Dispersive Shift of a Transmon–Resonator System with scQubits¶

:label: eq:transmon-resonator-hamiltonian-scq
\mathcal{H} = \omega_t\, a_t^\dagger a_t
+ \frac{\alpha}{2}\, a_t^{\dagger 2} a_t^{2}
+ \omega_r\, a_r^\dagger a_r
+ g\,(a_t + a_t^\dagger)(a_r + a_r^\dagger),

:label: eq:dispersive-shift-definition-scq
\chi = (E_{11} - E_{10}) - (E_{01} - E_{00}),

# uv pip install qpdk

import numpy as np

# Monkeypatch for NumPy 2.0 compatibility with scqubits.
# TODO: Remove once the minimum required scqubits version officially supports
# NumPy 2.x without relying on np.complex_ / np.float_ aliases.
if not hasattr(np, "complex_"):
    np.complex_ = np.complex128  # noqa: NPY201
if not hasattr(np, "float_"):
    np.float_ = np.float64  # noqa: NPY201

import polars as pl
import scipy
import scqubits as scq
from IPython.display import Math, display
from matplotlib import pyplot as plt

from qpdk import PDK
from qpdk.helper import display_dataframe
from qpdk.models.constants import c_0
from qpdk.models.cpw import cpw_parameters

PDK.activate()

# ruff: disable[E402]
from qpdk.models.perturbation import (
    dispersive_shift,
    dispersive_shift_to_coupling,
    ej_ec_to_frequency_and_anharmonicity,
    measurement_induced_dephasing,
    purcell_decay_rate,
    resonator_linewidth_from_q,
)
from qpdk.models.qubit import (
    coupling_strength_to_capacitance,
    ec_to_capacitance,
    ej_to_inductance,
)
from qpdk.models.resonator import resonator_frequency

# ruff: enable[E402]

\omega_t \approx \sqrt{8 E_J E_C} - E_C, \qquad \alpha \approx -E_C.

# Hamiltonian parameters (matching pymablock notebook for comparison)
EJ = 20.0  # Josephson energy in GHz
EC = 0.2  # Charging energy in GHz
omega_r_val = 7.0  # Resonator frequency in GHz
g_val = 0.1  # Coupling strength in GHz

transmon = scq.Transmon(
    EJ=EJ,
    EC=EC,
    ng=0.0,  # Offset charge
    ncut=30,  # Charge basis cutoff
    truncated_dim=6,  # Keep 6 transmon levels
)

resonator = scq.Oscillator(
    E_osc=omega_r_val,
    truncated_dim=6,  # Keep 6 Fock levels
)

omega_t_val, alpha_val = ej_ec_to_frequency_and_anharmonicity(EJ, EC)

display(
    Math(rf"""
\textbf{{Hamiltonian Parameters:}} \\
E_J = {EJ:.1f}\,\mathrm{{GHz}}, \quad
E_C = {EC:.1f}\,\mathrm{{GHz}} \\
\omega_t = \sqrt{{8 E_J E_C}} - E_C = {omega_t_val:.3f}\,\mathrm{{GHz}} \\
\alpha \approx -E_C = {alpha_val:.1f}\,\mathrm{{GHz}} \\
\omega_r = {omega_r_val:.1f}\,\mathrm{{GHz}} \\
g = {g_val:.1f}\,\mathrm{{GHz}} \\
\Delta = \omega_t - \omega_r = {omega_t_val - omega_r_val:.3f}\,\mathrm{{GHz}}
""")
)

eigenvals_t = transmon.eigenvals(evals_count=5)
f01 = eigenvals_t[1] - eigenvals_t[0]
f12 = eigenvals_t[2] - eigenvals_t[1]
anharmonicity = f12 - f01

display(
    Math(rf"""
\textbf{{Transmon Spectrum (scQubits):}} \\
0\rightarrow 1\ \text{{frequency:}}\ {f01:.3f}\,\mathrm{{GHz}} \\
1\rightarrow 2\ \text{{frequency:}}\ {f12:.3f}\,\mathrm{{GHz}} \\
\text{{Anharmonicity,}}\ \alpha =\ {anharmonicity:.3f}\,\mathrm{{GHz}}
""")
)

:label: eq:coupling-term-scq
\mathcal{H}_\text{int} = g\, n_t\, (a_r + a_r^\dagger),

hilbert_space = scq.HilbertSpace([transmon, resonator])

interaction = scq.InteractionTerm(
    g_strength=g_val,
    operator_list=[
        (0, transmon.n_operator),  # (subsystem_index, operator)
        (1, resonator.annihilation_operator() + resonator.creation_operator()),
    ],
)
hilbert_space.interaction_list = [interaction]

