Quantum to classical transition in an information ratchet

Recent years have seen a flurry of research activity in the study of minimal and autonomous information ratchets. However, the existing classical and quantum models are somewhat hard to compare and hence quantifying possible quantum supremacy in information ratchets has been elusive. We propose a step towards filling this void between quantum and classical ratchets by introducing a model with continuous variables: a quantum particle in a box coupled to a stream of qubits. The dynamics is solved exactly and we analyze the quantum to classical transition in terms of a natural timescale parameter for the model.

Figure 1

Schematic illustration of a continuous information ratchet, where the working medium is a quantum particle in a box and the information reservoir is realized as a stream of qubits.

Figure 2

Illustration of the dynamics described by Eq. (2). (a) In the beginning of the $n\mathrm{th}$ interval $\mathcal{D}$ is in some state (depicted by a blue solid line) and all $N-n$ qubits of $\mathcal{M}$ are in $|\downarrow \rangle$. (b) Same instant as in (a), but as represented in the reduced space of $\mathcal{D}$ and the $n\mathrm{th}$ qubit of $\mathcal{M}$. (c) Final quantum state in reduced space of $\mathcal{D}$ and the $n\mathrm{th}$ qubit of $\mathcal{M}$ at the end of the $n\mathrm{th}$ interval. (d) Same instant as in (c), but as represented from the point of view of $\mathcal{D}$ only; all $N-n-1$ qubits in $\mathcal{M}$ are still in the down state, but the $n\mathrm{th}$ qubit is now in a superposition of $|\uparrow \rangle$ and $|\downarrow \rangle$.

Figure 3

Energy (blue lower line) and von Neumann entropy (red upper line) of the demon as it undergoes repeated qubit interactions. We see that both increase in discrete steps at each new interaction with a qubit in state ${\rho }_n^1\left(0\right)$ [Eq. (11)].

Figure 4

Total current ${\mathrm{\Phi }}_{\text{SS}}\left(t\right)$ in the time periodic steady state with $\tau =\pi {\tau }_s/30$ for initial qubit preparations of ${\rho }_n^1\left(0\right)$ (blue solid line) and ${\rho }_n^2\left(0\right)$ (orange dashed line). Only the last five periods are shown.

Figure 5

Total current $\overline{\mathrm{\Phi }}$ for ${\rho }_n\left(0\right)={\rho }_n^1\left(0\right)$ as a function of qubit interaction time $\tau$ and the characteristic time ${\tau }_s$ with both $\tau$ and ${\tau }_s$ given in arbitrary units.

Figure 6

Total current $\overline{\mathrm{\Phi }}$ for ${\rho }_n\left(0\right)={\rho }_n^2\left(0\right)$ as a function of qubit interaction time $\tau$ and the characteristic time ${\tau }_s$ with both $\tau$ and ${\tau }_s$ given in arbitrary units.

Figure 7

Total current $\overline{\mathrm{\Phi }}$ for ${\rho }_n\left(0\right)={\rho }_n^4\left(0\right)$ as a function of qubit interaction time $\tau$ and the characteristic time ${\tau }_s$ with both $\tau$ and ${\tau }_s$ given in arbitrary units.

Figure 8

Trace distance between the demon state and the fixed point $D(\rho \left(n\tau \right),X)$ as a function of repeated qubit interactions in time. Here $M=10$ and ${\tau }_s=0.94\tau$.

Figure 9

Total current $\overline{\mathrm{\Phi }}$ for ${\rho }_n\left(0\right)={\rho }_n^3\left(0\right)$ as a function of qubit interaction time $\tau$ and the characteristic time ${\tau }_s$ with both $\tau$ and ${\tau }_s$ given in arbitrary units.