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

Author: Zlatko Minev
Purpose: Demonstrate the circuit QED conversion utilities in pyEPR.calcs: unit conversions, Josephson junction parameters, transmon physics, and physical constants.
File Status: Complete

References:

  • Koch et al., Charge-insensitive qubit design derived from the Cooper pair box, PRA 76, 042319 (2007) — the foundational transmon paper.

  • Minev et al., Energy-participation quantization of Josephson circuits, npj Quantum Information (2021); arXiv:2010.00620 — the EPR framework implemented in pyEPR.

  • Devoret & Schoelkopf, Superconducting Circuits for Quantum Information: An Outlook, Science 339, 1169 (2013) — broad review of circuit QED.

Introduction: The Key Players in Superconducting Qubit Design#

Superconducting qubits are built from two ingredients: linear electromagnetic resonators (modelled as LC oscillators) and Josephson junctions (JJs), which provide the essential nonlinearity. Without a nonlinear element, a purely harmonic oscillator has equally spaced energy levels and cannot be selectively addressed as a two-level system.

The Josephson Junction#

A Josephson junction is a thin insulating barrier sandwiched between two superconducting electrodes. Its constitutive relation is:

\[I = I_c \sin\varphi, \qquad V = \frac{\Phi_0}{2\pi}\dot{\varphi},\]

where \(\varphi\) is the gauge-invariant phase difference across the junction, \(I_c\) is the critical current, and \(\Phi_0 = h/(2e)\) is the (superconducting) flux quantum. These two equations together define a purely inductive, nonlinear element.

The potential energy stored in the junction is:

\[U_J(\varphi) = -E_J \cos\varphi,\]

where the Josephson energy \(E_J\) sets the depth of the cosine potential well:

\[E_J = \frac{\Phi_0}{2\pi} I_c = \frac{\hbar}{2e} I_c = \frac{\phi_0^2}{L_J}.\]

Here \(\phi_0 = \Phi_0/(2\pi) = \hbar/(2e)\) is the reduced flux quantum.

Linearised Josephson Inductance#

For small oscillations around \(\varphi = 0\), the cosine potential is well approximated as a parabola:

\[U_J \approx \frac{1}{2} E_J \varphi^2 + \mathcal{O}(\varphi^4).\]

This means the junction looks like a linear inductor with inductance:

\[L_J = \frac{\phi_0^2}{E_J} = \frac{(\Phi_0/2\pi)^2}{E_J}.\]

This linearised Josephson inductance \(L_J\) is the quantity that enters the electromagnetic simulation in HFSS. The EPR method extracts the zero-point fluctuations of the reduced flux \(\varphi_{\text{zpf}}\) across the junction, which then feeds into the full nonlinear analysis.

The Charging Energy#

Each electrode of the junction has a capacitance \(C_\Sigma\) (including the junction self-capacitance and any shunting capacitors). The charging energy is:

\[E_C = \frac{e^2}{2 C_\Sigma},\]

which is the energy cost of placing one Cooper pair (\(2e\)) on the capacitor. Note that some references define \(E_C = e^2/(2C)\) while others write \(4E_C\) for the charging energy — the convention used here follows Koch et al. (2007), where \(E_C = e^2/(2C_\Sigma)\).

The Transmon Hamiltonian#

A transmon qubit is a Cooper-pair box shunted with a large capacitor to increase \(C_\Sigma\) and suppress charge noise. The Hamiltonian (in the charge basis, ignoring offset charge) is:

\[\hat{H} = 4 E_C \hat{n}^2 - E_J \cos\hat{\varphi},\]

where \(\hat{n}\) is the Cooper-pair number operator conjugate to \(\hat{\varphi}\). The key dimensionless ratio is:

\[\frac{E_J}{E_C} \gg 1 \quad \text{(transmon regime)}.\]

Typically, \(E_J/E_C \sim 50\)–\(100\) for a transmon. In this limit:

  • The qubit transition frequency approaches the plasma frequency: \(f_{01} \approx \sqrt{8 E_J E_C}/h\).

