Internal Josephson Effect Overview

• Internal Josephson effect is the phase-coherent exchange between weakly coupled macroscopic quantum states mediated by differences in internal order parameters.
• It manifests in systems such as two-mode Bose condensates, exciton and electron–hole bilayers, and superconducting weak links, revealing oscillatory dynamics and phase-dependent transport phenomena.
• The phenomenon connects spontaneous symmetry breaking with practical observables like Rabi oscillations, critical tunneling enhancements, and temperature or current oscillations in various quantum platforms.

Searching arXiv for papers on internal Josephson effect and closely related formulations. Internal Josephson effect denotes phase-coherent exchange between weakly coupled macroscopic quantum states when the relevant difference is internal to the order parameter, internal quantum state, or symmetry sector rather than solely a spatially separated superconducting phase drop. In the literature, the term spans several closely related usages: coherent oscillations between internal states of Bose condensates, bright and dark excitonic components, or atomic and molecular condensates; quasiparticle or Noether currents between symmetry-broken magnetic or electron–hole condensates; phase-dependent heat transport in superconducting weak links; and coherent effects mediated by internal weak bonds, bound states, or Majorana modes in superconducting structures (Voronova et al., 2014, Combescot et al., 2012, Schilling et al., 2011, Efimkin et al., 2013, Giazotto et al., 2012, Beekman, 2019). Across these settings, the defining ingredients are a weak coupling between coherent subsystems, a relative phase or symmetry orientation, and an observable current or oscillation governed by the conjugate dynamics of phase and population or charge.

1. General formulation and relation to spontaneous symmetry breaking

A broad theoretical formulation identifies Josephson physics with weak coupling between two systems that each exhibit spontaneous continuous symmetry breaking. For left and right subsystems with order parameters related by a relative angle or phase in the manifold $G/H$, the current is the flow of the Noether charge associated with the broken generator. With weak coupling

$$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$

the generalized Josephson current is expressed as

$$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$

A chemical-potential asymmetry drives the a.c. effect through

$\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$

so the current oscillates at a frequency set by the energy difference between the subsystems (Beekman, 2019).

In internal bosonic Josephson junctions, the same structure appears in a two-mode Bose-Hubbard description,

${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$

where $J$ is the linear coupling between internal levels and $U$ is the nonlinear interaction. In the large-$N$ limit this maps to a 1D effective Schrödinger-like equation in the population-imbalance coordinate $z$, making the internal phase and imbalance explicit dynamical variables (Yuste et al., 2013).

A closely related mean-field formulation for a coupled two-component Bose condensate uses a Gross-Pitaevskii equation for one interacting component and a Schrödinger equation for one non-interacting component. For a homogeneous system, the dynamics reduce to coupled equations for population imbalance $\rho$ and relative phase $$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$0,

$$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$1

with conserved Hamiltonian

$$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$2

This is the internal analog of the classical nonrigid pendulum and the standard Josephson Hamiltonian (Voronova et al., 2014).

2. Internal bosonic Josephson junctions and multicomponent condensates

Internal bosonic Josephson junctions are realized when atoms occupy two coherently coupled internal states. In this setting the experimentally controlled parameter is the linear coupling $$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$3, and the objective can be not only to induce Josephson oscillations but also to steer the condensate adiabatically between many-body ground states. By mapping the large-$$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$4 dynamics to a time-dependent harmonic-oscillator problem, one obtains an Ermakov equation for a scaling factor $$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$5,

$$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$6

with boundary conditions chosen so that the initial and final states match the desired Bose-Hubbard ground states. The resulting shortcut-to-adiabaticity protocol improves the production of spin-squeezing relative to usually employed linear rampings and reaches the final ground state with almost unit fidelity in the numerical examples reported (Yuste et al., 2013).

The two-component condensate model with one interacting and one non-interacting branch exhibits a sequence of dynamical regimes controlled by the effective detuning $$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$7 and the blueshift parameter $$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$8. For $$H_K = -\frac{K}{2}\left(\mathcal{O}^\dagger_\mathrm{L}\mathcal{O}_\mathrm{R} + \mathcal{O}^\dagger_\mathrm{R} \mathcal{O}_\mathrm{L}\right),$$9 and negligible interactions, the imbalance obeys $$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$0, yielding harmonic Rabi oscillations. Nonzero interactions or large initial imbalance produce anharmonic Rabi oscillations. For nonzero detuning, the oscillation frequency becomes

$$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$1

the time-averaged imbalance becomes nonzero, and above a critical detuning the system enters a regime analogous to the internal a.c. Josephson effect, in which the relative phase runs monotonically rather than oscillating about a fixed value. At a specific detuning $$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$2, both population and phase can remain frozen at their initial values, so the internal oscillations are fully suppressed (Voronova et al., 2014).

