Junction Parameters
===================

Josephson Junction Parameters: Junction electrical and physical parameters extracted
                from or used as inputs to EPR analysis

A. Junction Electrical Parameters

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

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Josephson Inductance
     - L_J
     - nH
     - Linear (small-signal) inductance of Josephson junction; L_J = Φ₀ / (2π
                            I_c). Central EPR input.
     - 5 – 20 nH
     - 2 – 50 nH
     - 50 – 200 nH
     - > 500 nH
     - Sets qubit frequency via ω_q = 1 / √(L_J * C_S); too large → very low
                            frequency, thermally excited.
   * - Critical Current
     - I_c
     - µA
     - Maximum supercurrent through junction; I_c = Φ₀ / (2π L_J). Sets EJ =
                            I_c·Φ₀ / (2π).
     - 20 – 80 nA (transmon)
     - 5 – 200 nA
     - 200 nA – 2 µA
     - > 10 µA
     - Too high → small L_J → high frequency; too low → large L_J, strong flux
                            noise sensitivity.
   * - Critical Current Density
     - J_c
     - A/m²
     - Critical current per junction area; set by AlOx barrier thickness during
                            deposition.
     - 100 – 500 A/m²
     - 50 – 1000 A/m²
     - 1000 – 5000 A/m²
     - > 10⁴ A/m²
     - Reproducibility of J_c determines frequency spread; EPR sensitivity
                            analysis relates Jc to ω_q.
   * - Junction Capacitance
     - C_J
     - fF
     - Self-capacitance of the junction; contributes to total qubit capacitance
                            C_S.
     - 2 – 10 fF
     - 1 – 20 fF
     - 20 – 100 fF
     - > 200 fF
     - Large C_J reduces charging energy EC, lowering anharmonicity; EPR
                            partitions C_J from shunt.
   * - Junction Area
     - A_J
     - µm²
     - Physical overlap area of the junction; A_J = I_c/J_c. Fabrication
                            controlled.
     - 0.01 – 0.1 µm²
     - 0.005 – 0.5 µm²
     - 0.5 – 2 µm²
     - > 5 µm²
     - Larger area → larger C_J and lower EC; smaller area → harder
                            fabrication, larger variation.
   * - Josephson Energy
     - E_J / h
     - GHz
     - Josephson energy EJ = I_c·Φ₀ / (2π); governs tunneling energy.
     - 10 – 50 GHz
     - 5 – 100 GHz
     - 100 – 500 GHz
     - > 1 THz
     - With EC, determines qubit spectrum; EJ/EC > 50 for transmon regime.
   * - Charging Energy
     - E_C / h
     - MHz
     - Charging energy EC = e²/(2C_S); determines anharmonicity and charge
                            sensitivity.
     - 150 – 350 MHz
     - 100 – 500 MHz
     - 500 MHz – 1 GHz
     - > 2 GHz
     - EC ~ anharmonicity for transmon; high EC → charge qubit regime, high
                            noise sensitivity.

B. Junction Loss & Quality

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

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Junction Loss Tangent
     - tan delta_J
     - dimensionless
     - Intrinsic dielectric loss of AlOx tunnel barrier; limits junction Q and
                            T_1.
     - < 3×10⁻⁶
     - < 1×10⁻⁵
     - 1×10⁻⁵ – 1×10⁻⁴
     - > 1×10⁻³
     - TLS in AlOx barrier is historically the primary T_1 limit; improved by
                            ALD or crystalline barriers.
   * - Junction Subgap Resistance
     - R_sg
     - GΩ
     - Subgap resistance of junction; represents quasiparticle leakage channel.
     - > 100 GΩ
     - 10 – 100 GΩ
     - 1 – 10 GΩ
     - < 100 MΩ
     - Low R_sg indicates excess quasiparticle density; limits T_1 via
                            quasiparticle poisoning.
   * - Flux Noise Spectral Density
     - S_Φ(1Hz)
     - µΦ₀²/Hz
     - Amplitude of 1/f flux noise at 1 Hz; governs dephasing for
                            flux-sensitive qubits.
     - < 1 µΦ₀²/Hz
     - 1 – 5 µΦ₀²/Hz
     - 5 – 20 µΦ₀²/Hz
     - > 50 µΦ₀²/Hz
     - Arises from surface spin fluctuators; EPR current participation at
                            surfaces informs sensitivity.
   * - Charge Noise Spectral Density
     - S_q(1Hz)
     - e²/Hz
     - Amplitude of 1/f charge noise; relevant for charge-sensitive qubits.
     - < 10⁻⁷ e²/Hz
     - < 10⁻⁶ e²/Hz
     - 10⁻⁶ – 10⁻⁵ e²/Hz
     - > 10⁻⁴ e²/Hz
     - Exponentially suppressed in transmon regime; relevant for qubits with
                            EJ/EC < 20.