Statistical control of relaxation and synchronization in open anyonic systems

Abstract

Quantum statistics dictate how particles exchange and correlate—but in two-dimensional systems, these rules extend beyond bosons and fermions to anyons, quasiparticles with continuously tunable exchange phases. Here, we develop a Lindblad framework for anyonic oscillators and show that fractional statistics enable statistical control of decoherence in open quantum systems. By varying the anyonic phase and environmental correlations, we demonstrate tunable mode protection, identify exceptional points in the dissipative spectrum, and reveal temperature-dependent coherence bifurcations. We also demonstrate that signatures of the statistical phase should also be manifest in 2D coherent spectroscopic probes of these systems. These results establish the exchange phase as a functional control parameter for engineering dissipation-resilient quantum states.

Introduction

Anyons are quasiparticles in two-dimensional systems that obey fractional exchange statistics, interpolating continuously between bosonic and fermionic limits. These nontrivial exchange phases introduce unique dynamical signatures, particularly in open quantum systems where environmental coupling influences coherence and relaxation. Fractional quantization permits particles in two dimensions to exhibit statistics beyond the symmetric (bosonic) and antisymmetric (fermionic) wave- function classes allowed in three dimensions. The theoretical groundwork was established by Leinaas and Myrheim, who analyzed quantum mechanics in multiply connected configuration spaces¹. Wilczek later introduced a tangible realization in which charged particles bound to magnetic flux tubes acquire a phase upon exchange, providing the first physical model for anyons^2,3.

Goldin, Menikoff, and Sharp formalized this picture algebraically by classifying inequivalent representations of the local current algebra in non-simply connected spaces⁴. These developments laid the foundation for understanding topological phases of matter, particularly in the context of the fractional quantum Hall effect (FQHE), where quasiparticles exhibit fractional charge and anyonic statistics⁵.

Experimental signatures of anyonic behavior have been observed in FQHE systems through tunneling conductance, shot noise, and interferometric measurements. Fractional charge values such as e/3 and exchange-phase-dependent interference fringes provide compelling evidence for anyon statistics^6,7,8. Other platforms such as Kitaev’s toric code and topological superconductors also support Abelian and non-Abelian anyons, with recent focus on Majorana zero modes as a route to fault-tolerant topological quantum computation^9,10,11.

More recently, attention has shifted toward understanding how anyonic systems behave in open-system settings, where decoherence and dissipation play a central role. Recent studies have shown that dissipation can drive topological phase transitions, as in spin liquid systems undergoing anyon condensation¹². In Kitaev-type chains with boundary dissipation, anyon excitations display statistical phase-dependent asymmetries in relaxation and propagation¹³. These findings motivate a detailed investigation into how exchange statistics and environmental correlations jointly shape the coherence properties of open quantum systems.

Recent experiments have realized Abelian anyons in one-dimensional optical lattices with continuously tunable exchange statistics, using density-dependent tunneling in cold atomic systems¹⁴. This breakthrough establishes a new experimental platform for simulating and probing fractional statistics in highly controlled settings and motivates theoretical efforts to characterize coherence, dissipation, and statistical control in such systems.

We recently investigated noise-induced synchronization in open quantum systems consisting of two coupled quantum oscillators interacting with a common, correlated dissipative environment^15,16,17 where we assume that the environmental noise channels $\textbf{E}_1(t)$ and $\textbf{E}_2(t)$ evolve according to a coupled Ornstein–Uhlenbeck (OU) process written in differential form:

$$\begin{aligned} {\textrm{d}}{\textbf{E}}(t) = -\gamma \textbf{E}(t){\textrm{d}}{t} + \textbf{B} \cdot {\textrm{d}}{\textbf{W}}(t), \end{aligned}$$

where $\textbf{E}(t) = \begin{pmatrix} E_1(t) \\ E_2(t) \end{pmatrix}$, and ${\textrm{d}}{\textbf{W}}(t)$ is a vector-valued Wiener process with correlation structure:

$$\begin{aligned} \mathbb {E}\left[ {\textrm{d}}{\textbf{W}}(t){\textrm{d}}{\textbf{W}}^\intercal (t') \right] = \varvec{\Xi } \delta (t - t') {\textrm{d}}{t}, \quad \varvec{\Xi } = \begin{pmatrix} 1 & \xi \\ \xi & 1 \end{pmatrix}. \nonumber \end{aligned}$$

Here, the parameter $\xi \in [-1, 1]$ controls the degree of instantaneous correlation between the two noise channels. The matrix $\textbf{B}$ determines the noise amplitude and, together with $\varvec{\Xi }$, sets the stationary variance of the process. In the limit of perfect environmental correlation ($\xi = \pm 1$), the oscillators exhibit long-lived phase synchronization, particularly when they are spectrally near-resonant. The presence of nonzero quantum discord between subsystems indicated that correlated environmental noise can generate persistent quantum coherences even in the absence of direct coupling. We further showed that two spins can synchronize their relative phases through shared stochastic driving, akin to the escapement mechanism that regulates motion in classical pendulum clocks.

