Engineering long-lived vibrational states for an organic molecule

The optomechanical character of molecules was discovered by Raman about one century ago. Today, molecules are promising contenders for high-performance quantum optomechanical platforms because their small size and large energy-level separations make them intrinsically robust against thermal agitations. Moreover, the precision and throughput of chemical synthesis can ensure a viable route to quantum technological applications. The challenge, however, is that the coupling of molecular vibrations to environmental phonons limits their coherence to picosecond time scales. Here, we improve the optomechanical quality of a molecule by several orders of magnitude through phononic engineering of its surrounding. By dressing a molecule with long-lived high-frequency phonon modes of its nanoscopic environment, we achieve storage and retrieval of photons at millisecond time scales and allow for the emergence of single-photon strong coupling in optomechanics. Our strategy can be extended to the realization of molecular optomechanical networks.


optomechanical networks.
Molecules are usually considered in the realm of chemistry and as building blocks of organic matter. However, scientists have been increasingly turning their attention to molecules for their naturally rich and compact quantum mechanical settings, where a wide range of electronic, mechanical and magnetic degrees of freedom could be efficiently accessed and manipulated 2, 7-10 . A particularly intriguing promise of molecules is their use as quantum optomechanical platforms [2][3][4][5][11][12][13][14] , but this idea confronts the challenge that the various molecular degrees of freedom quickly lose their "quantumness" when coupled to the phononic bath of the environment in the condensed phase. In this theoretical work, we show how to create long-lived phononic states by tailoring the vibrational modes of organic crystals that embed impurity guest molecules. to support excellent quantum coherent optical transitions 6,8 . By nature, a dye molecule establishes a rich optomechanical system with large cross sections for transitions involving its electronic states (|g, e ) and vibrational levels (vibrons, |v ) 14 (see Fig. 1(b)). Vibrons can be long-lived in the gaseous state, but if the molecule is embedded in a solid matrix, the molecular levels also couple to the phonons (|b ) in the host 14,18 . The large phononic Ladder X M 6 6.5 Our strategy is to design the phononic landscape of the AC crystal and its substrate to create long-lived phonon modes for transferring and storing information from the guest molecule via external laser fields. To achieve this goal, we first reduce the crystal dimensions to gain access to discrete acoustic modes in the frequency range of a few GHz. This facilitates the selection of single modes in their ground state (n b ≈ 0 for T ∼ 0.1 K) and enhances the molecular electron-phonon coupling due to smaller phonon mode volumes (see estimations below). In a second step, we engineer the environment of the crystal to minimize the damping of its phonons. As a concrete working example, we place a hexagonal AC nanocrystal with  The dynamics of the interaction between a molecule and a single phonon mode of its environment can be described by the Hamiltonian 14, 21 Here, σ and b denote the electronic and vibrational annihilation operators, respectively, and the operators σ † and b † are their creation counterparts. The first two terms in Eq. (1)  A close look at the spectrum around the vibration frequency ω b (see Fig. 2 shows that this transition is three times more strongly excited on the PC and its linewidth is In Fig. 2(c), we plot the population of phonons against the laser frequency detuning.
We find that the generated signal is dramatically enhanced by 3 orders of magnitude in the presence of the PC. Moreover, comparing the heights of the two main resonances, we observe that the achieved phonon population is much higher if the excitation takes place through state |e, b , i.e., via the nanocrystal vibrational mode (see Fig. 2(d)). The linewidth of this transition remains limited by γ since its Einstein A-coefficient involves a similar frequency and dipole moment as for the decay of |e, b = 0 . These effects reflect the breakage of Kasha's rule, which states that fluorescence emission usually takes place from the lowest excited state 26 . The observed phenomena manifest that the molecule is dressed with the vibrational modes of its nanoscopic environment, acting on a par with the intrinsic molecular vibrational levels. We remark that stronger Rabi drivings or lower phonon decay rates would result in the appearance of optomechanical self-sustained oscillations 27 , the analysis of which goes beyond the scope of this paper. The onset of this regime is, however, highly suppressed for pulsed driving due to the finite duration of the excitation, as we exploit below for modes with longer phonon lifetimes.
A further illuminating way to investigate the fingerprints of long-lived phonons is to analyze the resonance fluorescence spectrum of the system driven at the ZPL (∆ = 0) 28 . The resulting emission spectrum in Fig. 2(e) shows very narrow peaks dominated by the phonon decay rate and a larger number of overtones. Remarkably, the spectrum also displays an anti-Stokes line at higher frequencies when coupled to the PC. We attribute this feature to coherent Raman scattering assisted by the vibrational levels of the electronic excited state in two steps. First, the vibrational manifold in the electronic ground state is coherently excited via Stokes processes through the ZPL. Second, its coupling to the upper manifold occurs via coherent anti-Stokes scattering (see Fig. 2(f)), under moderate Rabi frequencies for longlived vibrations (κ b γ). This phenomenon is the molecular analogue of the single-photon optomechanical strong-coupling regime, which yet remains to be reported experimentally in 8 the solid state 13 . In a nutshell, the molecule acts as an optical nanoantenna 29 that facilitates its coherent optomechanical coupling to phonons when g 0 is large enough to exceed the mechanical and optical losses, much similar to the role of a cavity in conventional quantum optomechanics 13 .
Next, we exploit the long coherence time of the proposed molecular platform to realize a quantum memory. Here, we employ a strong control pulse to coherently map (write) a weak signal to the long-lived phonon mode by means of stimulated Raman scattering (see Fig. 3(a)). The pulsed signal stored in the form of vibrations can then be coherently retrieved after a certain delay by applying a strong read pulse, as displayed in Fig. 3(b). The green symbols in this figure show an example of the numerical results for the generation and readout of the population for a phonon state with ms lifetime (κ b = 1.6 × 10 −6 γ). We note that the excitation of phonons in Fig. 3(b) is the result of a cooperative optomechanical driving of the molecule by both the signal and control pulses. Indeed, a single-beam excitation of phonons, either via control pulses without signal or vice versa, leads to two orders of magnitude smaller phonon population. We also estimated the efficiencies of the write and read steps to be η w = 40% and η r = 86%, respectively. To do this, we first normalized the power of the transmitted control laser beam to the incident power (average of 0.04 photons per pulse width of about 5 µs) and then integrated and compared the corresponding quantities obtained with and without the signal pulse (see SI for details).
To obtain a deeper insight into the dynamics of our molecular memory system, we also developed an analytical model to solve the equations for coherent vibrational and electronic mean fields (see SI for details). The excellent agreement between the numerical and analytical calculations (solid curve) in Fig. 3(b) confirms the coherent nature of the optical storage and read processes. The outcome of the analytical coherent model also yields compact efficiency expressions, η w ≈ 8 The resulting maximum write and read efficiencies for this model amount to 40% and 100%, respectively, close to our numerically found observations and show similar performance for a wide set of parameter, including smaller g 0 (see sweep maps in SI). A further benefit of the analytical model is that it allows us to directly and efficiently examine the full coherence time of the memory up to milliseconds, as presented in Fig. 3(c).
In conclusion, we have shown that by sculpting the nanoscopic environment of a molecule, one can dress it with new vibrational modes and, thus, engineer a novel optomechanical quantum memory with coherence times in the order of milliseconds. Furthermore, we demonstrated that the efficient coupling of the electronic and vibrational degrees of freedom of the composite system leads to a regime in which the conventional Kasha's rule no longer holds. These phenomena allow one to enter single-photon strong coupling in optomechanics 13 , a paradigm that has not yet been observed in the condensed phase. Our strategy can be readily generalized to the design of hybrid quantum optomechanical platforms in which the large optical cross section of a quantum emitter is combined with tailored vibrational modes of its environment to access long-lived ground states, providing an attrac-tive alternative to systems currently explored for quantum information processing based on spins 25,30 .

