00-notation.md

Before you start: notation and numerical habits

This book uses two kinds of numerical models: circuit spectra and time evolution. Keep their units separate. Read this page once, then keep it beside your editor.

Quantity	Meaning	Code convention
$E_C,E_J$	energies	Spectrum lab uses $E/h$ in GHz
$f$	cycles per second	MHz = cycles per microsecond
$\omega,\Omega,g,\alpha,\chi$	angular frequencies	Dynamics labs use rad/us, usually 2*np.pi*f_MHz
$H$ in the dynamics scripts	actually $H/\hbar$	Multiply by time in us to obtain a dimensionless phase
$\kappa$	photon-number decay rate	Population decays as $e^{-\kappa t}$; field as $e^{-\kappa t/2}$
$T_1,T_2,T_\phi$	time constants	us in labs 01 to 06; seconds in lab 11
$n$ in a charge basis	signed Cooper-pair number	Integer charge states, including negative integers
$a^\dagger a$	excitation number	Nonnegative oscillator Fock number; different from Cooper-pair number

For example, g = 2*np.pi*100 means $g/2\pi=100$ MHz. The excitation-swap time $\pi/(2g)$ is then 0.0025 us, or 2.5 ns. In a static diagonalization of $H/h$, no extra $2\pi$ is needed to obtain transition frequencies from energy differences.

Ground state and signs

All scripts use basis(2, 0) as the ground state and basis(2, 1) as the excited state. Computational $Z=\mathrm{diag}(1,-1)$ gives ground-state expectation $+1$, so the free qubit Hamiltonian is $-\omega_q Z/2$. The physical lowering operator is destroy(2), or $|0\rangle\langle1|$. QuTiP's sigmam() uses a spin convention and is not that matrix.

Chapter 05 uses the common atomic convention $\sigma_z=|e\rangle\langle e|-|g\rangle\langle g|=-Z$. Substitute $\sigma_z=-Z$ to translate its Hamiltonians into the code. Chapters 06 to 07 use computational $Z$ directly; Chapter 07 writes it as $\hat\sigma_z$ (with $\hat\sigma_z|0\rangle=+|0\rangle$), so its $\hat\sigma_z$ is $Z$, not Chapter 05's $\hat\sigma_z$.

Detuning is always a signed difference, but the pair being compared changes:

Context	Definition
Qubit-cavity (05 to 06, lab 10)	$\Delta=\omega_q-\omega_r$
Driven qubit (07, Ramsey lab)	$\Delta_d=\omega_d-\omega_q$
Two qubits (08, CZ lab)	$\Delta=\omega_1-\omega_2$
Driven cavity (lab 04)	$\delta=\widetilde\omega_r-\omega_d$

For a transmon, $\alpha=\omega_{12}-\omega_{01}<0$. Its signed half-separation of readout resonances is $\chi=g^2\alpha/[\Delta(\Delta+\alpha)]$ in second-order perturbation theory. Writing the resonances as $\widetilde\omega_r\mp\chi$ uses their dressed midpoint; it is not generally the uncoupled cavity frequency. Lab 04 chooses positive $\chi$ for a simple pointer-field example; the negative-detuning worked example in the text has negative $\chi$.

What makes a simulation trustworthy?

1. Predict a limiting case before running: no drive, no coupling, or no noise.
2. Compare a numerical result with an independent analytic identity.
3. Increase the Hilbert-space cutoff and tighten the integrator tolerances.
4. Separate numerical convergence from physical validity. A perfectly converged two-level model cannot predict transmon leakage.
5. Check the requested observable. Low final leakage does not establish an X gate; a conditional phase near pi does not establish CZ.

The lab runner implements checks of these kinds. For stochastic simulations, zero observed failures is not a zero failure probability: zero events in $N$ independent trials gives an approximate 95% upper bound $3/N$.

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