Fundamental limits to single-photon detection determined by quantum coherence and backaction

Single-photon detectors have achieved impressive performance and have led to a number of new scientific discoveries and technological applications. Existing models of photodetectors are semiclassical in that the field-matter interaction is treated perturbatively and time-separated from physical processes in the absorbing matter. An open question is whether a fully quantum detector, whereby the optical field, the optical absorption, and the amplification are considered as one quantum system, could have improved performance. Here we develop a theoretical model of such photodetectors and employ simulations to reveal the critical role played by quantum coherence and amplification backaction in dictating the performance. We show that coherence and backaction lead to trade-offs between detector metrics and also determine optimal system designs through control of the quantum-classical interface. Importantly, we establish the design parameters that result in a ideal photodetector with 100% efficiency, no dark counts, and minimal jitter, thus paving the route for next-generation detectors.

Figure 1

Single-photon detection in a fully coupled detector. (a) Illustration of a photodetector where photon wave packets interact with the detector (matter) degrees of freedom. The optically coupled excited state can decay into a number of optically inactive states. Information carriers, e.g., electrons, interact with the matter energy levels and scatter, causing a response that can be monitored through a measurement. (b) The conventional theory of photodetection assumes that the photon field, the absorption process, and the amplification process occur at different timescales and can therefore be treated separately. Alternatively, in this work we consider a fully coupled model where the three subsystems are treated as being part of one quantum system.

Figure 2

The two configurations under consideration. In both cases, we consider a single-photon Gaussian wave packet (left) incident on a two-level system, creating a resonant excitation. A quantum measurement element (vertical purple line) couples to the internal states and amplifies the signal to the classical domain (right). In panel (a), the quantum measurement element directly couples to the excited state and so $\left|X\right.〉\to \left|1\right.〉$, while in panel (b), the population in the excited state may decay into a third, optically inert state to which the quantum measurement element is coupled, and so $\left|X\right.〉\to \left|C\right.〉$.

Figure 3

Detection events and performance for Configuration 1. (a) The averaged population dynamics for Configuration 1 with amplification strengths $\chi =0$, $\chi =1.0{\mathrm{ns}}^{-1/2}$, and $\chi =3.3{\mathrm{ns}}^{-1/2}$. The instantaneous current is proportional to ${\chi }^2{\rho }_{11}$. (b) Sample trajectories and current outputs for $\chi =0.333{\mathrm{ns}}^{-1/2}$ (top panel) and $\chi =3.33{\mathrm{ns}}^{-1/2}$ (bottom two panels). The horizontal dashed lines are examples of current thresholds used to calculate the ROC curves. (c) ROC curve and jitter vs false positive rate, for several amplification strengths obtained by computing each for varying current thresholds for detection. The dashed line is the result for “detectors” that simply record random hits at varying rates, giving equal true positive and false positive rates. In the inset, the efficiency for a false positive rate of 0.01 is plotted as a function of $\chi$; optimal detection efficiency is obtained for intermediate amplification strength for a modest rate of false positives and strong amplification strength for minimal false positives. Stronger coupling also reduces jitter, which increases with the false positive rate.

Figure 4

Detection events and performance for Configuration 2. (a) Average population dynamics for different values of the incoherent transfer rate ${\mathrm{\Gamma }}^2$. (b) Sample trajectories and detector output for $\chi =0.5{\mathrm{ns}}^{-1/2}$ (top panel) and for $\chi =2.0{\mathrm{ns}}^{-1/2}$ (bottom two panels) where the photon is not absorbed and when the photon induces an excitation. The horizontal dashed lines are examples of current thresholds used to calculate the ROC curves. (c) ROC curve and jitter vs false positive rate for several amplification strengths. Very high detection efficiency is obtained for strong amplification at essentially no cost in terms of dark count rate or jitter. The efficiency is plotted against the parameter of relaxation into the dark state $\mathrm{\Gamma }$ (inset). There is a clear maximum; faster transfer collects excitation more efficiently but inhibits the excitation process.

Figure 5

The average dynamics of Configuration 2 with ${\gamma }^2={\mathrm{\Gamma }}^2=0.1{\mathrm{ps}}^{-1}$. Strong optical coupling means that excitation occurs rapidly compared with the pulse duration, and a matched rate of incoherent transfer converts excited-state population to dark-state population as soon as the former develops. Consequently, regardless of the strength of the quantum measurement, the measured state $\left|C\right.〉$ attains 100% probability and the excited state $\left|1\right.〉$ is nearly unoccupied throughout.

Figure 6

A schematic of a molecular photodetection device. Photons are absorbed by the molecular system (internal structure shown), and the detection event is amplified by electronic transport in the carbon nanotube. A quantum measurement master equation for the dynamics of this device is developed in this section.

Figure 7

The average dynamics of the three-state system with ${\gamma }^2={\mathrm{\Gamma }}^2=0.1{\mathrm{ps}}^{-1}$ for wave packets of widths (a) 1 ns, (b) 500 ps, (c) 250 ps, and (d) 100ps. We see that in all cases the photon is collected 100% of the time.