Finite element simulations
To gain insight into the acoustic phonon properties of AC crystals placed on unstructured and phononic crystal silicon substrates 31 , we study the linear elastodynamic wave equation Here, ω n is the resonance frequency of the normal mode with displacement field vector u n (r), s n (r) = ∇u n (r) represents its strain field tensor and a : b denotes double dot product between dyadics a and b. The studied system is characterized by the mass density ρ(r) and the spatial elastic stiffness tensor C(r). Since AC possesses an anisotropic stiffness tensor 32 , we use numerical simulations based on the finite-element method (COMSOL Multiphysics 33 ). To acquire access to the lifetime properties of the resulting modes, we apply absorbing perfectly matched layer (PML) and solve the resulting eigenvalue problem of Eq. 2 for the different materials (see more simulation details of the modes in Supplementary Information (SI)).

Open-systems dynamics
In order to account for the properties of an open system 34 in the molecular Hamiltonian of Eq. 1, we further include the Lindblad superoperators L{O}ρ = OρO † − 1/2{OO † , ρ} acting on the density matrix ρ, which account for the decay of the operators in the bath for the electronic transition L{ √ γ σ} and the vibrational AC crystal mode L{ √ κ b b}, with fullwidth decay rates γ and κ b , respectively. To pump the molecular Hamiltonian with one or two lasers, we include the driving terms Ω(σ † e −iω L t + σe iω L t ) with Rabi frequency Ω and driving frequency ω L . For the coherent pulse studies, we apply this term with timevarying contributions from the signal and control Rabi frequencies Ω s(c) , with laser frequencies ω s and ω c , respectively. We then solve the resulting master equation numerically via the QuTiP 35 to explore the main dynamical and steady-state properties of the system. To gain further insight into the simulations for the parameters considered in this work, we have also developed two analytical models based on the Quantum Langevin approach 14,36 . For the steady state averages, we noticed a fair approximation based on the decorrelated evolution between the electronic population and phononic displacement operator σ † σD(t)D † (t ) ≈ σ † σ(t) D(t)D † (t ) , which together with the dominance of the terms σb , σ † b † enable us to extend previous formulas 14, 36 to driving conditions near saturation.
For weak signals in the two-pulse driving for the memory study, we observe a significant supervised the project. All authors contributed to writing the paper.