In this study, we extend these concepts to anyonic systems and examine whether synchronization and phase-locking behavior persist when particles obey fractional statistics. We begin by developing an algebraic and dissipative framework for anyon oscillators subject to correlated noise. By generalizing the bosonic and fermionic algebras, we derive a Lindblad description that incorporates the anyonic exchange phase $\theta$ as a continuous control parameter. We analyze the thermal and dynamical properties of such systems, demonstrating that fractional statistics induce a continuous deformation in the dissipative structure. This deformation leads to tunable coherence bifurcations, exceptional points, and phase-sensitive mode protection. We conclude by showing how manipulation of the statistical phase and environmental correlations can be used to engineer dissipation-resilient quantum states, a key requirement for robust quantum information processing.

Anyonic oscillator dynamics and dissipation

We begin by developing a Lindblad framework for anyonic oscillators with fractional-statistics, where the algebra of creation and annihilation operators is deformed by an exchange phase $\theta$ that interpolates continuously between bosonic and fermionic limits. For a single anyon oscillator, the operators $a^\dagger$ and a satisfy the q-deformed commutation relation:

$$a a^\dagger - e^{i\theta } a^\dagger a = 1.$$

This deformation encodes the statistical phase acquired under exchange, with $\theta = 0$ and $\theta = \pi$ recovering the bosonic and fermionic cases, respectively. In a system of two anyon oscillators, the braided exchange relation becomes $a_1 a_2 = e^{i\theta } a_2 a_1$, ensuring that the many-body wavefunction acquires a phase $e^{i\theta }$ when particles are exchanged.

Abelian anyons, whose braiding statistics depend only on the number (but not order) of exchanges, can be treated algebraically in a relatively straightforward manner. Non-Abelian anyons, on the other hand, possess far richer and more complex exchange statistics. When two such particles are braided, the quantum state of the system evolves through a unitary transformation within a degenerate Hilbert space, meaning the outcome of braiding depends on the specific sequence of operations. This non-commutative behavior makes them especially promising for topological quantum computing, where information can be encoded and manipulated in a fault-tolerant way by controlling the braiding paths. Here, we focus on the open-system dynamics of Abelian anyons, reserving the non-Abelian case for future work.

Statistical algebra and deformed commutators

In our approach, the anyonic creation and annihilation operators obey a deformed commutation relation of the form

$${[}a, a^\dagger ] = \Phi (\hat{N}) = 1 + \left( e^{-i\theta } - 1\right) \hat{N},$$

where $\hat{N} = a^\dagger a$ is the number operator and $\theta$ is the statistical phase that interpolates between bosonic ($\theta = 0$) and fermionic ($\theta = \pi$) statistics. This form ensures that multiple occupation of a single mode is suppressed as $\theta \rightarrow \pi$, implementing a generalized Pauli exclusion principle. It is consistent with an underlying exchange algebra in which the wavefunction acquires a phase $e^{i\theta }$ upon particle exchange. The number-dependent commutator $\Phi (\hat{N})$ reflects the statistical interactions inherent to anyons, and captures the essential physics of exchange deformation at the operator level.

In the statistical evaluation of thermal expectation values, we adopt a related, but distinct, formulation where the ensemble average of the deformation function appears as $\langle \Phi (\hat{N})\rangle = \langle e^{i \theta \hat{N}}\rangle$, which naturally arises in the context of partition function evaluation and statistical weights for anyonic oscillators. This distinction between algebraic and statistical forms is discussed in various contexts throughout the anyon literature. The foundational basis for these generalized statistics was established by Leinaas and Myrheim¹, who demonstrated that particles in two spatial dimensions may exhibit intermediate exchange statistics when configuration space is properly quantized. Greenberg and Wigner further expanded the landscape of quantum statistics by formulating algebraic structures that interpolate between Bose and Fermi operators¹⁸, laying the groundwork for the operator-based approach we adopt here.

Because $\Phi (\hat{N})$ is operator-valued, it does not in general commute with a or $a^\dagger$, and care must be taken when evaluating expressions such as $[a^\dagger , a] a$ or $\Phi (\hat{N} + 1) a$ in the context of the Lindblad equation. When computing expectation values, we must retain the operator ordering until thermal averaging is performed.

Approximate unitary dynamics

In the absence of dissipation, the system evolves under the Hamiltonian $H = \hbar \omega \hat{N}$. For standard bosonic operators, the Heisenberg equation of motion yields $[H, a] = -\hbar \omega a$. For anyons, however, this result is not exact due to the operator-valued commutator structure. Formally,

$$\begin{aligned} {[}H, a] = \hbar \omega [a^\dagger a, a] = \hbar \omega (a^\dagger [a, a] + [a^\dagger , a] a). \end{aligned}$$

(1)

Since $[a, a] = 0$ and $[a^\dagger , a] = -\Phi (\hat{N} + 1)$, we find

$$\begin{aligned} {[}H, a] = -\hbar \omega \Phi (\hat{N} + 1) a. \end{aligned}$$

(2)

To simplify the resulting expressions, we adopt the approximation

$$\begin{aligned} {[}H, a] \approx -\hbar \omega a, \end{aligned}$$

(3)

which is exact in the bosonic limit and reasonable in the regime of weak deformation (small $\theta$) or low occupation ($\langle \hat{N} \rangle \ll 1$). This approximation allows us to isolate the effects of statistical deformation in the dissipative dynamics, where they appear most prominently. A full derivation using the deformed algebra $[a, a^\dagger ] = \Phi (\hat{N})$ shows that the Heisenberg equation acquires a number-dependent prefactor $\Phi (\hat{N})$, which introduces nonlinear corrections to the unitary dynamics. These corrections are neglected here to focus on the dissipative effects of fractional statistics. A detailed justification of this approximation is given in Section 1.3 of the Supporting Information, where we derive the exact Heisenberg equation of motion and demonstrate the conditions under which the approximation holds.

