---
title: "Transmon Artificial Atom Interactive Lab | Sandbox Physics"
description: "What do you give up when you make an artificial atom less sensitive to stray charge? Anharmonicity."
canonical_url: "https://sandboxphysics.com/labs/transmon-artificial-atom/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/transmon-artificial-atom/"
last_modified: "2026-10-27"
---

Q033 · Tuning a superconducting artificial atom

# Transmon Artificial Atom

**What do you give up when you make an artificial atom less sensitive to stray charge?** Anharmonicity. Raising the ratio of Josephson energy to charging energy makes the levels exponentially less sensitive to stray charge, but the level spacings grow more alike. Once the qubit transition sits close in frequency to the next one up, it is hard to drive one without the other.

Change the ratio of Josephson energy to charging energy. See how the charge dispersion and the uneven level spacing change, and compare the transitions recorded from two starting states.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model is the Hamiltonian of a single junction in the charge basis, with a check that the finite basis has converged. The chip is schematic, and the resonator stands for the readout chain; no cavity is simulated. The Lab does not predict lifetimes, gate fidelity or fabrication results.

02 / What did the readout record?

## Find the probe frequencies that change the state.

Show model reference

$h\nu/E_C$ · Blue: this sample. Orange: most recently saved other sample. Dashed: current model. Drag the cursor to tune.

At the selected frequency

$\text{—}$

Changed / preparations

— $\text{—}$

Uncertainty at this frequency

95% Wilson interval for independent yes-or-no readouts at this frequency, shown only after sampling. Each interval holds point by point; together they do not form a simultaneous band, and they do not cover optional stopping.

Latest 60 recorded outcomes

Blue means changed; gray means unchanged. Every trial starts freshly prepared.

Recorded sample

Energy levels and charge sensitivity

03 / Where do the lines come from?

## Flatten the charge response but keep the gaps unequal.

$n_g$ · $(E_j-E_0)/E_C$ · Model only. Drag horizontally; the matching slider also supports the keyboard.

Colored arrows: allowed transitions from the prepared level. Their thickness shows normalized strength. Gray ticks: other levels.

$\text{—}$

Inspect the prepared state and pulse

Model inspection · adds no counts

## A faster pulse can climb too far.

Prepared-state charge probabilities. Only the central seventeen charge states are plotted; all twenty-nine remain in the solver. $n$

Resonant pulse strength$\hbar\Omega/E_C$ $0.1$

$t/\tau,\quad\tau=\pi/\Omega$ · Blue: first excited state. Orange: higher-level leakage. Always starts in the ground state; independent of the spectroscopy preparation.

$\text{—}$

Saved comparisons

How saved samples work

Up to eight frozen samples. Restore to inspect; acquiring again starts a new run. CSV includes each bit, actual probe frequency and preparation. Reusing the seed reproduces draws, not independent evidence.

Model, basis convergence & sources

Energy is in the reference unit; probe frequency is ordinary frequency multiplied by Planck’s constant, not angular frequency. Curves and level arrows are model predictions. Recorded bits are simulated independent preparation–probe–readout trials.

$$
H/E_C=4(\hat n-n_g)^2-(E_J/E_C)\cos\hat\varphi
$$

Charge basis: 29 states centered on zero, checked against 33 states. The drive operator is charge. Lowest four energies are checked; higher levels are retained for transition-strength normalization. Charge dispersion is the absolute difference of the first gap at integer and half-integer offset charge. Larger junction energy reduces relative anharmonicity, not necessarily its absolute magnitude in charging-energy units.

Single fixed junction; no flux tuning law, quasiparticles, dielectric loss, cavity hybridization, multiphoton spectroscopy or lab-frame pulse dynamics. The chip and capacitive readout are a conceptual connection map, with junction size exaggerated. Ideal pure-state preparation and changed-state readout are assumed.

[Koch et al. · Transmon Hamiltonian and design tradeoffs](https://arxiv.org/abs/cond-mat/0703002) · [Schreier et al. · Spectroscopy and charge dispersion](https://arxiv.org/abs/0712.3581)

$\text{—}$

$$
p(\nu)=0.01+0.20\sum_{j\ne i}w_{ji}\left[1+4((\nu-\nu_{ji})/\Gamma)^2\right]^{-1}
$$

$$
w_{ji}=\frac{|\langle j|O|i\rangle|^2}{\sum_{k\ne i}|\langle k|O|i\rangle|^2}
$$

The readout uses an assumed incoherent Lorentzian response with a fixed one-percent false-change background and a twenty-percent contrast shared among the transitions. Width is the full width at half maximum. Strengths are normalized within the chosen preparation, so peak heights cannot compare absolute dipole moments across settings. All levels enter the normalization, including transitions outside the displayed band. This readout model does not solve coherent pulse dynamics and is not a fitted instrument calibration.

