Loss-resistant unambiguous phase measurement

Entangled multiphoton states have the potential to provide improved measurement accuracy, but are sensitive to photon loss. It is possible to calculate ideal loss-resistant states that maximize the Fisher information, but it is unclear how these could be experimentally generated. Here we propose a set of states that can be obtained by processing the output from parametric down-conversion. Although these states are not optimal, they provide performance very close to that of optimal states for a range of parameters. Moreover, we show how to use sequences of such states in order to obtain an unambiguous phase measurement that beats the standard quantum limit. We consider the optimization of parameters in order to minimize the final phase variance, and find that the optimum parameters are different from those that maximize the Fisher information.

Figure 1

A linear optical device (LOD) with two spontaneous parametric down-conversion (SPDC) states and vacuum modes as input. ${|\psi \rangle }_{\mathrm{out}}$ is the desired output state, and there is postselection on vacuum in the other modes. The two SPDC input states can be simplified to one.

Figure 2

(a) A triangular array of beam splitters and phase shifts fed with a two-mode input state ${|\psi \rangle }_{\mathrm{in}}$ and three vacuum modes $|0\rangle$. The phase shifts are included with the beam splitters (shown as thick black lines) for simplicity. Output photons are detected in two of the modes ${|\psi \rangle }_{\mathrm{out}}$, and vacuum is postselected in three of the output modes. All the arms below the dashed line are vacuum. (b) The five-port LOD can be simplified to a four-port one by keeping only the beam splitters above the dotted line.

Figure 3

The state preparation scheme for loss-resistant states. It is a three-port interferometer consisting of three beam splitters with reflectivities $R_1,R_2,$ and $R_3$ and two phase shifters ${\phi }_1$ and ${\phi }_2$. The initial dual Fock state $|n\rangle |n\rangle$ gives the $2n$-photon loss-resistant state in the output, as indicated in Eq. (12).

Figure 4

Adaptive phase measurement scheme. $\phi$ is the unknown phase, $\theta$ is the controlled phase, and $D_1$ and $D_2$ are the photon detectors in the two outputs. The diagonal lines in the two arms are the fictitious beam splitters with transmissivity $\eta$ modeling photon loss.

Figure 5

Maximum of classical Fisher information $F(\phi ,\theta )$ versus efficiency $\eta$. , four-photon exact optimal state Eq. (16), maximized over ${\chi }_1^{\prime }$, ${\chi }_2^{\prime }$, $\phi$, and $\theta$; , four-photon loss-resistant optimal states Eq. (15), maximized over $\chi$, $\phi$, and $\theta$; $+$, four single-photon states.

Figure 6

Classical Fisher information $F(\phi ,\theta )$ and Holevo phase variance $V_{\mathrm{H}}$ versus $\chi$ for $\eta =0.6$. Solid green line: Fisher information for $\phi =\pi /4$, $\theta =0$ versus $\chi$ calculated using Eq. (25) for the two-photon state given in Eq. (14). Dashed black line: phase variance versus $\chi$ for the sequence of seven single photons followed by one two-photon loss-resistant state in the adaptive measurement protocol. The Fisher information gives a lower bound to the phase variance, rather than an exact phase variance, so the value of $\chi$ that maximizes the Fisher information need not be the value that minimizes the phase variance.

Figure 7

The scaled phase variance $\mathcal{N}{V}_{\mathrm{H}}$ versus total number of photons $\mathcal{N}$ for $\eta =0.6$ for only single-photon states (SQL) (solid line) and for the optimal sequence of single-photon states combined with two- and four-photon loss-resistant states (shown by ).

Figure 8

The phase variance $V_{\mathrm{H}}$ versus the total number of photons $\mathcal{N}$ for $\eta =0.6$ for three different input states. : optimal sequence of single-photon states combined with two- and four-photon loss-resistant states. Solid line: only single-photon states (SQL). : the scheme proposed in [39].