Complex plasmon-exciton dynamics revealed through quantum dot light emission in a nanocavity

Plasmonic cavities can confine electromagnetic radiation to deep sub-wavelength regimes. This facilitates strong coupling phenomena to be observed at the limit of individual quantum emitters. Here, we report an extensive set of measurements of plasmonic cavities hosting one to a few semiconductor quantum dots. Scattering spectra show Rabi splitting, demonstrating that these devices are close to the strong coupling regime. Using Hanbury Brown and Twiss interferometry, we observe non-classical emission, allowing us to directly determine the number of emitters in each device. Surprising features in photoluminescence spectra point to the contribution of multiple excited states. Using model simulations based on an extended Jaynes-Cummings Hamiltonian, we find that the involvement of a dark state of the quantum dots explains the experimental findings. The coupling of quantum emitters to plasmonic cavities thus exposes complex relaxation pathways and emerges as an unconventional means to control dynamics of quantum states.

where ω e and γ e are the emitter resonance frequency and decay rate respectively, ω p and γ p are the plasmon frequency and plasmon decay rate, respectively and g is the coupling rate. , for situations in which the intrinsic decay rate of the dark exciton, γ gD, is modified. The value of the dark exciton decay rate used is ħγ gD = 3.25x10 -8 eV (a,b); ħγ gD = 3.25x10 -9 eV (c,d); ħγ gD = 3.25x10 -10 eV (e,f); and ħγ gD = 5x10 -9 eV (g,h). The main features of the spectra and emission statistics are very robust against this parameter. This is due to the fact that the dynamics are governed by the rate of energy transfer between the dark and the bright exciton rather that the intrinsic dark exciton lifetime itself.  Table 1 of the main text.

Supplementary
The spectral response is almost identical in all cases, but the form of g (2) (t) very sensitively depends on both γ Bg /γ Dg and g . The ratio γ Bg /γ Dg controls the relative contribution of the fast component of the decay with respect to the slow one. On the other hand, large values of D give rise to a Purcell effect that shortens the lifetime of the dark exciton, thus shortening the lifetime associated with the slow decay. By fitting an exponential function of the form

Supplementary Tables
Supplementary

Supplementary Note: Analytical treatment of the plasmon-exciton dynamics and photoluminescence.
To obtain an analytical insight into the exciton dynamics and the origin of the features appearing in the excitonic light-emission spectra, we use the separation of time scales that naturally emerges in the system under study. The bright exciton is coupled to the single mode of the plasmonic structure with a coupling strength B comparable to the intrinsic losses of the plasmon, and therefore the dynamics of the plasmon and the bright exciton must be treated on the same footing. The dark exciton, however, is coupled to the plasmon with a coupling strength D ≪ B , which allows us to treat this coupling perturbatively. This separation of time scales allows us to obtain an effective cavity-induced decay rate of the dark exciton Pur D (Purcell effect) and the resulting dark-exciton steady-state population.
Additional incoherent coupling between the dark exciton and the bright exciton with rates BD (dark-to-bright-exciton incoherent coupling rate) and DB (bright-to-dark-exciton incoherent coupling rate) are also included in the model. As we demonstrate below, their role is mainly to influence the relative distribution of excitonic population between the bright and the dark exciton and thus rescale the amplitudes of the features in the emission spectra arising from the bright-and the dark-exciton emission, respectively.

Effective decay rate of the dark exciton
In this section we provide details on the calculation of the effective dynamics (decay) of the dark exciton.
We calculate the effective decay rate of the dark exciton interacting with the cavity using the master equation. We first obtain approximate expressions for the following operator mean values: with gB = |g〉〈e B | , gD = |g〉〈e D | and DD = |e D 〉〈e D |. On the right-hand side of Supplementary Eq. (2) and (3) we have neglected D � † gB � ≈ 0 and D � † � ≈ 0, and used the adiabatic approximation to set the time derivatives in Supplementary Eqs. (2) and (3) equal to zero. From Supplementary Eqs. (2-3) we obtain: (4)

Supplementary Eqs. (1-3) also yield an effective equation of motion for ⟨ DD ⟩ upon insertion
of the approximate steady-state solution for � gD † � and � † gD � = � gD † � * : and With the effective decay rate in hand, we can obtain the steady-state populations of the dark exciton, ⟨ DD ⟩: Where Γ D = gD + BD + Pur D and we have assumed that the population of the bright state ⟨ BB ⟩ is small so that gD ≫ DB ⟨ BB ⟩ (with BB = |e B 〉〈e B |).