Physics tutorial

## Transmon Artificial Atom

**Background** Change the ratio of Josephson energy to charging energy. See how the charge dispersion and the uneven level spacing change, and compare the transitions recorded from two starting states.

**Why it matters** A spectrum tells you about both the Hamiltonian and the operator used to probe it.

Start with the essentials

- **Focus question**: What do you give up when you make an artificial atom less sensitive to stray charge?
- **One-sentence intuition**: Lowering the charge dispersion costs relative anharmonicity, so a device that is less sensitive to charge is not automatically a better qubit in every way.

### Core mathematical model

#### Charge-basis Hamiltonian

$$
H/E_C=4(\hat n-n_g)^2-(E_J/E_C)\cos\hat\varphi
$$

The charge hops only between neighboring integer numbers of Cooper pairs. The charge operator drives the transitions.

#### Approximation for a large Josephson energy

$$
h\nu_{01}\simeq\sqrt{8E_JE_C}-E_C,\qquad\alpha\simeq-E_C
$$

This is only the limit for a large ratio; the plotted spectrum comes from diagonalization. Relative anharmonicity falls as the first gap grows.

#### Charge dispersion

$$
\delta_{01}=|\Delta E_{01}(1/2)-\Delta E_{01}(0)|
$$

Compare the first transition at two charge offsets. A small dispersion is not a measured coherence time.

### Common difficulties

#### Not every level gives a peak

**Typical misconception** Every energy level must produce a visible peak.

**Better mental model** The prepared state, the probe operator, the line width, the frequency range and the finite number of counts all matter. Lines outside the plotted band still count in the strength normalization.

#### A model is not a calibrated instrument

**Typical misconception** A narrow simulated line establishes a long experimental lifetime.

**Better mental model** The line width and the low-contrast Lorentzian response are inputs set in the model. The separate pulse test uses a rotating-wave model; it does not infer pulse dynamics or hardware noise.

### Run the experiment

1. 01

   #### Predict before scanning

   Try the two charge-sensitive presets. Predict how much the first gap moves when the offset charge shifts.

   **What to observe:** The dashed curve comes from the model, not from recorded data. Hide it while you predict.

2. 02

   #### Record and save

   Scan, save the sample, change one preparation setting, then scan again.

   **What to observe:** Every frequency gets its own fresh, independent trials. The orange points keep the most recently saved preparation.

3. 03

   #### Explain the missing line

   Prepare the first excited state, and tell the downward transition apart from the next upward one.

   **What to observe:** A line's position is set by an energy difference, but the line shows up only if its matrix element is not zero.

Physical conclusion

The model is the Hamiltonian of a single junction in the charge basis, with a check that the finite basis has converged. The chip is schematic, and the resonator stands for the readout chain; no cavity is simulated. The Lab does not predict lifetimes, gate fidelity or fabrication results.

## Related experiments and articles

Condensed matter · Reading path

### Cold liquids and superconducting order

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How phase drives a supercurrent across a thin insulator**](https://sandboxphysics.com/stories/condensed-matter/josephson-phase-current/)
2. [Lab **SQUID: Flux into Voltage** SQUID flux response](https://sandboxphysics.com/labs/squid-flux-response/)
3. [Lab **Josephson Switching Statistics** Josephson switching statistics](https://sandboxphysics.com/labs/josephson-switching-statistics/)

Condensed matter · Reading path

### A collective circuit variable became quantum

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a circuit’s collective phase revealed tunneling and discrete levels**](https://sandboxphysics.com/stories/condensed-matter/macroscopic-quantum-circuit/)
2. [Lab **Josephson Switching Statistics** Josephson switching statistics](https://sandboxphysics.com/labs/josephson-switching-statistics/)
3. [Lab **SQUID: Flux into Voltage** SQUID flux response](https://sandboxphysics.com/labs/squid-flux-response/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)