Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers

The SU(1,1) interferometer can be thought of as a Mach-Zehnder interferometer with its linear beam splitters replaced with parametric nonlinear optical processes. We consider the cases of bright- and vacuum-seeded SU(1,1) interferometers using intensity or homodyne detectors. A simplified truncated scheme with only one nonlinear interaction is introduced, which not only beats conventional intensity detection with a bright seed, but can saturate the phase-sensitivity bound set by the quantum Fisher information. We also show that the truncated scheme achieves a sub-shot-noise phase sensitivity in the vacuum-seeded case, despite the phase-sensing optical beams having no well-defined phase.

Figure 1

Schematic of a Mach-Zehnder or SU(1,1) interferometer. A state $|\psi \rangle$ and a vacuum state $|0\rangle$ are inputs on either a linear beam splitter (BS) or a nonlinear optical process. Unseeded denotes $|\psi \rangle =|0\rangle$ and bright seeded denotes $|\psi \rangle =|\alpha \rangle$, a coherent state with ${\left|\alpha \right|}^2\gg 1$. One of the two output beams acquires a phase shift $\varphi$ with respect to the other beam and the two beams are recombined in either a BS or an NLO. The phase shift $\varphi$ is inferred via measurements from the detectors on the final output beams. Here ${\mathcal{F}}_{\text{Q}}$ and ${\mathcal{F}}_{\text{C}}$ refer to the Fisher information associated with the quantum state and entire apparatus, respectively (see the text).

Figure 2

The three experimental configurations considered in this paper. The thick green line is a strong classical pump beam. (a) Conventional SU(1,1) interferometer. (b) Conventional SU(1,1) with homodyne detection on the output. (c) Truncated SU(1,1) interferometer. Here LO indicates a local oscillator and beam splitters represent loss and are labeled by their transmission $\eta$.

Figure 3

Phase sensor that can be viewed either as an imbalanced Mach-Zehnder interferometer or as a balanced homodyne detector with a local oscillator that has a well-defined phase with respect to the weak beam. The weak beam passing through the optical phase shift $\varphi$ contains ${\overline{N}}_p$ photons on average.

Figure 4

Sensitivity as a function of operating point for bright-seeded configurations with a gain of 4. Sensitivity is shown for several experimental configurations: intensity detection (${\widehat{M}}_{\text{N}}$), gray dash-dotted line; intensity detection of mode $b_f$ only (${\widehat{M}}_{\text{Nb}}$), orange dashed line; homodyne detection for the truncated and conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q}}$) with LO phases fixed at ${\theta }_a={\theta }_b=\pi /2$, thick blue solid line; and homodyne detection for the truncated SU(1,1) interferometer with classical gain correction (${\widehat{M}}_{\lambda \text{Q}}$) optimized over the LO phase, black dotted line. The QCRB coincides with the black dotted line and the SQL the black solid line. The inset shows the same graph over a larger range.

Figure 5

Sensitivity as a function of gain for bright seeded configurations, optimized over $\varphi$ and LO phases. Sensitivity is shown for several experimental configurations: intensity detection (${\widehat{M}}_{\text{N}}$), gray dash-dotted line; intensity detection of mode $b_f$ only (${\widehat{M}}_{\text{Nb}}$), orange dashed line; homodyne detection for the truncated and conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q}}$), red solid line; and homodyne detection for the truncated SU(1,1) interferometer with classical gain correction (${\widehat{M}}_{\lambda \text{Q}}$), black dotted line. The QCRB coincides with the black dotted line and the SQL the black solid line.

Figure 6

Sensitivity as a function of operating point for vacuum-seeded schemes with a gain of 4. Sensitivity is shown for several experimental configurations: intensity detection (${\widehat{M}}_{\text{N}}$), gray dash-dotted line; homodyne detection of the truncated and conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q2}}$) with LOs set at ${\theta }_a={\theta }_b=\pi /2$, blue solid line; and homodyne detection of the truncated and conventional SU(1,1) (${\widehat{M}}_{\text{Q2}}$) optimized over LO phases, blue dashed line. The QCRB is the black dotted line and the SQL is the black solid line.

Figure 7

Sensitivity as a function of gain for homodyne measurement of the bright seeded truncated or conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q}}$) and no external loss. Gain is linear in the number of spontaneous photons: ${\overline{n}}_s=2g-2$. The blue circles, yellow squares, green diamonds, and orange triangles represent internal transmissions ${\eta }_{\text{ai}}={\eta }_{\text{bi}}=0.5$, 0.75, 0.95, and 1, respectively. The black lines are guides for the eye, with the dashed line having scaling $1/g$ and the solid $1/g^2$.

Figure 8

Sensitivity as a function of gain for vacuum seeded configurations, optimized over $\varphi$ and LO phases. Sensitivity is shown for several experimental configurations: homodyne detection of the truncated and conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q2}}$), orange dashed line; intensity detection (${\widehat{M}}_{\text{N}}$), gray dash-dotted line. The QCRB coincides with the gray dash-dotted curve and the SQL is the black solid curve.

Figure 9

Sensitivity as a function of gain for homodyne measurement of the vacuum seeded truncated or conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q2}}$) and no external loss. Gain is linear in the number of spontaneous photons: ${\overline{n}}_s=2g-2$. The blue circles, yellow squares, green diamonds, and orange triangles represent internal transmissions ${\eta }_{\text{ai}}={\eta }_{\text{bi}}=0.5$, 0.75, 0.95, and 1, respectively. The black lines are guides for the eye, with the dashed line having scaling $1/g$ and the solid $1/g^2$.

Figure 10

Sensitivity as a function of mean input seed photon number with $g=4$, optimized over $\varphi$ and LO phases. Sensitivity is shown for several experimental configurations: homodyne detection of the truncated SU(1,1) interferometer with classical gain correction (${\widehat{M}}_{\lambda \text{Q}}$), blue dotted line; homodyne detection of the truncated and conventional SU(1,1) interferometer (${\widehat{M}}_{\text{Q}2}$), purple circles; and intensity detection of conventional SU(1,1) (${\widehat{M}}_{\text{N}}$), orange dashed line. The QCRB is the black dashed curve and the SQL is the black solid curve. For small seeds, intensity detection is optimal and for large seeds homodyne detection is optimal.

Figure 11

Sensitivity versus gain of NLO 2 for intensity detection (${\widehat{M}}_{\text{N}}$) of the conventional SU(1,1) interferometer, optimized over $\varphi$. NLO 1 has $g=2$ and there is no internal loss. The blue circles, yellow squares, green diamonds, and orange triangles represent external transmissions of ${\eta }_{\text{ae}}={\eta }_{\text{be}}=0.5$, 0.75, 0.9, and 0.99, respectively.