Quantum Fluctuations in the Fröhlich Condensate of Molecular Vibrations Driven Far From Equilibrium

Fröhlich discovered the remarkable condensation of polar vibrations into the lowest frequency mode when the system is pumped externally. For a full understanding of the Fröhlich condensate one needs to go beyond the mean field level to describe critical behavior as well as quantum fluctuations. The energy redistribution among vibrational modes with nonlinearity included is shown to be essential for realizing the condensate and the phonon-number distribution, revealing the transition from quasithermal to super-Poissonian statistics with the pump. We further study the spectroscopic properties of the Fröhlich condensate, which are especially revealed by the narrow linewidth. This gives the long-lived coherence and the collective motion of the condensate. Finally, we show that the proteins such as bovine serum albumin and lysozyme are most likely the candidates for observing such collective modes in THz regime by means of Raman or infrared spectroscopy.

Figure 1

(a) Schematic of molecular vibrations driven into far-from-equilibrium regime; (b) Vibrational modes absorb energy from external source at rate of $r$ and surrounding medium acting as thermal bath causes the dissipation at rate of $\varphi$, corresponding to the 1st term in Eq. (1). Bath is also responsible for the energy redistribution (nonlinear) at rate of $\chi$, corresponding to the 2nd term in Eq. (1); (c) Feynman diagrams for those 1st and 2nd order processes.

Figure 2

Transition diagram in accordance to Eq. (2), showing the flow of probability in and out of the $|n_0⟩$ state from and to the neighboring $|n_0+1⟩$ and $|n_0-1⟩$ states. $(r+\varphi \overline{n})(n_0+1)$ and $(r+\varphi (\overline{n}+1))(n_0+1)$ represent the respective phonon emission and absorption, due to the one-phonon processes (pumping and dissipation); $\chi (\overline{n}+1)(N_T-n_0)(n_0+1)=\varphi \overline{n}(n_0+1)(\chi (\overline{n}+1)/\varphi \overline{n})(N_T-n_0)$ corresponds to the process in which phonons are emitted owing to both dissipation and energy redistribution; $\chi \overline{n}(N_T-n_0+D-1)(n_0+1)=\varphi (\overline{n}+1)(n_0+1)[\chi \overline{n}/\varphi (\overline{n}+1)](N_T-n_0+D-1)$ corresponds to the process in which phonons are absorbed owing to both dissipation and energy redistribution. Similar explanations exist for the other terms.

Figure 3

(a) Phonon numbers at the lowest mode ${\omega }_0$ and excited modes ${\omega }_l(l\ge 1)$ normalized by total phonon number $N$ vary with external energy supply. (b) Damping rate ${\gamma }_0$ of condensate varies with external energy supply. Lifetime $1/{\gamma }_0$ is plotted in small panel. In (b), dashed red line is for the case with no Fröhlich condensate when turning off the nonlinearity. Solvent temperatures $T=280\mathrm{K}$, corresponding to $\overline{n}=16$. Parameters are taken from Ref. [33] for BSA protein: ${\omega }_0=0.314\times 2\pi \mathrm{THz}$, $\varphi =6\mathrm{GHz}$, $\chi =0.07\mathrm{GHz}$, and $D=200$.

Figure 4

Phonon distribution of the lowest vibrational mode where the solvent temperature is $T=280\mathrm{K}$ corresponding to $\overline{n}=16$. Energy pump (a) $r=220\mathrm{GHz}$, (b) $r=0.55\mathrm{GHz}$; orange dots correspond to Poissonian distribution. Other parameters are the same as that in Fig. 3.