Non-Hermiticity-Induced Jumps in Open Systems

Updated 24 December 2025

Non-Hermiticity-induced jumps are abrupt transitions in open systems caused by non-Hermitian effects like quantum jumps and complex band structures.
These jumps shift topological invariants and phase boundaries, as seen in Lindbladian dynamics and modifications to effective Hamiltonians.
In non-Hermitian lattices, critical jump times and spatial shifts arise from the competition of Bloch modes under gain/loss imbalances, offering experimental insights.

Non-Hermiticity-induced jumps (NHJ) refer to discrete, abrupt transitions in the dynamical or topological properties of open quantum, classical, and lattice systems that originate from the effects of non-Hermitian terms—especially quantum jumps or complex band structure—in their generators. Unlike Hermitian systems, where evolution is unitary and transitions are parametrically smooth, inclusion of non-Hermitian effects (e.g., loss/gain, nonreciprocal couplings, quantum jump superoperators) produces regimes where system observables, phase boundaries, topological invariants, or spatial dynamical quantities undergo sharply non-analytic shifts. This phenomenon manifests both in the Lindbladian description of open quantum dynamics and in the wavepacket transport of spatially extended non-Hermitian lattices.

1. Open System Dynamics and the Lindblad Framework

Open quantum systems with Markovian dissipation are described by the Lindblad-Gorini-Kossakowski-Sudarshan master equation: $\partial_t\rho = -i[H_0,\rho] + \sum_i \gamma_i \left( L_i\rho L_i^\dagger - \frac{1}{2}\{L_i^\dagger L_i,\rho \} \right)$ Here, the $L_i$ are Lindblad (jump) operators, and $\gamma_i$ are decay or gain rates. The dynamics splits into "no-jump" evolution generated by a non-Hermitian effective Hamiltonian ( $H_{\mathrm{eff}}=H_0 - \frac{i}{2}\sum_i\gamma_i L_i^\dagger L_i$ ) and stochastic quantum-jump events which project and reset the state. The full Lindbladian evolution, including quantum jumps, admits a trajectory (Monte Carlo) unraveling and a superoperator (Liouville space) representation. Discarding the jump terms ("no-jump limit") accurately describes short-time or post-selected evolution but omits crucial long-time and topological effects inherent to the jump processes (Niu et al., 2022 Matsoukas-Roubeas et al., 2022 Niu et al., 2022).

2. Topological Phase Transitions Induced by Quantum Jumps

Non-Hermiticity-induced jumps are quantitatively manifest in the topological classification of open quantum lattice models, such as the dissipative Su-Schrieffer-Heeger (SSH) chain.

In the no-jump limit (i.e., $H_{\mathrm{eff}}$ ), the critical boundary for the winding number (under generalized Brillouin zone/GBZ formalism) is

$t_1 = \pm \sqrt{ t_2^2 + \left( \frac{\gamma}{2} \right)^2 }$

where $t_1$ and $t_2$ are intra/intercell hoppings, $\gamma = \gamma_l + \gamma_g$ the total (collective) loss and gain rates.

Full inclusion of jumps (via Liouvillian shape-matrix or third-quantization techniques) shifts the critical boundary to:

$t_1 = \pm\sqrt{ t_2^2 + \left( \frac{\gamma_l - \gamma_g}{2} \right)^2 }$

i.e., it becomes sensitive to the difference rather than the sum of loss and gain rates.

This non-analytic shift of the phase boundary defines the non-Hermiticity-induced jump in the topological invariant: the winding number changes discontinuously upon tuning system parameters across the new critical point determined by quantum jumps. The shift arises because the jump superoperator in the Lindbladian breaks particle-number block-diagonality and alters single-particle correlations through the difference in gain and loss rates, an effect invisible in $H_{\mathrm{eff}}$ dynamics (Niu et al., 2022).

3. Dynamical Spatial Jumps in Non-Hermitian Lattices

In one-dimensional non-Hermitian tight-binding lattices, NHJ appears as abrupt spatial jumps in the wavepacket center-of-mass, even in disorder-free systems under open boundary conditions. Consider a non-Hermitian lattice with complex hoppings ( $J^R \neq J^L$ ) and onsite potential: $H = \sum_{n} J^R a^\dagger_{n+1}a_n + J^L a^\dagger_n a_{n+1} + V \sum_n a^\dagger_n a_n$ The imaginary part of the band structure, $E(k) = E_R(k) + iE_I(k)$ , amplifies or suppresses different momentum components. For a broad wavepacket, there exists a well-defined "jump time"

$t_c = \frac{ \ln|\psi_{k_0}(0)/\psi_{k^*}(0)| }{ E_I(k^*)-E_I(k_0) }$

at which the initially subdominant momentum mode $k^*$ with maximal $E_I$ overtakes the dominant $k_0$ mode, abruptly relocating the packet's spatial center-of-mass by $\Delta x \sim v_g(k^*) t_c$ (He et al., 8 Dec 2025 Jana et al., 20 Dec 2025). This NH jump is not due to disorder or external fields but purely to non-Hermitian gain/loss imbalance and the competition among Bloch modes. The phenomenon is analytically described in the Hatano-Nelson model, where the onset and magnitude of the jump are precisely determined by initial state overlaps and the complex dispersion (Jana et al., 20 Dec 2025).

