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.

Lowest model gap $\text{—}$

Second gap minus first $\text{—}$

01 / Probe the atom at one frequency

Two islands joined by one junction have unequally spaced levels.

A simulated scan is ready below. Try the charge-sensitive presets, then raise the junction energy and watch the gaps respond less to charge.

Enlarged chip · slow drive markers; pad glow rescales model response, not physical light output

Probe frequency

h\nu/E_C

18.9

PrepareProbeChanged?

Ready. A scan covers 121 frequencies with 200 fresh preparations at 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

Presets load a simulated scan. Drag the drive port to tune, or the charge gate to change its bias.

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

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Changed / preparations

—

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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.

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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.

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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 · Schreier et al. · Spectroscopy and charge dispersion

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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.

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 misconceptionEvery energy level must produce a visible peak.

Better mental modelThe 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 misconceptionA narrow simulated line establishes a long experimental lifetime.

Better mental modelThe 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

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.
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.
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.

Transmon Artificial Atom

Find the probe frequencies that change the state.

Flatten the charge response but keep the gaps unequal.

A faster pulse can climb too far.