Observation of Perfect Absorption in Hyperfine Levels of Molecular Spins with Hermitian Subspaces

TL;DR

This work demonstrates Perfect Absorption (PA) in a passive open quantum system formed by molecular spins coherently coupled to a planar microwave resonator at milliKelvin temperatures, described by non-Hermitian Hamiltonians with emergent Hermitian subspaces. By rotating to the polariton basis, PA is shown to occur when the imaginary part of a polariton eigenfrequency vanishes ($\Im(\tilde{\Omega}_j)=0$), a condition tracked by the zeros of the reflection parameter $S_{11}(\omega)$ and tunable through resonator-spin detuning. The authors realize PA both in a single-spin ensemble (BDPA) and in a multi-spin ensemble (VOTPP) across strong and weak coupling regimes, revealing detuning-controlled balance between dressed cavity feeding and spin loss rates $\bar{\gamma}_j=(-\gamma_{\rm r}+\gamma_{\rm nr})|U_{j1}|^2+\gamma_{\rm s}|U_{j2}|^2$. The results illuminate how Hermitian subspaces shape coherent spectra in cavity QED and offer a flexible platform to explore non-Hermitian physics and potential microwave single-photon switching applications, extendable to other spin systems and frequency ranges.

Abstract

We investigate Perfect Absorption (PA) of radiation, in which incoming energy is entirely dissipated, in a system consisting of molecular spin centers coherently coupled to a planar microwave resonator operated at milliKelvin temperature and in the single photon regime. This platform allows us to fine tune the spin-photon coupling and to control the effective dissipation of the two subsystems towards the environment, thus giving us the opportunity to span over a wide space of parameters. Our system can be effectively described by a non-Hermitian Hamiltonian exhibiting distinct Hermitian subspaces. We experimentally show that these subspaces, linked to the presence of PA, can be engineered through the resonator-spin detuning, which controls the composition of the polaritons in terms of photon and spin content. In such a way, the required balance between the feeding and the loss rates is effectively recovered even in the absence of PT-symmetry. We show that Hermitian subspaces influence the overall aspect of coherent spectra of cavity QED systems and enlarge the possibility to explore non-Hermitian effects in open quantum systems. We finally discuss how our results can be potentially exploited for applications, in particular as single-photon switches and modulators.

Figures (6)

