# Applications for Microwave Kinetic Induction Detectors in Advanced Instrumentation
## 2 Microwave Kinetic Inductance Detectors: Basic Operation Principle  
Every charge carrier (electrons, holes, or Cooper pairs) has a finite mass, and due to interactions with the surrounding lattice its effective mass often differs from the mass of a free electron. The term 'kinetic inductance' describes that (caused by this final charge carrier mass) it always takes time and energy to reverse the direction of motion of charge carriers. Therefore, when an AC voltage is applied, current will always trail behind voltage, like it does in a coil due to magnetic inductance. The size of this additional inductance in a given circuit depends on the charge carrier's effective mass and velocity, and thus on charge carrier density. It is low for normal metals with high electron densities and a short free mean path, but high for superconductors (the Cooper pair density is low in comparison), or at high AC frequencies. In a superconductor, the interaction with the lattice leads to an attractive force between electrons, causing them to form pairs at very low temperatures. These Cooper pairs are no longer Fermions but Bosons, and thus will condense into the lowest state, allowing charge to be transported without resistance. The distance in energy to the next highest state is called the superconducting band gap \(\Delta\). Breaking a Cooper pair is equivalent to lifting its two electrons to the next higher state and thus requires at least twice this band gap energy. The resulting two unpaired electrons are called a quasi-particle and will recombine into a Cooper pair after the material specific quasi-particle lifetime. MKIDs use the effect of kinetic inductance in superconductors for photon or particle detection: For an MKID, a superconductor is lithographically patterned into an LC resonant circuit (see Figure 1a), a harmonic oscillator with a well-defined resonant frequency given by its capacitance C and total inductance \(L_{total}=L_{geometric}+L_{kinetic}\). (Figure 1a shows a schematic for an LC-resonator with dedicated and separated inductor \(L\) and capacitor C. For many applications MKIDs follow this design using an interdigitated capacitor and a long thin superconducting line as inductor. These MKIDs are often called Lumped Element KIDs, or LEKIDs to distinguish them from quarter-wave resonators. Quarter-wave MKIDs are preferred for some applications and use a superconducting line floating at one end and grounded at the other as resonator. They have their own advantages and disadvantages for detector array design and layout but follow the same basic operation principles.) If a photon with an energy of more than twice the superconducting band gap hits this resonator, some of the photon's energy will be transferred into heat and the rest will break Cooper pairs. This breaking of Cooper pairs by photon absorption reduces the LC-resonator's charge carrier density and thus increases the velocity of the remaining Cooper pairs that still have to carry the same current. This leads to an increase in kinetic inductance and thus a reduction of the LC-resonator's resonant frequency \(f_{r}\). This change in resonant frequency can be measured very precisely and is the signal that is monitored with an MKID to detect photons or particles. Figure 1: MKID operation principle. **(a)** LC resonant circuit; **(b)** amplitude of transmitted feedline signal before (blue) and after (red) photon absorption; **(c)** feedline signal phase response; **(d)** several resonators multiplexed with a single feedline. At resonance the LC circuit is effectively shorting the capacitively coupled microwave feedline to ground, resulting in a sharp dip in transmitted power. MKID resonators are capacitively coupled to a microwave feedline (which is why they are called'microwave' kinetic inductance detectors) and constantly driven at their resonant frequency. The change in \(f_{r}\) caused by photon absorption therefore shifts both the amplitude (Figure 1b) as well as the phase response (Figure 1c) of the signal transmitted on the feedline to lower frequencies.