Lindblad equation and coherence decay

We now consider a single anyonic oscillator weakly coupled to a thermal bosonic bath. The system evolves according to a standard Lindblad master equation:

$$\begin{aligned} \frac{d\rho }{dt} = -\frac{i}{\hbar }[H,\rho ] + \sum _k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right) , \end{aligned}$$

(4)

where the jump operators for emission and absorption are given by:

$$\begin{aligned} L_{\textrm{em}}&= \sqrt{\gamma (n_\theta + 1)}\, a, \end{aligned}$$

(5)

$$\begin{aligned} L_{\textrm{abs}}&= \sqrt{\gamma n_\theta }\, a^\dagger . \end{aligned}$$

(6)

The occupation number $n_\theta$ captures the statistical weighting of excited states in thermal equilibrium and reduces to the Bose–Einstein distribution for $\theta = 0$. The generalized occupation number for anyons with statistical angle $\theta$ is given by

$$\begin{aligned} n_\theta&= \text {Re} \left[ \frac{1}{e^{\beta \omega } - e^{i\theta }}.\right] \end{aligned}$$

(7)

$$\begin{aligned}&= \frac{z}{1 - 2z\cos \theta + z^2}, \end{aligned}$$

(8)

where $z = e^{-\beta \hbar \omega }$. This expression highlights the statistical suppression that occurs near $\theta = \pi$, where the denominator increases due to destructive interference in the statistical phase factor. This expression smoothly interpolates between bosonic and fermionic limits and reduces to the standard Bose-Einstein distribution $n = z / (1 - z)$ as $\theta \rightarrow 0$. The use of the real part ensures that $n_\theta$ remains a physically meaningful (real-valued) occupation number for all $\theta \in [0, \pi ]$. To compute the time evolution of $\langle a \rangle$, we use the adjoint Lindblad equation:

$$\begin{aligned} \frac{d}{dt}\langle a \rangle = -i\omega \langle a \rangle + \langle \mathcal {D}^\dagger [a] \rangle , \end{aligned}$$

(9)

where the adjoint dissipator is given by:

$$\begin{aligned} \mathcal {D}^\dagger [a] = \frac{1}{2}\sum _k ([L_k^\dagger , a] L_k + L_k^\dagger [a, L_k]). \end{aligned}$$

(10)

Evaluating this for each jump operator yields:

$$\begin{aligned} \mathcal {D}^\dagger [a]&= \frac{\gamma (n_\theta + 1)}{2} [a^\dagger , a] a + \frac{\gamma n_\theta }{2} [a, a^\dagger ] a \end{aligned}$$

(11)

$$\begin{aligned}&= -\frac{\gamma (n_\theta + 1)}{2} \Phi (\hat{N} + 1) a + \frac{\gamma n_\theta }{2} \Phi (\hat{N}) a. \end{aligned}$$

(12)

Thus, the full equation of motion becomes:

$$\begin{aligned} \frac{d}{dt} \langle a \rangle = -i\omega \langle a \rangle - \Gamma _\theta \langle a \rangle , \end{aligned}$$

(13)

where

$$\begin{aligned} \Gamma _\theta = \frac{\gamma }{2}\left[ (n_\theta + 1) \langle \Phi (\hat{N} + 1) \rangle - n_\theta \langle \Phi (\hat{N}) \rangle \right] . \end{aligned}$$

(14)

The key subtlety lies in treating $\Phi (\hat{N})$ and $\Phi (\hat{N}+1)$ as operator-valued quantities that do not commute with a. These expressions must be inserted into thermal expectation values with proper operator ordering. The average values $\langle \Phi (\hat{N}) \rangle$ and $\langle \Phi (\hat{N}+1) \rangle$ can be evaluated analytically using the partition function for anyons:

$$\begin{aligned} \langle \Phi (\hat{N}) \rangle&= \text {Re}\left( \sum _{n=0}^{\infty } e^{-\beta \hbar \omega n} e^{i\theta n} \right) \nonumber \\&= \frac{1 - z \cos \theta }{1 - z}, \end{aligned}$$

(15)

$$\begin{aligned} \langle \Phi (\hat{N} + 1) \rangle&= \text {Re}\left( \sum _{n=0}^{\infty } e^{-\beta \hbar \omega n} e^{i\theta (n+1)} \right) \nonumber \\&= \frac{1 - z \cos \theta }{1 - 2z \cos \theta + z^2}, \end{aligned}$$

(16)

where $z = e^{-\beta \hbar \omega }$. This yields a fully analytic expression for the coherence decay rate $\Gamma _\theta$, which recovers the expected limits:

