Power of an optical Maxwell's demon in the presence of photon-number correlations

We study how correlations affect the performance of the simulator of a Maxwell's demon demonstrated in a recent optical experiment [M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, Phys. Rev. Lett. 116, 050401 (2016)]. The power of the demon is found to be enhanced or hindered, depending on the nature of the correlation, in close analogy to the situation faced by a thermal demon.

Figure 1

For the original “thermal” demon, one does not expect correlations between the fluctuations of the two halves. But were one to play that game for the sake of it, the effect of the correlations on the demon's power is intuitive. If at the trapdoor each time a fast molecule comes from the left there is a fast molecule coming from the right, the demon's action will be hindered. If each time a fast molecule comes from the left there is a slow molecule coming from the right, the power of the demon is enhanced.

Figure 2

The setup under study, explained in detail in the text. It consists of the optical source and modes $\rho$, the electronic detection circuit with the linear photodiode (PD) and capacitor (C), as well as the devices of the demon with the beam splitter (BS) and the avalanche photodiode (APD). In the color version, the demon is in red, the optical elements are in black, and the electronic circuitry is in blue.

Figure 3

Graph of maximum photon-number difference that can be created by Maxwell's demon when the average photon number in the two independent thermal states at the same temperature is varied. The plateau value is $16/27$, reached for $\overline{n}\approx 50$.

Figure 4

Uncorrelated thermal states: efficacy of the demon over different values of ${\overline{n}}_A$ with a fixed ratio of ${\overline{n}}_B/{\overline{n}}_A$. Here we consider the photon-number difference gained due to the action of the demon only, which is given by $\langle \mathrm{\Delta }{n}_{\mathrm{D}}\rangle =\langle \mathrm{\Delta }n\rangle -({\overline{n}}_B-{\overline{n}}_A)(1-R)$, since the latter subtracted quantity would be the photon-number difference achievable even without the demon's action. The denominator of ${\overline{n}}_A$ is merely there for scaling purposes. ${\langle \mathrm{\Delta }{n}_{\mathrm{D}}\rangle }_{\max }$ is the maximum of $\langle \mathrm{\Delta }{n}_{\mathrm{D}}\rangle$ achieved by the optimization of the parameters.

Figure 5

Uncorrelated thermal states: demon help is maximal when ${\overline{n}}_A={\overline{n}}_B$, even at low ${\overline{n}}_A$.

Figure 6

Uncorrelated thermal states: the blue dotted line is the protocol being implemented without the demon, which is just the difference of ${\overline{n}}_B-{\overline{n}}_A$. Shown also for comparison is the green solid line, which is the optimal protocol by changing the charge bias when ${\overline{n}}_B<{\overline{n}}_A$, but still without the action of the demon. The red dashed curve shows the maximum work that can be extracted with the demon while keeping the charging bias as if ${\overline{n}}_B\ge {\overline{n}}_A$. Notice that by doing so, even in the regime of ${\overline{n}}_B<{\overline{n}}_A$, the demon achieves ${\langle \mathrm{\Delta }n\rangle }_{\max }>0$. This shows that the demon can create a backflow, overcoming an unfavorable bias. The plot is done by varying ${\overline{n}}_B$ while keeping ${\overline{n}}_A={10}^4$. The black dash-dotted line shows the contribution of the demon only, which is the difference between the red dashed curve and the blue dotted line.

Figure 7

Graph of maximum photon-number difference that can be created by Maxwell's demon when the average photon number in the number-correlated state is varied.

Figure 8

Summary of the effect of the demon over all the cases with thermal marginals. We kept to the limit $0\le \overline{n}\le 1$ because the mixed NOON states are only valid in that regime (refer to Sec. 3e for details). For ease of viewing, we only restricted ourselves to ${\overline{n}}_B={\overline{n}}_A$ in this graph, therefore ${\langle \mathrm{\Delta }n\rangle }_{\text{max}}$ here is solely from the contribution of the demon. It is clear from this graph that both the quantum correlated states (TMSS and mixed NOON) fare better than the classical (both uncorrelated and classically correlated) states under low $\overline{n}$. In the large $\overline{n}$ regime, however (not shown in this graph), the classical (uncorrelated) states fare the best if we require the marginals to remain thermal.