imaging,” Phys. Rev. Lett. 109, 123601 (2012). [CrossRef] [PubMed]

19. B. P. Abbott, and LIGO Scientific Collaboration, and Virgo Scientific Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016). [CrossRef] [PubMed]

20. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784 (2007). [CrossRef] [PubMed]

21. M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5309–5317 (2013). [CrossRef] [PubMed]

22. A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006). [CrossRef] [PubMed]

23. M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111, 173601 (2013). [CrossRef] [PubMed]

24. B. Escher, R. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011). [CrossRef]

25. G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett. 94, 220405 (2005). [CrossRef] [PubMed]

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

27. R. Demkowicz-Dobrzański and L. Maccone, “Using entanglement against noise in quantum metrology,” Phys. Rev. Lett. 113, 250801 (2014). [CrossRef]

28. Z. Huang, C. Macchiavello, and L. Maccone, “Usefulness of entanglement-assisted quantum metrology,” Phys. Rev. A 94, 012101 (2016). [CrossRef]

29. J. Dressel, K. Lyons, A. N. Jordan, T. M. Graham, and P. G. Kwiat, “Strengthening weak-value amplification with recycled photons,” Phys. Rev. A 88, 023821 (2013). [CrossRef]

30. K. Lyons, J. Dressel, A. N. Jordan, J. C. Howell, and P. G. Kwiat, “Power-recycled weak-value-based metrology,” Phys. Rev. Lett. 114, 170801 (2015). [CrossRef] [PubMed]

31. Y.-T. Wang, J.-S. Tang, G. Hu, J. Wang, S. Yu, Z.-Q. Zhou, Z.-D. Cheng, J.-S. Xu, S.-Z. Fang, Q.-L. Wu, C.-F. Li, and G.-C. Guo, “Experimental demonstration of higher precision weak-value-based metrology using power recycling,” Phys. Rev. Lett. 117, 230801 (2016). [CrossRef] [PubMed]

32. M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. X 1, 621–639 (2016).

33. P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013). [CrossRef] [PubMed]

34. T. Baumgratz and A. Datta, “Quantum enhanced estimation of a multidimensional field,” Phys. Rev. Lett. 116, 030801 (2016). [CrossRef] [PubMed]

35. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983). [CrossRef] [PubMed]

36. C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002). [CrossRef]

37. C. C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010). [CrossRef]

38. L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22, 11945–11953 (2014). [CrossRef] [PubMed]

39. W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12, 113025 (2010). [CrossRef]

40. M. Takeoka, R.-B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17, 043030 (2015). [CrossRef]

41. J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008). [CrossRef]

42. C. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976).

43. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012). [CrossRef]

44. J. Cheng, “Quantum metrology for simultaneously estimating the linear and nonlinear phase shifts,” Phys. Rev. A 90, 063838 (2014). [CrossRef]

45. 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, 052118 (2017). [CrossRef]

46. B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4, 4 (2017). [CrossRef]

47. 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, 053837 (2017). [CrossRef]

48. P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014). [CrossRef]

49. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, C. You, S. Adhikari, J. P. Dowling, and Y. Zhao, “Orbital-angular-momentum-enhanced estimation of sub-heisenberg-limited angular displacement with two-mode squeezed vacuum and parity detection,” Opt. Express 26, 16524–16534 (2018). [CrossRef] [PubMed]

50. T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81, 033819 (2010). [CrossRef]

51. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Deterministic super-resolved estimation towards angular displacements based upon a sagnac interferometer and parity measurement,” arXiv preprint arXiv:1809.04830 (2018).

52. B. T. Gard, D. Li, C. You, K. P. Seshadreesan, R. Birrittella, J. Luine, S. M. H. Rafsanjani, M. Mirhosseini, O. S. Magaña-Loaiza, and B. E. Koltenbah, “Photon added coherent states: nondeterministic, noiseless amplification in quantum metrology,” arXiv preprint arXiv:1606.09598 (2016).

53. J. Beltrán and A. Luis, “Breaking the Heisenberg limit with inefficient detectors,” Phys. Rev. A 72, 045801 (2005). [CrossRef]

Simultaneously estimating two phase shifts with a power-recycled tri-port interferometer fed by coherent states

Abstract

1. Introduction

2. Fundamental principle

3. Quantum Fisher information and the sensitivity limit

4. Effects of photon loss

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (4)

Equations (41)

OSA Continuum