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 $\varepsilon(t) = A\sin(\Omega t)$.

The system evolves according to the Lindblad master equation,

$$\frac{d\rho}{dt} = -i[H(t), \rho] + \mathcal{D}(\rho),$$

with dissipator

$$\mathcal{D}(\rho) = \frac{\gamma}{N}\left(V\rho V^\dagger - \frac{1}{2}\{V^\dagger V, \rho\}\right), \quad V = (b_1^\dagger + b_2^\dagger)(b_1 - b_2),$$

and $\gamma$ 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 ($H(t+T) = H(t)$) induces a stroboscopic evolution governed by the Floquet superoperator: $$\mathcal{F} = \mathcal{T}\exp\left[\int_0^T dt\, \mathcal{L}(t)\right], \quad \mathcal{L}(\rho) = -i[H(t),\rho] + \mathcal{D}(\rho),$$ so that

$$\rho_{n+1} = \mathcal{F}[\rho_n], \quad \rho_n = \rho(nT).$$

Spectral decomposition yields eigenmodes $\chi_k$ and Floquet multipliers $\mu_k$: $$\mathcal{F}[\chi_k] = \mu_k \chi_k, \quad k=1,\ldots,(N+1)^2.$$ Dissipation enforces $|\mu_k| \leq 1$, with the unique steady state $\mu_1 = 1$. The quantum Neimark–Sacker bifurcation is detected when the subdominant eigenvalue pair $\mu_{2,3}\approx re^{\pm i\theta_0}$ approaches the unit circle ($r \to 1$), with $\theta_0$ 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,

$$|\vartheta,\varphi\rangle = \sum_{j=0}^N \sqrt{\binom{N}{j}} \Bigl[\cos\left(\tfrac{\vartheta}{2}\right)\Bigr]^j \Bigl[e^{i\varphi}\sin\left(\tfrac{\vartheta}{2}\right)\Bigr]^{N-j} |j\rangle,$$

parameterized by $(\vartheta, \varphi)$. The Husimi Q-function,

$$Q(\vartheta, \varphi) = \langle\vartheta, \varphi|\rho|\vartheta, \varphi\rangle,$$

exhibits a unimodal peak for $U < U^*$, centered around the limit-cycle fixed point ("uniREdit"). At $U \approx U^*$, 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): $$i\frac{d}{dt}|\psi\rangle = \tilde{H}(t)|\psi\rangle,\quad \tilde{H}(t) = H(t) - \frac{i}{2}V^\dagger V,$$ with quantum jumps $|\psi\rangle \to V|\psi\rangle/\|V|\psi\rangle\|$ at rates set by $\gamma$. Recording observables $n(mT)=\langle\psi|b_1^\dagger b_1|\psi\rangle$, $e(mT)=\langle\psi|H|\psi\rangle$ at stroboscopic times yields a 2D cloud in $(n,e)$ phase space: unimodal below $U^*$, 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 $(n(mT),e(mT))$, one defines the phase

$$\theta_m = \mathrm{atan2}(e(mT) - \bar{e},\; n(mT) - \bar{n}),$$

where $(\bar{n},\bar{e})$ is the cloud's center. The instantaneous winding number is

$$\omega_m = \frac{\theta_m - \theta_{m-1}}{2\pi} \bmod 1,$$

and the long-term average

$$\omega = \lim_{M\to\infty}\frac{1}{M}\sum_{m=1}^M \omega_m.$$

For $U^*$, typically $\omega\approx0.58$, indicating an irrational rotation (dense coverage). As $U$ increases, $\omega$ shifts to simple rationals (e.g., $3/5$), 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 $N$. In the classical limit ($N\to\infty$), the bifurcation occurs at $U_c\approx0.11$. For finite $N$, the threshold shifts: for $U=0.1125$, the bagel forms only for $N \gtrsim 100$; for $N\lesssim50$, distributions remain unimodal. The bagel's diameter $D$, estimated from the maximum separation in $Q(\vartheta,\varphi)$ at $\varphi = \pi/2$, follows for $U$ near criticality,

$$D \sim (U - U^*_N)^{\alpha},\quad \alpha(N)\to\tfrac{1}{2}\ \text{as}\ N\to\infty.$$

Simultaneously, the closing of the spectral gap $1 - |\mu_{2,3}|$ with increasing $N$ mirrors the classical scenario of Floquet multipliers crossing the unit circle.

7. Physical Mechanism and Quantum-Classical Correspondence

For $U < U^*_N$, dissipation and periodic driving confine the system to a unique stroboscopic limit-cycle ("uniREdit"). Increasing $U$ 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 $(n,e)$ winding around a ring with a well-defined rotation number $\omega$.

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 $N$, govern the sharpness and onset of this transition, with larger $N$ 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 $(n,e)$	Blob around fixed point	Ring (torus)
Rotation number	Zero/undefined	Well-defined, typically irrational
Particle number dependence	No ring for small $N$	Sharper transition at large $N$

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).