Microwave-field quantum metrology with inherent robustness against detection losses enabled by Rydberg interactions

TL;DR

This work tackles robust microwave-field sensing with Rydberg-atom ensembles by exploiting intrinsic dipolar interactions to implement a premeasurement processing step that protects encoded information against detection losses. The authors develop a two-particle interaction model, extend to multi-particle regimes, and validate the concepts with an ultracold $^{87}$Rb experiment that demonstrates a ~3.3-fold FI enhancement under lossy readout. A simple two-excitation toy model with Kraus operators captures the essential FI improvement, and rigorous bounds plus numerical optimization prove the protocol is optimal for the considered loss model. Experimentally, they achieve a Fisher-information-per-detected-photon of about $3.6$ with a normalized FI of $ ilde{\\mathcal{F}}\approx 3.3$, translating to a per-shot microwave-field sensitivity of $\Delta E_{MW} \approx 44$ $\mu$V cm$^{-1}$ and indicating a practical path to loss-robust quantum metrology using inherent sensor interactions.

Abstract

Quantum sensing and metrology present one of the most promising near-term applications in the field of quantum technologies, with quantum sensors enabling unprecedented precision in measurements of electric, magnetic or gravitational fields and displacements. Experimental loss at the detection stage remains one of the key obstacles to achieving a truly quantum advantage in many practical scenarios. Here, we combine the capabilities of Rydberg atoms to both sense external fields and be used for quantum information processing, thereby largely overcoming the issue of detection losses. While utilising the large dipole moments of Rydberg atoms in an ensemble to achieve a $\SI{39}{\nV\per\cm \hertz\tothe{-1/2}}$ sensitivity, we employ inter-atomic dipolar interactions to take advantage of an error-prevention protocol that protects information against conventional losses at the detection stage. Counterintuitively, the protocol's idea is based on introducing an additional non-linear, lossy quantum channel, which results in a 3.3-fold enhancement of Fisher information. The presented results pave the way for broader adoption of quantum-information-inspired enhancements enabled by intrinsic interactions present in a sensor system, and more broadly in practical quantum metrology and communication, without the need for a general-purpose quantum computer.

Figures (9)

• Figure 1: (a) Ideational scheme of the presented experiment. Initially, photons are stored in the atomic ensemble as a collective Rydberg excitation $\ket{\psi_0}$ in the mode $d$. Afterwards, the spinwaves created that way are placed in the superposition between modes $d$ and $p$ representing the two different Rydberg atomic levels with rotation $\theta$. This allows for the dipolar Rydberg-Rydberg interactions, denoted as $\Lambda$, to scatter the collective excitations in different modes (depicted with green area). LAI (light-atoms interface) is applied to convert the stored excitations to photons, allowing for the detection with the overall efficiency $\eta$. (b) Pair interaction strength for Rydberg states $\ket{d} = \ket{49^2D_{5/2}\;m_J=5/2}$ and $\ket{p} = \ket{50^2P_{3/2}\;m_J=3/2}$. Energy of symmetric superposition $\ket{dp}+\ket{pd}$ is denoted with a blue line, whereas the red line represents the energy of anti-symmetric superposition. (c) Two-excitation spinwave wavefunction $\Psi(x_p,x_d)$ with one excitation in $\ket{p}$ and position $x_p$ and the other in state $\ket{d}$ and position $x_d$. Depending on the distance between the excitations, a part of the wavefunction will acquire additional phase due to the pair interactions (depicted with a hatched region). (d) Elements of the two-excitation collective superposition. Blue and orange circles represent atoms in state $\ket{d}$ and $\ket{p}$, respectively. The grey circles in the background represent the rest of the atoms in the ensemble. (d) Elements of the two-excitation collective superposition with fixed position of the atom in state $\ket{p}$. The blurred blue region represents an atom in state $\ket{d}$ in spatial superposition. Below, the plot shows the absolute value and phase of a slice of the wavefunction. The region around the fixed atom in state $\ket{p}$ acquires an additional phase.
• Figure 2: (a) Scheme of the experimental setup based on the ultracold atoms in the magneto-optical trap and geometric arrangement of the beams. The signal beam and counter-propagating coupling beam are exciting the spinwave in the atomic ensemble. Coherence transfer (CT) beams shining from the side of the chamber are cancelling the wavevector of the induced coherence. A microwave field is applied from the side with a horn antenna. (b) Experimental sequence for the interaction-enhanced metrology via increasing the steepness of the Rabi oscillation slope. (c) Relevant $^{87}\mathrm{Rb}$ energy level configuration for the experiment.
• Figure 3: (a) Experimental sequence for the measurement of the interaction-induced decay rate as a function of the super Rabi oscillation angle $\theta$. (b) Data for calibration of interaction-induced decay rate with fitted exponential decays for different super Rabi oscillation angles. (c) Interaction-induced decay rates with fitted theoretical predictions. Inset of (c): (d) Interaction-induced decay rates as a function of the ratio of population in the $p$ mode with fitted linear function. (e) Rabi oscillation between mode $d$ and $p$ modified by interaction-induced decay. Blue points represent oscillations measured at mode $d$. Green points represent oscillations measured at mode $p$. Dashed lines are theoretical predictions that were fitted to experimental data. Dotted lines represent Rabi oscillation without Rydberg dipolar interactions. The experimental data were normalised with the normalisation factor $\bar{n}$. Plots (b) and (c) correspond to the experimental sequence depicted in (a), and plot (e) corresponds to the experimental sequence depicted in Fig. \ref{['fig:mot_lvls_seq']}(b).
• Figure 4: (a) Estimated oscillation angle calculated using maximum likelihood estimation is represented with blue triangles. Set oscillation angles are represented with orange dots. Blue and red shaded areas represent the variance of the maximum likelihood estimator and inverse square root of FI, respectively, per $N = 100$ experimental shots. (b) Normalized theoretical FI and inverse of normalized variance of the estimator. The black line represents the value of the normalized FI without the interactions, and the shaded area above represents the interaction-induced metrological improvement. (c) Difference between the set oscillation angle and the estimated value.
• Figure 5: (a) Scheme of the precision enhancement protocol. After encoding the parameter $\theta$, the system undergoes a pre-measurement processing operation denoted with $\Lambda$. Later, both modes $p$ and $d$ are measured with efficiency $\eta$. (b) Action of a lossy channel on the states. The proposed interaction $\Lambda$ transfers the red component $\ket{1,1}$, which introduces overlapping decay paths to state $\ket{0,0}$ and leaves black components with reversible decay paths. (c) Increase of FI with the proposed interaction for a range of detection efficiencies $\eta$. The dotted line corresponds to the value of $\eta$ presented in (e). (d) The plot shows expectation values of excitation number operators on the state. The dashed lines represent expectation values without the operation. (e) The plot shows the normalized FI and its enhancement for some values of the encoded parameter. The dashed line represents FI without the operation. Normalized FI compares the performance with and without the operation $\Lambda$.