Optimal measurement scheme with a proper choice of data processing is important to realize high-precision phase measurement.^[1–3] For the commonly used particle-number difference measurement (or the measurement) over quasi-classical coherent states, the phase sensitivity is constrained by the shot-noise limit (SNL) , where N is the number of particles of the input state. Various schemes have been proposed and demonstrated to improve the phase estimation precision beyond the SNL by using quantum entangled states,^[4,5] e.g., the so-called NOON states . This is a maximally entangled state with all the particles being either in the mode a or in the mode b, leading to the Heisenberg limited phase sensitivity δ θ = 1/N.^[3–6] However, the NOON states are difficult to prepare and are easily subject to the loss-induced decoherence.^[7,8]

Compared with various kinds of entangled states, the twin-Fock states of photons (or atoms) |n, n⟩_a,b are believed to be more robust to the decoherence and hence result in a better phase estimation precision under typical experimental noise.^[9] With the twin-Fock states as the input of the Mach–Zehnder interferometer (MZI), the interferometric signal is vanishing if one performs the measurement at the output ports.^[10] Therefore, various measurement schemes such as the parity detection,^[11–17] the measurement,^[18–20] and the so-called single-fringe detection^[21] have been proposed to realize high-precision phase measurement beyond the SNL. These phase estimation protocols are based on the inversion estimators with their uncertainties predicted by the error-propagation formula.

According to the general theory of quantum metrology, the ultimate estimation precision is determined by the Cramér–Rao lower bound (CRB)^[22–25] , where is the classical Fisher information (CFI), dependent on the measurement probabilities. Furthermore, the CRB can be saturated asymptotically by maximum likelihood estimation (MLE) or Bayesian estimation.^[26,27] In this paper, we investigate high-precision phase measurement with the twin-Fock states of light. We show that the measurement is optimal to achieve a global phase estimation at the Heisenberg limit. More specially, we find that the CRB saturates the quantum Cramér–Rao lower bound (QCRB) for any value of the phase shift, i.e., , where N = 2n is the number of particles and F_Q = N(N + 2)/2 denotes the quantum Fisher information (QFI). This result remains valid for the measurement. To demonstrate it clearly, we use the six-photon twin-Fock state as the input and consider experimental imperfections in the measurement probabilities, similar to a recent experimental work.^[21] Our results indicate that both the precision and the sensing region of the measurement are better than those of the single-fringe measurement. Finally, we numerically simulate the phase estimation protocol using the asymptotically unbiased maximum likelihood estimator.

For the input twin-Fock states |n, n⟩_a,b, the probabilities for detecting particle-number difference (i.e., the so-called measurement) are given by

Fig. 1.

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Fig. 1. The conditional probabilities for detecting particle-number difference: pμ(θ) for (a) μ = 0, (b) μ = ± 1, (c) μ = ± 2, and (d) μ = ± 3. The red dashed lines show the ideal results; the blue solid lines give the occurrence probabilities with the experimental imperfections, obtained from Eq. (6). Numerical simulations give the statistical average of the occurrence frequency (the solid circles) and its standard derivation (the bars) of each outcome, simulated by M = 20 replicas of random numbers at each θ ∈ (−π, π). Hereinafter, denotes the statistical average.

The amount of phase information that is encoded by an arbitrary pure state is given by the quantum Fisher information , where the expectation value ⟨···⟩_in is taken with respect to the input state. For the N-photon twin-Fock states, it is easy to obtain the QFI and hence the QCRB . To saturate it, we consider the measurement and calculate the CFI^[22–25]

Fig. 2.

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Fig. 2. The CFI for the measurement (thick solid line), the single-fringe detection (thin solid line), and the multi-fringe detection (red dashed line), obtained respectively, from Eqs. (2), (5), and (7), with . The dashed horizontal lines are the classical limit of the QFI FQ = N and the ideal result of the QFI with the twin-Fock input state FQ = N(N + 2)/2 for N = 6. The shaded area is a region where the CFI of the measurement outperforms that of the single-fringe detection.

