Abstract
Quantum metrology can approach measurement precision of Heisenberg Limit using an ideal quantum source, which has attracted a great interest in fundamental physical studies. However, the quantum metrology precision is impressionable to the system noise in experiments. In this paper, we analyze the influence of multiphoton events on the phase estimation precision when using a nondeterministic single photon source. Our results show there are an extra bias and quantum enhanced region restriction due to multiphoton events, which declines the quantum phase estimation precision. A limitation of multiphoton probability is obtained for quantum enhanced phase estimation accuracy under different experimental model. Our results provide beneficial suggestions for improving quantum metrology precision in future experiments.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum metrology can improve the sensing performance beyond classically achievable precision using quantum optical sources, as was demonstrated in squeezed state enhanced gravitational wave detection for instance [1,2]. Moreover, quantum metrology has attracted widespread attentions in quantum imaging [3], atom clock measuring [4,5] or biological sample probing [6], etc. In theory, the standard quantum limit is the upper limitation of metrology based on a classical laser source due to the intrinsic noise. Using an ideal quantum source, the metrology precision could approach Heisenberg limit ($1/n$, where $n$ is the total photon number) for a quantum metrology [7–14].
However, the Heisenberg limit is very difficult to reach due to the noise influence in quantum metrology experiments [14–20]. Many studies have concerned the influences of imperfect interferometers and detectors on quantum metrology precision [21–23]. For example, a lossy interferometer induces an extra information loss, which leads to the asymmetric photon distribution on the output arms of the interferometer [23–28]. The phase fluctuations between different modes result in the variation of off-diagonal elements of the interferometer density matrix [28–31]. Meanwhile, the limited detection efficiency and dark counts of single photon detectors result in photon event recording error [22,23]. And bunching events cannot be distinguished by commonly used single photon detectors [32,33]. To distinguish multiphoton events, transition-edge-sensor (TES) number-resolving detectors are developed for bunching events detection in visible and telecom bands [34–37], and pseudo photon number-resolving detectors (PNRDs) with on-off photon detector arrays are invented to detect bunching events with fundamental components [21,38–41]. However, the multiphoton events are rarely considered in quantum metrology to date [42,43]. For a nondeterministic single photon sources based on spontaneous parametric down-conversion (SPDC), multiphoton phenomena are commonly existing [44]. These imperfect components can deviate the probability distribution during projective measurements, leading to an extra information loss in phase estimation process.
In this paper, we theoretically analyze the influence of imperfect single photon source on quantum phase estimation. The classical Fisher information (FI) and quantum Fisher Information (QFI) are analyzed to evaluate the phase uncertainty bound of a quantum metrology system. Multiphoton events induce a stronger restriction of the quantum enhanced bound, and the upper limitation of multiphoton probability for quantum enhanced precision is obtained. Our results express the information loss caused by multiphoton events and provide a limitation of the multiphoton probability in quantum metrology.
2. Phase estimation process and Fisher information
A phase estimation system includes an initial probe state $\rho _0$ preparation, an unitary evolution $U(\boldsymbol {\theta })$, and a positive-operator valued measure (POVM) $\hat {E}$ acting on the evolved state $\rho (\boldsymbol {\theta } ) = U(\boldsymbol {\theta } ){\rho _0}{U^{\dagger} }(\boldsymbol {\theta } )$. The phases $\boldsymbol {\theta }=\{\theta _1,\theta _2,\ldots,\theta _M\}$ under test are estimated using a suitable estimator $\Lambda (x)$ based on measurement results $x = \{ {x_1},{x_2}, \ldots,{x_\nu }\}$, where $\nu$ is the number of measurements.
The accuracy of an estimation process is quantified by the Mean Square Error (MSE), which is defined by
The transformation of annihilation operators can be calculated by
For multiphase estimation processes, QFIM can be calculated with the output state $\psi _{\boldsymbol {\theta }}$ [13]:
Maximum likelihood estimation (MLE) approach is the most commonly used estimator by maximizing the likelihood function
where $x_n$ is the $n$-th coincidence event, $\boldsymbol {\theta }$ is the $M$-phase sequence [45]. The logarithm of the likelihood function is defined as the log-likelihood function:MLE is an asymptotically unbiased estimator, i.e. $\mathop {\lim }_{\nu \to \infty } {\Lambda ^{{\text {MLE}}}}(\boldsymbol {\theta }) = \boldsymbol {\theta }$ for any $\boldsymbol {\theta }$ to be estimated. Thus Eq. (2) is satisfied for $\nu \gg 1$ under MLE.