An even more differentiated internal Josephson dynamics appears in atomic–molecular condensates in a square optical lattice with a staggered gauge field. The Bose-Hubbard model contains species-dependent hopping amplitudes and fluxes together with an atom–molecule conversion term $$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$3. Two parameter regimes support coherent oscillations. In scenario I, both atomic and molecular condensates are zero-momentum condensates. In scenario II, the atomic condensate is at finite momentum $$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$4 and carries a vortex-antivortex structure, जबकि the molecular condensate remains at zero momentum. The internal Josephson oscillation is then between two condensates with qualitatively distinct order parameters, and the corresponding out-of-phase atom–molecule collective mode is a gapped “Leggett” mode whose gap sets the oscillation frequency $$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$5 (Lim et al., 2010).

3. Excitonic and electron–hole realizations

In excitonic systems, the internal Josephson effect can couple different components of the same condensate. A dense exciton condensate is predicted to become “gray,” meaning that a dark condensate acquires a coherent bright component through carrier exchange. In a simplified bright–dark model, the effective mean-field Hamiltonian is

$$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$6

A bright component appears only above the threshold

$$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$7

with

$$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$8

Near equilibrium, the conjugate variables $$\langle \partial_t Q^\mathrm{r}_\mathrm{L} \rangle = -\frac{K}{\hbar} m_\mathrm{L} m_\mathrm{R} \mathbf{m}_0^\dagger T^\mathrm{r} \sin(\alpha T^\mathrm{r})\mathbf{m}_0.$$9 perform Josephson oscillations with

$\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$0

Because only the bright component couples to photons, the predicted observable is an oscillation of the photoluminescence intensity and phase, proposed as a strong proof of the “gray” BCS-like exciton condensate (Combescot et al., 2012).

In topological-insulator thin films with electrons on one surface and holes on the opposite surface, the internal Josephson language is used for tunneling between the two layers when electron–hole pairing is incipient or established. The tunneling Hamiltonian is

$\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$1

and Cooper pair fluctuations are encoded in the vertex

$\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$2

The central result is a critical enhancement of zero-bias tunneling conductivity by Cooper electron–hole pair fluctuations. Near the transition temperature $\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$3,

$\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$4

and near the disorder-driven quantum critical point $\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$5 at $\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$6,

$\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$7

The critical exponents are $\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$8 and $\alpha(t) = \alpha^0 + \frac{2\mu}{\hbar} t,$9. The effect is interpreted as a fluctuational internal Josephson effect and described as a general phenomenon for electron–hole bilayers (Efimkin et al., 2013).

4. Magnetic systems and generalized internal currents

For magnetic insulators with field-induced magnetic-ordering transitions interpreted as Bose-Einstein condensation of magnetic bosonic quasiparticles, the internal Josephson effect takes the form of quasiparticle transfer across a weak magnetic link. Each condensate is represented by

${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$0

and Feynman-type coupled equations yield

${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$1

For identical systems, the formal d.c. Josephson current is constant but unobservable because the macroscopic phase difference is expected to vanish. For different critical fields, a spontaneous a.c. quasiparticle current develops, and the proposed experimental signature is the appearance of sidebands in ESR or Raman spectra separated by ${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$2 (Schilling et al., 2011).

The generalized framework extends the same logic to order parameters beyond ${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$3. For a Heisenberg magnet, the coupling ${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$4 produces a spin Josephson current such as

${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$5

For superfluid helium-3, the order parameter is a ${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$6 matrix, and the current is written in terms of ${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$7. For crystalline solids, the predicted “crystalline Josephson effect” is a force periodic in relative displacement,

${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$8

These cases support the interpretation that internal Josephson phenomena are not restricted to charge transport but encompass any Noether current associated with a broken continuous symmetry (Beekman, 2019).