Decomposition of the emission spectra into the bright-and dark-state contribution
Next, we calculate the approximate photoluminescence spectrum of the light emitted from the plasmonic cavity and directly link the light emission to the underlying excitonic dynamics.
To that end, we approximate the time evolution of the plasmon annihilation operator in the adiabatic approximation and express the emission spectrum in terms of the excitonic operators. In particular, we assume that we can split the total photoluminescence spectrum, em ( ) = em(B) ( ) + em(D) ( ), into the contributions that emerge due to the bright exciton, em(B) ( ), and dark exciton, em(D) ( ), respectively: Here we have used the lower index B and D to explicitly mark the dynamics of the bright and the dark exciton, respectively. For brevity we omit this index in the following discussion of the individual spectral contributions, unless it is needed for clarity.

Photoluminescence spectrum due to the bright exciton, ( ) ( )
To obtain em(B) ( ) we use the quantum regression theorem (QRT), assuming again that the dark exciton does not significantly influence the dynamics of the coupling between the plasmon and the bright exciton: After inserting the result into Supplementary Eq. (7) we obtain: with Where B = gB + DB (i.e. it does not contain pure dephasing processes but only decay processes). The expressions for � † �, � † gB �, ⟨ BB ⟩ were obtained from the following system of steady-state equations derived from the master equation: Emission spectrum due to the dark exciton ( ) ( ) The contribution to the emission spectrum arising due to the dark state can be obtained using the eigenvector perturbation theory to approximate the two-time correlation function ⟨ † (0) ( )⟩. We can obtain from the QRT the following system of differential equations for the two-time correlation functions: The first matrix in the parenthesis on the right-hand side, denoted as 0 , is responsible for the dynamics of the unperturbed system, whereas the second matrix in the parenthesis, denoted as , represents the perturbative coupling of the dark exciton to the strongly interacting system composed by the bright exciton and the plasmon. The differential equation can be formally solved using the eigenvalue decomposition of matrix = 0 + . If has non-degenerate eigenvalues and corresponding left (right) eigenvectors ( ), the solution of the differential equation is: and the coefficients are determined from the initial condition: where , is the -th component of the -th left eigenvector. In the perturbative approach we first obtain the exact eigenvectors 0 and 0 of 0 . We further assume that the eigenvectors 01 , 02 ( 01 , 02 ) belong to the subspace describing the dynamics of the bright exciton interacting with the plasmon, and the vector 03 ( 03 ) belongs to the dark-exciton subspace of 0 . The approximate eigenvectors of can be found as: To obtain the approximate emission spectrum we further assume that the eigenvectors corresponding to the dynamics of the plasmon coupled with the bright exciton, 1 , 2 ( 1 , 2 ), remain approximately unchanged (decoupled from the dark exciton): 1 ≈ 01 , 2 ≈ 02 , but we apply perturbation theory to obtain the eigenvector 3 : The solution then separates into two independent contributions that give rise to: (i) the emission from the bright exciton coupled with the plasmon, � B † (0) B ( )� = ∑ ,1 =1,2 (corresponding to the spectrum em(B) ( ) shown above for ≈ 0 , with the unperturbed eigenvalues 0 and where , is the -th component of the -th right eigenvector), and (ii) the emission due to the dark exciton, � D † (0) D ( )� = 3 3,1 3 . After performing the algebraic manipulations and inserting the result into Supplementary Eq. (8) we obtain: with