3. Single phase estimation
For an ideal initial input state ${|\psi _0 \rangle } = |1{\rangle _{{a_1}}}|0{\rangle _{{a_2}}}$ as shown in Fig. 11(a), the expectation value $P$ of photon projector $\hat {S}_{b_1}$ is
Where $|\psi \rangle$ is the output state. Based on the inversion method, the phase can be estimated using the probability distribution $P$ as ${\theta _{{\text {est}}}} = \arccos (P)$ [53]. The phase uncertainty is deduced from the error propagation formula [7,54]Another single phase estimation model is represented in Fig. 11(b), where idler and signal photons are injected into the two ports of an MZI simultaneously. The twin-Fock state $|\psi \rangle = |N/2,N/2\rangle$ is used as an input state which can improve the phase estimation uncertainty to the Heisenberg limit with the total input photon number $N$ [18,56]. The lower bound of the phase uncertainty using twin-Fock states is
A photon number measurement is required to saturate the lower bound for $N>2$. Quantum state projection method has been developed to simulate number resolving detectors called PNRDs [14,17,39], where multiple photons are separated and detected by a single photon detector respectively. Unfortunately, it still can’t saturate QCRB with a set of projectors over the ideal output cases, which will be discussed later.Using the simplest form of twin-Fock state $|1_{a_1},1_{a_2}\rangle$ as an input photon state, the coincidence count rate has the form
The minimal phase variance $\Delta \theta = 0.2512$ is achieved at $\theta \approx 0.28\pi$, as shown in Fig. 33(b). However, due to the multi-photons information loss during measurement, it still cannot reach the theoretical lower phase variance bound ${(\Delta \theta )^2} = 0.2450$ for an imperfect photon source, as the QFI of the initial state $\rho _0 = (1 - k)|11\rangle \langle 11| + k|22\rangle \langle 22|$ is $F_Q=4+8k$. The necessary and sufficient conditions to achieve the lower bound of QFI are
and4. Multiphase estimation
Compared with two-mode MZIs, multimode interferometers have higher phase sensitivity in phase estimation [58,59]. Previous studies have shown an adaptive phase estimation strategy to optimize the precision of multiphase estimation [60–62]. The three-beam extension of a beam splitter, named tritter, has the mathematical description form [63]
For a three-mode interferometer, since the unitary process without phase information does not effect the result of QFIM, the last tritter does not need to take into consideration [57]. The output state after the first tritter and phase encoded shifters can be described as
MLE requires a prior knowledge about the probability distribution $P(\mu )$ of the output measurement events $\mu$. For example, the likelihood function is $L(X;\boldsymbol {\theta } ) = \prod \nolimits _{p,q} {{P_i}{{(23 \to pq)}^{{n_{pq}}}}}$, where $n_{pq}$ is the number of measurements on output $(p, q)$, and ${P_i}(23 \to pq)$ is the ideal probability distribution. However, the probability distribution variation caused by imperfect single photon sources introduces an extra bias in the estimation process. The bias distribution results are shown in Figs. 55(a) and 55(b) under the assumption that the input state is a mixed state ${\rho _0} = (1 - k)|11\rangle \langle 11| + k|22\rangle \langle 22|$. At the point $({\theta _1},{\theta _2}) = \left ( {0.299, - 2.370} \right )$, the estimation of MLE is deviated from the real value due to the multiphoton influence. In order to eliminate the bias, multiphoton effects have to be considered. The probability distribution should be normalized as the multiphoton input states may trigger more than one coincident record at the same time. The modified likelihood function can be written as $L(X;\theta ) = \prod \nolimits _{p,q} {P{{(23 \to pq)}^{{n_{pq}}}}/[\sum \nolimits _{i,j} {P(23 \to ij)} ]}$ where $P(23 \to ij)$ is the real output probability to eliminate the bias. One might use post-selection to exclude multiphoton detecting events to neglect four-photon coincidence events. However, this method excludes separated single photon events that arrived at the same time resolution in error [65], and maintains some four-photon states unfiltered such as $|400\rangle$ or $|220\rangle$, etc. Furthermore, inefficiency TES detectors may misjudge the multiphoton states as single photon outputs. These states also lead to a probability deviation which is similar to our demonstration, and not discussed in this paper.
Same as the case of the two-mode interferometer, extra multiphoton events also decrease the phase accuracy. The necessary and sufficient condition for saturating QCRB is [57]
With the imperfect input state, one can obtain thatThe diagonal elements are always nonnegative in the above formula, thus the total phase variance satisfies
To achieve a phase uncertainty lower than separate photons, one needs to limit the above information loss in a small range. For two-mode interferometer model, the multiphoton probability $k$ should be lower than 0.5345. Three-mode interferometer requires a stricter region, $k$ cannot surpass 0.0406 which means only a close-to-pure state can generate quantum enhanced accuracy. The normalized relationship between phase variance and multiphoton probability $k$ are shown in Fig. 66. The multiphoton probability limitation is under the assumption that other parts of the measurement process is ideal. However, in realistic experiments, due to the photon loss and imperfect detector efficiency, the multiphoton states are more easily detected [42]. The limitation would be more restrict once other noises in the measurement system are considered.
5. Conclusion
In this paper, the influence of multiphoton events of single photon sources on quantum enhanced phase measurement is demonstrated. The multiphoton events are spontaneously and nondeterministic generated in parametric conversion single photon sources. It induces an extra bias which declines the phase estimation accuracy compared with ideal case. Meanwhile, the quantum enhanced phase estimation is restricted in a limited region for both single-phase and multi-phase scenarios. The phase estimator is also improved through considering the multiphoton events. Our results provide suggestions to eliminate multiphoton bias and analyzing information losses with imperfect single photon sources.
Funding
National Natural Science Foundation of China (62075238).
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
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