In the general setup, we consider some initial probe in a state |ψ 0 〉 which is subject to an unknown phase shift described by a unitary \(U(\phi ) = e^{i\phi \hat n}\), where \(\hat n\) is the photon number operator. The goal is to estimate the phase ϕ. After the interaction, an entangled state is used to teleport the output state U(ϕ)|ψ 0 〉 back to interact with the phase shift again. This process is then iterated m times. If the teleportation is perfect, this would correspond to the transformation |ψ 0 〉 → (U(ϕ))(m+1)|ψ 0 〉 = U((m + 1)ϕ)|ψ 0 〉 of the input state where m is the number of teleportations. By coherently applying the phase (m + 1) times, the signal can have both super-resolution and super-sensitivity since it will now depend on (m + 1)ϕ instead of just ϕ.22,23

As a physical realization of this protocol, we consider the setup illustrated in Fig. 1 where consecutive two-mode squeezed vacuum states are supplied by interferring the output of two single-mode squeezed vacuum sources on a balanced beam splitter.27 One mode is delayed in a fiber and will subsequently be subject to an unknown phase shift described by the unitary U(ϕ). Feedback based on previous measurements is applied before the phase shift. This feedback can be implemented by mixing in an auxiliary laser field using a high transmission beam splitter.27 The delay (T) is chosen such that the phase shifted mode is interfered with the first mode of the subsequent two-mode squeezed vacuum state on a balanced beam splitter before measurement. The setup, which is similar to a Mach–Zehnder interferometer with feedback, is inspired by ref. 28 where the generation of continuous variable (CV) cluster states is demonstrated. As demonstrated in ref. 28, the squeezed light can be produced using optical parametric oscillation (OPO). Binning the temporal modes such that T is larger than the inverse bandwidth of the OPO results in independent squeezed temporal modes. We choose the measurements and the feedback such that the CV teleportation protocol of ref. 29 is realized. In this teleportation protocol, the momentum quadrature of one of the output modes and the position quadrature of the other is measured, which can be achieved with homodyne detection in a single-shot measurement. The feedback consists of quadrature displacement based on the measurement outcomes. For perfect teleportation, infinitely many photons are, in principle, needed in the two-mode squeezed vacuum states. The number of photons actually obtaining the phase shift will nonetheless only depend on the initial input state. For situations where the phase shift is obtained by interaction with a photosensitive system,8,9,10,11 the effective number of probe photons interacting with the system is the limited resource. This number will be \(\sim (m + 1)n_0\) where n 0 is the number of photons in the initial state. We will show that Heisenberg-limited sensitivity in terms of probe photons can be reached with a simple coherent state as input state. Furthermore, the phase resolution can be enhanced by a factor of m + 1.

We consider a coherent state |−iα〉 as the initial probe state |ψ 0 〉 \(\left( {\alpha \in {\Bbb R}} \right)\). In the setup of Fig. 1, we can input the initial state by displacing the initial vacuum mode of the lower arm using the feedback laser. After the interaction of U(ϕ), the state will be |−iαeiϕ〉. This state is now teleported back to the second mode of the first two-mode squeezed vacuum state following the CV protocol of ref. 29 The two-mode squeezed vacuum state has squeezing parameter r such that \(\left\langle {\left( {\hat x_2 - \hat x_3} \right)^2} \right\rangle = e^{ - 2r}/2\), where \(\hat x_2\), \(\hat x_3\) are the position quadratures for the two modes. The first mode of the two-mode squeezed vacuum state is mixed with the probe state on the balanced beam splitter before measurement. The output modes of the beam splitter have position quadratures \(\hat x_1^\prime = (\hat x_1 + \hat x_2)/\sqrt 2\) and \(\hat x_2^\prime = (\hat x_1 - \hat x_2)/\sqrt 2\) with similar expressions for the momentum quadratures. Here \(\hat x_1\) is the position quadrature of the probe state. The quadratures \(\hat p_1^\prime\) and \(\hat x_2^\prime\) are measured giving measurement outcomes \(\{ p_1^\prime ,x_2^\prime \}\). The feedback then implements the displacements \(\hat x_3 \to \hat x_3^\prime = \hat x_3 + g_x\sqrt 2 x_2^\prime\) and \(\hat p_3 \to \hat p_3^\prime = \hat p_3 + g_p\sqrt 2 p_1^\prime\), which concludes the teleportation protocol of ref. 29

