ENGINEERING AND OPERATING A SUPERCONDUCTING WAVEGUIDE QED SYSTEM
3.1 Circuit design
C h a p t e r 3
ENGINEERING AND OPERATING A SUPERCONDUCTING
Z control line provides the current for flux biasing the qubit frequency, where the tuning rate is determined by a combination of Z-line-qubit distance, the area of the SQUID loop, and the position of the air bridge to balance the asymmetric current.
We usually use the Z line design that provide a tuning of 1 mA per frequency sweep period. Note that the qubit can also have radiative loss through the Z line if the qubit shape does not exhibit reflection symmetry with respect to the axis along the Z line [69] (Fig.3.1c), which needs to be taken into account in the qubit design.
XY line
Z line 50 μm
a b c
Figure 3.1: Superconducting qubit and read-out resonator circuit design. a, False colored optical image of an Xmon qubit (orange) and a lumped element read- out resonator (green) with the XY line (pink) and Z line (navy). This panel is adapted from [132] with reprint permission. The right panel shows the zoomed-in view of a Z line before the junctions and airbridges are fabricated. b, Circuit diagram of the Xmon qubit (upper panel) and the read-out resonator (lower panel). The metal island of the qubit and the read-out resonator is colored orange and green, respectively.
c, Cartoon showing the inductive coupling with the Z line (represented by a “T”).
The green shading represents the desired coupling between the SQUID loop and the Z line, and the red (blue) shading represents the spurious qubit coupling to the Z line with positive (negative) amplitude. The spurious coupling vanishes for the left panel, but is still present for the right panel.
Resonator
The commonly used resonator in the community is the λ/4 or λ/2 transmission line resonator, where λ is the microwave wavelength. The fundamental mode of standing-wave resonances formed by open/short-circuit boundary conditions is employed as, e.g., the read-out mode. This type of resonator is easy to design and predict its resonance frequency. The drawback of this design is the large footprint:
at 6 GHz, the wavelengthλ≈15mm on a high-resistance silicon substrate, meaning aλ/4resonator can take more than600×600µm2.
Another design for the resonator is the lumped element inductance-capacitance (LC) resonator design, e.g., the ones used in [170]. The metal wing gives rise to the capacitance to ground, and the thin meandering wire connected to the ground provides the inductance (Fig.3.1a and b). This design can reduce the footprint by half, whereas the resonance frequency needs to be obtained from EM simulations.
Waveguide
Coplanar waveguide (CPW), shown in Fig. 3.2a, is the basic transmission line waveguide used as on-chip control and read-out lines for superconducting circuits, which also serves as the linearly-dispersioned waveguide in Chapter4. The theoret- ical modeling of a transmission line waveguide is an important subject, discussed in detail in microwave engineering textbooks such as [171]. The key parameter in design is the characteristic impedanceZ0 =p
Lu/Cu, whereLuandCuare the unit length inductance and capacitance of the waveguide. For a CPW, the impedance is controlled by the width of the center trace (colored orange in Fig.3.2a) as well as the gap between the center trace and the ground (dark gray gap between the orange and the light gray region in Fig. 3.2a). Usually, we design the CPW to have impedanceZ0 = 50 Ωto achieve impedance matching to external transmission lines, thus minimizing the reflection at the boundary.
500 μm 100 μm
• • •
• • •
d
• • • • • •
a b
Figure 3.2: Waveguide circuit design. a, False colored optical image (upper panel) of a CPW waveguide (orange) and its circuit diagram (lower panel) showing elements representing the distributed inductance and capacitance with the unit length value ofLu andCu. The optical image is adapted from [47] with reprint permission. b, False colored optical image (upper panel) of a metamaterial waveguide (blue) with the unit cell size ofdconnected via a tapering section (purple) to a CPW waveguide (red). The circuit diagram of the metamaterial waveguide (lower panel) shows that a unit cell (shaded in gray) consists of a lumped element LC resonator with capacitive coupling to neighboring cells. The optical image is adapted from [132] with reprint permission. In both optical images, the light grey area is the ground plane.
To engineer the dispersion of a microwave waveguide, a simple way is to start from a CPW, and periodically modulate the impedance [120], which realizes a photonic crystal [154] in the microwave domain. Another method is to create periodic cells of microwave structures much smaller than the microwave wavelength (see, e.g., Fig. 3.2b), i.e., a metamaterial [155]. For example, in [172], the metamaterial is constructed by periodically loading a CPW waveguide with resonant structures to create a waveguide with band-stop spectrum. Another example is a waveguide with band-pass spectrum, which can be built by an array of coupled lumped element resonators detailed in Chapter5 and6. Additionally, the metamaterial waveguide can also be used to achieve both energy and momentum matching in traveling wave amplifiers to achieve wide-band amplification gains [173,174].