• For $\theta = 0$ (bosons): $\Gamma _\theta = \gamma /2$ (temperature-independent)

• For $\theta = \pi$ (fermions): $\Gamma _\theta \rightarrow \gamma (n + 1/2)$ (Pauli blocking)

Fig. 1

Statistical coherence decay rate $\Gamma _\theta$ for an anyonic oscillator coupled to a thermal bosonic bath, plotted as a function of exchange angle $\theta$ for several values of $kT/\hbar \omega$ (indicated on the curves). For $\theta = 0$, the rate is constant ($\gamma /2$) and independent of temperature, as expected for bosons. For finite $\theta$, the decay rate acquires strong temperature dependence due to the operator-valued commutators and statistical occupation numbers. In the fermionic limit $\theta \rightarrow \pi$, Pauli blocking suppresses the relaxation rate. The non-monotonic behavior at intermediate $\theta$ reflects the nontrivial interplay between deformation statistics and thermal fluctuations.

The behavior of $\Gamma _\theta$ as a function of $\theta$ is illustrated in Fig. 1. For small $\theta$, the decay rate increases with temperature, while for larger values, it exhibits a turnover as the statistical suppression of accessible states outweighs the thermal population. This competition leads to an optimal intermediate $\theta$ at which coherence decay is maximized, before falling off in the fermionic regime.

Coupled anyon oscillators and exceptional points

To further generalize the analysis, consider the case of two anyon modes $a_1$ and $a_2$, coupled by an exchange term J, with Hamiltonian

$$\begin{aligned} H/\hbar&= \omega ( a_{1}^{\dagger } a_{1} + a_{2}^{\dagger }a_{2}) + J(a_{1}^{\dagger }a_{2} + a_{2}^{\dagger }a_{1}). \end{aligned}$$

(17)

We can bring H into diagonal form by defining new anyon operators: ${\tilde{b}}_\pm = (a_1 \pm e^{\pm i\theta } a_2)/\sqrt{2}$ such that their commutation relation preserves

$$\begin{aligned} {[}{\tilde{b}}_\pm ,{\tilde{b}}_\pm ^\dagger ] \approx (e^{i\theta N_1} + e^{i\theta N_2})/2 \end{aligned}$$

(18)

reflecting the non-local averaging over the anyonic occupation numbers and ${\hat{N}}_1$ and ${\hat{N}}_2$ are the local occupations. These bring the Hamiltonian into diagonal form

$$\begin{aligned} {\tilde{H}}/\hbar =\sum _{s=\pm } \omega _s {\tilde{b}}_s^\dagger {\tilde{b}}_s, \end{aligned}$$

(19)

with normal mode frequencies

$$\begin{aligned} \omega _\pm = \omega \pm J \cos (\theta ). \end{aligned}$$

(20)

The notion of parity-time (PT) symmetry plays an important role in the analysis of dissipative quantum systems. In the bare oscillator basis, the parity operator $\mathcal {P}$ acts to exchange the modes $a_1 \leftrightarrow a_2$, while the time-reversal operator $\mathcal {T}$ performs complex conjugation. The combined PT operator leaves the Hamiltonian invariant when $\theta = 0$, and more generally preserves the structure of the non-Hermitian evolution when the system remains balanced. Importantly, the normal mode operators ${\tilde{b}}_\pm$ are eigenstates of the parity operator:

$$\begin{aligned} \mathcal {P}\, {\tilde{b}}_\pm = \pm {\tilde{b}}_\pm , \end{aligned}$$

(21)

meaning they correspond to symmetric and antisymmetric combinations of the local anyon modes. In the absence of dissipation, the evolution of these modes remains decoupled.

Following along the lines of our previous works^15,16,17, we define Lindblad jump operators in terms of the deformed normal mode operators as

$$\begin{aligned} L_k = \lambda _k^{(+)} {\tilde{b}}_{+} + \lambda _k^{(-)} {\tilde{b}}_{-} \end{aligned}$$

(22)

with explicit coefficients $\lambda _k^{(\pm )}$ for the emission and absorption channels. (A derivation of these equations is presented in the Supplementary Information).

Using these, we derive the equations of motion for the deformed anyonic operators which we write in matrix form as

$$\begin{aligned} \frac{d}{dt} \begin{pmatrix} \langle \tilde{b}_{+} \rangle \\ \langle \tilde{b}_{-} \rangle \end{pmatrix} = W_{\textrm{eff}} \cdot \begin{pmatrix} \langle \tilde{b}_{+} \rangle \\ \langle \tilde{b}_{-} \rangle \end{pmatrix}, \end{aligned}$$

(23)

in which $W_{\textrm{eff}}$ is given by

$$\begin{aligned} W_{\textrm{eff}} = \begin{pmatrix} -i\omega _{+} - \Gamma _\theta & -\xi \Gamma _\theta \\ -\xi \Gamma _{\theta } & -i\omega _{-} - \Gamma _\theta \end{pmatrix}, \end{aligned}$$

(24)

where $\Gamma _\theta$ is the statistical damping rate derived in Eq. (14). Here, the diagonal terms correspond to the normal dissipative dynamics of the mode, as modified by the statistical contribution. However, the off-diagonal terms explicitly break parity symmetry by coupling states of opposite parity. When the correlation strength $\xi$ exceeds a critical threshold, the non-Hermitian structure of $W_{\textrm{eff}}$ leads to coalescing eigenvalues and $\mathcal{P}\mathcal{T}$ symmetry breaking.

