The experimental investigation of the noise is presented in chapter 5, which is the focus of this thesis. It is likely that the noise is related to the dielectric constant fluctuation caused by state transitions (by absorption or emission of thermal phonons) or the energy level fluctuations in the TLS.
Microwave kinetic inductance detectors
- Introduction to low temperature detectors
- Principle of operation
- Technical advantages
- Applications and ongoing projects
From an energy point of view, the inductanceLki accounts for the energy stored in the supercurrent as Coopers kinetic energy. The Al quasiparticles then diffuse to the edges of the CPW ground plane, where a narrow strip of the lowest-gap superconductor (Ti or W) overlaps the Al ground planes.
Other applications of superconducting microwave res- onatorsonators
Microwave frequency domain multiplexing of SQUIDs
Because of the flux-dependent Josephson inductance, the SQUID loop acts as a variable flux inductor. Therefore, a change in flux in the SQUID circuit will modify the total inductance, leading to a readable resonance frequency shift.
Coupling superconducting qubits to microwave resonators
Panels A, B, and C show the entire device, consisting of the CPW resonator and power line, the coupling capacitor, and the Cooper pair box, respectively. Therefore, this dispersive measurement scheme performs a quantum non-demolition readout of the qubit state.
Coupling nanomechanical resonators to microwave resonators
It can be shown that for weak coupling g or large detuning ∆ = ε/h−fr (ε is the two-level energy of the qubit and fr is the resonance frequency), g ≪∆, the reactive loading effect of the qubit will increase the resonator frequency to shift by ±g2/∆ depending on the quantum state of the qubit. Because the Al beam is electrically connected to the center strip, the local center strip-to-ground capacitance depends on the position of the beam.
Non-local electrodynamics of superconductor and the Mattis-Bardeen theory
As with the classical skin effect, both local and non-local behavior can occur in the calculation of λ. From the BCS theory, Mattis and Bardeen derived a non-local equation between the total current density J~ (including the supercurrent and the normal current) and the vector potential A[39]:~.
Surface impedance of bulk superconductor
- Solution of the Mattis-Bardeen kernel K(q)
- Asymptotic behavior of K(q)
- Surface impedance Z s and effective penetration depth λ eff for spec- ular and diffusive surface scattering
- Surface impedance in two limits
- Numerical approach
- Numerical results
2.48, the local limit occurs when the characteristic length scale of the Mattis-Bardeen nucleus (the smallest of ξ0andl) is much smaller than the length scale of the penetrating magnetic field (λeff). In "Supermix", the reduced energy gap ∆(T)/∆0 as a function of T /Tc is interpolated from a table of experimentally measured values given by Muhlschlegel[46] for T /Tc>0.18, and from the temperature low approximate expression (see Eq. 2.88).
Surface impedance of superconducting thin films
Equations for specular and diffusive surface scattering
40], we find that the frequency dependence calculated by "surimp" is identical to that calculated by Popel. For specular scattering boundary on both sides of the film, the problem can be solved by mirroring the field and current repeatedly to fill the entire space (see Fig. 2.7) and applying the equation.
Numerical approach
2.6, both sides of the film are connected to free space and the electromagnetic wave is incident from z = 0 into the film. In the case that the electromagnetic wave is incident from one side of the thin film (d < λeff) and the other side is exposed to free space, we have Hy(d)≈ 0 (see the previous discussion of Eq. 2.65) and Hy(0) = K = Rd.
Numerical results
Finally, if the film is very thin, Z11 and Z21 become equal, implying that the film has been fully penetrated. We also notice that the impedance is 1/d2, which will be explained later in this chapter.
Surface impedance Z s in various limits expressed by σ 1 and σ 2
In the extreme anomalous limit, both Zs and λeff are only related to the value of K∞(ξ0, l, T) according to Eq. In the local limit, both Zsandλeff are only related to the value of K0(ξ0, l, T) according to Eq.
Change in the complex conductivity δσ due to temperature change and pair breakingand pair breaking
If the film thickness t is less than the electron mean free path in the large case l∞, l will be limited by surface scattering and l ≈d. By introducing µ∗, the total QP density nqp (including thermal and excess QPs), the superconducting gap ∆ and the complex conductance σ can again be derived by substituting f(E;µ∗, T) for f(E; T) in the corresponding BCS formula and the Mattis-Bardeen formulas.