# Diagonalize the full Hamiltonian
evals, evecs = hilbert_space.eigensys(evals_count=12)
# Stack into (evals_count, hilbert_dim) matrix where each row is an eigenvector
evecs = np.array([np.array(v.full()).flatten() for v in evecs])

dim_t = transmon.truncated_dim
dim_r = resonator.truncated_dim


def bare_state_vec(qt_idx: int, res_idx: int) -> np.ndarray:
    """Return the bare state |qt_idx, res_idx⟩ in the full Hilbert space."""
    vec = np.zeros(dim_t * dim_r)
    vec[qt_idx * dim_r + res_idx] = 1.0
    return vec


def find_dressed_index(bare_vec: np.ndarray) -> int:
    """Return the dressed-state index with maximum overlap with bare_vec."""
    return int(np.argmax(np.abs(evecs @ bare_vec) ** 2))


idx_00 = find_dressed_index(bare_state_vec(0, 0))
idx_10 = find_dressed_index(bare_state_vec(1, 0))
idx_01 = find_dressed_index(bare_state_vec(0, 1))
idx_11 = find_dressed_index(bare_state_vec(1, 1))

E_00, E_10, E_01, E_11 = evals[idx_00], evals[idx_10], evals[idx_01], evals[idx_11]
chi_scqubits = (E_11 - E_10) - (E_01 - E_00)

# Compare to the analytical formula from qpdk.models.perturbation
chi_analytical = dispersive_shift(omega_t_val, omega_r_val, alpha_val, g_val)

display(
    Math(rf"""
\chi_{{\mathrm{{scQubits}}}} = {chi_scqubits * 1e3:.3f}\,\mathrm{{MHz}} \\
\chi_{{\mathrm{{analytical}}}} = {chi_analytical * 1e3:.3f}\,\mathrm{{MHz}} \\
\text{{Relative error: }}
{abs(chi_scqubits - chi_analytical) / abs(chi_analytical) * 100:.1f}\,\%
""")
)

:label: eq:coupling-capacitance
g &\approx \frac{1}{2} \frac{C_\text{c}}{\sqrt{C_{\Sigma} \left( C_\text{r} - \frac{C_\text{c}^2}{C_\text{q}} \right) }}  \sqrt{\omega_\text{q}\omega_\text{r}} \\
&\approx \frac{1}{2} \frac{C_\text{c}}{\sqrt{C_{\Sigma} C_\text{r}}}  \sqrt{\omega_\text{q}\omega_\text{r}}, \quad \text{for } C_\text{c} \ll C_\text{q}

_a_plus_adag = resonator.annihilation_operator() + resonator.creation_operator()
g_sweep = np.linspace(0.01, 0.3, 200)
chi_sweep_analytical = np.array(
    [dispersive_shift(omega_t_val, omega_r_val, alpha_val, gi) for gi in g_sweep]
)


def chi_scqubits_at_g(g: float) -> float:
    """Numerically compute χ via scQubits at a given coupling g.

    Returns:
        The dispersive shift value.
    """
    hs = scq.HilbertSpace([transmon, resonator])
    hs.interaction_list = [
        scq.InteractionTerm(
            g_strength=g,
            operator_list=[(0, transmon.n_operator), (1, _a_plus_adag)],
        )
    ]
    ev, evc = hs.eigensys(evals_count=12)
    # Stack into (evals_count, hilbert_dim) matrix where each row is an eigenvector
    evc = np.array([np.array(v.full()).flatten() for v in evc])

    def di(qi: int, ri: int) -> int:
        v = np.zeros(dim_t * dim_r)
        v[qi * dim_r + ri] = 1.0
        return int(np.argmax(np.abs(evc @ v) ** 2))

    i00, i10, i01, i11 = di(0, 0), di(1, 0), di(0, 1), di(1, 1)
    return float((ev[i11] - ev[i10]) - (ev[i01] - ev[i00]))


g_sweep_coarse = np.linspace(0.01, 0.3, 20)
chi_sweep_scqubits = np.array([chi_scqubits_at_g(gi) for gi in g_sweep_coarse])

fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(
    g_sweep * 1e3,
    chi_sweep_analytical * 1e3,
    "-",
    linewidth=2,
    label="Analytical (qpdk)",
)
ax.plot(
    g_sweep_coarse * 1e3,
    chi_sweep_scqubits * 1e3,
    "o",
    markersize=6,
    label="scQubits (numerical)",
)
ax.axhline(0, color="gray", linestyle="--", alpha=0.5)
ax.set_xlabel("Coupling strength $g$ (MHz)")
ax.set_ylabel(r"Dispersive shift $\chi$ (MHz)")
ax.set_title(
    rf"Dispersive shift vs. coupling ($\omega_t={omega_t_val:.2f}$ GHz, "
    rf"$\omega_r={omega_r_val:.1f}$ GHz)"
)
ax.legend()
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()