  • The anharmonicity (the difference between the \(|1\rangle\to|2\rangle\) and \(|0\rangle\to|1\rangle\) transition frequencies) is approximately \(\alpha \approx -E_C\), which is small but sufficient to address the qubit selectively.

  • The qubit is exponentially insensitive to charge noise for \(E_J/E_C \gtrsim 50\).

Connection to EPR#

The EPR method (arXiv:2010.00620) links the HFSS eigenmode simulation directly to the quantum Hamiltonian. The zero-point flux fluctuation across junction \(j\) in mode \(m\) is:

\[\varphi_{\text{zpf}}^{(j,m)} = \sqrt{\frac{\hbar}{2} \cdot \frac{\omega_m}{E_J^{(j)}}},\]

or equivalently \(\varphi_{\text{zpf}} = (2E_C/E_J)^{1/4}\) in the transmon limit. The participation ratio \(p_{jm}\) is the fraction of the mode’s inductive energy stored in junction \(j\). Together these give the anharmonicity and cross-Kerr couplings directly from the HFSS solution.

The utilities in this tutorial let you convert between all the representations (\(L_J\), \(I_c\), \(E_J\), \(E_C\), \(C\)) and compute the transmon spectrum, complementing the HFSS-based EPR workflow.

%load_ext autoreload
%autoreload 2

1. Unit Conversions#

Why unit conversions matter#

pyEPR works internally in SI units (metres, henries, farads, joules, radians/second), but circuit QED practitioners invariably think in convenient scaled units:

Quantity

Practical unit

SI equivalent

Inductance

nH (nano-henry)

\(10^{-9}\) H

Capacitance

fF (femto-farad)

\(10^{-15}\) F

Frequency / Energy

GHz

\(h \times 10^9\) J

Frequency / Energy

MHz

\(h \times 10^6\) J

Current

nA (nano-ampere)

\(10^{-9}\) A

The Convert class in pyEPR.calcs provides bidirectional conversion between SI and these practical units, as well as conversions between physically equivalent representations (e.g., \(L_J \leftrightarrow E_J \leftrightarrow I_c\), and \(C_\Sigma \leftrightarrow E_C\)).

Note: These conversions are intentionally kept as a lightweight internal implementation — no external dependency is introduced for what are essentially a handful of scaling factors and physical constants.

Elementary SI conversions#

Convert.toSI(value, unit) scales a value into SI (multiply by the unit’s SI prefix factor). Convert.fromSI(value, unit) scales a value out of SI (divide by the prefix). The round-trip (identity) test confirms correctness.

import pyEPR.calcs
from pyEPR.calcs import Convert
print("Convert.toSI(1,'nH')     = ", Convert.toSI(1,'nH'), "H")
print("Convert.fromSI(1.0,'nH') = ", Convert.fromSI(1.0,'nH'), "nH")
print("Identity: ",                  Convert.toSI(Convert.fromSI(1.0,'nH'),'nH'))

2. Josephson Junction Parameters: \(E_J\), \(L_J\), \(E_C\), and \(C_\Sigma\)#

Josephson energy from inductance: \(E_J \leftrightarrow L_J\)#

The Josephson energy and the linearised Josephson inductance are related by:

\[E_J = \frac{\phi_0^2}{L_J} = \frac{(\Phi_0/2\pi)^2}{L_J} = \frac{\hbar^2}{4e^2 L_J},\]

or in practical units:

\[E_J \, [\text{GHz}] = \frac{\phi_0^2}{L_J \, [\text{H}] \cdot h \, [\text{J}\cdot\text{s}]} \times 10^{-9}.\]

For a typical transmon junction with \(L_J = 10\,\text{nH}\):

\[E_J \approx \frac{(3.29 \times 10^{-16}\,\text{Wb})^2}{10 \times 10^{-9}\,\text{H} \times 6.626 \times 10^{-34}\,\text{J}\cdot\text{s}} \approx 16.35\,\text{GHz},\]

which is a canonical value you will see throughout the transmon literature.