Theoretical calculation of α from quasi-static analysis and conformal mapping technique
- Quasi-TEM mode of CPW
- Calculation of geometric capacitance and inductance of CPW using conformal mapping techniqueconformal mapping technique
- Theoretical calculation of α for thick films (t ≫ λ eff )
- Theoretical calculation of α for thin films (t < λ eff )
- Partial kinetic inductance fraction
To evaluate the integral, we use the same two-step SC mapper as used in Section 3.1.2.3: the. We can see that the center strip accounts for more than half of the kinetic inductance.
Experimental determination of α
- Principle of the experiment
- α-test device and the experimental setup
- Results of 200 nm Al α-test device (t ≫ λ eff and t ≪ a)
- Results of 20 nm Al α-test device (t < λ eff )
- A table of experimentally determined α for different geometries and thicknesses.thicknesses
Since the temperature dependence of the surface impedance is an intrinsic property of the superconductor, Xs(T) and Rs(T) are common for resonators of all geometries made from the same superconducting film. Unfortunately, the sheet resistance of the film at 4 K has not been measured for this device and λL cannot be checked using the procedures described in Section 3.1.4.
Quarter-wave transmission line resonator
Input impedance and equivalent lumped element circuit
The Zl = 0 case corresponds to the simple shorted λ/4 resonator and the Zl 6= 0 case corresponds to the hybrid resonator, which has a short sensor strip section near the shorted end made of a different type of superconductor or a different geometry than the rest of the resonator. Thus, a short-circuited transmission line of length λ/4 is equivalent to a parallel RLC resonant circuit, with the equivalent lumped elements ˜R, ˜L and ˜C related to the distributed R, L and C of the transmission line with
Voltage, current, and energy in the resonator
Network model of a quarter-wave resonator capacitively coupled to a feedlinecoupled to a feedline
- Network diagram
- Scattering matrix elements of the coupler’s 3-port network
- Scattering matrix elements of the extended coupler-resonator’s 3- port networkport network
- Transmission coefficient t 21 of the reduced 2-port network
- Properties of the resonance curves
In this diagram, the coupler is modeled as a 3-port network block whose port 3 is connected to one end of the λ/4 transmission line. The scatter matrix of a 3-port coupler network can be easily derived from its unitary equivalent circuit (Fig.
However, in photon detection applications, the quasiparticles are usually generated only in the center strip, and so Eq. However, the noise in MKIDs turns out to be almost exclusively in the phase direction.
- Hybrid resonators
- Static response
- Power dissipation in the sensor strip
- Dynamic response
For example, in the submm hybrid MKID the sensor strip is made of Al thin film (~40 nm) and the rest of the resonator is made of Nb thick film (> 100 nm). In the case that Qi is set by the loss of the superconductor in the sensor strip, we find
A historical overview of the noise study
Nevertheless, based on TLS theory and experimental observations of excess noise, we propose a semi-empirical model that is practically applicable for noise prediction in resonators. They provide direct experimental evidence that TLSs are distributed on the surface of the resonator, but not in the bulk substrate.
Noise measurement and data analysis
For this reason, we provide a review of the standard TLS theory in the first half of Section 5.4. The next two experiments, which explore the geometric scaling of TLS-induced frequency shift and noise, are the two critical experiments of this chapter.
HEMT (4 K)
General properties of the excess noise
- Pure phase (frequency) noise
- Power dependence
- Metal-substrate dependence
- Temperature dependence
- Geometry dependence
The temperature dependence of the excess frequency noise is best demonstrated by the experiment in which the noise of a Nb in Si resonator is measured at temperatures below 1 K. To better determine the temperature and power dependence of the frequency noise, we take the values of noise at 1 kHz from the phase noise spectrum (red curve) at each reading power and each temperature.
