Spectral splitting of a stimulated Raman transition in a single molecule

The small cross-section of Raman scattering poses a great challenge for its direct study at the single-molecule level. By exploiting the high Franck-Condon factor of a common-mode resonance, choosing a large vibrational frequency difference in electronic ground and excited states and operating at $T<2\mathrm{K}$, we succeed at driving a coherent stimulated Raman transition in individual molecules. We observe and model a spectral splitting that serves as a characteristic signature of the phenomenon at hand. Our study sets the ground for exploiting the intrinsic optomechanical degrees of freedom of molecules for applications in solid-state quantum optics and information processing.

Figure 1

(a) Level scheme for model calculations including the electronic ground and excited states $|g\rangle$ and $|e\rangle$ with no vibrational excitation as well as the vibronic states $|v\rangle$ and $|w\rangle$. Arrows indicate which transitions can be reached with the excitation laser (blue) and Stokes laser (orange). ${\mathrm{\Gamma }}_i$: Relaxation rates, ${\mathrm{\Omega }}_i$: Rabi frequencies, ${\mathrm{\Delta }}_i$: frequency detunings. (b) Population in state $|e\rangle$ for ${\mathrm{\Gamma }}_w=2{\mathrm{\Gamma }}_v=1000{\mathrm{\Gamma }}_e, {\mathrm{\Omega }}_{\mathrm{exc}}=2\sqrt{{\mathrm{\Gamma }}_e{\mathrm{\Gamma }}_w}, ({\nu }_{vw}-{\nu }_{ge})/\left(2\pi \right)=-10{\mathrm{\Gamma }}_w, {\mathrm{\Omega }}_{ge}={\mathrm{\Omega }}_{\mathrm{stk}}, {\mathrm{\Delta }}_{\mathrm{stk}}=0$, and various values of the Stokes laser Rabi frequency ${\mathrm{\Omega }}_{\mathrm{stk}}=a·{\mathrm{\Gamma }}_v$. The values are plotted as a function of the excitation laser detuning ${\mathrm{\Delta }}_{\mathrm{exc}}$. For ${\mathrm{\Omega }}_{\mathrm{stk}}=5{\mathrm{\Gamma }}_v$, the profile is not centered around ${\mathrm{\Delta }}_{\mathrm{exc}}=0$ because the Stokes laser exerts a noticeable AC-Stark shift on the $|g\rangle \leftrightarrow |e\rangle$ transition. (c) Same as in (b) but for ${\rho }_{vv}$ (solid curves) and its coherent part $|{\rho }_{gv}{|}^2$ (dashed curves).

Figure 2

(a) Chemical structures of DBT and pDCB. (b) Sketch of the sample and focusing optics in the cryostat. Exc: excitation laser, stk: Stokes laser, fluo: fluorescence, SIL: solid-immersion lens. (c) Level scheme of a DBT molecule with several vibronic states associated with the most prominent vibronic transitions from the vibrational ground states $|g\rangle$ and $|e\rangle$. Transitions $|g\rangle \leftrightarrow |w\rangle$ and $|v\rangle \leftrightarrow |w\rangle$ addressed by the two lasers are marked by solid double-headed arrows. The dashed double-headed arrow indicates that the Stokes laser is in the spectral vicinity of the 00ZPL transition. The curvy red arrow indicates the fluorescence transition used to measure the molecular excited state population. CMR: common mode resonance.

Figure 3

Scan of the Stokes laser frequency ${\nu }_{\mathrm{stk}}$ at various power levels of the resonant excitation laser (${\mathrm{\Delta }}_{\mathrm{exc}}=0$). The level schemes on top of the plot indicate the processes underlying the spectral features in the data. PSB: phonon sideband, STED: stimulated emission depletion, 00ZPL: 00-zero-phonon line.