Diagonalization of $W_{\textrm{eff}}$ yields eigenvalues:

$$\begin{aligned} \lambda _\pm = -i\omega - \Gamma _{\theta } \pm \sqrt{\left( \frac{\delta }{2}\right) ^2 - \xi ^{2} {\Gamma _{\theta }}^2}, \end{aligned}$$

(25)

where $\delta = \omega _{+} - \omega _{-} = 2J \cos \theta$ is the splitting between the normal modes.

The onset of symmetry breaking and exceptional points (EPs) occurs when the square root becomes degenerate and non-Hermitian—that is, when the eigenvalues coalesce. This defines the critical correlation strength:

$$\begin{aligned} \xi _{\text {crit}} = \frac{|\delta |}{2\Gamma _\theta } = \frac{|J \cos \theta |}{\Gamma _\theta }. \end{aligned}$$

(26)

To visualize the onset of PT symmetry breaking, we plot the eigenvalues of $W_{\textrm{eff}}$ in Fig. 2 as a function of the correlation parameter $\xi$ for several values of the statistical angle $\theta$. In the unbroken PT phase, the eigenvalues remain complex-conjugate pairs with degenerate real parts and distinct imaginary components. As $\xi$ increases, the real parts bifurcate and the imaginary parts coalesce, signaling a spontaneous breaking of PT symmetry. For bosonic systems ($\theta = 0$), the exceptional point occurs at the analytical threshold $\xi = \Delta \omega / (2\Gamma _\theta )$. Increasing $\theta$ suppresses $\Gamma _\theta$ and therefore raises the critical value of $\xi$, delaying the onset of symmetry breaking. In the fermionic limit ($\theta \rightarrow \pi$), the statistical suppression of coherence becomes complete ($\Gamma _\theta \rightarrow 0$), and no exceptional point appears within the physical domain $\xi \le 1$.

The PT symmetry-breaking threshold depends on the exchange interaction J, the statistical phase $\theta$, and the coherence decay rate $\Gamma _\theta$. Notably, this definition of $\xi _{\text {crit}}$ aligns with the expression derived in our earlier work on bosonic systems in Ref.¹⁷, thereby generalizing the concept of exceptional points to fractional-statistics systems. For $|\xi |< \xi _{\text {crit}}$, both modes decay at the same rate given by $\Gamma _\theta$ but at oscillate at different frequencies. For values of $|\xi | > \xi _{\text {crit}}$, the two modes oscillate the same frequency ($\omega$); however, one mode will become overdamped and one mode will become undamped—effectively decoupled from the environment. Consequently, the presence of a correlated noise leads to spontaneous synchronization (or anti-synchronization) of the two oscillators in a direct analogy of Huygens’ double pendulum clock^16,17,19.

Fig. 2

Bifurcation of eigenvalues under correlated dissipation. Real and imaginary parts of the eigenvalues of the effective evolution operator $W_{\textrm{eff}}$ as a function of correlation strength $\xi$ for several values of the statistical angle $\theta$. Parameters are set to $J = 0.1$, $\gamma = 1$, and $kT/\hbar \omega = 1$. For small $\theta$, the bifurcation occurs near $\xi \approx 0.1$, consistent with the bosonic limit. As $\theta$ increases, the onset of PT symmetry breaking is delayed, requiring stronger correlations to induce the transition. In the fermionic limit ($\theta \rightarrow \pi$), relaxation is suppressed due to Pauli blocking and the exceptional point disappears.

The critical value $\xi _{\textrm{crit}}$ defined in Eq. (23) depends both on the frequency splitting $\Delta \omega$ and the statistical coherence damping $\Gamma _\theta$. Figure 3 shows how this critical threshold varies with detuning and interaction strength. In the bosonic limit, our result reduces to that derived analytically in Ref.¹⁷, as expected. However, for anyonic systems, the dependence on the statistical phase $\theta$ enters implicitly through $\Gamma _\theta$, which modifies the curvature of the phase boundary. The figure illustrates how both microscopic coupling (via J) and statistical interactions conspire to shape the onset of PT symmetry breaking.

Fig. 3

Critical correlation strength $\xi _{\textrm{crit}}$ as a function of detuning $\Delta \omega$ and exchange coupling J. Phase diagram showing the threshold $\xi _{\textrm{crit}} = \Delta \omega / (2\Gamma _\theta )$ for the onset of PT symmetry breaking as a function of oscillator detuning and interaction strength. The coherence decay rate $\Gamma _\theta$ depends on the statistical phase and temperature, thereby modulating the location of the exceptional point. For bosonic systems, this expression recovers the analytical result derived in Ref.¹⁷; for fractional statistics, the phase-dependent $\Gamma _\theta$ reshapes the threshold contour.

