Tailoring the superradiant and subradiant nature of two coherently coupled quantum emitters

The control and manipulation of quantum-entangled states is crucial for the development of quantum technologies. A promising route is to couple solid-state quantum emitters through their optical dipole-dipole interactions. Entanglement in itself is challenging, as it requires both nanometric distances between emitters and nearly degenerate electronic transitions. Here we implement hyperspectral imaging to identify pairs of coupled dibenzanthanthrene molecules, and find distinctive spectral signatures of maximally entangled superradiant and subradiant electronic states by tuning the molecular optical resonances with Stark effect. We demonstrate far-field selective excitation of the long-lived subradiant delocalized state with a laser field tailored in amplitude and phase. Optical nanoscopy of the coupled molecules unveils spatial signatures that result from quantum interferences in their excitation pathways and reveal the location of each emitter. Controlled electronic-states superposition will help deciphering more complex physical or biological mechanisms governed by the coherent coupling and developing quantum information schemes.

: Single qubit rotation. a, Sketch of the experimental setup: A CW dye laser, optically chopped with an optical modulator produces optical pulses with a duration of 100 ns and a rise time/fall time of 500 ps (10%--90%) at a repetition rate of 100 kHz. The laser frequency is locked on a wavemeter with a digital PID to the frequency of a single emitter ZPL within 10 MHz. An avalanche photodiode combined with a start--stop acquisition card synchronized with the excitation pulses records the arrival times of the red--shifted fluorescence photons stemming from the molecule. b, Temporal evolution of the fluorescence signal from an isolated single DBATT molecule upon pulsed resonant excitation (Integration time 25 min, bin width 64 ps). The fluorescence background stemming from out--of--focus molecules has been subtracted and the fluorescence signal is normalized to 1/2 at long times in order to mimic the evolution of the excited state population. The black curve is a theoretical simulation of the excited state population, which reproduces the damped Rabi oscillations following the leading edge of the excitation pulse, with a Rabi angular frequency 11.7 ! and an optical coherence lifetime ! = 2 ! . The excited state lifetime ! = 7.39 ± 0.02 ns is deduced from the exponential decay following the trailing edge of the pulse. Fig. 2: Characteristics of the sample and electrodes. a, Schematic of the sample and electrodes. A comb of gold interdigitated electrodes with a spacing of 20 µm is sputtered onto a glass coverslip over which a 0.7 µm-thick SiO 2 layer is deposited by plasma-enhanced chemical vapor deposition. b, Distribution of electrical potential around the electrodes. In between the electrodes and for an applied voltage of 150 V, the electric field reaches a value of 25 MV m -1 , which is close to the breakdown value in silica 1 .   Fig. 4 when the Gaussian-shaped laser beam of intensity 200 W cm -2 is tuned at resonance with the symmetric state (a), the two-photon transition (b) and the antisymmetric state (c). Photon bunching is evidenced in (c) as a signature of two-photon emission from the doubly excited state . The second-order correlation functions (red curves) are computed for nearly parallel dipoles in the J-configuration, = −17 ! , !" = 0.3 ! , and = 10 ! . The black arrows indicate the correspondence between the coincidence histograms and the selected ZPLs in the fluorescence excitation spectrum (d) recorded at the same intensity. Supplementary Fig. 11: Tailoring the selection rules by laser field structuration. a, Schematics of two sets of 3D locations of coupled molecules with parallel dipoles (along the x-axis) leading to the same coupling strength = −17 ! . Configuration 1 (left scheme): One molecule is located at x = -9 nm, y = 0 nm (coordinates in the focal plane) and z 0 = 0 nm (out-of-plane component), while the other is located at x = 9 nm, y = 0 nm and z 1 = -13.4 nm. Configuration 2 (right scheme): One molecule is located at x= -6.4 nm, y= 6.4 nm and z 0 = 0 nm, while the other is located at x = 6.4 nm, y = -6.4 nm, and z 2 = -1.3 nm. The in-plane separation distance between the coupled molecules is 18 nm in both configurations. The fluorescence excitation spectra are computed in the cases of an illumination with a Gaussian (b,c) or a doughnut (d,e) beam shape, taking a detuning = 10 ! and a cross damping rate !" = 0.35 ! , i.e. α = 0.35. They are presented for three different excitation intensities: 6×10 !! I ! (blue curves), 4 I ! (purple curves) and 16 I ! (green curves). The spectra in b,d (resp. c,e) correspond to the configuration 1 (resp. 2) depicted in a.