Figure 4

Resonant stimulated Raman transition in a single DBT molecule. (a) Excitation spectra of the vibronic transition $|g\rangle \leftrightarrow |w\rangle$ monitored via fluorescence from $|e\rangle$ as a function of the excitation laser frequency detuning (${\mathrm{\Delta }}_{\mathrm{exc}}$). Spectra were recorded at different powers ($P_{\mathrm{stk}}$) of the Stokes laser at ${\mathrm{\Delta }}_{\mathrm{stk}}=0$ and $P_{\mathrm{exc}}=0.6\mathrm{mW}$ (${\mathrm{\Omega }}_{\mathrm{exc}}=\sqrt{1.5{\mathrm{\Gamma }}_e{\mathrm{\Gamma }}_w}$). Green curve depicts a fit to the sum of two Lorentzian functions. (b) Same as in (a) but for different ${\mathrm{\Delta }}_{\mathrm{stk}}$ and at $P_{\mathrm{stk}}=1.63\mathrm{mW}$. The data are plotted with an offset. (c) Total populations ${\rho }_{vv}$ (solid orange line) and ${\rho }_{ee}$ (solid black line), and level of coherence in $|v\rangle$ ($|{\rho }_{vg}{|}^2/{\rho }_{vv}$, dashed blue line) at ${\mathrm{\Delta }}_{\mathrm{exc}}=0$ as predicted by the model fitted to the data in (a).

Figure 5

Theoretical predictions for total (a,b) and coherent (c,d) population in $|v\rangle$ for various combinations of the excitation and Stokes laser Rabi frequencies (${\mathrm{\Omega }}_{\mathrm{exc}}$ and ${\mathrm{\Omega }}_{\mathrm{stk}}$) as well as the excitation laser frequency detuning. ${\mathrm{\Delta }}_{\mathrm{stk}}=0$ for all cases. The red markers and lines indicate the (maximum) values used in our experiments.

Figure 6

Sketch of the experimental setup. Abbreviations: 4f: 4f telecentric system – AOM: acousto-optic modulator – APD: avalanche photodiode – ASPH: aspheric lens – BP: bandpass filter – BS: beam splitter – CRYOSTAT: dilution cryostat – $\lambda /2$: half-wave plate – $\lambda /4$: quarter-wave plate – ND: variable neutral density filter – PD: photodiode – PID: PID-controller – PM: polarization-maintaining fiber – P: linear polarizer – SIL: solid immersion lens – SM: scanning mirror – SP: shortpass filter – SUB: substrate with etched nano-channel – Ti:Sa: tunable Ti:sapphire laser – WM: wavemeter.

Figure 7

Model used to fit the experimental data. (a) Level scheme including seven molecular states and their spontaneous decay rates as well as several laser-induced transitions. The states $|g\rangle , |v\rangle , |e\rangle$, and $|w\rangle$ correspond to those used in Fig. 2 of the main text. $|G\rangle$ and $|E\rangle$ are phonon sideband ‘states’ and $|x\rangle$ is an additional vibronic level in the spectral vicinity of $|w\rangle$. (b) Vibronic line profile of the transitions $|g\rangle \leftrightarrow |w\rangle$ and $|g\rangle \leftrightarrow |x\rangle$ for various values of the Stokes laser Rabi frequency and model parameters that resemble the experimental data: ${\mathrm{\Delta }}_{\mathrm{stk}}=0, {\mathrm{\Gamma }}_w/\left(2\pi \right)=25\mathrm{GHz}, {\mathrm{\Gamma }}_x={\mathrm{\Gamma }}_E={\mathrm{\Gamma }}_w, {\mathrm{\Gamma }}_v={\mathrm{\Gamma }}_w/2, {\mathrm{\Gamma }}_e/\left(2\pi \right)={\mathrm{\Gamma }}_{eg}/\left(2\pi \right)=23\mathrm{MHz}, {\mathrm{\Gamma }}_{ev}=0, {\mathrm{\Gamma }}_G=1/\left(1\mathrm{ps}\right), {\delta }_{gw,gx}=7·{\mathrm{\Gamma }}_w, {\delta }_{vw,ge}=-10·{\mathrm{\Gamma }}_w, {\mathrm{\Omega }}_{\mathrm{exc}}^2/\left({\mathrm{\Gamma }}_e{\mathrm{\Gamma }}_w\right)=1.5, {\beta }_E=0.25{\beta }_x=0.5, {\beta }_g=1$, and ${\beta }_G=0.1$. The model assumes that the FC factor of the transition $|v\rangle \leftrightarrow |x\rangle$ vanishes. As the Stokes laser Rabi frequency increases, the line profile of $|w\rangle$ splits and the amplitude of the spectral profile of $|w\rangle$ as well as the background level decrease due to stimulated emission along $|e\rangle \to |G\rangle$.

