Junction Parameters

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

1. Junction Electrical Parameters

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.

2. Junction Loss & Quality

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.