Supplementary Note 1 : Eigen energies of the molecular system and steady state populations of , , and
Starting from the Hamiltonian H specified in equation (1), which is valid in the rotating wave approximation, we perform the unitary transformation U = !! ! !( ! ! ! ! ! ) to remove the explicit temporal dependence of the driving field. In this basis, the Hamiltonian H writes : where U ! denotes the Hermitian adjoint of U, which leads to the expression: In the following we assume that the two emitters are maximally entangled (i.e. = ) and the laser frequency ! is set to ! .
When the focused laser spot is swept along the inter-emitter (x) axis, both Rabi frequencies Ω ! and Ω ! are real numbers and the diagonalization of H is straightforward, with eigen-energies are given by : and the corresponding eigen-vectors by where ! (i = 1,2,3,4) are normalization coefficients: When the focused laser spot is swept along the orthogonal direction (y-axis), the two Rabi frequencies are complex numbers with identical modulus and different phases: For a laser beam with a Laguerre Gaussian mode LG 01 , this phase difference corresponds to the angle separating the segments stemming from the doughnut center and connecting each emitter (see Fig. 5a). The eigenenergies are solutions of the following polynom obtained by diagonalization of the Hamiltonian expressed in the The solution = 0 corresponds to the ground state energy and does not depends on y. Note that two of the four eigen-energies become degenerate when the center of the doughnut beam is located at one of the two specific positions where = ± ! ! . This results in a dip in the fluorescence image due to destructive interference between the two excitation paths from to . The eigen-energies are in this case: For any other position of the beam along the y-axis, the derivation of the steady state populations of , , and is done numerically.

Supplementary Note 2 : ESSat images of coupled or uncoupled emitters.
In the following, we consider two coupled molecules having identical resonance frequencies ω ! and emission rates ! , located 60 nm apart in the x-direction with parallel dipoles oriented along the y-axis (H-type dipole configuration). We use the combined Debye-Waller/Franck-Condon factor to take into account the presence of vibrational states of the electronic ground state of the molecules.
The ESSat images are obtained by computing the fluorescence signal proportional to 2 !,! + !,! + !,! for each position of the doughnut beam, where !,! , !,! , !,! are the steady state populations of , , and derived from the master equation for the density operator 3 : where the first term describes the coherent evolution of the density matrix and the second term the dissipative contribution, with the following notations : • Γ !" = Γ !" is the cross damping rate and Γ !! = Γ !! = ! is the radiative decay rate of a molecule, • ℋ = ℋ ! + ℋ !! + ℋ !"#$% is the total Hamiltonian of the system, ! is the Hamiltonian of the two bare molecules, is the Hamiltonian of the non retarded dipole-dipole interaction with an amplitude (valid for a separation distance between molecules less than the optical wavelength), . is the interaction between the two emitters and the coherent laser field. Neglecting the vectorial nature of light, we use the following in-plane distribution of electric field, expressed in polar coordinates, to compute the two Rabi frequencies Ω ! and Ω ! : We first consider uncoupled molecules by setting = 0 ( Supplementary Fig. 12). At low intensity ( 5 ! ), where ! is the saturation intensity of a molecule, and for a laser frequency set to ω ! , we observe a single fluorescence dip centered between the two emitters without super resolving them (Supplementary Fig. 12a). When increasing the excitation intensity, the fluorescence dip splits into two minima centered on each emitter ( Supplementary Figs. 12b-d). At high saturations, the amplitudes of these dips reach half of the maximal signal, as expected when one molecule is excited while the other one is not.
In the case of coupled molecules ( = 1), this picture drastically changes. Supplementary Fig. 13 shows ESSat images of a coupled pair for various laser intensities ranging from 5 ! to 8.2 × 10 ! ! . As in the case of uncoupled molecules, the ESSat image fails to super-resolve the emitters at low excitation intensities (Supplementary Fig. 13a). For intermediate intensities (Supplementary Fig. 13b), two fluorescence dips show up along the yaxis (the direction orthogonal to the segment connecting the two emitters). These dips correspond to destructive interference between the two excitation pathways ⟶ → and ⟶ → , which occurs at the two locations where Ω ! = Ω ! ±! ! ! . Indeed, the two-photon transition has a vanishing effective Rabi frequency: Upon increasing intensities, a structured fluorescence landscape develops along x, with the onset of two additional dips at high saturation, as shown in Supplementary Fig.   10c,d. In the regime of extremely high saturations, four dips appear along the x-axis ( Supplementary Fig. 13e,f)