5. Superconducting structures, internal weak links, and bound-state mediation

In superconducting systems, “internal” can refer either to internal degrees of freedom or to weak links and bound states located inside a material or device architecture. A granular YBaCuO–normal metal point contact provides an example of the latter usage. Electrical measurements on submicron point contacts with direct conductivity reveal a narrow minimum in the differential resistance near zero bias, interpreted as arising from internal weak Josephson bonds in the YBaCuO near the contact. Under a 7.5 GHz microwave field, blurred Shapiro-like oscillations appear in ${\cal H}_{\rm BH}= -2J \hat{J}_x + U \hat{J}_z^2,$9, shifted to finite bias because of substantial series resistance. At higher bias, additional oscillations are attributed to phase-slip-center-type weak bonds formed by significant current injection, and in magnetic fields stronger than the first critical field, flux-quantization effects are observed for separate granules (Rybal'chenko et al., 2017).

A distinct superconducting mechanism is the vortex-state-mediated Josephson effect in an SNS junction supporting vortices. Besides conventional Andreev bound states, the supercurrent includes coherent tunneling of vortex bound states and hybridized vortex–Andreev states. The current is written as

$J$0

For short junctions $J$1, vortex-mediated channels enhance the supercurrent and produce distinct steps in the current–phase relation. When $J$2 becomes comparable to $J$3, the steps evolve into sawtooth oscillations, and for sufficiently long junctions a supercurrent reversal can occur. The paper presents these structures as smoking-gun signatures of vortex bound states in superconductors (Zeng et al., 2014).

Topological superconducting junctions introduce another internal channel structure. In a Fano-Josephson device formed by two topological superconducting wires coupled directly and indirectly via a quantum dot, the low-energy Hamiltonian contains both direct Majorana coupling and dot-mediated coupling,

$J$4

Fano interference then drives Josephson phase transitions among topological-$J$5, $J$6, $J$7, and $J$8 regimes. For the tuned condition $J$9 with $U$0, the fractional Josephson effect disappears and the junction becomes a $U$1-phase normal Josephson junction. If Majorana bound states within the same wire overlap, the current is always $U$2-periodic and robustly in the $U$3 phase (Zhen et al., 2014).

6. Heat transport, observables, and alternative formulations

Internal Josephson terminology also appears in phase-dependent energy transport. In a temperature-biased superconducting interferometer, the total heat current through a DC-SQUID is

$U$4

where $U$5 is the phase-independent quasiparticle contribution and $U$6 is the interference contribution due to the interplay between quasiparticles and the Cooper-pair condensate. Experimentally, the device exhibits magnetic-flux-dependent temperature oscillations of amplitude up to $U$7 mK and a flux-to-temperature transfer coefficient exceeding $U$8 mK$U$9 at 235 mK; the modulation disappears above $N$0 mK (Giazotto et al., 2012). This result is important because it shows that internal Josephson coherence can govern dissipative heat flux, not only nondissipative charge currents.

A different reformulation proposes that Josephson oscillations arise from periodic tunneling of bound electrons described by a two-level Hamiltonian

$N$1

with superconducting charge transfer

$N$2

Within that picture, the condition

$N$3

is presented as a prerequisite for the Josephson effect, persistent currents, thermal equilibrium, and superconductivity, and negative dynamic resistance is attributed to the interplay between normal and superconducting currents (Szeftel et al., 2020). This suggests an alternative microscopic emphasis compared with the order-parameter and broken-symmetry formulations used in the other works.

The experimentally relevant signatures reported across the literature are diverse:

Platform	Internal variable or current	Reported signature
Two-component and internal BJJs	Relative phase and population imbalance	Harmonic or anharmonic Rabi oscillations, running phase, suppression at specific detuning
Exciton and electron–hole condensates	Bright–dark transfer or interlayer tunneling	Photoluminescence oscillations, sharp zero-bias tunneling peak, critical conductivity enhancement
Magnetic and superconducting devices	Quasiparticle, heat, spin, or supercurrent	ESR sidebands, temperature oscillations, Shapiro-like structures, sawtooth current–phase relations

Taken together, these results show that internal Josephson effect is not a single narrowly defined phenomenon but a family of phase-coherent transport and oscillation effects. What remains common is the conversion between conjugate phase and population dynamics, or more generally the flow of a Noether quantity through a weak coupling, whether the observable is charge, spin, heat, displacement, or the occupation of distinct condensate components.