Probing biological light-harvesting phenomena by optical cavities

We propose a driven optical cavity quantum electrodynamics (QED) setup aimed at directly probing energy transport dynamics in photosynthetic biomolecules. We show that detailed information concerning energy transfer paths and delocalization of exciton states can be inferred (and exciton energies estimated) from the statistical properties of the emitted photons. This approach provides us with a spectroscopic tool to interrogate biological systems in terms of quantum optical phenomena which have usually been studied in solid-state or atomic systems (e.g., semiconductor quantum dots and trapped atoms) and which are now extended to a broader range of spectroscopy experiments.

Figure 1

Artistic impression of proposed setup. A sample of light-harvesting complexes is confined in an optical cavity (resonance frequency ${\omega }_c$) with a quality factor $Q$. The sample is subjected to an orthogonal driving laser field of frequency ${\omega }_l$. Photons leaking out of the cavity are collected by a photon detector.

Figure 2

Stationary mean photon number of the cavity mode as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (vertical axis) and the pump laser frequency ${\omega }_l$ (horizontal). Both frequencies are in cm${}^{-1}$ but shifted by 12 195 cm${}^{-1}\sim 820$ nm. The dephasing rate is $\gamma =10$ cm${}^{-1}$. The diagonal peaks correspond to the FMO exciton energies, while the off-diagonal peaks are related to energy transfer between different excitons. Top inset: computed spectrum of electronic excitations the FMO complex. The exciton peaks are broadened by the $\gamma =10$ cm${}^{-1}$ linewidth. The fine structure inside the red box is further analyzed in Fig. 3.

Figure 3

Stationary mean number of photons in the cavity mode as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (bottom axis) and the pump laser frequency ${\omega }_l$ (right), both in units of cm${}^{-1}$, but shifted by 12 195 cm${}^{-1}$. The frequency range corresponds to the red box in Fig. 2. One can clearly observe an anticrossing effect, which is associated with a strong driving field.

Figure 4

Stationary mean number of photon in the cavity mode as a function of the laser field coupling given in cm${}^{-1}$/D (horizontal axis) and the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (vertical) given in cm${}^{-1}$. The pump laser frequency is ${\omega }_l=-{\omega }_c+510$ cm${}^{-1}$, which corresponds to an cross-diagonal cut of a diagonal peak (red box in Fig. 2).

Figure 5

Stationary mean photon number of the cavity mode as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (vertical axis) and the pump laser frequency ${\omega }_l$ (horizontal), both are in cm${}^{-1}$, but shifted by 12 195 cm${}^{-1}$. The dephasing rates are $\gamma =1,50,100$ cm${}^{-1}$ (from top to bottom).

Figure 6

Stationary mean photon number of the cavity mode as a function of the cavity ($Q={10}^3$) resonance frequency ${\omega }_c$ (vertical axis) and the pump laser frequency ${\omega }_l$ (horizontal), both in units of cm${}^{-1}$—shifted by 12 195 cm${}^{-1}$. The dephasing rate is $\gamma =10$ cm${}^{-1}$.

Figure 7

Stationary mean photon number of the cavity mode as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (vertical axis) and the pump laser frequency ${\omega }_l$ (horizontal), both in units of cm${}^{-1}$—shifted by 12 195 cm${}^{-1}$. The dephasing rate is $\gamma =50$ cm${}^{-1}$, the pump field is $E=0.01$ cm${}^{-1}$/D, and the cavity coupling coefficient is $g=0.01$ cm${}^{-1}$/D.

Figure 8

Stationary mean photon number of the cavity mode as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (vertical axis) and the pump laser frequency ${\omega }_l$ (horizontal), both in units of cm${}^{-1}$—shifted by 12 195 cm${}^{-1}$—in the presence of an inhomogeneous broadening induced by static disorder in the site energies, set to 50 cm${}^{-1}$, in the case of a dephasing rate $\gamma =10$ cm${}^{-1}$.

Figure 9

Second-order coherence function at time delay zero $g^{\left(2\right)}\left(0\right)$ for a stationary photon state in the cavity [$g=0.1$ cm${}^{-1}$/D and $Q={10}^4$ (top), $Q={10}^3$ (middle) $g=0.01$ cm${}^{-1}$/D, $E=0.01$ cm${}^{-1}$/D, and $Q={10}^4$ (bottom)] as a function of the cavity resonance frequency ${\omega }_c$ (vertical axis) and the pump laser frequency ${\omega }_l$ (horizontal axis). Both frequencies are in cm${}^{-1}$—shifted by 12 195 cm${}^{-1}$. The dephasing rate is $\gamma =10$ cm${}^{-1}$. The black lines refer to $g^{\left(2\right)}\left(0\right)=1$.

Figure 10

Photon mode population (rescaled by a factor ${10}^{-3}$) and second-order coherence function at zero time delay $g^{\left(2\right)}\left(0\right)$ of the stationary cavity mode as a function of the dephasing rate, for a diagonal (circles) and off-diagonal (squares) peak in the mean photon number 2D map in Fig. 2.

Figure 11

Second-order coherence function at time delay zero $g^{\left(2\right)}\left(0\right)$ of the cavity ($Q={10}^4$) mode versus time for different values of dephasing rates and for diagonal and off-diagonal peaks, in the case of dephasing rate $\gamma =10$ cm${}^{-1}$.

Figure 12

Top: Measure of quantum correlation (logarithmic negativity) between the cavity mode and the FMO sample after about 800 fs, as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (bottom axis) and the pump laser frequency ${\omega }_l$ (right), both in units of cm${}^{-1}$, but shifted by 12 195 cm${}^{-1}$, in the case of dephasing rate $\gamma =10$ cm${}^{-1}$. Bottom: Time evolution of the logarithmic negativity measure for ${\omega }_c={\omega }_l=200$ cm${}^{-1}$.

Figure 13

Trapping force (in units of gravity force) vs position (nm) for a single FMO complex, when subject to a cw laser field with a spatial Gaussian profile peaked in the center of the cavity, $E_0\left(x\right)={\overline{E}}_0{e}^{-x^2/\left(2\lambda \right)}$ with ${\overline{E}}_0=5$ cm${}^{-1}$/D $\approx 30\times {10}^6$ J/(C$·$m), carrier frequency set to ${\omega }_l=-5000$ cm${}^{-1}$, and an optimal orientation of polarization axis. Top inset: trapping potential as a function of the position. Bottom inset: in the presence of such laser irradiation (even applied for about $1\mathrm{ps}$), the amount of total excitation in the FMO system is essentially vanishing (i.e., smaller than $3\times {10}^{-4}$).

Figure 14

We consider a setup of two equal spatially separate optical cavities in which two LHC samples are confined, while subjected to two similar pump laser fields. Two auxiliary photodetectors are also introduced to detect photons, leaking out of the cavities, after being mixed on a $50/50$ beam splitter.

Figure 15

Top: Amount of quantum correlations (measured by logarithmic negativity) between the two FMO samples trapped in two spatially separate cavities after $\sim$140 fs as a function of the cavity ($Q={10}^4$) resonance frequency ${\omega }_c$ (bottom axis) and the pump laser frequency ${\omega }_l$ (right), both in units of cm${}^{-1}$, in the case of dephasing rate $\gamma =10$ cm${}^{-1}$. Bottom: entanglement vs time for ${\omega }_c={\omega }_l=200$ cm${}^{-1}$.