Resonator & Coupling
========================

Resonator & Coupling Parameters: Readout and bus resonator characteristics plus
                qubit-resonator coupling extracted via EPR

A. Resonator Properties

.. list-table::
   :header-rows: 1

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Resonator Frequency
     - omega_r / 2pi
     - GHz
     - Bare resonator frequency; should be detuned from qubit to remain in
                            dispersive regime.
     - 6.5 – 8.5 GHz
     - 5 – 10 GHz
     - 3 – 5 GHz
     - < 2 GHz
     - Must satisfy \|Δ\| = \|ω_q - ω_r\| >
                            > g for dispersive approximation; EPR gives bare frequency.
   * - Resonator Internal Q
     - Q_int
     - dimensionless
     - Internal (material-limited) quality factor of the readout resonator.
     - > 10⁵
     - 10⁴ – 10⁵
     - 10³ – 10⁴
     - < 500
     - Low Q_int adds photon loss increasing measurement back-action and
                            reducing readout SNR.
   * - Resonator External Q
     - Q_ext
     - dimensionless
     - External quality factor set by coupling to transmission line; determines
                            measurement bandwidth.
     - 10³ – 10⁴ (fast readout)
     - 500 – 2×10⁴
     - 50 – 499
     - < 50
     - Too high → slow readout; too low → Purcell-enhanced qubit decay.
                            Optimized with Purcell filter.
   * - Resonator–Qubit Detuning \|Δ\|
     - \|Δ\|/2π
     - GHz
     - Frequency detuning between qubit and resonator; must be large compared
                            to coupling g.
     - 1.0 – 3.0 GHz
     - 0.5 – 4.0 GHz
     - 0.1 – 0.5 GHz
     - < 0.05 GHz
     - Small detuning violates dispersive approximation; EPR hybridization
                            factor chi tracks this.
   * - Purcell Decay Rate
     - γ_P/2π
     - kHz
     - Qubit decay rate via resonator Purcell channel; κ * (g/Δ)². Must be 
                            <
                            < 1/T1_target.
     - < 1 kHz
     - 1 – 10 kHz
     - 10 – 100 kHz
     - > 500 kHz
     - Dominant T_1 limit in many designs without Purcell filter; directly
                            predicted by EPR p_J^res.

B. Coupling Strengths

.. list-table::
   :header-rows: 1

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Transverse Coupling g / 2pi
     - g / 2pi
     - MHz
     - Qubit-resonator coupling strength (Jaynes-Cummings); extracted from EPR
                            as g = √(p_J^res * ω_r * ω_q / 2).
     - 50 – 150 MHz
     - 20 – 200 MHz
     - 5 – 19 MHz
     - < 2 MHz
     - Sets χ and readout speed; g/\|Δ\| < 0.1 required for dispersive
                            regime.
   * - Dispersive Coupling g/Δ ratio
     - g/\|Δ\|
     - dimensionless
     - Dimensionless ratio quantifying proximity to strong-coupling limit; must
                            be <
                            < 1 for dispersive readout.
     - 0.01 – 0.05
     - 0.005 – 0.09
     - 0.09 – 0.15
     - > 0.20
     - Ratio > 0.1 causes photon-number-dependent qubit dephasing and
                            higher-order dispersive corrections.
   * - Bus Coupler Coupling (2Q)
     - J / 2pi
     - MHz
     - Exchange coupling between two qubits via bus resonator or direct
                            capacitance.
     - 5 – 20 MHz
     - 2 – 30 MHz
     - 0.5 – 1.9 MHz
     - < 0.1 MHz
     - Too weak → slow two-qubit gates; too strong → residual ZZ. EPR predicts
                            J from geometry.
   * - Residual ZZ (static)
     - zeta_ZZ / 2pi
     - kHz
     - Always-on longitudinal (ZZ) qubit–qubit interaction; source of
                            conditional phase errors.
     - < 10 kHz
     - < 100 kHz
     - 100 – 500 kHz
     - > 1 MHz
     - Limits two-qubit gate fidelity via conditional phase accumulation;
                            minimized by tunable couplers.
   * - Stray Coupling (nearest-neighbor)
     - J_stray / 2pi
     - kHz
     - Unintended coupling between non-adjacent qubits; extracted from
                            full-chip EPR simulation.
     - < 10 kHz
     - < 50 kHz
     - 50 – 200 kHz
     - > 500 kHz
     - Degrades multi-qubit gate fidelity; EPR full-chip simulation essential
                            to identify stray modes.