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Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns

Toshiyuki Fujii, Fumio Hiroshima, Satoshi Tanda

TL;DR

This work rigorously constructs the Josephson junction Hamiltonian first on $\ell^2_{\mathbb{N}}\otimes\ell^2_{\mathbb{N}}$ and then realizes it as $H_{S^1}$ on $\mathcal{H}_{S^1}=L^2(S^1)\otimes L^2(S^1)$, establishing a fiber-wise reduction. On each fiber, the JJ dynamics collapse to a Mathieu operator of the form $\tfrac{2}{C}(-i\partial_\theta)^2-2\alpha\cos\theta$, linking the lattice model to phase-cosine tunneling phenomena; a magnetic phase $\Phi$ induces a current $I_{S^1}(\Phi)$ and, under a constant field, yields the Fraunhofer diffraction pattern via a linear phase gradient across the junction. The analysis also identifies a no-interference regime where no Mathieu operator appears, and proves a spectral decomposition $H_{S^1}=\bigoplus_k M_k$, with the spectrum given by $\sigma(H_{S^1})=\overline{\bigcup_k \sigma(M_k)}$; the Aharonov-Bohm effect is encoded through unitary phase shifts $U(\Phi)$ that relate $H_{S^1}(\Phi)$ to $H_{S^1}(0)$. Together, these results provide a rigorous operator-theoretic bridge between JJ physics, Mathieu operators, and Fraunhofer-type interference, with clear pathways to multi-junction arrays and richer symmetry structures.

Abstract

The Hamiltonian J of the Josephson junction is introduced as a self-adjoint operator on l2 tensor l2. It is shown that J can also be realized as a self-adjoint operator HS1 on L2(S1) tensor L2(S1), from which a Mathieu operator given by "-d^2/dθ^2 - 2α cos θ" is derived. A fiber decomposition of HS1 with respect to the total particle number is established, and the action on each fiber is analyzed. In the presence of a magnetic field, a phase shift defines the magnetic Josephson junction Hamiltonian HS1(Φ) and the Josephson current IS1(Φ). For a constant magnetic field inducing a local phase shift Φ(x), the corresponding local current IS1(Φ(x)) is computed, and it is proved that the Fraunhofer pattern arises naturally.

Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns

TL;DR

This work rigorously constructs the Josephson junction Hamiltonian first on and then realizes it as on , establishing a fiber-wise reduction. On each fiber, the JJ dynamics collapse to a Mathieu operator of the form , linking the lattice model to phase-cosine tunneling phenomena; a magnetic phase induces a current and, under a constant field, yields the Fraunhofer diffraction pattern via a linear phase gradient across the junction. The analysis also identifies a no-interference regime where no Mathieu operator appears, and proves a spectral decomposition , with the spectrum given by ; the Aharonov-Bohm effect is encoded through unitary phase shifts that relate to . Together, these results provide a rigorous operator-theoretic bridge between JJ physics, Mathieu operators, and Fraunhofer-type interference, with clear pathways to multi-junction arrays and richer symmetry structures.

Abstract

The Hamiltonian J of the Josephson junction is introduced as a self-adjoint operator on l2 tensor l2. It is shown that J can also be realized as a self-adjoint operator HS1 on L2(S1) tensor L2(S1), from which a Mathieu operator given by "-d^2/dθ^2 - 2α cos θ" is derived. A fiber decomposition of HS1 with respect to the total particle number is established, and the action on each fiber is analyzed. In the presence of a magnetic field, a phase shift defines the magnetic Josephson junction Hamiltonian HS1(Φ) and the Josephson current IS1(Φ). For a constant magnetic field inducing a local phase shift Φ(x), the corresponding local current IS1(Φ(x)) is computed, and it is proved that the Fraunhofer pattern arises naturally.

Paper Structure

This paper contains 28 sections, 37 theorems, 241 equations, 4 figures.

Table of Contents

  1. Introduction
  2. JJ-Hamiltonian and magnetic JJ-Hamiltonian
  3. JJ-Hamiltonian $H_{\rm JJ}$
  4. Magnetic JJ-Hamiltonian $H_{\rm JJ}(\Phi)$ and Josephson current
  5. JJ-Hamiltonian $H_{\rm JJ}^f$ on $\ell^2_{{\fam\msyfam Z}\times {\fam\msyfam N}}$
  6. Alternative complete orthonormal system $\{\Phi_{n,m}\}_{(n,m)\in{\fam\msyfam Z}\times {\fam\msyfam N}}$ in $\ell^2_{{\fam\msyfam N}\times {\fam\msyfam N}}$
  7. Tunneling Hamiltonian in terms of $\{\Phi_{n,m}\}_{(n,m)\in{\fam\msyfam Z}\times {\fam\msyfam N}}$
  8. JJ-Hamiltonian $H_{\rm JJ}^{f}$ on $\ell^2_{{\fam\msyfam Z}\times {\fam\msyfam N}}$
  9. JJ-Hamiltonian $H_{\rm JJ}^u$ on $\ell^2_{\fam\msyfam Z}\otimes \ell^2_{\fam\msyfam Z}$
  10. Representation on $\ell^2_{\fam\msyfam Z}\otimes\ell^2_{\fam\msyfam N}$
  11. Representation on $(\ell^2_{\fam\msyfam Z}\otimes\ell^2_{\fam\msyfam N})\oplus (\ell^2_{\fam\msyfam Z}\otimes\ell^2_{\fam\msyfam N})$
  12. Representation on $\ell^2_{\fam\msyfam Z}\otimes\ell^2_{\fam\msyfam Z}$
  13. JJ-Hamiltonian $H_{S^1}$ on $\mathcal{H}_{S^1}$
  14. Representation on $\mathcal{H}_{S^1}$
  15. Symmetric JJ-Hamiltonian
  16. ...and 13 more sections

Key Result

Lemma 2.1

It follows that $\sigma( H_C)=\{\frac{1}{2C}(n-q)^2\}_{n\in{\fam\msyfam N}}$ and the multiplicity of each eigenvalue $\frac{1}{2C}(n-q)^2$ is infinity.

Figures (4)

  • Figure 1: ${\fam\msyfam N}\times {\fam\msyfam N} \cong {\fam\msyfam Z}\times {\fam\msyfam N}$ by $f=i_X\circ i$
  • Figure 2: $U=({\rm 1 l}\otimes \kappa)\circ J\circ \tau$, $\mathscr{U}=U\circ(\rho\otimes{\rm 1 l}) \circ u\circ S_f$ and $\mathcal{U}=\mathscr{F} \circ \mathscr{U}$
  • Figure 3: $K_{ij}$: action of $I_{S^1}(\Phi)$ on $p_i\otimes p_j \mathcal{H}_{S^1}$
  • Figure 4: $C(x,y;s,t)$

Theorems & Definitions (43)

  • Lemma 2.1
  • Definition 2.2: JJ-Hamiltonian
  • Definition 2.3: Magnetic JJ-Hamiltonian
  • Lemma 2.4
  • Lemma 2.5: Gauge transformation
  • Example 2.6: Constant magnetic field
  • Definition 2.7: Josephson current
  • Lemma 2.8
  • Lemma 3.1
  • Lemma 4.1
  • ...and 33 more