Charging energy from capacitance: \(E_C \leftrightarrow C_\Sigma\)#

The charging energy follows from:

\[E_C = \frac{e^2}{2 C_\Sigma},\]

or in practical units:

\[E_C \, [\text{MHz}] = \frac{(1.602 \times 10^{-19}\,\text{C})^2}{2 \times C_\Sigma \, [\text{F}] \times h \, [\text{J}\cdot\text{s}]} \times 10^{-6}.\]

For a typical transmon shunting capacitance \(C_\Sigma = 65\,\text{fF}\), this gives \(E_C \approx 300\,\text{MHz}\), so \(E_J/E_C \approx 54\) — solidly in the transmon regime.

The code below exercises both conversions and their inverses.

from IPython.display import Latex
Lj = 10
display(Latex(r"$E_J = %.2f \text{ GHz} \qquad \text{for } L_J=%.2f\text{ nH}$" % (\
        Convert.Ej_from_Lj(Lj, 'nH', "GHz"),Lj)))

print('\nConvert back %.2f nH' % Convert.Lj_from_Ej(16.35E3, 'MHz', 'nH'),'\n')

display(Latex(r"$E_C = %.2f \text{ MHz} \qquad \text{for } C_J=%.2f\text{ fF}$" % (\
        Convert.Ec_from_Cs(65., 'fF', "MHz"),65.)))

display( 'Convert back:',Latex(r"$C_J = %.2f \text{ fF} \qquad \text{for } E_C=%.2f\text{ MHz}$" % (\
        Convert.Cs_from_Ec(300, 'MHz', "fF"),300)))

3. Critical Current: \(I_c \leftrightarrow L_J\)#

The critical current \(I_c\) is the maximum supercurrent the junction can carry. It is related to the linearised inductance by:

\[I_c = \frac{\phi_0}{L_J} = \frac{E_J}{\phi_0},\]

where \(\phi_0 = \Phi_0/(2\pi) \approx 3.29 \times 10^{-16}\,\text{Wb}\) is the reduced flux quantum. Numerically, for \(L_J = 10\,\text{nH}\):

\[I_c = \frac{3.29 \times 10^{-16}}{10 \times 10^{-9}} \approx 32.9\,\text{nA}.\]

This is consistent with the Ambegaokar–Baratoff relation at \(T=0\):

\[I_c = \frac{\pi \Delta}{2 e R_N},\]

where \(\Delta\) is the superconducting gap and \(R_N\) is the junction’s normal-state resistance. For aluminium (\(\Delta \approx 170\,\mu\text{eV}\)), a critical current of \(30\,\text{nA}\) corresponds to a normal resistance \(R_N \approx 9\,\text{k}\Omega\), which is a typical fabrication target for transmon junctions.

Key point: \(I_c\) is directly measurable and is the primary specification used in junction fabrication. The pyEPR conversion functions let you move fluently between \(I_c\) (a fabrication parameter) and \(L_J\) (the simulation parameter).

print(Convert.Ic_from_Lj(10))
Convert.Lj_from_Ic(32)

4. Physical Constants#

All the conversion formulas above involve a small set of fundamental constants. pyEPR exposes these directly from pyEPR.calcs.convert so that you can write raw physical calculations without relying on external libraries.

Symbol

Name

Value

\(h\) (Planck)

Planck constant

\(6.626 \times 10^{-34}\) J\(\cdot\)s

\(\hbar\) (hbar, ħ)

Reduced Planck constant

\(h/(2\pi) = 1.055 \times 10^{-34}\) J\(\cdot\)s

\(e\) (e_el, elementary_charge)

Elementary charge

\(1.602 \times 10^{-19}\) C

\(\Phi_0\) (fluxQ)

Flux quantum (\(h/2e\))

\(2.068 \times 10^{-15}\) Wb

\(\phi_0\) (ϕ0)

Reduced flux quantum (\(\Phi_0/2\pi = \hbar/2e\))

\(3.291 \times 10^{-16}\) Wb

\(\pi\) (π, pi)

Pi

\(3.14159\ldots\)

The reduced flux quantum \(\phi_0 = \hbar/(2e)\) is particularly important: it is the natural unit for the phase variable \(\varphi\), and it determines both the Josephson inductance (\(L_J = \phi_0^2/E_J\)) and the flux quantisation condition.