For the twin-Fock input states, the output signal of the measurement is always vanishing,^[32] implying that the inversion estimator based on the average signal cannot infer the true value of the phase shift. To overcome it, other measurement schemes have been proposed, e.g., the parity measurement^[11–17] and the measurement.^[18–20] With the error propagation formula, it has been shown that the phase sensitivity of the two measurements can only saturate the QCRB at a certain value of the phase shift (e.g., θ = 0).^[21] We show that the CFI of the measurement is the same with that of the measurement, leading to a global phase estimation (see Appendix A).

Recently, Xiang et al.^[21] have demonstrated a very simple and efficient single-fringe detection. Their measurement scheme has been proposed originally by Sun et al.^[33] and is equivalently to a projection measurement over the output state, with the signal , corresponding to Eq. (1) for μ = 0. Note that p₀(θ) describes the occurrence probability for detecting equal numbers of particles at the two output ports n₁ = n₂ = n, i.e., μ = (n₁ − n₂)/2 = 0, denoted hereafter by the outcome “0”. All the other detection events can be treated as a single outcome “∅”, with the occurrence probability .

Obviously, the single-fringe detection is indeed a binary-outcome measurement scheme,^[34–36] where the two outcomes “0” and “∅” can be obtained by coarse-graining over the original (N + 1) measurement outcomes of the measurement. For any binary-outcome measurement, it has been shown that the inversion estimator based on the average signal always saturates the CRB ,^[34–36] with the CFI given by

The experimental imperfections can be taken into account artificially.^[37] Similar to Ref. [38], we adopt a simple method by introducing the peak heights and the visibility in the probabilities {p_μ(θ)}, i.e., replacing the occurrence probabilities by

To simulate the measurement, we first generate random numbers according to at a given phase shift (see Appendix B). The occurrence numbers of all the outcomes and hence the occurrence frequencies can be obtained from the random numbers, which comprise a single phase measurement.^[27] Next, we repeat the above process to obtain the average occurrence frequencies (see the circles of Fig. 1) for any value of θ ∈ (−π, π), which can be fitted by . In a real experiment, the phase-dependent occurrence probabilities can be obtained from the interferometer calibration.

Using the occurrence probabilities , we obtain the CFI of the measurement and that of the single-fringe detection , as depicted by Fig. 2. One can see that both and cannot reach the ideal result of the QFI F_Q = 24. For the single-fringe detection (see the thin solid line), the phase-dependence of shows a quite good agreement with that of Ref. [21]. Maximum of the CFI is found at the optimal working point θ_max = 0.29 (16.5°), with . While for the measurement (the solid thick line), appears at θ_max = 0.35 (20°).

The shaded area in Fig. 2 is a region where the amount of phase information extracted by the measurement is larger than that of the single-fringe detection. Obviously, this is straightforward since the former one contains (N + 1) measurement contributions and hence . However, to find out which measurement outcomes dominate in the CFI is still nontrivial. We now consider the CFI of a multi-outcome measurement, where besides the outcome μ = 0, additional contributions from the outcomes μ = ± 2 are taken into account,

Compared with the single-fringe case, is indeed the CFI of a multi-fringe detection.^[36] Using the occurrence probabilities , we calculate the phase-dependence of , as illustrated by the red dashed line of Fig. 2. One can see that the maximum of also appears at θ_max = 0.35 (20°), with . Obviously, both and are obtained by {coarse-graining} over the original (N + 1) measurement outcomes, leading to . Indeed, the coarse-graining method over a general multi-outcome measurement (i.e., combining several different outcomes into a single measurement outcome) will reduce the CFI; see Appendix C. Our results in Fig. 2 show that the outcomes μ = 0, ± 2 are of importance to achieve the best sensitivity of the measurement .