Coupling between circuit elements
The coupling between two capacitively-coupled LC resonators (Fig.3.3a) is given by
g = Cg
pC1,ΣC2,Σ
√ω1ω2
2 , (3.1)
whereCg is the coupling capacitance between the two resonators,Cj,Σ ≡Cj+Cg
with j = 1,2 is the total capacitance of each resonator, and ωj is the resonance frequency. This formula can be obtained by writing down the Hamiltonian of the circuit and performing the second quantization to extract the coupling coefficientg (see, e.g., App. D of [175]). The formula is widely used to design the coupling between a qubit and a read-out resonator, between a qubit and a waveguide resonator, and between a read-out resonator and a waveguide resonator in Chapter5and6. The capacitance values can be extracted from EM simulations given the circuit geometry.
The coupling of a qubit/resonator to a waveguide can be quantified by the waveguide- induced decay rate. The open environment of the waveguide can be modeled as a 50-Ωimpedance on each end (Fig.3.3b). The decay rate can be obtained from the classical external Q factor of a parallel LC circuit
Qe=ω0average energy stored in the resonator atω0
average power lost to the external circuit = ω0C
Re[Y(ω0)], (3.2) whereω0 = 1/√
LC is the resonance frequency andY(ω)is the admittance seen from the external port (Fig.3.3b). Assuming the external load can be treated as a perturbation, i.e.,Qe≫1, the decay rate into a waveguide can be deduced as
Γ1D = Cg
C 2
Z0
2L, (3.3)
L1 C1 L2 C2 Cg
L C
Cg
Z(ω) Z0 Z0 Z0
Y(ω) Z(ω)
a b
Figure 3.3: Coupling between circuit elements. a, Circuit diagram of two res- onators with inductanceLjand capacitanceCjcoupled via the coupling capacitance Cg. b, Left: Circuit diagram of an LC resonator capacitively (Cg) coupled to an open environment with frequency-dependent characteristic impedance Z(ω). The admittance Y(ω) seen from the resonator is shaded in blue. Right: Examples of Z(ω) showing the impedance of a double-ended waveguide (parallel Z0) and the impedance of a single-ended waveguide (Z0).
where the resistance associated with the waveguide is Z0/2. In order to increase Γ1D, we can raise the coupling capacitance by increasing the width of the waveguide center trace or eliminating the ground plane between the Xmon capacitor and the center trace, giving rise toΓ1D/2π≈100MHz in Chapter4. This formula can also be used to estimate the decay rate into the XY line, in which case the associated resistance isZ0 (Fig.3.3b, single-ended waveguide).
System-level circuit design
Equipped with the above basic knowledge and formula, we are able to design the geometry of the metal for each element, confirming that the parameters satisfy our design goals in the EM simulation of the parts. However, it is inefficient to simulate the entire waveguide QED system with geometry spreading over a 1 cm×1 cm (or 2 cm) substrate. Especially, this hampers the design of the metamaterial waveguide (Fig. 3.2b) where strong coupling between unit cells is assumed and the precise dispersion relation requires the simulation of an infinite structure. A solution is provided in [144]: we use the EM simulation of a single unit cell to extract the S-parameters, convert it into the dispersion relation of an infinite structure [171], and extract the lumped element inductance and capacitance values from fitting the dispersion relation.
Another challenge lies in the finite number of unit cells we can fit on a physical device, meaning the passband always consists of discrete modes instead of a continuum.
This becomes a problem if we perform waveguide QED experiments in the passband
(Chapter5) or use the passband as the feedline for reading out the qubits (Chapter6).
We overcome the challenge by designing tapering sections [144] (purple section in Fig.3.2b), adapting the Bloch impedance of the periodic structure [171] to the50-Ω port impedance. Another way to view it is that the tapering section increases the external coupling of each mode to the ports, thus creating a quasi-flat passband by the overlap of modes with large linewidths. In the example of the waveguide in Fig. 3.2b, this intuition leads to the design principle of gradually increasing the coupling capacitance between the unit cells while maintaining the resonance frequency of the cell by decreasing the capacitance to ground. To obtain the best parameters, we start from the design of a bandpass filter (iFilterTM Module) in Cadence®Microwave Office® which minimizes the ripples in the passband. Using these initial values for the tapering section containing, e.g., 4 tapering cells, we simulate the transmission from the waveguide and further optimize with the target of passband transmission achieving unity. The optimization gives us the circuit parameters we can use to design individual tapering cells.
Besides designing the tapering section, we also use Microwave Office® to extract parameters involving the entire circuit, such as the decay rate into the metamaterial waveguide by simulating the real part of the admittance in Eq.3.2.
3.2 Device fabrication and packaging