The cell below performs a raw calculation of \(E_J\) using these constants, as a cross-check of the formula and the Convert functions.

from pyEPR.calcs.convert import π, pi, ϕ0, fluxQ, Planck, ħ, hbar, elementary_charge, e_el
print("Test EJ raw calculation = %.2f"%( ϕ0**2 / (10E-9 * Planck) *1E-9 ) ,'GHz')

The result \(E_J \approx 16.35\,\text{GHz}\) for \(L_J = 10\,\text{nH}\) is confirmed, matching the Convert.Ej_from_Lj result above. This is the fundamental cross-check: when you type ϕ0**2 / (10E-9 * Planck) you are directly evaluating the defining formula \(E_J = \phi_0^2 / L_J\), expressed in joules, and then converting to GHz by dividing by \(h\) and scaling by \(10^{-9}\).

5. Transmon Qubit Physics#

From nonlinear oscillator to weakly anharmonic qubit#

The transmon Hamiltonian \(\hat{H} = 4E_C \hat{n}^2 - E_J\cos\hat{\varphi}\) can be analysed in the phase basis. In the transmon limit \(E_J/E_C \gg 1\), the junction phase is well localised near \(\varphi = 0\), and the cosine can be expanded:

\[\hat{H} \approx 4E_C \hat{n}^2 + \frac{1}{2}E_J\hat{\varphi}^2 - \frac{1}{24}E_J\hat{\varphi}^4 + \ldots\]

The first two terms are a harmonic oscillator (the plasma oscillator) with angular frequency:

\[\omega_p = \sqrt{8 E_J E_C} / \hbar, \quad \text{or} \quad f_p = \frac{\sqrt{8 E_J E_C}}{h}.\]

The third term (\(-E_J\hat{\varphi}^4/24\)) is the leading nonlinear correction. In second-order perturbation theory it shifts the \(|1\rangle \to |2\rangle\) transition relative to \(|0\rangle \to |1\rangle\) by:

\[\alpha \equiv f_{12} - f_{01} \approx -E_C / h,\]

where \(f_{01}\) is the qubit frequency and \(\alpha\) is the anharmonicity. For a typical transmon with \(E_C/h \approx 300\,\text{MHz}\), the anharmonicity is about \(-300\,\text{MHz}\), which is sufficient to selectively drive the \(|0\rangle \leftrightarrow |1\rangle\) transition with nanosecond pulses.

Zero-point fluctuations#

Treating the transmon as a quantum harmonic oscillator, the zero-point fluctuation of the reduced flux \(\hat{\varphi}\) is:

\[\varphi_{\text{zpf}} = \left(\frac{2E_C}{E_J}\right)^{1/4}.\]

This quantity is central to the EPR analysis: it sets the strength of the nonlinearity. In the transmon limit, \(\varphi_{\text{zpf}} \ll 1\) (small oscillations), which is consistent with the perturbative treatment. The full EPR formula for the Kerr nonlinearity is \(\chi \approx -E_C (\varphi_{\text{zpf}})^4 / 2\) per mode.

Charge insensitivity#

The exponential suppression of charge noise in the transmon regime is one of its most important features. The charge dispersion (sensitivity of \(f_{01}\) to offset charge) decays exponentially as:

\[\varepsilon_m \propto e^{-\sqrt{8 E_J/E_C}}.\]

For \(E_J/E_C = 50\), this suppression is enormous (\(\sim e^{-20}\)), making the transmon essentially charge-insensitive even though it is still formally a charge qubit.

The transmon_print_all_params function#

CalcsTransmon.transmon_print_all_params(Lj_nH, Cs_fF) is a convenience function that takes the junction inductance (in nH) and shunting capacitance (in fF), computes \(E_J\) and \(E_C\), and prints the full set of transmon parameters in a formatted display. This is useful for a quick sanity check during device design.

For the canonical transmon parameters \(L_J = 13\,\text{nH}\), \(C_\Sigma = 65\,\text{fF}\):

pyEPR.calcs.CalcsTransmon.transmon_print_all_params(13, 65);

The displayed quantities include:

  • \(E_J\), \(E_C\) (in MHz and GHz): the fundamental circuit parameters.

