Switchable Dissipative Ising Coupling

• Switchable dissipative Ising coupling is a programmable interaction mediated by engineered dissipative channels in open systems, allowing dynamic tuning of ferro- and antiferromagnetic order.
• It leverages non-Hermitian evolution and tailored Lindblad dynamics across magnonic, polaritonic, and mechanical platforms to simulate complex optimization tasks.
• Real-time reconfiguration of coupling graphs via reservoir engineering and feedback control offers robust performance in non-equilibrium quantum simulators and neuromorphic devices.

Switchable dissipative Ising coupling refers to engineered interactions between effective Ising spins (binary or continuous degrees of freedom) that arise from, and are mediated by, dissipative channels in driven open quantum or classical systems, and whose sign, strength, or functional form can be dynamically tuned or switched. This paradigm is foundational for the realization of programmable Ising machines, quantum simulators, and neuromorphic devices, with experimental platforms spanning magnonic, photonic, mechanical, and polaritonic systems. Dissipative Ising couplings differ from purely coherent exchange via their intrinsic energy non-conserving nature, often emerging from reservoir engineering, feedback, or bath-mediated processes, and demonstrate robust, rapid, and flexible reconfigurability, including dynamical reversibility between ferro- and antiferromagnetic orderings.

1. Fundamental Principles and Mathematical Formalism

The archetypal dissipative Ising model consists of a network of nodes, each encoding a spin variable $\sigma_i \in \{\pm1\}$ or a continuous angle $\theta_i$, subject to effective pairwise couplings $J_{ij}(t)$ mediated by dissipative processes. The general objective Hamiltonian for discrete Ising and continuous XY spins is

$$H_{Ising} = -\sum_{i<j} J_{ij}^{(I)} s_i s_j, \qquad H_{XY} = -\sum_{i<j} J_{ij}^{(XY)} \cos(\theta_i - \theta_j).$$

Crucially, the $J_{ij}$ can be time-dependent and completely programmable in both amplitude and sign, achieved by material, drive, or circuit-level control of the underlying dissipative channels (Kalinin et al., 2020, Dou et al., 27 Nov 2025).

Coupling mechanisms rely on non-Hermitian evolution, typically modeled by Lindblad or classical master equations, rather than solely via Hamiltonian exchange. In open quantum networks, the effective Liouvillian may contain symmetric and antisymmetric collective loss channels, with corresponding rates $\Gamma_+, \Gamma_-$, yielding a dissipative Ising interaction $J_{diss} = (\Gamma_- - \Gamma_+)/2$ (Dou et al., 27 Nov 2025). In classical bistable oscillators, spin-flip rates inherit exponential sensitivity to the states of coupled neighbors, controlled via dissipative drive or feedback (Han et al., 2023).

2. Physical Implementations

2.1 Magnonic–Mechanics Hybrid Platform

In magnon-based systems, switchable dissipative Ising coupling is realized via three-body interactions among photon (microwave), magnon (spin wave), and phonon (mechanical) modes. Two magnon modes $(a_1, a_2)$ are coupled via lossy transmission-line photons, with an auxiliary mechanical mode subjected to a two-phonon pump $s_b = |s_b|e^{i\varphi}$ that implements squeezing. Adiabatic elimination of the cavity photons gives rise to Lindblad jump operators $\mathcal{D}[a_1 \pm a_2]$, with the dominance of either channel determined by the squeezing parameter $r = \tanh^{-1}(2|s_b|/\Gamma_b)$. The effective dissipative Ising coupling is

$$J_{diss} = \Gamma_{cc} \sinh(2r), \qquad \Gamma_{cc} = \lambda_c^2/\Gamma_{cav}$$

and its sign is switched by adjusting the pump phase $\varphi$, dynamically inverting between ferro- and antiferromagnetic effective interactions (Dou et al., 27 Nov 2025).

2.2 Polaritonic XY-Ising Machine

Networks of spatially separated polariton condensates implement switchable dissipative Ising coupling via controllable re-injection of emission between nodes, with programmable amplitude and phase weights achieved optically (spatial light modulators, SLMs) or via hybrid electronic–photonic feedback. Both "relative" (proportional to the emitter intensity) and "absolute" (photon-number preserving) coupling schemes are supported, and the phase of the coupling channel directly sets the sign of $J_{ij}$. Full switchability is achieved by dynamically reconfiguring the SLM or electronic modulation parameters, enabling arbitrary graph connectivity and rapid sign changes (Kalinin et al., 2020).

2.3 Parametric Mechanical Oscillators

Pairs of noise-driven, parametrically modulated micromechanical oscillators function as bistable spins, where state-dependent dissipative electrostatic coupling modifies the noise-activated switching rates in a Glauber-like fashion. The effective Ising coupling is realized through cross-driving, and switchability is achieved by modulating the inter-oscillator voltage $V_{ij}(t)$, the relative phase $\phi_j$, or by sweeping the resonator frequency to change the susceptibility $\chi_i$, providing electrical and dynamical control over both amplitude and sign of $J_{ij}$ (Han et al., 2023).

3. Dynamical and Switchability Mechanisms

Switchability of dissipative Ising couplings exploits tunable dissipator engineering or feedback-mediated control. In the magnonic scheme, switching is achieved via control of the mechanical pump phase $\varphi$, which rotates the squeezing axis and exchanges dominant collective dissipative modes. This switches the sign of $J_{diss}$ between positive (ferromagnetic) and negative (antiferromagnetic), as explicitly reflected in the structure of the effective Lindbladian and non-Hermitian Hamiltonian.

