UniREdit-Bagel: Quantum Transition in Open Systems

• The paper demonstrates that increasing interaction strength induces a quantum Neimark–Sacker bifurcation, evidenced by Floquet spectral analysis and Husimi Q-function deformation.
• The topic is defined by a transition from a unimodal phase-space peak to a toroidal (bagel) structure in driven, dissipative quantum systems, with dynamics governed by U and particle number.
• The analysis highlights practical insights on spectral multipliers, rotation numbers, and quantum trajectory statistics, illustrating clear quantum-classical correspondence in bifurcation behavior.

The UniREdit-Bagel transition refers to a qualitative change in the asymptotic behavior of an open, periodically driven quantum system—a transition from a unimodal ("uniREdit") to a “bagel”-shaped distribution in phase space. This phenomenon, observed in the context of the open quantum dimer model coupled to a Markovian bath, constitutes a quantum analogue of the classical Neimark–Sacker bifurcation, where a stable fixed point loses stability and gives birth to an invariant torus. The transition is characterized through spectral properties of the Floquet map, phase-space analysis via the Husimi Q-function, and the analysis of quantum trajectories and their rotation number. The control parameter for this transition is the interaction strength $U$, with the sharpness of the transition governed by the particle number $N$.

1. The Open Quantum Dimer Model and Lindblad Dynamics

The open quantum dimer model consists of $N$ interacting bosons hopping between two sites ("dimer") with Hamiltonian

$$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$

where $b_j$, $b_j^\dagger$ are bosonic annihilation/creation operators at site $j$, $n_j = b_j^\dagger b_j$, $J$ is set to unity (energy unit), $U$ is scaled interaction strength, and the periodic drive $N$0.

The system evolves according to the Lindblad master equation,

$N$1

with dissipator

$N$2

and $N$3 the dissipation rate. The dissipative term recycles antisymmetric modes back to the symmetric manifold, effecting phase synchronization.

2. Floquet Map, Spectral Analysis, and Bifurcation Signature

Periodicity ($N$4) induces a stroboscopic evolution governed by the Floquet superoperator: $N$5 so that

$N$6

Spectral decomposition yields eigenmodes $N$7 and Floquet multipliers $N$8: $N$9 Dissipation enforces $N$0, with the unique steady state $N$1. The quantum Neimark–Sacker bifurcation is detected when the subdominant eigenvalue pair $N$2 approaches the unit circle ($N$3), with $N$4 tied to the emergent rotation number.

3. Phase-Space Structure: Husimi Q-Function and Ring (Bagel) Formation

Phase-space analysis leverages SU(2) coherent states,

$N$5

parameterized by $N$6. The Husimi Q-function,

$N$7

exhibits a unimodal peak for $N$8, centered around the limit-cycle fixed point ("uniREdit"). At $N$9, this peak destabilizes and deforms into a closed ring ("bagel"), the quantum analog of a toroidal attractor seen in the classical mean-field Poincaré section.

4. Quantum Trajectories and Emergence of the Quantum Torus

Quantum state evolution can also be interrogated via the Monte Carlo wave-function method (quantum jumps): $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$0 with quantum jumps $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$1 at rates set by $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$2. Recording observables $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$3, $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$4 at stroboscopic times yields a 2D cloud in $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$5 phase space: unimodal below $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$6, but ring-shaped above, confirming the formation of a quantum torus.

5. Rotation Number and Classification of Motion on the Torus

The ring structure supports a well-defined phase dynamics. For points $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$7, one defines the phase

$$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$8

where $$H(t) = -J(b_1^\dagger b_2 + b_2^\dagger b_1) + \frac{2U}{N}\sum_{g=1}^2 n_g(n_g-1) + \varepsilon(t)(n_2 - n_1),$$9 is the cloud's center. The instantaneous winding number is

$b_j$0

and the long-term average

$b_j$1

For $b_j$2, typically $b_j$3, indicating an irrational rotation (dense coverage). As $b_j$4 increases, $b_j$5 shifts to simple rationals (e.g., $b_j$6), leading to multimodal stroboscopic structure.

6. Particle Number as Control Parameter and Transition Characteristics

The quantum UniREdit–Bagel transition is strongly dependent on particle number $b_j$7. In the classical limit ($b_j$8), the bifurcation occurs at $b_j$9. For finite $b_j^\dagger$0, the threshold shifts: for $b_j^\dagger$1, the bagel forms only for $b_j^\dagger$2; for $b_j^\dagger$3, distributions remain unimodal. The bagel's diameter $b_j^\dagger$4, estimated from the maximum separation in $b_j^\dagger$5 at $b_j^\dagger$6, follows for $b_j^\dagger$7 near criticality,

$b_j^\dagger$8

Simultaneously, the closing of the spectral gap $b_j^\dagger$9 with increasing $j$0 mirrors the classical scenario of Floquet multipliers crossing the unit circle.

7. Physical Mechanism and Quantum-Classical Correspondence

For $j$1, dissipation and periodic driving confine the system to a unique stroboscopic limit-cycle ("uniREdit"). Increasing $j$2 leads to loss of stability of this limit-cycle via a Neimark–Sacker bifurcation, resulting in an invariant torus in the classical mean-field model. In the quantum regime, this manifests as:

• Transformation of the Husimi Q-function from a single peak to a toroidal ring,
• The appearance of a complex-conjugate Floquet eigenvalue pair with magnitude approaching unity,
• Quantum trajectories in $j$3 winding around a ring with a well-defined rotation number $j$4.

The "bagel" is thus the quantum analog of the classical invariant curve, characterized by Q-function topology, Floquet spectral features, and trajectory statistics. Quantum fluctuations, controlled by $j$5, govern the sharpness and onset of this transition, with larger $j$6 driving more pronounced classical behavior.

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Summary Table: Key Features of the UniREdit–Bagel Transition

Feature	Unimodal (“uniREdit”)	Bagel (Quantum Torus)
Husimi Q-function	Single peak	Toroidal ring
Floquet spectrum	Real subdominant eigenvalues	Complex-conjugate multipliers
Trajectory cloud in $j$7	Blob around fixed point	Ring (torus)
Rotation number	Zero/undefined	Well-defined, typically irrational
Particle number dependence	No ring for small $j$8	Sharper transition at large $j$9

The UniREdit–Bagel transition provides a paradigmatic example of how dissipative, periodically driven quantum systems realize quantum analogs of classical bifurcations, with clear spectral, phase-space, and trajectory-based signatures (Yusipov et al., 2019).