:label: eq:flux-tunable-ej
E_J(\Phi) = E_{J,\mathrm{max}}
  \left|\cos\!\left(\pi\frac{\Phi}{\Phi_0}\right)\right|
  \sqrt{1 + d^2 \tan^2\!\left(\pi\frac{\Phi}{\Phi_0}\right)},

tunable_transmon = scq.TunableTransmon(
    EJmax=40.0,  # Maximum Josephson energy in GHz
    EC=0.2,  # Charging energy in GHz
    d=0.1,  # Junction asymmetry
    flux=0.0,  # External flux in units of Φ₀
    ng=0.0,
    ncut=30,
)

tunable_transmon.plot_evals_vs_paramvals(
    "flux", np.linspace(-0.5, 0.5, 201), subtract_ground=True, evals_count=5
)
plt.show()

# Design targets
chi_target_mhz = -1.0  # Target dispersive shift in MHz
chi_target = chi_target_mhz * 1e-3  # Convert to GHz

# Qubit and resonator parameters
EJ_design = 20.0  # GHz
EC_design = 0.2  # GHz
omega_r_design = 7.0  # GHz

omega_t_design, alpha_design = ej_ec_to_frequency_and_anharmonicity(
    EJ_design, EC_design
)

# Find g using the qpdk analytical inversion
g_design = dispersive_shift_to_coupling(
    chi_target, omega_t_design, omega_r_design, alpha_design
)

# Verify analytically and with scQubits
chi_verify_analytical = dispersive_shift(
    omega_t_design, omega_r_design, alpha_design, float(g_design)
)
chi_verify_scq = chi_scqubits_at_g(float(g_design))

display(
    Math(rf"""
\textbf{{Step 1–3: Hamiltonian Design}} \\
\text{{Target:}}\quad \chi = {chi_target_mhz:.1f}\,\mathrm{{MHz}} \\
\text{{Required coupling:}}\quad g = {float(g_design) * 1e3:.1f}\,\mathrm{{MHz}} \\
\text{{Verification (analytical):}}\quad \chi = {chi_verify_analytical * 1e3:.3f}\,\mathrm{{MHz}} \\
\text{{Verification (scQubits):}}\quad \chi = {chi_verify_scq * 1e3:.3f}\,\mathrm{{MHz}}
""")
)

# Total qubit capacitance
C_sigma = float(ec_to_capacitance(EC_design))

# Josephson inductance
L_J = float(ej_to_inductance(EJ_design))

# Determine resonator length for the target frequency
ep_eff, z0 = cpw_parameters(width=10, gap=6)


def _resonator_objective(length: float) -> float:
    """Minimize the squared frequency error.

    Returns:
        The squared error between the calculated and target frequency.
    """
    freq = resonator_frequency(
        length=length,
        epsilon_eff=float(np.real(ep_eff)),
        is_quarter_wave=True,
    )
    return (freq - omega_r_design * 1e9) ** 2


result = scipy.optimize.minimize(_resonator_objective, 4000.0, bounds=[(1000, 20000)])
resonator_length = result.x[0]

# Total resonator capacitance from CPW impedance and phase velocity
# {cite:p}`gopplCoplanarWaveguideResonators2008a`:
# C_r = l / Re(Z_0 * v_p)
C_r = 1 / np.real(z0 * (c_0 / np.sqrt(ep_eff))) * resonator_length * 1e-6  # F

# Coupling capacitance
C_c = float(
    coupling_strength_to_capacitance(
        g_ghz=float(g_design),
        c_sigma=C_sigma,
        c_r=C_r,
        f_q_ghz=omega_t_design,
        f_r_ghz=omega_r_design,
    )
)

display(
    Math(rf"""
\textbf{{Step 4: Circuit Parameters}} \\
C_\Sigma = {C_sigma * 1e15:.1f}\,\mathrm{{fF}} \\
L_J = {L_J * 1e9:.2f}\,\mathrm{{nH}} \\
C_r = {C_r * 1e15:.1f}\,\mathrm{{fF}} \\
C_c = {C_c * 1e15:.2f}\,\mathrm{{fF}}
""")
)