Figure 4 presents the PT symmetry-breaking phase diagram in the $(\theta , \xi )$ plane for several fixed temperatures. Each isotherm marks the critical boundary above which the eigenvalues of $W_{\textrm{eff}}$ bifurcate and the system enters the PT-broken regime. At low temperatures ($kT \ll \hbar \omega$), the system is highly sensitive to correlations: even weak coupling $\xi$ can induce symmetry breaking near the bosonic limit ($\theta \approx 0$). As temperature increases, thermal fluctuations suppress coherent dynamics, requiring stronger correlation to overcome statistical damping. Interestingly, the effect saturates at high temperature ($kT \gg \hbar \omega$), reflecting the asymptotic behavior of $\Gamma _\theta$. This structure mirrors the behavior of $\langle \Phi (N) \rangle$ and illustrates how thermal and statistical effects jointly control the transition into the non-Hermitian PT-broken regime.

Fig. 4

PT phase diagram: critical isotherms in the $(\theta , \xi )$ plane. Each curve corresponds to a critical boundary at fixed temperature $kT/\hbar \omega$, above which the system enters the PT-broken regime. At low temperatures, any amount of correlation ($\xi > 0$) leads to PT symmetry breaking, especially near the bosonic limit ($\theta = 0$). As temperature increases, stronger correlations are needed to overcome thermal noise and induce the transition. For $kT \gg \hbar \omega$, the statistical deformation saturates and $\Gamma _\theta$ approaches its fermionic asymptote, suppressing the influence of $\xi$.

In summary, the two-oscillator anyonic system exhibits rich non-Hermitian dynamics governed by the interplay of fractional statistics, coherent coupling, and noise correlation. By transforming to a normal-mode basis and analyzing the effective evolution matrix $W_{\textrm{eff}}$, we have shown that correlated dissipation mixes parity eigenstates and leads to PT symmetry breaking above a critical correlation strength $\xi _{\textrm{crit}}$. This threshold depends sensitively on the statistical phase $\theta$ and temperature through the coherence decay rate $\Gamma _\theta$, which encodes both thermal and statistical suppression. Our analytical results for $\xi _{\textrm{crit}}$ generalize known expressions for bosonic systems and reveal a tunable route to exceptional-point behavior in fractional-statistics systems. The resulting phase diagrams underscore the potential for using coherent spectroscopy to probe topological and statistical signatures in open quantum matter.

Detection via non-linear spectroscopy

Having established the dynamical framework for coupled anyon oscillators and characterized the onset of PT symmetry breaking due to correlated dissipation, we now examine how these effects manifest in spectroscopic observables. In particular, two-dimensional (2D) coherent spectroscopy, and specifically single-quantum coherence (SQC) response functions, offer a sensitive window into the interplay between fractional statistics, coherence decay, and noise-induced mode mixing. As demonstrated in prior work¹⁶, such spectroscopic methods are uniquely capable of resolving phase-sensitive signatures that would be obscured in linear response. Moreover, recent studies by the present author and collaborators¹⁷ have shown how topological and statistical phases can distort the Kramers–Kronig relations in nonlinear response functions, providing a direct probe of geometric and symmetry-breaking effects in open quantum systems. In what follows, we compute and analyze the 2D SQC spectra for the coupled anyon model introduced above, elucidating how the phase $\theta$, the correlation parameter $\xi$, and the effective coherence damping $\Gamma _\theta$ reshape the spectral features.

Coherent multidimensional spectroscopy offers a powerful means to resolve the correlations and dynamics of anyonic systems, where exchange statistics deviate from standard bosonic or fermionic behavior. Unlike linear techniques, which conflate homogeneous and inhomogeneous broadening, coherent probes using phase-locked pulse sequences can isolate intrinsic dephasing and reveal pathways that may be sensitive to fractional statistics.^20,21,22

Recent theoretical studies have established nonlinear spectroscopy as a powerful tool for detecting the signatures of anyonic statistics in topologically ordered systems. In particular, pump–probe protocols have been shown to reveal braiding-induced phase shifts in the dynamical response functions of two-dimensional systems hosting anyons^23,24. These braiding effects introduce universal features in the nonlinear signal that persists under realistic conditions, including finite temperature and non-universal interactions. In the toric code, for example, the nonlinear response distinguishes trivial from non-trivial anyon braiding, while in broader classes of topological phases, anomalous thermal relaxation and robust phase coherence provide experimentally accessible indicators of fractional statistics²⁵. These findings support the feasibility of using multidimensional coherent spectroscopies to probe and control statistical phases, aligning with our proposal to resolve mode-selective decoherence pathways in anyonic oscillator networks.

In a typical three-pulse 2D spectroscopy experiment, the system is driven by a sequence of phase-stabilized laser pulses with well-defined wave vectors $\textbf{k}_1$, $\textbf{k}_2$, and $\textbf{k}_3$. The emitted nonlinear signal appears in specific directions determined by the phase-matching condition, which enforces conservation of photon momentum among the optical fields. For the rephasing signal—often associated with a photon echo—the signal wave vector is given by $\textbf{k}_{\text {sig}} = -\textbf{k}_1 + \textbf{k}_2 + \textbf{k}_3$. This particular combination isolates Liouville pathways that involve evolution with negative time-ordering in the first interval, enabling partial cancellation of inhomogeneous broadening. It is important to emphasize that these wave vectors refer to the directions of the applied laser fields and are not related to any Bloch wave vectors or crystal momentum within the sample. The phase-matching geometry provides a versatile means of isolating specific dynamical pathways in the nonlinear response, independent of the underlying translational symmetry. For a comprehensive discussion of these techniques and phase-matching rules, we refer the reader to the foundational texts by Mukamel²⁶ and Hamm and Zanni²⁷.