In polariton and oscillator platforms, switchability is achieved by (i) electrically or optically gating coupling channels (on/off or $0/\pi$ phase shifts) and (ii) dynamically updating the network connectivity via real-time control elements (SLMs, EOMs, FPGA/PC feedback loops). In mean-field Floquet-dissipative Ising models, tuning the drive frequency $\Omega$ relative to the structure (e.g., derivative zero crossings) of the bath spectral density $\nu(\omega)$ dynamically switches the effective coupling sign $J_{eff}(\Omega,h)$, thus driving the system between distinct thermodynamic phases (Goldstein et al., 2015).

4. Master Equation Modeling and Non-Equilibrium Stationarity

Dissipative Ising platforms are modeled by master equations whose transition rates explicitly encode the influence of dissipative coupling. In the classical micromechanical context, the spin-flip rates are

$$W_i(\sigma_i \rightarrow -\sigma_i) = W_i^0 \exp \left[-\sigma_i \sum_j J_{ij} \sigma_j / D \right],$$

where $D$ is the noise intensity. Asymmetric or state-dependent rates break detailed balance and produce stationary circulating probability currents, characteristic of non-equilibrium dissipative systems (Han et al., 2023).

In the driven-dissipative quantum Ising model, the mean-field master equation couples populations via Floquet-dressed transition rates determined by drive parameters, bath hybridization $\nu(\omega)$, and temperature. The steady-state magnetization (order parameter) is the solution to a non-equilibrium self-consistency equation,

$$\phi = \frac{R_{\downarrow \rightarrow \uparrow}(\phi) - R_{\uparrow \rightarrow \downarrow}(\phi)}{R_{\downarrow \rightarrow \uparrow}(\phi) + R_{\uparrow \rightarrow \downarrow}(\phi)}.$$

Analysis in this framework reveals that the dissipative Ising coupling $J_{eff}$, and consequently the critical temperature and universality class, can be completely reprogrammed by external driving (Goldstein et al., 2015).

5. Performance, Programmability, and Robustness

Switchable dissipative Ising architectures offer substantial advantages in connectivity, rapidity, and reconfigurability over systems constrained by geometrical coupling. Benchmarks in polaritonic networks show that, for Max-Cut problems of size $N=25$ and $N=49$ at edge density $0.5$ and $J_0=0.04$, median ground-state approximation accuracies for XY models exceed $99\%$ and for Ising models reach $88\%$ using relative coupling. Absolute coupling achieves somewhat lower Ising performance but maintains high XY accuracy (see Table below) (Kalinin et al., 2020).

Network Size	XY (Relative)	Ising (Relative)	XY (Absolute)	Ising (Absolute)
25	99.3%	87.8%	96.8%	72.9%
49	98.2%	81.7%	93.3%	52.3%

In magnonic devices, the effective $J_{diss}$ is robust against strong unswitchable background dissipation by adjusting the magnon pump, as confirmed by numerical simulation showing persistent sign reversibility and strong coherent peak formation in phase-space distributions under high loss (Dou et al., 27 Nov 2025). In mechanical oscillator experiments, control over $V_{cpl}$, phase, and susceptibility parameterizes $J_{ij}$ across a wide range with readout fidelity $>99\%$ and rates tunable over $0.01-1$ Hz (Han et al., 2023).

6. Experimental Parameter Regimes and Realization

Realistic implementation of switchable dissipative Ising couplings in magnonic, photonic, or mechanical systems rests on viable parameter regimes. For magnon systems: magnon frequency $\omega_m/2\pi \sim 10$ GHz, mechanical mode $\omega_b/2\pi \sim 10$ MHz, and transmission line photon frequency $\omega_c/2\pi \sim 10$ GHz, with pump strengths $s_a \sim 1$ MHz, coupling rates $\lambda_c \sim 1$ MHz, and cavity linewidths $\Gamma_{cav} \sim 10$ MHz. Under these conditions, $J_{diss}$ is electrically tunable over $0.1$–$10$ MHz and supports rapid, real-time switching (Dou et al., 27 Nov 2025).

For mechanical oscillators: noise $D\approx 3\times10^{-6}$ (N$^2$kg$^{-2}$Hz$^{-1}$), coupling voltages $V_{cpl}$ up to $0.5$ V, and drive amplitudes $F_d\sim10^{-17}$ Nm yield dimensionless coupling $K\approx 0-1.2$, and switching speeds from $0.01$–$1$ Hz (Han et al., 2023).

7. Implications and Applications

Switchable dissipative Ising coupling provides the basis for architectures capable of efficiently solving combinatorial optimization tasks (e.g., Max-Cut, ground-state search), simulating non-equilibrium many-body physics (including novel steady states with stationary currents), and exploring universal behavior in systems driven far from equilibrium, such as induced tricritical points and drive-tunable critical exponents (Goldstein et al., 2015). The ability to toggle interaction graphs and sign dynamically realizes neuromorphic and quantum information processing devices with “on-the-fly” programmable Hamiltonians, uniquely enabled by the dissipative–reservoir engineering paradigm (Kalinin et al., 2020, Dou et al., 27 Nov 2025).

A plausible implication is that, as scalable fabrication and integration mature, switchable dissipative Ising couplings will underpin realizations of large-scale analog Ising machines and expand access to novel quantum computational and simulation regimes.