The feedback displaces the quadratures such that the teleported state, ψ 1 will be close to |−iαeiϕ〉. The quality of the teleportation will depend on the amount of squeezing contained in the two-mode squeezed vacuum state and the feedback strength quantified by the gains g x and g p . In the limit of high squeezing, perfect teleportation is obtained for g x = g p = 1. The protocol now repeats itself m times corresponding to m teleportations being performed. At the end of the protocol, the squeezed light sources should be switched off such that the final teleported state |ψ m 〉 is not mixed with any two-mode squeezed states before measurement. The teleported state |ψ m 〉 is subject to the phase shift resulting in state \(\left| {\psi _{\mathrm{m}}^\prime } \right\rangle = U(\phi )\left| {\psi _{\mathrm{m}}} \right\rangle\). This state is split by the 50:50 beams splitter (with vacuum in the other input port) and the position quadratures of the output modes are measured and classically added with equal weight of \(1/\sqrt 2\). This is equivalent to measuring the position quadrature \(\left( {\hat x_m^\prime } \right)\) of \(\left| {\psi _{\mathrm{m}}^\prime } \right\rangle\) before the beam splitter and we can therefore simply consider this situation. Assuming gains of g x = g p = 1, the mean and variance of \(\hat x_m^\prime\) is

$$\left\langle {\hat x_{\mathrm{m}}^\prime } \right\rangle = \left\langle {\psi _{\mathrm{m}}^\prime } \right|\hat x_{\mathrm{m}}^\prime \left| {\psi _{\mathrm{m}}^\prime } \right\rangle = \alpha \,{\mathrm{sin}}((m + 1)\phi )$$ (1)

$${\mathrm{Var}}\left( {\hat x_{\mathrm{m}}^\prime } \right) = \left\langle {\psi _{\mathrm{m}}^\prime } \right|\left( {(\hat x_{\mathrm{m}}^\prime )^2 - \left\langle {\hat x_{\mathrm{m}}^\prime } \right\rangle ^2} \right)\left| {\psi _{\mathrm{m}}^\prime } \right\rangle = \frac{{1 + 2me^{ - 2r}}}{4}.$$ (2)

It is clear from Eq. (1) that the signal exhibits super-resolution in ϕ by a factor of (m + 1). The sensitivity of the measurement can be quantified as6

$$\sigma _m = \frac{{\sqrt {{\mathrm{Var}}(\hat x_{\mathrm{m}}^\prime )} }}{{|\delta \langle \hat x_{\mathrm{m}}^\prime \rangle /\delta \phi |}} = \frac{{\sqrt {1 + 2me^{ - 2r}} }}{{2(m + 1)\alpha |{\mathrm{cos}}((m + 1)\phi )|}}.$$ (3)

Note that the sensitivity exhibits a linear decrease in the number of teleportations m as long as |cos((m + 1)ϕ)| ≈ 1 and the squeezing is sufficiently strong such that \(2me^{ - 2r} \ll 1\). For a classical (SQL) strategy with m independent coherent states |−iα〉, the sensitivity would have a scaling of \(\propto 1/(\sqrt m \alpha )\). The average number of probe photons, n j contained in the state |ψ j 〉 is

$$n_j = \alpha ^2 + je^{ - 2r},$$ (4)

thus the total average number of probe photons that have interacted with the phase shift system will be

$$n_{{\mathrm{total}}} = \mathop {\sum}\limits_{j = 0}^m n_j = (m + 1)\alpha ^2 + \frac{1}{2}m(m + 1)e^{ - 2r}.$$ (5)

If the coherent state contains one photon (α = 1) on average, we have that \(n_{{\mathrm{total}}} = (m + 1)\left( {1 + \frac{1}{2}me^{ - 2r}} \right)\) and the sensitivity is

$$\sigma _m = \frac{{\sqrt {\left( {1 + \frac{1}{2}me^{ - 2r}} \right)^2\left( {1 + 2me^{ - 2r}} \right)} }}{{2n_{{\mathrm{total}}}|{\mathrm{cos}}((m + 1)\phi )|}}.$$ (6)