Figure 8

(a) Saturation measurement of the prominent vibronic transition $|S_0,0\rangle \leftrightarrow |S_1,291{\mathrm{cm}}^{-1}\rangle$ in $S_1$ (laser frequency: ${\nu }_{\mathrm{exc}}=411290\mathrm{GHz}$). $R$ denotes the count rate registered at the APD in the detection path (see Fig. 6). The measurement was performed on resonance with the vibronic transition and with the same settings of the detection bandpass as used for the vibronic splitting experiments. The fit of the function $R_{\infty }P/P_{\mathrm{sat}}/(1+P/P_{\mathrm{sat}})$ yields $R_{\infty }=145\mathrm{kcps}$. (b) Raw APD count rate associated with the data shown in Fig. 4 of the main text. The colors indicate different powers of the Stokes laser as specified in (c). (c) Raw APD count rate recorded while the excitation laser was blocked and the Stokes laser was tuned on resonance with the transition $|v\rangle \leftrightarrow |w\rangle$. The dashed line is a linear fit to data points with $P_{\mathrm{stk}}\le 0.1\mathrm{mW}$. The offset at $P_{\mathrm{stk}}=0$ is due to the dark counts of the APD and background stray light.

Figure 9

Measurements for the independent determination of the parameters of the model shown in Fig. 7. (a) Fluorescence excitation scan of the transition $|g\rangle \leftrightarrow |w\rangle$. The black dashed line shows the fit of a rate equation model to the data (see Ref. [22] for more details). (b) STED spectroscopy measurement of the transition $|e\rangle \leftrightarrow |v\rangle$. The black dashed line shows the fit of a rate equation model to the data (see [22] for more details). A comparison of (a) and (b) shows that the vibrational wave number of the mode(s) associated with $|w\rangle$ and $|v\rangle$ is considerably higher in $S_0$. (c) Scan of the excitation laser over the resonances of the transitions $|g\rangle \leftrightarrow |w\rangle$ and $|g\rangle \leftrightarrow |x\rangle$ under the same experimental conditions used to record the data of Figs. 8 and 4 (main text) and $P_{\mathrm{stk}}=0$. The solid black line shows a fit of ${\rho }_{\mathrm{ee}}$ resulting from a Lindblad master equation model using the Hamiltonian (C1). The black dashed line shows the ‘background’ absorption via level $|E\rangle$. See Appendix pp5-s3 for more details. We remark that (c) shows a similar scan as (a) but with a broader spectral range and a larger saturation parameter.

Figure 10

Saturation parameter of the excited state depletion process via $|e\rangle \leftrightarrow |G\rangle$ resulting from the fit of the model discussed in Appendix pp3 to the data of Fig. 4 in the main text. The black line is a linear fit to the data points.

Figure 11

Level of coherence in the population transferred to the vibronic state $|v\rangle$. (a) Total population (solid lines, right axis) and the level of the coherently transferred population (dashed lines, left axis) in level $|v\rangle$ predicted by the model of Fig. 7 and the parameters determined from the fit to the experimental data [shown in Fig. 4 of the main text] and ${\mathrm{\Delta }}_{\mathrm{exc}}=0$. With an increasing fraction of spontaneous decay ${\mathrm{\Gamma }}_{ev}/{\mathrm{\Gamma }}_e$ along $|e\rangle \to |v\rangle$, the fraction of coherently transferred population drops. (b) Fraction of the coherently transferred population in $|v\rangle$. For the maximum value of the Stokes laser Rabi frequency ${\mathrm{\Omega }}_{\mathrm{stk}}/\left(2\pi \right)=17\mathrm{GHz}$ achieved in the experiment of Fig. 4 in the main text, the model predicts that about 80% of the total population is coherent with the driving laser fields.