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Tutorial 2: Dissipative Participation Ratios and Dielectric Loss#

Author: Zlatko Minev

Series navigation: ← Tutorial 1: Startup Example | Tutorial 3: Circuit QED Parameters →

Introduction: Dielectric Loss in Superconducting Qubits#

One of the central challenges in building high-coherence superconducting qubits is minimising energy loss to the electromagnetic environment. Among the dominant loss channels is dielectric loss: energy absorbed by lossy dielectric materials (substrate, surface oxides, resist residues) that are threaded by the electric field of the qubit mode.

The dissipative EPR#

The energy-participation ratio (EPR) method generalises naturally to dissipative elements. For a lossy dielectric volume $V$ with loss tangent $\tan\delta$, the dielectric loss rate of mode $m$ is:

\[\kappa_m^{(\mathrm{diel})} = \sum_k p_k^{(\mathrm{diel})} \cdot \tan\delta_k \cdot \omega_m\]

where the dielectric participation ratio is:

\[p_k^{(\mathrm{diel})} = \frac{\mathcal{E}_{\mathrm{elec},k}}{\mathcal{E}_{\mathrm{elec},\mathrm{total}}}\]

This is the fraction of the mode’s electric energy stored in volume $k$. A small $p_k$ means the dielectric barely interacts with the qubit mode; reducing $p_k$ (by geometry optimisation) directly improves coherence time $T_1 = 1/\kappa$.

Energy definitions (from Appendix E of arXiv:2010.00620)#

The time-averaged electric and magnetic energies stored in volume $V$ are:

\[\mathcal{E}_{\mathrm{elec}} = \frac{1}{4}\,\mathrm{Re}\int_V \mathrm{d}v\;\vec{E}_{\max}^* \cdot \overleftrightarrow{\epsilon} \cdot \vec{E}_{\max}\]

\[\mathcal{E}_{\mathrm{mag}} = \frac{1}{4}\,\mathrm{Re}\int_V \mathrm{d}v\;\vec{H}_{\max}^* \cdot \overleftrightarrow{\mu} \cdot \vec{H}_{\max}\]

where $\vec{E}_{\max}$ and $\vec{H}_{\max}$ are the complex peak-amplitude field distributions from the HFSS eigenmode solution. pyEPR computes these integrals by invoking the HFSS fields calculator through its COM interface.

Typical loss channels#

Channel	Location	Typical $p$	$\tan\delta$
Bulk substrate (Si, sapphire)	Volume	5–40 %	$\sim 10^{-6}$
Substrate surface (native oxide)	Interface sheet	0.1–2 %	$\sim 10^{-3}$
Metal–substrate interface	Sheet	0.1–5 %	$\sim 10^{-3}$
Seam (gap in ground plane)	Line	—	current-dependent

Reducing the qubit’s electric-field participation in the substrate is a major driver of 3D cavity architectures, flip-chip designs, and substrate etching strategies.

Reference: Z. K. Minev et al., Energy-participation quantization of Josephson circuits, npj Quantum Information 7, 131 (2021) (arXiv:2010.00620), Appendix E.

The simple way#

Following Tutorial 1, let us load the pyEPR package and connect to the example HFSS design. We will then compute the dielectric participation of the silicon substrate at three levels of abstraction, from the highest-level convenience function down to direct use of the HFSS fields calculator.

import pyEPR as epr

Load the Ansys HFSS tutorial file#

As in Tutorial 1, let us find the path to the example project.

from pathlib import Path
path_to_project = Path(epr.__file__).parent.parent / '_example_files'
print(f'Example project located in:\n {path_to_project}')

Connect to HFSS and create the distributed-analysis object eprh:

pinfo = epr.ProjectInfo(project_path = path_to_project, 
                         project_name = 'pyEPR_tutorial1',
                         design_name  = '1. single_transmon')
eprh = epr.DistributedAnalysis(pinfo)

Calculating the substrate participation for mode 0 (the qubit mode)#

The fundamental eigenmode (mode index 0) is the qubit mode — a standing wave with maximum electric field concentrated near the Josephson junction and the capacitor pads. We first select this mode and then ask: what fraction of the mode’s total electric energy is stored in the silicon substrate?

This single number, $p_{\mathrm{Si}}$, directly determines how strongly the qubit loses energy to substrate two-level-system (TLS) defects: $$T_1^{-1}(\text{substrate TLS}) \approx p_{\mathrm{Si}} \cdot \tan\delta_{\mathrm{Si}} \cdot \omega_q$$

eprh.set_mode(0) 

# Calculate the dielectric energy-participation ratio of the substrate
# relative to the total electric energy of all objects.
#
# Returns: p_dielectric, (E_object, E_total)
p_dielectric, (E_substr, E_total) = eprh.calc_p_electric_volume('substrate', 'AllObjects')

print(f'Energy in silicon substrate  = {100*p_dielectric:.1f}%')

Now compute the electric energy stored in the vacuum (cavity enclosure). We pass in the already-computed E_total to avoid recomputing it — field integrals in HFSS can be slow for large meshes.