Thus, if \(me^{ - 2r} \ll 1\), the sensitivity exhibits Heisenberg scaling in the number of probe photons for |cos((m + 1)ϕ)| ≈ 1. This sensitivity is similar to what could be obtained using NOON states of (m + 1) photons and expresses the ultimate scaling allowed by quantum mechanics.1

Limited squeezing

One of the dominant experimental limitations of the proposed protocol will arguably be the amount of squeezing in the two-mode squeezed vacuum states. This will limit how many teleportations can be performed before the extra noise from the imperfect teleportations will dominate the signal. We therefore consider what the optimum strategy is given a constraint on the amount of squeezing. In addition, we also limit the total average number of photons that can interact with the phase shift system. We then optimize over the number of teleportations m and the size of the coherent probe state α to find the strategy that provides the maximum sensitivity for these limitations. Furthermore, we also allow for arbitrary gains g x and g p . The result of the optimization is shown in Fig. 2 where we illustrate the performance relative to a standard coherent state protocol with matched average photon number. For such an approach, the sensitivity is simply \(\sigma _{{\mathrm{coh}}} = 1/\left( {2\sqrt {n_{{\mathrm{total}}}} |{\mathrm{cos}}(\phi )|} \right)\) where n total is the average number of probe photons. For |cos(ϕ)| ≈ 1, the coherent state approach exhibits sensitivity at the SQL. Figure 2 shows the two effects of the imperfect teleportation; noise is added in the \(\hat x\)-quadrature (see Eq. (2)) and more photons are added to the probe state (see Eq. (5)). In the minimization, the error from the extra photons added by an imperfect teleportation has smaller weight for higher n total . In the limit where \(n_{{\mathrm{total}}} \gg e^{2r}\), the enhancement is \(\sim e^r/\sqrt 2\) and equal gains of g x = g p = 1 are optimal. This is the limit where the extra photons added to the probe state do not have any significant effect on the optimum performance. We note that a similar enhancement in sensitivity could be obtained by using a squeezed coherent state as probe.15,30 For such protocols, the squeezed photons, however, interact with the phase shift system, which is not the case here. Consequently, our protocol also works in the limit \(n_{{\mathrm{total}}} \ll e^{2r}\) where an enhancement of \(\sim \left( {n_{{\mathrm{total}}}e^{2r}/2} \right)^{\frac{1}{4}}\) can be obtained for g x = g p = 1. Note that our numerical optimization shows that larger enhancement can also be obtained for optimized gains in this limit (see Fig. 2b). Finally, we note that no enhancement is possible when \(r \lesssim 0.35\) for g x = g p = 1 since the extra amplitude noise added by the teleportation cancels the gain in the signal (see Eq. (3)). For optimized gains, however, there can be a small enhancement.

Fig. 2 Performance for finite squeezing. Maximum gain in sensitivity by using the teleportation scheme compared to a classical coherent state protocol for limited amount of squeezing (r) and fixed average number of total photons n total . We have assumed that |cos((m + 1)ϕ)| ≈ 1. The performance is better for high n total because here the photons added to the probe states by imperfect teleportation have smaller weight compared to the extra noise added to the quadrature. We have assumed gains of g x = g p = 1 in (a) while we have numerically optimized the gains in (b). It is seen that in the limit \(n_{{\mathrm{total}}} \lesssim e^{2r}\) the optimal gains are different from g x = g p = 1. The optimal number of teleportations m found in the optimizations are indicated with circles, squares and diamonds. These indicate the transitions to m ≥ 10, 100, and 1000, respectively, on the curves Full size image

Photon loss

One of the technological challenges of using highly entangled quantum states for enhanced phase measurements is that they are very fragile to losses. Multi-pass protocols share this fragility since losses grow exponentially with the number of passes through the sample.24 This means that if the losses are too high, the sensitivity enhancement of the multi-pass protocol proposed here will vanish. Note however that while approaches based on NOON states rely on single photon detection, this protocol is based on homodyne detection, which in practice is much more efficient. Since imperfect photon detection will add to the overall loss, this means that the effective loss may be substantially reduced with this protocol.

