Quantum resource-theoretical analysis of the role of vibrational structure in photoisomerization

Abstract

Thermodynamical systems at the nanoscale, such as single molecules interacting with highly structured vibrational environments, typically undergo non-equilibrium physical processes that lack precise microscopic descriptions. Photoisomerization is such an example which has emerged as a platform on which to study single-molecule ultrafast photochemical processes from a quantum resource theoretic perspective. However, upper bounds on its efficiency have only been obtained under significant simplifications that make the mathematics of the resource-theoretical treatment manageable. Here we generalize previous models for the photoisomers, while retaining the full vibrational structure, and still get analytical bounds on the efficiency of hotoisomerization. We quantify the impact of such vibrational structure on the optimal photoisomerization quantum yield both when the vibrational coordinate has no dynamics of its own and when we take into account the vibrational dynamics. This work serves as an example of how to bridge the gap between the abstract language of quantum resource theories and the open system formulation of nanoscale processes.

Figures (5)

• Figure 1: Energy landscape for a typical photoisomer. The system starts in the electronic ground state at $\varphi=0$, and it is photoexcited (wavy arrow) by a light source. It can then relax to the cis ground state $\mathcal{E}(\varphi=\pi)$, while in contact with its environment. Our results are independent of the actual intermediate dynamics of the process. The four dots represent the states that are considered in the four-levels model of halpern2020fundamental. As described in the main body, the two intervals $\mathcal{B}(0)$ and $\mathcal{B}(\pi)$ are introduced, defined as balls of radius $\Phi_0<\pi/2$ centered in $\varphi=0$ and $\varphi=\pi$. Then, the third interval $\mathcal{R}$ is considered, such that $\mathcal{B}(0)\cup\mathcal{R}\cup\mathcal{B}(\pi)=[0,2\pi]$.
• Figure 2: Visual representation of the approximations mentioned in the main body. The two intervals $\mathcal{B}(0)$ and $\mathcal{B}(\pi)$ are introduced, defined as balls of radius $\Phi_0<\pi/2$ centered in $\varphi=0$ and $\varphi=\pi$. Then, the third interval $\mathcal{R}$ is considered, such that $\mathcal{B}(0)\cup\mathcal{R}\cup\mathcal{B}(\pi)=[0,2\pi]$. The initial and final states are assumed to have angular distributions $\xi_i(\varphi)$ and $\xi_f(\varphi)$ respectively. Then, $\xi_i(\varphi)$ is assumed to only have support in $\mathcal{B}(0)$, while $\xi_f(\varphi)$ is assumed to only have support in $\mathcal{B}(0)\cup\mathcal{B}(\pi)$. The initial and final states considered in previous models are recovered in the limit $\Phi_0\to 0$.
• Figure 3: Schematics for the symbols used in the main body. Near each energy landscape minima, the curves $\mathcal{E}_k(\varphi)$ are approximately quadratic. Then, each of these harmonic wells (corresponding to $\varphi=0,\pi/2,\pi$) gives rise to a sequence of well-localised energy levels, with gaps denoted by $\omega_0, \omega_b, \omega_\Delta$ respectively. The harmonic oscillator eigenstates corresponding to the cis ground state are denoted by $\ket{0,n}$ for $n=0,1,2,\dots$, and the same for the other wells. In the limit of the oscillator frequencies being much smaller than the barrier energy $E_b$, the number of levels in each well is very large, and the thermomajorization curves associated to states of the system can be approximated by smooth concave functions.
• Figure 4: Comparison of the two solutions $\gamma^*_{\rm stat}$ and $\gamma^*_{\rm dyn}$ with the optimal yield $\gamma^*$ obtained in the 3-level model. Here $W=5$, $q=0.5$, $\omega_0=\omega_\Delta=0.1$, $\beta=1$, and $\tilde{\beta}=3$.
• Figure 6: Illustrative sketch for the construction of the thermomajorization curves and the computation of the optimal yield. The curve $L_i(x)$ corresponding to the initial state displays the partition in blue and green segments, as described in the main body. The total lengths $Q_1$ and $Q_2$ are indicated on the horizontal axis. A possible final thermomajorization curve $L_f(x)$ is shown in orange. It touches the initial curve in one point and then it can have any functional behaviour, provided that it stays under $L_i(x)$. The optimal yield $\gamma^*$ is then computed as the value of the initial state curve on the horizontal coordinate $x=Q_2$.