A tensor network approach to sensing quantum light-matter interactions

Abstract

We present the fundamental limits to the precision of estimating parameters of a quantum matter system probed by light, even when some of the light is lost. This practically inevitable scenario leads to a tripartite quantum system of matter, and light -- detected and lost. Evaluating fundamental information theoretic quantities such as the quantum Fisher information of only the detected light was heretofore impossible. We succeed by expressing the final quantum state of the detected light as a matrix product operator. We apply our method to resonance fluorescence and pulsed spectroscopy. For both, we quantify the sub-optimality of continuous homodyning and photo-counting measurements in parameter estimation. For the latter, we find that single-photon Fock state pulses allow higher precision per photon than pulses of coherent states. Our method should be valuable in studies of quantum light-matter interactions, quantum light spectroscopy, quantum stochastic thermodynamics, and quantum clocks.

Figures (5)

• Figure 1: (a) Schematic representation of our tripartite setup. The emitter (M), a matter system with Hamiltonian $H^\mathrm{M}_{\theta}$, interacts with propagating light (L), initially in a coherent state, through an operator $J_\theta$, and also with the environment (E), resulting in the Lindblad operators $L_j$ acting on M. Only L can be detected after the interaction,from which the parameter $\theta$ is to be estimated. (b) Tripartism: The tripartite Hilbert space structure $\mathcal{H}^\mathrm{M} \otimes \mathcal{H}^\mathrm{L} \otimes \mathcal{H}^{\mathrm{E}}$. The impact of E is purified to an Hamiltonian interaction with operators $L_j$. The tripartite state $\ket{\Psi^{\mathrm{MLE}}(t)}$ is a pure continuous matrix product state. (c) After coarse-graining into time-bins of size $\Delta t,$ the reduced state of the light $\rho^{\mathrm{L}}_\theta$ is approximately described by a matrix product operator. (d) The QFI of a state in MPO form is computed variationally Chabuda2020 using Eq. \ref{['eq:QFI_variational']}; the tensors $\mathcal{B}_{\theta,[n]}$ are the MPO representation of the derivative $\partial_\theta \rho^{\mathrm{L}}_\theta$. (e) Two case studies considered in this work: (I) Rabi frequency $\Omega$ estimation in resonance fluorescence with detectors of efficiency $\eta <1$; and (II) dipole-moment $\Gamma$ estimation of a two-level system with pulsed classical light, when not all emitted light can be detected. $\Gamma_{\!\perp}$ denotes emission into the lost or undetected modes.
• Figure 2: QFIs and CFIs for Rabi frequency ($\Omega$) estimation, generically denoted as $\mathrm{FI}_\Omega(t)$ rescaled by the evolution time. The markers correspond to different quantities, as in (a); colors correspond to values of the efficiency $\eta$, as in (b). Panel (a) shows the dynamics of the various quantities as a function of the evolution time, for a fixed $\Omega = 0.1\gamma$. Panel (b) shows the the same figures of merit evaluated at the time $t_\mathrm{fin}=10/\gamma$, as a function of the Rabi frequency; the dash-dotted lines represents the sub-QFI. The MPO-QFI is $\mathcal{Q}[ \rho_\Omega^{\mathrm{L}}]$, the TSME-QFI is $\mathcal{Q}[ \rho_\Omega^{\mathrm{LE}} ]$ which for our problem is also the QFI of $\rho^{\mathrm{LE}}_\Omega$ for $\eta=1$, the PD-CFI is $\mathcal{C}[ p_{\Omega}^{\mathrm{pd}} ]$ and the HD-CFI is $\mathcal{C}[p_\Omega^{\mathrm{hd}}]$. The HD-CFI is for a homodyine angle $\varphi = \pi /2$, which is known to be optimal for $\Omega$ estimation Kiilerich2016.
• Figure 3: QFIs and CFIs for $\Gamma$ estimation with pulsed light, generically denoted as $\mathrm{FI}_\Gamma(t),$ multiplied by $\Gamma^2$ to be adimensional and rescaled by the average number of photons $\bar{n}$ . Circles represent the MPO-QFI, stars the PD-CFI, the dash-dotted line the sub-QFI. The colors represent different average number of photons $\bar{n}$ for the coherent states; the black dashed line represents single photons. The light-blue shapes is a visual aid that represents the pulse profile $|\phi(t)|^2$ (not to scale on the y-axis). The centre of the pulse is at $t_c = 6.5 T$. Each panel corresponds to the parameter values shown on top. To avoid clutter, we only plot the PD-CFI, since HD was found to always perform worse.
• Figure 4: QFIs and CFIs for Rabi frequency ($\Omega$) estimation, generically denoted as $\mathrm{FI}_\Omega(t)$ rescaled by the evolution time. The markers correspond to different quantities, as in (a); colors correspond to values of the efficiency $\eta$, as in (b). Panel (a) shows the dynamics of the various quantities as a function of the evolution time, for a fixed $\Omega = 0.1\gamma$. Panel (b) shows the the same figures of merit, except the CFIs, evaluated at the time $t_\mathrm{fin}=10/\gamma$, as a function of the Rabi frequency; the dash-dotted lines represents the sub-QFI. The MPO-QFI is $\mathcal{Q}[ \rho_\Omega^{\mathrm{L}}]$, the TSME-QFI is $\mathcal{Q}[ \rho_\Omega^{\mathrm{LE}} ]$ which for our problem is also the QFI of $\rho^{\mathrm{LE}}_\Omega$ for $\eta=1$, the PD-CFI is $\mathcal{C}[ p_{\Omega}^{\mathrm{pd}} ]$ and the HD-CFI is $\mathcal{C}[p_\Omega^{\mathrm{hd}}]$. The HD-CFI is for a homodyine angle $\varphi = \pi /2$, which is known to be optimal for $\Omega$ estimation Kiilerich2016.
• Figure 5: QFIs and CFIs for $\Gamma$ estimation with pulsed light, generically denoted as $\mathrm{FI}_\Gamma(t),$ multiplied by $\Gamma^2$ to be adimensional and rescaled by the average number of photons $\bar{n}$ . Circles represent the MPO-QFI, stars the PD-CFI, the dash-dotted line the sub-QFI. The colors represent different average number of photons $\bar{n}$ for the coherent states. The light-blue shapes is a visual aid that represents the pulse profile $|\phi(t)|^2$ (not to scale on the y-axis). The centre of the pulse is at $t_c = 6.5 T$. Each panel corresponds to the parameter values shown on top.