Q033 · Charge / junction / spectrum

Transmon Artificial Atom

Tune the Josephson-to-charging energy ratio. Compare charge dispersion, unequal level spacings and recorded transitions from two prepared states.

Lowest model gap $\text{—}$

State / design reference $\text{—}$

01 / ASK THE ATOM WITH A FREQUENCY.

Two islands. One junction. Unequal steps.

Compare the first two charge-sensitive presets, saving a scan of each. Then increase the junction energy: charge shifts matter less, but adjacent gaps become relatively closer.

Enlarged single-junction chip · separate drive, charge gate and readout · resonator dynamics omitted

Probe frequency

h\nu/E_C

18.9

PrepareProbeChanged?

Ready. A scan records 121 frequencies, 200 fresh preparations each.

MAKE A COMPARISON

Junction / charging energy

E_J/E_C

50

Offset charge

n_g

0.25

Prepared energy level

i

0

Prepared level: zero is the ground state, one is the first excited state. Downward and upward transitions both count as a change.

Line width, coupling & sampling

Line width (FWHM)

h\Gamma/E_C

0.18

Sampling seed

s_0

24133

Tuning frequency keeps the records. Changing preparation starts a fresh sample; save first. Presets and camera views do not acquire data.

A single-junction charge-basis Hamiltonian with a finite-basis convergence check. The chip is schematic; the resonator represents the readout chain, not a simulated cavity. No lifetime, gate fidelity or fabrication prediction.

02 / WHAT DID THE READOUT RECORD?

Build a spectrum, one bit at a time.

Show model reference

$h\nu/E_C$ · Blue: this sample. Orange: most recently saved other sample. Dashed: current model. Drag the cursor to tune; curves do not create records.

AT THE SELECTED FREQUENCY

\text{—}

Changed / preparations

—

\text{—}

95% Wilson interval for independent binary records at this frequency. No interval before sampling. These are pointwise, not simultaneous bands or optional-stopping guarantees.

Latest 60 recorded outcomes

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

Recorded sample

03 / WHERE DO THE LINES COME FROM?

Flatten the charge response. Keep unequal gaps.

$n_g$ · $(E_j-E_0)/E_C$ · 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{—}

MODEL INSPECTION · NO COUNTS ADDED

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{—}

04 / HOLD ON TO THE COMPARISON

Save a sample before changing the atom.

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 · Schreier et al. · Spectroscopy and charge dispersion

\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}

Declared incoherent Lorentzian response: fixed one-percent false-change background and twenty-percent contrast budget. Width is FWHM. Strengths are normalized within the chosen preparation, so heights cannot compare absolute dipole moments across settings. All levels enter the normalization, including transitions outside the displayed band. This readout model is not a coherent pulse solver or a fitted instrument calibration.

Start with the essentials

Focus question: Make an atom less sensitive to charge. What do you give up?
One-sentence intuition: Suppressing charge dispersion costs relative anharmonicity. A less charge-sensitive device is not automatically a better qubit in every respect.

Core mathematical model

Charge-basis Hamiltonian

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

Nearest-neighbor hopping between integer Cooper-pair charge states. The charge operator drives transitions.

Large-junction approximation

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

Only a large-ratio asymptote; the displayed spectrum comes from diagonalization. Relative anharmonicity decreases as the first gap grows.

Charge dispersion

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

Compare the same first transition at two charge offsets. Do not confuse small dispersion with a measured coherence time.

Common difficulties

Peaks and states

Typical misconceptionEvery energy level must produce a visible peak.

Better mental modelPreparation, the probe operator, line width, frequency range and finite counts all matter. Lines outside the shown band remain in the strength normalization.

A model is not a calibrated instrument

Typical misconceptionA narrow simulated line establishes a long experimental lifetime.

Better mental modelThe line width and low-contrast Lorentzian response are declared inputs. The separate pulse test is a rotating-wave model, not inferred pulse dynamics or hardware noise.

Run the experiment

01
Predict before scanning
Try the two charge-sensitive presets. Predict how much the first gap changes when the charge shifts.
What to observe: The dashed curve is a model reference, not recorded data. Hide it for the prediction.
02
Record and save
Scan, save the sample, change one preparation setting, then scan again.
What to observe: Each frequency has fresh independent trials. Orange points retain the most recently saved preparation.
03
Explain the missing line
Prepare the first excited state. Separate the downward transition from the next upward transition.
What to observe: Line positions are energy differences; visibility also requires a nonzero matrix element.