To saturate the CRB, we consider the well-known phase-estimation protocol based on the MLE, which is unbiased as the number of independent measurements (see e.g. Ref. [22]). Numerically, the MLE θ_est can be solved by maximizing the likelihood function (i.e., a multinomial distribution)

In Fig. 3, we plot the scaled phase distribution as a function of θ for θ₀ = π/4. From the inset, one can see that there are four peaks of over the entire phase interval θ ∈(−π, π), and only one of them can be used to predict θ₀. This leads to the so-called phase ambiguity.^[39–41] To avoid it, we simply introduce a prior knowledge about θ₀ using the phase distribution , where P(θ) = 1 for θ ∈ (0,π/2), and 0 outside. For large enough , one can find that the phase distribution can be well approximated by a Gaussian with respect to the phase shift θ,

Fig. 3.

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Fig. 3. Maximum likelihood estimator. For a given true value of phase shift θ0 = π/4, the phase distribution as a function of θ (the red dashed line), which approaches to a Gaussian (solid blue line) and hence can be used to determine the MLE θest and its variance σ (i.e, 68.3% confidence interval). The inset: four peaks of over the entire phase interval θ ∈ (−π, π). The occurrence number of each outcome is obtained from random numbers.

In Fig. 4, we plot the statistical average of the phase uncertainty (see the circles) and its standard derivation (the bars) for each true value of phase shift θ₀ ∈ (0,π/2). From the inset, one can note that the standard derivation of the MLE ≫ (θ₀ − ⟨θ_est⟩_s), indicating that the MLE is unbiased.^[27] As a result, the phase uncertainty per measurement shows a good agreement with the CRB , see the blue solid line. Remarkably, the CRB of the measurement can beat the shot-noise limit within a phase interval, e.g., 0 < θ₀ < arctan 2 (∼ 63°); while for the single-fringe detection, it is the interval (∼ 39°). This means that the quantum-enhancement sensing region of the measurement is about {twice} wider than that of the single-fringe detection (see the vertical grid lines of Fig. 4). The enlarged sensing region is important in quantum multiparticle microscopy that uses nonclassical states of light for illumination.^[42]

Fig. 4.

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Fig. 4. For each given true value of the phase shift θ0 ∈ (0, π/2), statistical average of (the solid circles) and its standard derivation (the bars), simulated by M = 20 replicas of random numbers. Blue solid line is the CRB of the measurement; and black dashed line is the CRB of the single-fringe measurement. The inset: statistical average of (θ0 − θest) as a function of θ0 and the standard derivation. Horizontal grid lines are the shot-noise limit and the ideal result of the QCRB for N = 6. The vertical grid lines give and arctan 2.

Finally, it should be mentioned that compared with other entangled states, the twin-Fock states are more robust to decoherence and give better phase estimation precision under typical experimental noise.^[9] To date, small N twin-Fock states of light have been prepared using spontaneous parametric down-conversion. Nevertheless, it is still a significant challenge to generate high photon-number states with a deterministic way. Fortunately, the deterministic preparations of atomic twin-Fock states have been proposed and demonstrated using spin-1 Bose–Einstein condensates.^[20,43,44] In particular, Luo et al.^[44] have shown that the atomic twin-Fock states can be deterministically generated by controlled quantum phase transition. These results give a renewed interest to realize phase measurements near the Heisenberg limit with large N twin-Fock states.^[45]

In summary, we have investigated an optimal phase estimation protocol with the twin-Fock states as the input of a two-mode Mach–Zendner interferometer. Our analytical results indicate that the CFI of the measurement is the same as that of the measurement and both of them saturate the QFI for any true value of phase shift, i.e., , which leads to a global phase estimation . To demonstrated it clearly, we consider the six-particle twin-Fock state as the input and numerically simulate the (N + 1) occurrence probabilities of the measurement, where the experimental imperfections are taken into account phenomenologically. Our results show that the maximum value of the CFI is about 20.1, approaching to the ideal result of the QFI F_Q = 24. Furthermore, the quantum-enhancement sensing region of the measurement is about twice wider than that of the single-fringe detection, which is important to improve the global imaging quality of a quantum multiphoton microscopy.^[42] Finally, we numerically simulate the phase estimation protocol using the asymptotically unbiased maximum likelihood estimator.