We investigate the performance of the proposed protocol in the presence of both loss acting on the probe state corresponding to a lossy phase shift system and loss acting on the two-mode squeezed vacuum states. Any (symmetric) detection loss would add directly to both of these losses. We model the losses with fictitious beam splitters where the unused output port is traced out. To model the lossy phase shift system, a fictitious beam splitter of transmission η 1 is inserted after the phase shift U(ϕ) (see Fig. 1). For the loss in the two-mode squeezed vacuum state, fictitious beam splitters both with transmission η 2 are inserted for each of the modes. For simplicity, we have assumed equal losses for both modes. Assuming equal gains of g x = g p = 1, the signal and sensitivity after m teleportations for {η 1 , η 2 } < 1 is

$$\langle \hat x_m^\prime \rangle = \alpha \eta _1^{\frac{{m + 1}}{2}}{\mathrm{sin}}((m + 1)\phi )$$ (7)

$$\sigma _m = \frac{{\sqrt {1 + 2\eta _1\frac{{1 - \eta _1^m}}{{1 - \eta _1}}\left( {\eta _2e^{ - 2r} + 1 - \eta _2} \right)} }}{{2(m + 1)\alpha \eta _1^{\frac{{m + 1}}{2}}|{\mathrm{cos}}((m + 1)\phi )|}}.$$ (8)

As expected, the loss on the probe state (η 1 ) enters in the expression for the sensitivity exponentially in m, while loss on the two-mode squeezed vacuum states (η 2 ) only has a linear effect in m. The effect of η 2 < 1 on the sensitivity is equivalent to having a limited squeezing of \(r_{{\mathrm{lim}}} = - \frac{1}{2}ln\left( {\eta _2e^{ - 2r} + 1 - \eta _2} \right)\). This also holds when considering the average number of total probe photons incident on the phase shift system. For m teleportations and gains of g x = g p = 1, we have that

$$n_{{\mathrm{total}}} = \frac{{1 - \eta _1^{m + 1}}}{{1 - \eta _1}}\alpha ^2 + \frac{{m(1 - \eta _1) - \eta _1(1 - \eta _1^m)}}{{(1 - \eta _1)^2}}\left( {\eta _2e^{ - 2r} + 1 - \eta _2} \right).$$ (9)

Note that by taking the limit η 1 → 1 for η 2 = 1, Eqs. (7)–(9) reduce to Eqs. (1), (3) and (5). If excess noise on the squeezed states is included by mixing in thermal states of average photon number \(\bar n\) instead of vacuum in the fictitious beam splitters (η 2 ), one would have that \(r_{{\mathrm{lim}}} = - \frac{1}{2}ln\left( {\eta _2e^{ - 2r} + (1 + 2\bar n)(1 - \eta _2)} \right)\). We will assume that \(\bar n \ll 1\) such that excess noise can be neglected. To see the effect of finite losses, we again compare the protocol to the simple coherent strategy for which \(\sigma _{{\mathrm{coh}}} = 1/\left( {2\sqrt {\eta _1n_{{\mathrm{total}}}} |{\mathrm{cos}}(\phi )|} \right)\) in the presence of loss. The result of the optimization is shown in Fig. 3. While a small improvement was found by optimizing the gains, near optimal performance is reached for g x = g p = 1. The error from losses in the two-mode squeezed vacuum state limits the gain in the same way as finite squeezing does for the lossless case. Consequently, when these losses dominate the error, the enhancement is \(\sim 1/\sqrt {2(1 - \eta _2)}\) and no enhancement is possible for η 2 ~1/2. When losses in the probe state limit the enhancement, the optimum performance is effectively found as a tradeoff between the \(\sqrt {m + 1}\) enhancement due to the teleportation and the exponential reduction due to the loss. As a result, we find that the enhancement is \(\sim \sqrt {2/(3(1 - \eta _1))}\) and no enhancement is possible for η 1 ~ 1/3. We note that while losses quickly reduce the enhancement, the scheme still exhibits enhanced sensitivity compared to the standard coherent state probe even for substantial losses.