Focusing our attention on the rephasing third-order response, we write

$$\begin{aligned} \tilde{R}^{(3)}(\omega _3, t_2, \omega _1) \propto \left( \frac{i}{\hbar }\right) ^3 \, \text {Tr} \left\{ \mu ^{(+)} \tilde{\mathcal {G}}(\omega _{3}) \mu ^{(-)} \mathcal {G}(\tau _2) \mu ^{(+)} \tilde{\mathcal {G}}(\omega _t) \mu ^{(-)} \rho _{\text {eq}} \right\} \end{aligned}$$

(27)

Here, $\mu ^{(+)}$ and $\mu ^{(-)}$ are superoperators acting on the right (+, ket-side) or left (-,bra-side) side of the density matrix, respectively. $\mathcal {G}(t)$ denotes the time-evolution superoperator that includes both coherent Hamiltonian evolution and dissipative dynamics. Similarly, $\tilde{\mathcal {G}}(\omega )$ is the resolvent of the Liouvillian, with $\omega _t$ and $\omega _\tau$ being the Fourier-conjugate variables of $t_3$ and $t_1$ respectively, and $\mathcal {G}(\tau _2) = e^{-i \mathcal {L} \tau _2}$ remains in the time domain and $\mu ^{(\pm )}$ are superoperators representing the light-matter interaction.²⁶ However, we have to point out that the dipole interaction

$$\mu \propto a_1^\dagger + a_1 + a_2^\dagger + a_2$$

also caries information concerning the statistical phase since it is constructed from the distorted creation/annihilation operators. Consequently, we anticipate signatures of this to be present in the 3rd-order rephasing signals.

To interpret the physical content of Eq. (27), we recall that the rephasing pathway corresponds to a specific time-ordered sequence of interactions between the light field and the system. Starting from the thermal equilibrium density matrix $\rho _{\textrm{eq}} = |0\rangle \langle 0|$, the system evolves through a sequence of coherence and population states induced by successive light–matter interactions. In superoperator notation, which tracks the action on the ket and bra sides of the density matrix, the rephasing pathway can be expressed as:

$$\begin{aligned} \rho ^{(0)} \xrightarrow {\mu ^{(-)}} |a\rangle \langle 0| \xrightarrow {\mathcal {G}(t_1)} \xrightarrow {\mu ^{(+)}} |a\rangle \langle b| \xrightarrow {\mathcal {G}(t_2)} \xrightarrow {\mu ^{(-)}} |0\rangle \langle b| \xrightarrow {\mathcal {G}(t_3)} \xrightarrow {\mu ^{(+)}} |0\rangle \langle 0| \end{aligned}$$

(28)

in which the system is return to the ground state. The kets $|a\rangle$ and $|b\rangle$ represent excited states generated by the action of the $\mu ^{(\pm )}$ operators and are generally superpositions of the symmetric and antisymmetric normal modes. The structure of these transitions and the resulting lineshape are directly influenced by the statistical phase factors that modify the matrix elements of $\mu$ in the anyonic basis. These phase-dependent modifications are expected to produce measurable signatures in the rephasing 2D spectra.

To probe the dynamical consequences of anyonic statistics and noise correlation beyond population dynamics, we computed the rephasing two-dimensional (2D) coherent response function $\chi ^{(3)}(\omega _3, \tau _2 = 0, \omega _1)$ across a grid of statistical angles $\theta$ and noise correlation strengths $\xi$. Importantly, the structure of the 2D spectrum reveals how coherence lifetimes and dephasing pathways are reshaped by exchange statistics and environmental symmetry. In the bosonic ($\theta = 0$) and fermionic ($\theta = \pi$) limits, the system supports exact decomposition into symmetric and antisymmetric normal modes. At $\xi = -1$ (fully anti-correlated noise), the symmetric mode is decoherence-free, yielding a single sharp (delta-function) spectral feature aligned along the $\omega _3 = \omega _1$ diagonal, while the antisymmetric mode is heavily damped and spectroscopically silent. Conversely, when $\xi = +1$ (fully correlated noise), the antisymmetric mode becomes protected while the symmetric mode rapidly decays. These dynamical symmetries underpin the emergence of coherence protection under extreme bath correlation. However, the form of the transition dipole operator only creates (or removes) excitations within the symmetric (or bright) mode of the system. Hence, as correlation is increased towards $\xi \rightarrow 1$, protecting the anti-symmetric mode, the line-shape of the right mode becomes increasingly broadened.