4. Quantum Jumps and Sudden Transitions in Time-Dependent Dynamics

In slowly varying non-Hermitian Hamiltonians, quantum jump processes or, more generally, non-Hermitian components can induce abrupt "sudden transitions" between instantaneous eigenstates. For a time-dependent non-Hermitian $H(t)$ , the breakdown of the adiabatic theorem manifests as non-adiabatic, non-perturbative transfers (“jumps”) between decaying/amplifying eigenstates when coherence factors and exponential integrals of eigenvalue separations reach unity: $| \rho_a(t_*)\, e^{i \Theta_{12}(t_*)} | \sim 1, \hspace{8pt} \Theta_{12}(t) = \int_0^t [E_1(t')-E_2(t')] dt'$ with characteristic transition timescales set by the inverse imaginary gap. The mathematical structure follows from Schur decompositions and the temporal evolution of eigenmode amplitudes. Physically, the sudden switch reflects the dominance of the most amplifying (or least decaying) eigenvector once the time-evolved overlaps become exponentially separated (Wang et al., 2018).

5. Quantum-Jump–Induced Shifts in Nonequilibrium Many-Body Systems

In open many-body systems, NHJs induced by jump terms may shift the location of nonequilibrium phase transitions. In dissipative fermionic BCS superfluids, inclusion of two-body loss jump processes modifies the BCS gap equation and shifts the critical interaction strength $U_c$ for superfluidity: $U_c(\kappa) = U_c(0) \sqrt{1 + \left( \frac{\kappa}{2 U_c(0)} \right)^2 }$ This shift arises from first-order corrections to the ground-state condensation energy, which are absent if only $H_{\mathrm{eff}}$ is considered. Similarly, in the non-Hermitian Kondo effect, the RG flow can "jump" back to a non-interacting fixed point at a critical value of the imaginary part of the Kondo exchange, corresponding to a non-Hermitian quantum phase transition (Nakagawa et al., 2018 Niu et al., 2022).

6. Stochastic and Statistical Signatures: Anderson Attractors and Dephasing-Induced Jumps

Non-Hermiticity-induced jumps also govern statistical properties of quantum-jump trajectories in disordered and/or dephased systems. In non-Hermitian Anderson chains, the quantum-jump statistics exhibit non-Poissonian (power-law) waiting-time distributions and sudden transitions between localized spatial attractors (Anderson modes). The inclusion of non-Hermitian, phase-sensitive dissipators leads to regimes exhibiting either diffusive or ballistic motion, with abrupt changes in spatial propagation and jump interval statistics (Yusipov et al., 2017).

A related mechanism is found in dephased non-Hermitian disordered lattices, where abrupt jumps of wavefunction intensity between distant sites arise due to stochastic stabilization of localized eigenmodes of the incoherent propagator $S(l) = |e^{-i H l}|^2$ . Under sufficiently rapid dephasing, these eigenmodes are sharply localized, and population dynamics is marked by predictable "jumps" between localized centers, absent in Hermitian or weakly disordered non-Hermitian systems (Kokkinakis et al., 2024).

7. Physical Interpretation and Experimental Outlook

Non-Hermiticity-induced jumps are fundamentally tied to the breakdown of unitarity and reciprocity in open quantum dynamics:

Discrete transitions (e.g., topological winding reversal, RG flow jumps, spatial center-of-mass jumps) are not captured by approximations that omit jump superoperators.
In topological systems, quantum jumps can replace nonreciprocal gain/loss parameters in critical formulas with their rate differences, modifying phase diagrams even at the single-particle level.
Dynamical spatial jumps provide direct and robust experimental signatures in photonic lattices, synthetic dimensions, and quantum walks.
The interplay of jump-induced decoherence and non-Hermitian band topology enables control and engineering of abrupt transport and phase-transition phenomena in both quantum and classical platforms.

Non-Hermiticity-induced jumps provide a unifying framework for understanding discontinuous, topologically, and dynamically non-analytic changes in open systems, highlighting the essential role of both continuous non-unitary drift and discrete quantum-jump back-action in non-Hermitian physics (Niu et al., 2022 He et al., 8 Dec 2025 Jana et al., 20 Dec 2025 Niu et al., 2022 Nakagawa et al., 2018 Kokkinakis et al., 2024 Yusipov et al., 2017 Wang et al., 2018).