  • \(f_{01}\) (in GHz): the qubit transition frequency \(\approx \sqrt{8 E_J E_C}/h\).

  • \(\alpha\) (anharmonicity, in MHz): \(\approx -E_C/h\).

  • \(\varphi_{\text{zpf}}\) (dimensionless): zero-point fluctuation of the reduced phase, \((2E_C/E_J)^{1/4}\).

  • \(Z\) (in \(\Omega\)): characteristic impedance of the linearised junction-capacitor circuit, \(Z = \sqrt{L_J/C_\Sigma}\).

Raw dictionary output#

The underlying function transmon_get_all_params(Ej_MHz, Ec_MHz) returns the same quantities as a Python dictionary, which is convenient for programmatic access and post-processing.

pyEPR.calcs.CalcsTransmon.transmon_get_all_params(Convert.Ej_from_Lj(13, 'nH', 'MHz'), Convert.Ec_from_Cs(65, 'fF', 'MHz'))

Some key values from the output to understand:

  • Phi_ZPF (in Wb): the dimensional zero-point flux fluctuation \(\Phi_{\text{zpf}} = \phi_0 \cdot \varphi_{\text{zpf}}\).

  • Q_ZPF (in C): the zero-point charge fluctuation \(Q_{\text{zpf}} = \hbar / (2\Phi_{\text{zpf}})\).

  • phi_ZPF (dimensionless): the reduced flux ZPF \(\varphi_{\text{zpf}} = (2E_C/E_J)^{1/4}\).

  • n_ZPF: the Cooper-pair number ZPF \(n_{\text{zpf}} = 1/(4\varphi_{\text{zpf}})\).

  • Omega_MHz: the angular plasma frequency \(\omega_p/(2\pi)\) in MHz… wait — this is actually \(\omega_p\) in rad/s divided by \(2\pi\) and converted, i.e. the plasma frequency \(f_p = \sqrt{8E_JE_C}/h\).

  • f_MHz: the qubit frequency \(f_{01} \approx f_p\) in GHz (note units in key name are legacy; value is in GHz).

  • Z_Ohms: the impedance \(Z = \sqrt{L_J/C_\Sigma}\) in ohms, which determines the magnitude of vacuum fluctuations and hence coupling strengths to microwave cavities.

For the parameters here (\(L_J = 13\,\text{nH}\), \(C_\Sigma = 65\,\text{fF}\)): \(E_J/E_C \approx 12574/298 \approx 42\), which is in the transmon regime but on the lower end. Increasing \(C_\Sigma\) (adding more shunting capacitance) would push \(E_J/E_C\) higher, improving charge insensitivity at the cost of reduced anharmonicity.

Summary: Connecting Circuit Parameters to the EPR Workflow#

The conversion utilities in this tutorial sit at the interface between fabrication/design and simulation:

Fabrication target
      I_c  ──────────────────────────────► L_J  (simulation input)
                                             │
                                             ▼
                                    HFSS eigenmode simulation
                                             │
                                     EPR analysis extracts
                                      φ_zpf per mode/junction
                                             │
                                             ▼
                                    Quantum Hamiltonian
                                    ω_01, α, χ (cross-Kerr)

The workflow is:

  1. Choose \(E_J\) and \(E_C\) to hit a target qubit frequency and anharmonicity, using transmon_print_all_params.

  2. Convert \(E_J \to L_J\) with Convert.Lj_from_Ej to get the HFSS junction parameter.

  3. Convert \(E_J \to I_c\) with Convert.Ic_from_Lj to specify the junction to your fabrication team.

  4. Run HFSS with the linearised inductance \(L_J\), then use DistributedAnalysis (Tutorial 1) to extract \(p_{jm}\) and \(\varphi_{\text{zpf}}\).

  5. Compute the full Hamiltonian with QuantumAnalysis (Tutorial 1), recovering \(\alpha\), \(\chi\), etc.

For more details, see the EPR paper (arXiv:2010.00620) and Koch et al., PRA 76, 042319 (2007).