• Figure 1: Implementing and modeling the open passive quantum system.a Sketch of the lumped element resonator with all sample positions investigated in this work. The large light-yellow rectangle represents the BDPA sample, while the smaller rectangles represent the different positions of the VOTPP crystal (#A to #D, from red to light-blue). The distance between the antenna (yellow) and the chip can be adjusted at room temperature to vary the radiative relaxation rate of the resonator $\gamma_{r}$. The position of the sample controls the coupling strength $g_{\mu}$ with the resonator. The red arrow shows the direction of the applied static magnetic field, $B_{0}$. b Model adopted in this work for the open quantum system in a. The resonator is coupled to both spins and its input/output line (antenna). Only a single spin $\mu$-th ensemble is shown for clarity. c Molecular structure for BDPA and VOTPP. Labels indicate their electronic $S=1/2$ and nuclear $I=7/2$ spins. Images reproduced from yamabayashiJACS2018Azuma2006 with permission. d Easyspin simulation of the $\{S_z,I_z\}$ energy levels of VOTPP obtained with the parameters and the Hamiltonian reported in yamabayashiJACS2018bonizzoniNPJQUANT2020 at 9.88 GHz. Vertical arrows show the eight allowed $\ket{-\frac{1}{2},I_z} \leftrightarrow \ket{\frac{1}{2},I_z}$ transitions giving the $\mu$-th (sub)ensembles, denoted with different colors. The vertical energy scale is cut for better clarity.
• Figure 2: Perfect Absorption for the Single Ensemble Case.a Normalized reflection map ($|S_{11}(\omega)|$) measured for the BDPA sample at 25 mK as a function of the static magnetic field $B_0$. b Simulated reflection map obtained using the fit parameters extracted from the map in panel a, according to Eq. (\ref{['eq:meth-S11']}) in Methods (fit parameters are reported in Table~1 of the Supplementary Information). c Theoretical (green and blue lines) and experimental (orange and red dots) normalized reflection spectra extracted from the maps in a and b, showing two dips with nearly zero reflection (see vertical lines $B_1$ and $B_2$ in panel a). d (Upper panel) Imaginary parts of $\tilde{ \Omega}_{1,2}$, $[\Im(\tilde{\Omega}_{1,2})]$, as a function of $B_0$, calculated using Eq. (\ref{['eq:S11_zeros']}) and the fit parameters obtained from a. Perfect absorption (blue crosses) occurs at the polariton resonances when $\Im(\tilde{\Omega}_{1,2})$ crosses zero. d (Lower panel) Dressed cavity feeding rate $\bar{\gamma}_{{\rm c}i}$ and dressed spin loss rate $\bar{\gamma}_{{\rm s}i}$ as functions of $B_0$, computed according to Eq. (\ref{['eq:gamma_bar_strong']}). Perfect absorption (blue crosses) is achieved when $\bar{\gamma}_{{\rm c}i}=\bar{\gamma}_{{\rm s}i}$ for the $i$-th polariton (see main text).
• Figure 3: Perfect Absorption During the Transition from Strong to Weak Coupling Regime.a, b Normalized reflection ($\left \lvert S_{11} \right \rvert$) maps as a function of the static magnetic field $B_0$, simulated using Eq. (\ref{['eq:meth-S11']}) in Methods with the parameters fitted from the data in Fig. \ref{['fig:single_spin']}, except for using lower coupling strength values of $g/2\pi = 1.5\,$MHz (a) and $g/2\pi = 1\,$MHz (b), respectively. c Imaginary parts of $\tilde{ \Omega}_{1,2}$, $[\Im(\tilde{\Omega}_{1,2})]$, calculated as a function of the static magnetic field using the relaxation rates fitted from the data in Fig. \ref{['fig:single_spin']} and four different values of $g/2\pi$. Perfect absorption occurs when $\Im(\tilde{\Omega}_{1,2})$ crosses zero and cannot be realized for $g/2\pi <\,1.4\,$MHz (see main text).
• Figure 4: Perfect Absorption for the Multiple Spin Case. a, d Normalized reflection ($\left \lvert S_{11} \right \rvert$) maps measured as a function of the static magnetic field $B_0$ at 30 mK for the VOTPP crystal. PA is observed in proximity of multiple hyperfine levels. The strongest (position $\#A$) and the weakest (position $\#D$) coupling regimes are shown, respectively. b, e Simulated reflection maps obtained by fitting the maps in a,d according to Eq. (\ref{['eq:meth-S11']}) in Methods. c, f (Upper panels) Imaginary parts of the complex frequencies, $\Im{(\tilde{\Omega}_j)}$, as a function of the magnetic field $B_0$, for the $j$-th polariton frequency. The horizontal black dotted line at $\Im{(\tilde{\Omega})}=0$ corresponds to the PA condition. c, f (Lower panels) Real parts of the polariton frequencies, $\Re{(\Omega_j)}$, with the predicted PA points, showing excellent agreement with the experimental data. Both $\Im{(\tilde{\Omega}_j)}$ and $\Re{(\Omega_j)}$ are calculated using Eq. (\ref{['eq:S11_zeros']}) supported by Eq. (\ref{['eq:meth-S11']}) in Methods. g, h, i Experimental (red dots) and theoretical (blue lines) normalized reflection obtained according to the vertical lines shown in b, e ($B_1$, $B_2$, and $B_3$), displaying perfect absorption dips and one not satisfying this condition. All fit parameters are given in tables 5 and 8 of Supplementary Information.
• Figure 5: Comparison with PT symmetry.a Surface plot showing the imaginary part of the complex eigenvalues of the effective non-Hermitian Hamiltonian $\hat{H}_{\rm RZ}$ as a function fo the detuning $\Delta$ and the spins decay rate $\gamma_{\rm s}$. The simulation is obtained using $N=1$ and the values reported in Table S1. The two red lines indicate where an Hermitian subspace is realized, i.e., the intersection with the $\Im (\tilde{\Omega}) = 0$ plane (in blue), where one of the eigenvalues becomes real. These lines coalesce at the point $(\gamma_{\rm s}, \Delta) = (\gamma_{\rm r} - \gamma_{\rm nr}, 0)$, marked by the red triangle, corresponding to the PT symmetry condition. b Normalized reflection map as a function of the detuning for a PT-symmetric system in the strong coupling regime. PA is achieved simultaneously for both polariton branches at $\Delta = 0$, conversely to the result presented in this work. c Imaginary parts of the eigenfrequencies $\Im (\tilde{\Omega})$ as a function of the detuning for different coupling strengths, ranging from the weak to the strong coupling regime, in a PT-symmetric system.