Merge pull request #218 from angelPappas/JJ_model

Adding Josephson Junction model based on the RCSJ model.

Author: joamatab

docs/bibliography.bib

@@ -138,6 +138,20 @@ @book{m.pozarMicrowaveEngineering2012
   publisher = {John Wiley \& Sons, Inc.},
   isbn = {978-0-470-63155-3}
 }
+@article{McCumber1968,
+  title = {Effect of ac Impedance on dc Voltage-Current Characteristics of Superconductor Weak-Link Junctions},
+  volume = {39},
+  issn = {1089-7550},
+  url = {http://dx.doi.org/10.1063/1.1656743},
+  doi = {10.1063/1.1656743},
+  number = {7},
+  journal = {Journal of Applied Physics},
+  publisher = {AIP Publishing},
+  author = {McCumber,  D. E.},
+  year = {1968},
+  month = jun,
+  pages = {3113–3118}
+}
 @article{norrisImprovedParameterTargeting2024,
   title = {Improved Parameter Targeting in {{3D-integrated}} Superconducting Circuits through a Polymer Spacer Process},
   author = {Norris, Graham J. and Michaud, Laurent and Pahl, David and Kerschbaum, Michael and Eichler, Christopher and Besse, Jean-Claude and Wallraff, Andreas},

qpdk/models/__init__.py

@@ -7,6 +7,7 @@
     capacitor,
     gamma_0_load,
     inductor,
+    josephson_junction,
     open,
     short,
     short_2_port,
@@ -39,6 +40,7 @@
         coupler_straight,
         gamma_0_load,
         inductor,
+        josephson_junction,
         open,
         rectangle,
         quarter_wave_resonator_coupled,

qpdk/models/generic.py

@@ -194,6 +194,57 @@ def inductor(
     return single_impedance_element(z=Zᵢ, z0=z0)
 
 
+@partial(jax.jit, inline=True)
+def josephson_junction(
+    f: ArrayLike = jnp.array([5e9]),
+    ic: float = 1e-6,
+    capacitance: float = 50e-15,
+    resistance: float = 10e3,
+    ib: float = 0.0,
+    z0: int | float | complex = 50,
+) -> sax.SType:
+    r"""Josephson junction (RCSJ) small-signal Sax model.
+
+    Linearized RCSJ model consisting of a bias-dependent Josephson inductance
+    in parallel with a capacitance and resistance.
+
+    Valid in the superconducting (zero-voltage) state and for small AC signals.
+
+    See :cite:`McCumber1968` for details.
+
+    Args:
+        f: Array of frequency points in Hz
+        ic: Critical current I_c in Amperes
+        capacitance: Junction capacitance C in Farads
+        resistance: Shunt resistance R in Ohms
+        ib: DC bias current I_b in Amperes (|ib| < ic)
+        z0: Reference impedance in Ω
+
+    Returns:
+        sax.SType: S-parameters dictionary
+    """
+    # Flux quantum [Wb]
+    PHI0 = 2.067833848e-15
+
+    ω = 2 * jnp.pi * jnp.asarray(f)
+
+    # Bias-dependent phase factor
+    cos_phi0 = jnp.sqrt(1.0 - (ib / ic) ** 2)
+
+    # Josephson inductance
+    LJ = PHI0 / (2 * jnp.pi * ic * cos_phi0)
+
+    # Admittances (parallel RCSJ)
+    Y_R = 1 / resistance
+    Y_C = 1j * ω * capacitance
+    Y_L = 1 / (1j * ω * LJ)
+
+    # Total impedance
+    Z_JJ = 1 / (Y_R + Y_C + Y_L)
+
+    return single_impedance_element(z=Z_JJ, z0=z0)
+
+
 if __name__ == "__main__":
     from matplotlib import pyplot as plt