Two-level system model
- Tunneling states
- Two-level dynamics and the Bloch equations
- Solution to the Bloch equations
- Relaxation time T 1 and T 2
- Dielectric properties under weak and strong electric fields
- A semi-empirical noise model assuming independent surface TLS fluctuatorsfluctuators
Similarly, the TLS contribution to the real part of the dielectric constant is given by ǫ′TLS(ω). Ec for TLS dissipation saturation (Eq. 5.66) and noise spectral density coefficient κ(ν, ω, T) can vary with (microwave) frequency ω and temperature T[63].
Experimental study of TLS in superconducting resonators
- Study of dielectric properties and noise due to TLS using super- conducting resonatorsconducting resonators
- Locating the TLS noise source
This implies that the measured excess noise from the SiNx-covered device, as shown in Fig. The markers represent different resonator geometries, as indicated by the values of the center strip width sr in the legend.
Method to reduce the noise
- Hybrid geometry
- Removing TLS
- Amplitude readout
In the zoom view of the pulse response, the pulse and the noise ellipse can be identified. From the average pulse trajectory and the major axis of the noise ellipse, we determine ψ ≈15◦.
The signal chain and the noise propagation
- Quasiparticle density fluctuations δn qp under an optical loading p
When the sensor strip is under optical loading, the total quasiparticle density nqp is the sum of the thermal quasiparticle density nthqp (generated by thermal phonons) and the excess quasiparticle density nqp (generated by optical photons). The detector is single mode (AΩ =λ2, where A is the area of the detector and Ω is the diffraction limited solid angle) and is only sensitive to one of the two polarizations;.
Noise equivalent power (NEP)
- Background loading limited NEP
- Detector NEP limited by the HEMT amplifier
- Requirement for the HEMT noise temperature T n in order to achieve BLIP detectionBLIP detection
- Detector NEP limited by the TLS noise
The HEMTTn noise temperature is equivalent to a voltage swing δV2− seen at the input gate of the HEMT, with the noise power spectrum given by. We find that the values of Tnamp listed in Table 6.1 are all less than 5 K, while Tnpha are all greater than 5 K, the noise temperature of the HEMT currently in use.
Derivation of one-dimensional Mattis-Bardeen kernel K(η) and K (q)
A one-dimensional kernel in Fourier space K(q) can be developed using the Fourier transform of Eq. Instead of working on the final result, we start from one of the intermediate results in the equation.
R(a, b) and S (a, b)
Separate expressions of the real and imaginary parts are also available, given by Popel[40]. Separate expressions of the real and imaginary parts are also available and given by Popel.
Dimensionless formula
Singularity removal
Evaluation of K(η)
From the solution to the magnetostatic problem, Iz divides into equal halves at the bottom and top of the center strip and jz is symmetrical on the two sides. Therefore, the ratio ofjztojtis is in the order of the ratio of the wavelength to the transverse dimension of the transmission line.
The fitting model
In this appendix we discuss how to determine the resonance parameters for, Qr and Qc by fitting to the measured complex21 data from the network analyzer.
The fitting procedures
- Step 1: Removing the cable delay term
- Step 2: Circle fit
- Step 3: Rotating and translating to the origin
- Step 4: Phase angle fit
- Step 5: Retrieving other parameters
In this step we determine the center zc = xc+jyc and the radius r of the circle z′ resulting from the previous step. This is a restricted nonlinear minimization problem that can be solved using the standard Lagrange multiplication method.
Fine-tuning the fitting parameters
The result from the stepwise fitting procedure is shown in (a) and (b), while the result from the refined fitting procedure is shown in (c) and (d). This method of estimating σ2z works quite well because the first m data points are usually at off-resonance frequencies with the Gaussian distributed noise from the measurement system.
The output frequencies of the two synthesizers are set to be 1 kHz apart and the IQ ellipses are digitized at a sampling rate of 2 kHz for 1 second (two circles are recorded). The IQ ellipses measured at a range of frequencies and RF input powers by beating two synthesizers are plotted in Figure 1.
Integrating χ res (ω) over TLS parameter space
In fact, for the entire range of a, F(a) can be well approximated by a simpler function, as shown in Fig.
Leduc, "Equivalence of the effects on the complex conductivity of superconductor due to temperature change and external pair breaking," Journal of Low Temperature Physics. Leduc, “Experimental study of the kinetic inductance fraction of superconducting coplanar waveguide,” Nuclear Instruments and Methods in Physics Research, Section A.