f_resonator_achieved = resonator_frequency(
    length=resonator_length,
    epsilon_eff=float(np.real(ep_eff)),
    is_quarter_wave=True,
)

display(
    Math(rf"""
\textbf{{Step 5: Layout Parameters}} \\
\text{{Resonator length:}}\quad l = {resonator_length:.0f}\,\mathrm{{\mu m}} \\
\text{{Resonator CPW width:}}\quad w = 10\,\mathrm{{\mu m}}, \quad
\text{{gap}} = 6\,\mathrm{{\mu m}} \\
\text{{Resonator frequency achieved:}}\quad
f_r = {f_resonator_achieved / 1e9:.3f}\,\mathrm{{GHz}}
""")
)

# Resonator quality factor and linewidth
Q_ext = 10_000  # External quality factor
kappa = resonator_linewidth_from_q(omega_r_design, Q_ext)

# Purcell decay rate
gamma_purcell = purcell_decay_rate(
    float(g_design), omega_t_design, omega_r_design, kappa
)
T_purcell = 1 / (gamma_purcell * 1e9) if gamma_purcell > 0 else float("inf")

# Measurement-induced dephasing
n_bar = 5.0  # Mean photon number during readout
gamma_phi = measurement_induced_dephasing(chi_verify_analytical, kappa, n_bar)

# SNR estimate: the signal-to-noise ratio scales as χ/κ
chi_over_kappa = abs(chi_verify_analytical) / kappa

display(
    Math(rf"""
\textbf{{Readout System Parameters:}} \\
Q_\text{{ext}} = {Q_ext:,} \\
\kappa = \omega_r / Q_\text{{ext}} = {kappa * 1e6:.1f}\,\mathrm{{kHz}} \\
\gamma_\text{{Purcell}} = \kappa (g/\Delta)^2
  = {gamma_purcell * 1e6:.2f}\,\mathrm{{kHz}}
  \quad \Rightarrow \quad T_\text{{Purcell}}
  = {T_purcell * 1e6:.0f}\,\mathrm{{\mu s}} \\
\Gamma_\phi(\bar{{n}}={n_bar:.0f})
  = 8\chi^2 \bar{{n}} / \kappa
  = {gamma_phi * 1e6:.2f}\,\mathrm{{kHz}} \\
|\chi| / \kappa = {chi_over_kappa:.1f}
  \quad (\text{{want}} \gtrsim 1 \text{{ for high-fidelity readout}})
""")
)

data = [
    ("Target", "Target dispersive shift $\\chi$", f"{chi_target_mhz:.1f}", "MHz"),
    ("Hamiltonian", "$E_J$", f"{EJ_design:.1f}", "GHz"),
    ("Hamiltonian", "$E_C$", f"{EC_design:.1f}", "GHz"),
    ("Hamiltonian", "$\\omega_t$", f"{omega_t_design:.3f}", "GHz"),
    ("Hamiltonian", "$\\alpha$", f"{alpha_design:.1f}", "GHz"),
    ("Hamiltonian", "$\\omega_r$", f"{omega_r_design:.1f}", "GHz"),
    ("Hamiltonian", "$g$", f"{float(g_design) * 1e3:.1f}", "MHz"),
    ("Circuit", "$C_\\Sigma$", f"{C_sigma * 1e15:.1f}", "fF"),
    ("Circuit", "$L_J$", f"{L_J * 1e9:.2f}", "nH"),
    ("Circuit", "$C_r$", f"{C_r * 1e15:.1f}", "fF"),
    ("Circuit", "$C_c$", f"{C_c * 1e15:.2f}", "fF"),
    ("Layout", "Resonator length", f"{resonator_length:.0f}", "µm"),
    ("Layout", "Resonator CPW width", "10", "µm"),
    ("Layout", "Resonator CPW gap", "6", "µm"),
    ("Layout", "Resonator freq", f"{f_resonator_achieved / 1e9:.3f}", "GHz"),
    ("Readout", "$Q_{\\mathrm{ext}}$", f"{Q_ext:,}", ""),
    ("Readout", "$\\kappa$", f"{kappa * 1e6:.1f}", "kHz"),
    ("Readout", "$T_{\\mathrm{Purcell}}$", f"{T_purcell * 1e6:.0f}", "µs"),
    ("Readout", "$|\\chi|/\\kappa$", f"{chi_over_kappa:.1f}", ""),
]

df = pl.DataFrame(data, schema=["Category", "Parameter", "Value", "Unit"], orient="row")

display_dataframe(df)

:filter: docname in docnames