Fundamental limitations on photoisomerization from thermodynamic resource theories

Small, out-of-equilibrium, and quantum systems defy simple thermodynamic expressions. Such systems are exemplified by molecular switches, which exchange heat with a bath. These molecules can photoisomerize, or change conformation, or switch, on absorbing light. The photoisomerization probability depends on kinetic details that couple the molecule's energetics to its dissipation. Therefore, a simple, general, thermodynamic-style bound on the photoisomerization probability seems out of reach. We derive such a bound using a resource theory. The resource-theory framework is a set of mathematical tools, developed in quantum information theory, used to generalize thermodynamics to small and quantum settings. From this toolkit has been derived a generalization of the second law, the thermomajorization preorder. We use thermomajorization to upper-bound the photoisomerization probability. Then, we compare the bound with an equilibrium prediction and with a Lindbladian model. We identify a realistic parameter regime in which the Lindbladian evolution saturates the thermomajorization bound. We also quantify the energy coherence in the electronic degree of freedom, and we argue that this coherence cannot promote photoisomerization. This work illustrates how quantum-information-theoretic thermodynamics can elucidate complex quantum processes in nature, experiments, and synthetics.

Figure 1

Photoisomer's energy landscape and conformations. Two representative potential-energy surfaces for the ground and excited electronic states of an isomer, as well as the cis and trans configurations associated with the ground-state minima. The black curves represent adiabatic states, or instantaneous energy eigenstates. The dashed red curves represent diabatic states, which approximately equal adiabatic states at $\phi =0,\pi$.

Figure 2

Thermomajorization bound on the photoisomerization yield ${\rho }_{-}\left(\pi \right)$ and comparisons with equilibrium statistical mechanics. The red dashed curve shows the predicted equilibrium yield, and the blue solid curve shows the resource-theory bound. Possible optimal yields (gray region) are shown for an initially excited state (a), an initial superposition (b), and an unexcited state (c). The insets illustrate the molecule's energy levels. The shaded dots show the initial state's probability weights.

Figure 3

Comparison of thermomajorization bound with time-dependent Lindblad dynamics. Calculations are performed on the four-level system shown on the right. The filled circles illustrate the initial probability weights, and the arrows signify the possible transitions. Each energy level's population evolves as the correspondingly colored curves in the plots: $|{\mathcal{E}}_{+}\left(0\right)\rangle$ (dotted black), $|{\mathcal{E}}_{-}\left(\pi \right)\rangle$ (solid red), $|{\mathcal{E}}_{-}\left(0\right)\rangle$ (dashed blue), and $|{\mathcal{E}}_{+}\left(\pi \right)\rangle$ (solid gray). The grayed area denotes the region accessible to $|{\mathcal{E}}_{-}\left(\pi \right)\rangle$ according to thermomajorization. Population dynamics are shown following (a) full excitation to $|{\mathcal{E}}_{+}\left(0\right)\rangle$, (b) half-excitation to a state of energy diagonal $\frac{1}{2}|{\mathcal{E}}_{+}\left(0\right)\rangle \langle {\mathcal{E}}_{+}\left(0\right)|+|{\mathcal{E}}_{-}\left(0\right)\rangle \langle {\mathcal{E}}_{-}\left(0\right)|$, and (c) failure to excite the state significantly: ${\rho }_i=(1-\epsilon )|{\mathcal{E}}_{-}\left(0\right)\rangle \langle {\mathcal{E}}_{-}\left(0\right)|+\epsilon |{\mathcal{E}}_{+}\left(0\right)\rangle \langle {\mathcal{E}}_{+}\left(0\right)|$, wherein $\epsilon \ll 1$. The parameters used are $\beta \mathrm{\Delta }E=1.5, \beta E_1=30, \beta \hslash {\mathrm{\Gamma }}_{{\mathcal{E}}_{+}\left(0\right),{\mathcal{E}}_{-}\left(\pi \right)}=1$, and $\beta \hslash {\mathrm{\Gamma }}_{{\mathcal{E}}_{+}\left(0\right),{\mathcal{E}}_{-}\left(0\right)}=\beta \hslash {\mathrm{\Gamma }}_{{\mathcal{E}}_{+}\left(\pi \right),{\mathcal{E}}_{+}\left(0\right)}=0.01$.

Figure 4

Dissipative Landau-Zener model for the lower trans state's population and for coherence, relative to the energy eigenbasis, in the electronic state. The initial state, $|{\psi }_1\rangle$, was evolved from an initial time of $-t_f$ to $t_f=50\hslash /\lambda$. (a) Final lower-trans-level population as a function of transition speed, $v$, for different dephasing rates $\mathrm{\Gamma }$. (b) Reduced Fisher information, as a function of $v$ for different dephasing rates $\mathrm{\Gamma }$. The grayed area represents the monotone bound (16).