Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relative phase and dynamical phase sensing in a Hamiltonian model of the optical SU(1,1) interferometer

T. J. Volkoff

TL;DR

This work presents a first-principles Hamiltonian model for the optical SU(1,1) interferometer formed by two downconversion processes with opposite pump phases, highlighting φ as a kinematic relative phase and θ as a dynamical interaction time. It demonstrates that the optimal sensitivity to φ occurs at φ = π with Heisenberg-like QFI scaling in energy, while the optimal readout is a weighted two-mode shift operator that saturates the bound asymptotically, unlike total photon-number readouts. For the dynamical phase θ, the Hamiltonian model exhibits a logarithmically modified Heisenberg scaling at θ = 0 within a specific parameter domain, with an asymptotically optimal readout derived from a two-mode squeezed-state picture and a φ–θ duality in the asymptotics. The results reveal fundamental differences from circuit-based models, motivate a first-principles approach to multi-downconversion sensors, and point to experimental tests and applications in Gaussian-state generation and CV quantum sensing.

Abstract

The SU(1,1) interferometer introduced by Yurke, McCall, Klauder is reformulated starting from the Hamiltonian of two identical optical downconversion processes with opposite pump phases. From the four optical modes, two are singled out up to a relative phase by the assumption of exact alignment of the interferometer (i.e., mode indistinguishability). The state of the two resulting modes is parametrized by the nonlinearity $g$, the relative phase $φ$, and a dynamical phase $θ$ resulting from the interaction time. The optimal operating point for sensing the relative phase (dynamical phase) is found to be $φ= π$ ($θ=0$) with quantum Fisher information exhibiting Heisenberg scaling $E^{2}$ (logarithmically modified Heisenberg scaling $\left({E\over \ln E}\right)^{2}$). Compared to the predictions of the circuit-based model, we find in that in the Hamiltonian model: 1. the optimal operating points occur for a non-vacuum state inside the interferometer, and 2. measurement of the total photon number operator does not provide an estimate of the relative or dynamical phase with precision that saturates the quantum Cramer-Rao bound, whereas an observable based on weighted shift operators becomes optimal as $g$ increases. The results indicate a first-principles approach for describing general optical quantum sensors containing multiple optical downconversion processes.

Relative phase and dynamical phase sensing in a Hamiltonian model of the optical SU(1,1) interferometer

TL;DR

This work presents a first-principles Hamiltonian model for the optical SU(1,1) interferometer formed by two downconversion processes with opposite pump phases, highlighting φ as a kinematic relative phase and θ as a dynamical interaction time. It demonstrates that the optimal sensitivity to φ occurs at φ = π with Heisenberg-like QFI scaling in energy, while the optimal readout is a weighted two-mode shift operator that saturates the bound asymptotically, unlike total photon-number readouts. For the dynamical phase θ, the Hamiltonian model exhibits a logarithmically modified Heisenberg scaling at θ = 0 within a specific parameter domain, with an asymptotically optimal readout derived from a two-mode squeezed-state picture and a φ–θ duality in the asymptotics. The results reveal fundamental differences from circuit-based models, motivate a first-principles approach to multi-downconversion sensors, and point to experimental tests and applications in Gaussian-state generation and CV quantum sensing.

Abstract

The SU(1,1) interferometer introduced by Yurke, McCall, Klauder is reformulated starting from the Hamiltonian of two identical optical downconversion processes with opposite pump phases. From the four optical modes, two are singled out up to a relative phase by the assumption of exact alignment of the interferometer (i.e., mode indistinguishability). The state of the two resulting modes is parametrized by the nonlinearity , the relative phase , and a dynamical phase resulting from the interaction time. The optimal operating point for sensing the relative phase (dynamical phase) is found to be () with quantum Fisher information exhibiting Heisenberg scaling (logarithmically modified Heisenberg scaling ). Compared to the predictions of the circuit-based model, we find in that in the Hamiltonian model: 1. the optimal operating points occur for a non-vacuum state inside the interferometer, and 2. measurement of the total photon number operator does not provide an estimate of the relative or dynamical phase with precision that saturates the quantum Cramer-Rao bound, whereas an observable based on weighted shift operators becomes optimal as increases. The results indicate a first-principles approach for describing general optical quantum sensors containing multiple optical downconversion processes.

Paper Structure

This paper contains 7 sections, 1 theorem, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Warm up: no difference between a circuit-based and Hamiltonian-based $SU(2)$ interferometer
  3. Hamiltonian and circuit-based models of two-mode $SU(1,1)$ interferometer
  4. Estimation of $\phi$ in the Hamiltonian model
  5. Estimation of the dynamical phase $\theta$
  6. Discussion
  7. Phase space properties of the Hamiltonian model

Key Result

Lemma 5.1

Let $\lambda^{2}>\theta^{2}$. Then, as $\lambda\rightarrow \infty$, converges to zero in $\ell^{2}(\mathbb{C})^{\otimes 2}$, where the range $[{\pi\over 2},\pi]$ is taken for $\cos^{-1}$.

Figures (2)

  • Figure 1: $SU(1,1)$ interferometer with downconversion modes labeled.
  • Figure 2: $\text{QFI}(\theta)$ relative to the total energy on the dynamical phase domain $\theta = [0,2g + 6]$ for various $g$. Black dots show the boundary of Domain 1 and Domain 2 at $\theta = 2g$, and blue horizontal lines show the relative QFI corresponding to the analytical result $\text{QFI}(\theta=0)={\sinh^{4}2g\over g^{2}}$. Analogous data for (\ref{['eqn:thqficirc']}) are not shown because the $\text{QFI}(\theta)$ is independent of $\theta$ and $E(g)=1$ at $\theta=0$.

Theorems & Definitions (2)

  • Lemma 5.1
  • proof