# Pass in the precomputed E_total so we don't recompute it
p_vac, (E_vac, E_total) = eprh.calc_p_electric_volume('cavity_enclosure', E_total=E_total)

print(f'Energy in vacuum = {100*p_vac:.1f}%')
print(f'\nSince there are no other volumes, the two should sum to one: {p_dielectric + p_vac:.6f}')

Let us inspect the full function signature. Note the E_total optional argument, useful for caching expensive field integrals:

? eprh.calc_p_electric_volume

Level 2: Calculating the energies directly#

The calc_p_electric_volume convenience function internally calls calc_energy_electric, which performs a single volume integral of $\vec{E}^* \cdot \overleftrightarrow{\epsilon} \cdot \vec{E}$ using the HFSS fields calculator. We can call this directly:

E_total  = eprh.calc_energy_electric(obj='AllObjects')
E_substr = eprh.calc_energy_electric(obj='substrate')
print(f'Energy in substrate = {100*E_substr/E_total:.1f}%')

?eprh.calc_energy_electric

Level 3: Using the HFSS Fields Calculator directly#

For complete transparency and for custom field integrals not built into pyEPR, you can use the HFSS fields calculator object CalcObject directly. This is also useful for computing surface integrals (seam loss), line integrals (current density), or any other derived field quantity.

The HFSS Fields Calculator#

The HFSS Fields Calculator enables post-processing of arbitrary derived quantities from the EM field solution. It operates lazily — the computation is deferred until a numerical result is requested — which makes it efficient for chaining operations. The calculator is accessed via pyEPR’s CalcObject wrapper.

Direct computation of the total electric energy#

We compute: $$\mathcal{E}_{\mathrm{elec}} = \mathrm{Re}\int_V \mathrm{d}v\;\vec{E}_{\max}^* \cdot \overleftrightarrow{\epsilon} \cdot \vec{E}_{\max}$$

step by step: retrieve $\vec{E}$, smooth it, form $\vec{E}^* \cdot \epsilon \cdot \vec{E}$, take the real part, and integrate over the volume. Each step pushes an operation onto the calculator stack.

from pyEPR.core import *
from pyEPR.ansys import CalcObject

self, volume = eprh, 'AllObjects'

calcobject = CalcObject([], self.setup)
vecE = calcobject.getQty("E").smooth()
A = vecE.times_eps()
B = vecE.conj()
A = A.dot(B)
A = A.real()
A = A.integrate_vol(name=volume)

E_total = A.evaluate(lv=self._get_lv()) 
E_total

from pyEPR.core import *

self, volume = eprh, 'substrate'

calcobject = CalcObject([], self.setup)
vecE = calcobject.getQty("E").smooth()
A = vecE.times_eps()
B = vecE.conj()
A = A.dot(B)
A = A.real()
A = A.integrate_vol(name=volume)

E_subs = A.evaluate(lv=self._get_lv()) 
E_subs

print(f'Energy in substrate: {100*E_subs/E_total:.1f}%')

Summary#

We demonstrated three equivalent paths to compute the dielectric participation ratio $p_{\mathrm{Si}}$, in increasing order of transparency:

Level	Function	Best for
High	`eprh.calc_p_electric_volume(...)`	Quick participation of standard volumes
Mid	`eprh.calc_energy_electric(obj=...)`	Custom volume ratios, reusing `E_total`
Low	`CalcObject` fields calculator directly	Custom integrals, surface/line loss, debugging

All three give the same answer. The choice depends on how much control you need over the calculation.

Next steps#

With the participation ratios in hand, you can compute the dielectric quality factor: $$Q_m^{(\mathrm{diel})} = \frac{1}{p_{\mathrm{Si}} \cdot \tan\delta_{\mathrm{Si}}}$$

For silicon, $\tan\delta \approx 5 \times 10^{-7}$ at millikelvin temperatures and low fields. A participation of 10 % would give $Q \approx 2 \times 10^7$, or $T_1 \approx 500\,\mu s$ at 5 GHz — consistent with state-of-the-art 3D transmon experiments.

Surface and seam losses can be computed analogously using calc_p_electric_surface and related functions. See the pyEPR documentation for the full API.

Reference: Appendix E of Minev et al., arXiv:2010.00620.

Up next: Tutorial 3 — Circuit QED Parameters: Josephson Junctions, Transmons, and Unit Conversions