Rabi and Josephson Oscillations

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The dynamics of two coupled quantum anharmonic oscillators is a fundamental problem in quantum physics that applies to a number of different physical systems. These include light-matter interaction (polaritons), as well as macroscopic systems defined by an order parameter, such as superconductors, superfluids, Bose–Einstein condensates in traps, and so on. In the latter case, there are three regimes, depending on the ratio of the magnitudes of interactions and coupling , following the classification of Leggett. He describes the problem as "the dynamics of bosons restricted to occupy the same two-dimensional single-particle Hilbert space'' [1]. These regimes are, respectively,


1. The Rabi regime (, weakly interacting oscillators)

2. The Josephson regime ()

3. The Fock regime (, dominated by quantum effects)

The Rabi and Josephson regimes are the ones most often investigated, and their behavior is exhibited in this Demonstration. In the mean-field approximation, their dynamics are determined largely by the population imbalance between the oscillators , with the total number of particles and their relative phase. The trajectories are typically plotted in a phase space, as shown in the diagram on the lower right.

A complete and more insightful representation makes use of the Bloch sphere, on the lower left. We show here the two representations side by side, as functions of the initial conditions and of the population imbalance and relative phase at . We show both: the general solution for arbitrary interaction strength in red and the non-interacting case , the pure Rabi regime, which is represented by a circle on the sphere) in blue. We also take account of detuning between the oscillators [2].

The Demonstration shows how phenomena such as self-trapping or running phases also occur in the pure Rabi regime (no interactions). An unambiguous and general criterion to distinguish the regimes is provided by a stability analysis of the dynamical equations that identifies the Josephson regime by the presence of at least one fixed saddle point. The positions of the fixed points are also shown, as are the zeros of the steady-state equations. More details on this regime, as well as a discussion of the case with finite lifetime and interactions between the modes can be found in [3].


Contributed by: Amir Rahmani and Fabrice P. Laussy (April 2016)
Open content licensed under CC BY-NC-SA



Snapshot 1: A case of self-trapping. Due to very large interactions, , the system remains trapped in one mode (red) rather than Rabi-oscillating between them (blue).

Snapshot 2: In the presence of detuning, the trajectory of the noninteracting system (blue) may resemble that of a self-trapped system. In red, the trajectory on the Bloch sphere in the interacting case (red) is seen to deviate strongly from a circle.

Snapshot 3: Detuning makes the Josephson dynamics less robust. Here, the system is in the Rabi regime, as shown by the stability analysis (three centers for the fixed points).

[1] A. J. Leggett, "Bose-Einstein Condensation in the Alkali Gases: Some Fundamental Concepts," Review of Modern Physics, 73(307), 2001. doi:10.1103/RevModPhys.73.307.

[2] N.  S. Voronova, A. A. Elistratov, and Yu. E. Lozovik, "Detuning-Controlled Internal Oscillations in an Exciton-Polariton Condensate," Physical Review Letters, 115(186402), 2015. doi:10.1103/PhysRevLett.115.186402.

[3] A. Rahmani and F. P. Laussy, "Polaritonic Rabi and Josephson Oscillations," arxiv.org/abs/1603.05971.

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