Problem 4 Superconducting nanowire single-photon detectors (SNSPDs) are extremely sensitive devices that can detect individual particles of light, called photons, with nearly 100% detection efficiency. They were invented about 25 years ago and have become the world's fastest and most efficient detectors for low-light applications, such as quantum communications and computing, lidar, and astronomy. A circuit diagram for an SNSPD is shown in the figure below. It consists of a constant current source Io, which is used to bias a load resistor, R1, and the SNSPD, which is modeled as an inductor in series with a resistor R2 that is in parallel with a switch. The SNSPD operates as follows: The detector circuit is cooled to ultracold temperatures (<4 Kelvin) so that the SNSPD circuit superconducts (this is modeled as the switch, S, being closed, as shown). Because the SNSPD is superconducting (i.e. zero resistance), the bias current is routed through it instead of the load resistor R1. When a single photon incident on the SNSPD is absorbed, the SNSPD becomes a normal conductor with a resistance R2 >> R1 (modeled as the switch open and current flowing through L and R2). This diverts the bias current through the load resistor R1. Once the majority of the current is diverted to R1, the SNSPD can dissipate the absorbed photon's energy and return back to the superconducting state (switch closes again). Current will begin to flow back through the zero-resistance SNSPD, resetting the device for detection of another photon. We read out the voltage across R1, which is our signal that indicates we've detected a photon. Each photon absorbed by the SNSPD = a voltage pulse signal vs. Let's determine what this pulse looks like. SNSPD 150 nH 10 R1 10 ΜΑ 50 Ω R2 3ΜΩ Vs 1. No photons have been incident on the detector for a long time, and thus the switch has been closed for t<0 as shown in the figure. Find i, (0-) and v₂ (0¯). 2. At time t = 0, the SNSPD absorbs an incident photon. This can be modeled by the switch opening. Find i,(0+) and v(t). Sketch v(t), labeling the maximum and time constant. (Hint #1: R2 >> R1. What can we assume about iz(oo)?) (Hint #2: Vs(t) = VR1(t). What is VRI(0)? VRI(00)? What is the time constant ta?) 3. Let us assume that enough current has been diverted from the SNSPD and that it can return back to the superconducting state (the switch closes). Let's redefine time t = 0 here. a. With the switch closed, find v:(00) and the new time constant, which we'll call fr, the detector reset time. b. Find vs(t). Sketch vs(t), labeling the maximum and time constant. c. It takes a time t = 3t, for an SNSPD circuit to fully reset and be able to detect an incoming photon with high probability. What is the maximum rate at which we could detect incoming photons (in units of Hz)? 4. We can reduce L or increase the load resistance, R1, in order to minimize the reset time tr to detect incoming photons at a higher rate. But there's a lower bound on tr - the time that it takes for the superconducting nanowire (which is normally conducting due to absorbing the photon) to cool and return to the superconducting state. Let's define this time as ta. If the reset time t, is faster than this, current begins to divert back to the SNSPD before it has fully returned to the superconducting state. This keeps the SPSND from superconducting. and it "latches" into the normal conducting state and won't be able to absorb more photons. a. Choose the optimal L ifta is equal to 200 ps (assume R1 and R2 have not changed). b. What is the corresponding maximum count rate? Is this better, worse, or similar to current state-of-the-art?