As the statistical angle $\theta$ increases from 0 to $\pi$, anyonic cross terms in the dissipator induce phase-sensitive leakage between modes. This results in progressive distortion of the protected coherence, visible as peak elongation, tilt, and loss of diagonal symmetry in the 2D response. The statistical phase therefore tunes the degree of mode isolation and coherence lifetime. Notably, at intermediate values $\theta = \pi /2$, the rephasing signal exhibits mixed-mode characteristics, indicating a breakdown of the exact symmetry protection. The interplay between $\theta$ and $\xi$ leads to a continuous deformation of the 2D lineshape—from sharply localized diagonal peaks to broadened, off-diagonal responses—highlighting a form of statistical control over coherence dynamics. These features suggest that fractional statistics not only alter population relaxation but also reshape the quantum pathways contributing to nonlinear response. The ability to encode these effects in a measurable 2D signal opens the door to experimental validation using advanced coherent spectroscopy, such as 2D Fourier-transform methods or time-domain Ramsey probes in programmable quantum simulators.

Fig. 5

Two-dimensional rephasing spectra $S^{(3)}(\omega _3, \omega _1)$ for a system of two anyonic oscillators as a function of statistical angle $\theta$ (vertical axis) and environmental noise correlation $\xi$ (horizontal axis). Each panel shows the real part of the rephasing response function computed at fixed values of $\theta \in \{0, \pi /2, \pi \}$ and $\xi \in \{-1, -1/2, 0, +1/2, +1\}$. The contours highlight the formation of long-lived coherence pathways as a function of fractional statistics and bath correlation. Leftmost panels (red) correspond to purely absorptive responses in the absence of coherence. As $\xi$ approaches ±1, mode protection emerges—enhanced or suppressed depending on the match between normal mode symmetry and noise structure. Exchange statistics ($\theta \ne 0$) break the symmetry between protected and unprotected modes, leading to asymmetric line shapes and phase-sensitive decoherence. Simulation parameters: oscillator frequencies $\omega _1 = \omega _2 = 1$, inter-oscillator coupling $J = 0.1$, temperature $T = 0.1$, and bare dissipation rate $\gamma = 0.1$.

Discussion

The integration of anyonic fractional statistics alters the structure of dissipation and influences the stability of protected modes. In bosonic systems ($\theta = 0$), exact mode protection is achieved when the correlation strength reaches $\xi = \pm 1$, fully decoupling one mode from the environment. In contrast, for anyonic oscillators ($\theta \ne 0$), the dissipation structure is modified such that the nominally protected mode acquires weak coupling to the bath via cross terms introduced by the statistical phase. The dissipation no longer decomposes cleanly into symmetric and antisymmetric modes, resulting in partial leakage of the protected state. Consequently, exact mode protection is preserved only in the bosonic ($\theta = 0$) or fermionic ($\theta = \pi$) limits.

Fractional statistics deform the dissipative dynamics of coupled oscillators, enabling tunable mode protection and coherence control. Unlike in purely bosonic or fermionic systems, anyonic oscillators exhibit phase-sensitive decoherence rates and exceptional-point behavior that depends on temperature. This suggests a strategy for engineering dissipation-resilient quantum states in topological devices by tuning the exchange statistics and environmental correlations. Our findings show that the statistical phase $\theta$ can be treated as a functional control parameter for modulating decoherence pathways, opening a route to coherence control through statistical engineering.

The bifurcation between symmetric and antisymmetric mode lifetimes observed in our model reflects a dynamical symmetry-breaking process akin to spontaneous PT-symmetry breaking in non-Hermitian systems, as the eigenvalues of the dynamical matrix lose their complex-conjugate symmetry. When the environmental coupling becomes strongly correlated ($\xi \rightarrow \pm 1$), the Liouvillian spectrum exhibits exceptional points—non-Hermitian degeneracies where both eigenvalues and eigenvectors coalesce—leading to the emergence of dominant, long-lived modes. This phenomenon manifests as synchronization or antisynchronization of the coupled oscillators, depending on the sign of the bath correlation. The statistical phase enhances this symmetry breaking by modulating the off-diagonal terms in the effective evolution matrix $W_{\textrm{eff}}$, thereby shifting the location and character of the exceptional points. These findings connect naturally to broader efforts to characterize noise-induced phase locking in open quantum systems^16,17,28, and echo the non-Hermitian transitions studied in optical and condensed matter platforms²⁹. Fractional statistics thus provide a novel and tunable axis for controlling dissipative phase transitions and coherence lifetimes in topological matter.

Recent experimental platforms have begun to exploit passive phase tuning to harness correlated thermal noise in transducers for enhanced quantum capacity³⁰, which echoes theoretical predictions of coherence stabilization under structured noise we have observed in anyonic or two-qubit systems. Moreover, recent experimental work by Kwan et al.¹⁴ demonstrated the realization of Abelian anyons with continuously tunable statistical phase in one-dimensional optical lattices. Using density-dependent Peierls phases, they achieved precise control over the exchange angle, sweeping from the bosonic to fermionic limits. This capability mirrors the framework developed in our model, where $\theta$ deforms both the algebra and the dissipation. Their platform offers a promising route for testing key predictions of our work—such as coherence bifurcations, statistical suppression of relaxation, and noise-induced synchronization—using cold-atom systems with programmable statistics. The results in Fig. 5 strongly suggest that coherent probes such as Ramsey interferometry or two-dimensional spectroscopy may offer viable detection strategies^22,31.