Superconducting Qubit Feature Lab

This note is a rendering test for Quartz features using superconducting qubits as the theme. It includes long equations, a derivation, Mermaid diagrams, callouts, tables, footnotes, code blocks, task lists, and internal links.

Inline And Block Math

The qubit state can be written inline as $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ with normalization $|\alpha {|}^2+|\beta {|}^2=1$.

A driven two-level Hamiltonian in the lab frame is often modeled as

$$H/\hslash ={\omega }_q\frac{{\sigma }_z}{2}+\Omega (t)\cos ({\omega }_{d}t+\varphi ){\sigma }_x.$$

Moving into a frame rotating at the drive frequency and applying the rotating-wave approximation gives

$$H_{\mathrm{rot}}/\hslash =\Delta \frac{{\sigma }_z}{2}+\frac{\Omega (t)}{2}\left(\cos \varphi {\sigma }_x+\sin \varphi {\sigma }_y\right),$$

where $\Delta ={\omega }_q-{\omega }_d$.

Control intuition

The microwave phase $\varphi$ chooses the rotation axis in the equatorial plane of the Bloch sphere. Pulse area sets the rotation angle:

$$\theta =\int \Omega (t)dt.$$

LC Oscillator Quantization

Start with the classical energy of an LC circuit:

$$E=\frac{Q^2}{2C}+\frac{{\Phi }^2}{2L}.$$

Promote charge $Q$ and flux $\Phi$ to operators with commutator

$$[\hat{\Phi },\hat{Q}]=i\hslash .$$

This gives the quantum harmonic oscillator Hamiltonian

$${\hat{H}}_{LC}=\frac{{\hat{Q}}^2}{2C}+\frac{{\hat{\Phi }}^2}{2L}=\hslash {\omega }_r\left(a^{†}a+\frac{1}{2}\right),{\omega }_r=\frac{1}{\sqrt{LC}}.$$

That alone is not a qubit because adjacent transitions are degenerate:

$$E_{n+1}-E_n=\hslash {\omega }_r.$$

A Josephson junction adds nonlinearity, replacing the linear inductive energy with

$$U(\phi )=-E_J\cos \phi .$$

For a transmon, the approximate Hamiltonian is

$${\hat{H}}_{\mathrm{transmon}}=4E_C(\hat{n}-n_g{)}^2-E_J\cos \hat{\phi }.$$

See Transmon Anharmonicity Derivation for a longer worked derivation.

Derivation Sketch

When $E_J/E_C\gg 1$, expand the cosine around $\phi =0$:

$$-E_J\cos \phi \approx -E_J+\frac{E_J}{2}{\phi }^2-\frac{E_J}{24}{\phi }^4+O({\phi }^6).$$

The quadratic term creates an oscillator with plasma frequency

$$\hslash {\omega }_p=\sqrt{8E_J{E}_C}.$$

The quartic term gives a weak anharmonic correction. To first order, the transition frequency is approximately

$${\omega }_{01}\approx \frac{1}{\hslash }\left(\sqrt{8E_J{E}_C}-E_C\right),$$

and the anharmonicity is approximately

$$\alpha ={\omega }_{12}-{\omega }_{01}\approx -\frac{E_C}{\hslash }.$$

Approximation boundary

This expansion is useful for intuition, but real devices require corrections from charge dispersion, higher levels, coupling to readout resonators, packaging modes, and calibration drift.

Mermaid Diagram

flowchart TD
  A[Fabricate superconducting circuit] --> B[Cool to millikelvin temperature]
  B --> C[Find qubit and resonator frequencies]
  C --> D[Calibrate microwave pulses]
  D --> E[Measure Rabi and Ramsey experiments]
  E --> F[Estimate T1 and T2]
  F --> G{Good enough?}
  G -- yes --> H[Run gates and readout]
  G -- no --> I[Adjust pulse shape, bias, filtering, or design]
  I --> C

Sequence Diagram

sequenceDiagram
  participant C as Controller
  participant Q as Transmon Qubit
  participant R as Readout Resonator
  participant A as Amplifier Chain

  C->>Q: shaped microwave pulse
  Q-->>Q: state rotates on Bloch sphere
  C->>R: readout tone
  R-->>A: state-dependent microwave response
  A-->>C: digitized I/Q signal
  C->>C: classify 0 or 1

Comparison Table

Quantity	Meaning	Typical Role
$E_J$	Josephson energy	Sets nonlinear inductive scale
$E_C$	Charging energy	Controls anharmonicity and charge sensitivity
$T_1$	Relaxation time	Limits excited-state lifetime
$T_2$	Dephasing time	Limits phase memory
$\chi$	Dispersive shift	Sets readout contrast in [[topics/circuit-qed-readout

Callouts

Note

Superconducting qubits are macroscopic electrical circuits, but their low-energy dynamics are quantum mechanical.

Collapsible example: Ramsey phase

During a Ramsey experiment, detuning causes phase accumulation:

$$\varphi (t)=\Delta t.$$

Oscillations in measured population reveal detuning and dephasing.

Calibration question

If leakage to $|2⟩$ increases after a faster pulse, should the pulse be longer, more carefully shaped, or DRAG-corrected?

Code Block

import numpy as np
 
# Toy estimate: transmon frequency in GHz when EJ and EC are in GHz units.
def transmon_f01(EJ, EC):
    return np.sqrt(8 * EJ * EC) - EC
 
for EJ_over_EC in [25, 50, 75]:
    EC = 0.25
    EJ = EJ_over_EC * EC
    print(EJ_over_EC, round(transmon_f01(EJ, EC), 3))

Task List

• Render inline math.
• Render block equations.
• Render Mermaid flowchart.
• Render Mermaid sequence diagram.
• Render callouts.
• Add a real source note for a transmon paper.
• Add a calibration notebook note.

Footnote

The transmon was introduced to reduce charge-noise sensitivity while retaining enough anharmonicity for control.¹

Related

• Quantum Notes
• Superconducting Qubits
• Transmon Qubits
• Qubit Coherence
• Circuit QED Readout

Footnotes

1. Placeholder citation note. Replace this with a full paper reference in Source Notes. ↩