Handwrite， step by step solutions Superconducting nanowire single-photon detectors (SNSPDs) are extremely sensitive devices thatcan detect individual particles of light, called photons, with nearly 100% detection efficiency. Theywere invented about 20 years ago and have become the world's fastest and most efficient detectorsfor 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 currentsource Io, which is used to bias a load resistor, R1, and the SNSPD, which is modeled as an inductorin series with a resistor R2 that is in parallel with a switch. The SNSPD operates as follows: Thedetector circuit is cooled to ultracold temperatures ( R1 (modeled as the switch open and current flowingthrough L and R2). This diverts the bias current through the load resistor R1. Once the majority ofthe current is diverted to R1, the SNSPD can dissipate the absorbed photon's energy and returnback to the superconducting state (switch closes again). Current will begin to flow back throughthe zero-resistance SNSPD, resetting the device for detection of another photon. We read out thevoltage across R1, which is our signal that indicates we've detected a photon. Each photonabsorbed by the SNSPD = a voltage pulse signal vs. Let's determine what this pulse looks like.SNSPD1010 μαwwR1100 ΩmmL100 nHR22ΜΩ0Vs1. No photons have been incident on the detector for a long time, and thus the switch has beenclosed for t R1. What can we assume about it()?)(Hint #2: Vs(t) = VRI(t). What is VR1(0)? VR1(0)? What is the time constant tel?)3. Let us assume that enough current has been diverted from the SNSPD and that it can returnback to the superconducting state (the switch closes). Let's redefine time t = 0 here.With the switch closed, find vs(oo) and the new time constant, which we'll call tr,the detector reset time.b.Find vs(t). Sketch vs(t), labeling the maximum and time constant.c. It takes a time t = 3tr for an SNSPD circuit to fully reset and be able to detect anincoming photon with high probability. What is the maximum rate at which wecould 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 trto detect incoming photons at a higher rate. But there's a lower bound on tr- the time thatit takes for the superconducting nanowire (which is normally conducting due to absorbingthe photon) to cool and return to the superconducting state. Let's define this time as td. Ifthe reset time t, is faster than this, current begins to divert back to the SNSPD before it hasfully 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 if ta 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 tocurrent state-of-the-art?

Question

Handwrite， step by step solutions Superconducting nanowire single-photon detectors (SNSPDs) are extremely sensitive devices thatcan detect individual particles of light, called photons, with nearly 100% detection efficiency. Theywere invented about 20 years ago and have become the world's fastest and most efficient detectorsfor 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 currentsource Io, which is used to bias a load resistor, R1, and the SNSPD, which is modeled as an inductorin series with a resistor R2 that is in parallel with a switch. The SNSPD operates as follows: Thedetector circuit is cooled to ultracold temperatures (< 4 Kelvin) so that the SNSPD circuitsuperconducts (this is modeled as the switch, S, being closed, as shown). Because the SNSPD issuperconducting (i.e. zero resistance), the bias current is routed through it instead of the loadresistor R1. When a single photon incident on the SNSPD is absorbed, the SNSPD becomes anormal conductor with a resistance R2 >> R1 (modeled as the switch open and current flowingthrough L and R2). This diverts the bias current through the load resistor R1. Once the majority ofthe current is diverted to R1, the SNSPD can dissipate the absorbed photon's energy and returnback to the superconducting state (switch closes again). Current will begin to flow back throughthe zero-resistance SNSPD, resetting the device for detection of another photon. We read out thevoltage across R1, which is our signal that indicates we've detected a photon. Each photonabsorbed by the SNSPD = a voltage pulse signal vs. Let's determine what this pulse looks like.SNSPD1010 μαwwR1100 ΩmmL100 nHR22ΜΩ0Vs1. No photons have been incident on the detector for a long time, and thus the switch has beenclosed for t<0 as shown in the figure. Find i₁(0¯) and vs (0¯). 2. At time t = 0, the SNSPD absorbs an incident photon. This can be modeled by the switchopening. Find i, (0+) and vs(t). Sketch vs(t), labeling the maximum and time constant.(Hint #1: R2 >> R1. What can we assume about it()?)(Hint #2: Vs(t) = VRI(t). What is VR1(0)? VR1(0)? What is the time constant tel?)3. Let us assume that enough current has been diverted from the SNSPD and that it can returnback to the superconducting state (the switch closes). Let's redefine time t = 0 here.With the switch closed, find vs(oo) and the new time constant, which we'll call tr,the detector reset time.b.Find vs(t). Sketch vs(t), labeling the maximum and time constant.c. It takes a time t = 3tr for an SNSPD circuit to fully reset and be able to detect anincoming photon with high probability. What is the maximum rate at which wecould 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 trto detect incoming photons at a higher rate. But there's a lower bound on tr- the time thatit takes for the superconducting nanowire (which is normally conducting due to absorbingthe photon) to cool and return to the superconducting state. Let's define this time as td. Ifthe reset time t, is faster than this, current begins to divert back to the SNSPD before it hasfully 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 if ta 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 tocurrent state-of-the-art?

Accepted Answer

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10 10 μα R1 100 Ω | SNSPD L 100 nH R2 2ΜΩ S V 1. No photons have been incident on the detector for a long time, and thus the switch has bee closed for t <0 as shown in the figure. Find i, (0) and vs (07).