Exceptional point structure and chiral state transfer in a rate-operator-engineered PT-symmetric Bose-Hubbard dimer

arXiv:2610.30836 · quant-ph · Submitted 2026-10-22 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Exceptional point structure and chiral state transfer in a rate-operator-engineered PT-symmetric Bose-Hubbard dimer".

Mira: Here's the gist of the paper. The gist The rate operator quantum jumps (ROQJ) allow for versatile modification of this non-Hermitian evolution, with each non-Hermitian Hamiltonian corresponding to a different continuous measurement scheme > page 1.

Kai: The research applies the ROQJ formalism to modify the non-Hermitian dynamics of the open three-particle Bose-Hubbard dimer model under environmental non-local dephasing to construct a PT-symmetric non-Hermitian Bose-Hubbard Hamiltonian with a general two parameter PT-symmetric imaginary diagonal part > page 1.

Mira: The ROQJ allows for versatile modification of this non-Hermitian evolution, with each non-Hermitian Hamiltonian corresponding to a different continuous measurement scheme > page 1.

Kai: The formalism unravels the GKSL ME into stochastic pure state trajectories, and by post-selecting on no jumps occurring one obtains a trajectory generated by a non-Hermitian Hamiltonian > page 1.

Paper summary: Kai: So, we're looking at this paper, "Exceptional point structure and chiral state transfer in a rate-operator-engineered PT-symmetric Bose-Hubbard dimer." It tackles how you can use something called the rate operator quantum jumps, or ROQJ, to change the dynamics of an open system.

Mira: Basically, they're taking a master equation for an open three-particle Bose-Hubbard dimer and using ROQJ to turn it into a stochastic path of pure states. The main goal seems to be constructing these non-Hermitian Hamiltonians, which lets them explore different measurement schemes in one place.

Lev: I'm curious if this is something we can actually run on real hardware. They are building a PT-symmetric non-Hermitian Bose-Hubbard Hamiltonian with a general two parameter imaginary diagonal part, generalizing the usual gain-loss model. That sounds like it would require some pretty precise control over the system parameters.

Kai: Exactly, and they do something really interesting by characterizing the symmetry structure of the spectrum for different on-site interaction strengths. They find an unbroken PT-symmetry region enclosed by order two exceptional lines of different character and four order four exceptional points.

Mira: That's a specific feature they found that goes beyond what you'd expect from a standard gain-loss Bose-Hubbard dimer, especially when the on-site interaction strength is zero. They also noted regions with degenerate imaginary parts of the eigenvalues, which isn't seen in that usual model.

Lev: So if we are looking at vanishing on-site interaction, they find these two types of exceptional lines and four order four exceptional points where two of those order four points aren't present in the standard gain-loss dimer. That tells us something about the topology of the parameter space for this system.

Kai: And that parameter space also contains regions with degenerate imaginary parts, which is another thing they found not found in that prior work on the gain-loss Bose-Hubbard dimer. It suggests a richer structure in this specific model when we look at how these parameters interact.

Mira: They then use time-dependent rate operator transformations to study state transfer under slowly varying parameter loops. This is where they get into dynamical consequences, specifically chiral state transfer when loops encircle an order four exceptional point.

Lev: Chiral state transfer means the final state after a slow loop depends on the direction you traversed around that loop, rather than just where you started. That's a pretty strong statement about controlling the system dynamics through path geometry.

Kai: They also found chiral state transfer even when there is a non-zero homogeneous on-site interaction, provided the trajectory crosses over broken PT-symmetry regions of different character. That shows this effect isn't just limited to the zero interaction case.

Mira: And they also found direction dependent state transfer between eigenstates when using inhomogeneous on-site interactions, but they managed to do this without having to cross or encircle any exceptional points at all. That's a neat distinction from the previous findings.

Lev: So what does this mean for someone just listening? It means you can have state transfer that depends on which way you move your system parameters, and it happens in different ways depending on whether the interaction is uniform or not.

Kai: And they showed that instead of only changing the physical parameters of the system or the environment, we can change the measurement scheme itself to create time-dependent non-Hermitian Hamiltonians using ROQJ. That opens up a new way to control dynamics without physically tuning every bit of hardware.

Mira: That versatility in changing the continuous measurement scheme allows them to engineer vastly different jump trajectories and non-Hermitian dynamics, which is a lot of control over what happens next.

Lev: From an engineering standpoint, that's promising because it suggests we might be able to design complex evolution paths by choosing how we measure the system, rather than just manipulating the external fields directly.

Kai: The conclusion of this paper highlights that ROQJ lets them design time-dependent PT-symmetric non-Hermitian Hamiltonians that can freely traverse parts of the parameter space, passing between unbroken and broken PT-symmetry phases and circling exceptional points over time.

Mira: It’s important because it shows how you can use measurement scheme switching to navigate the complex phase structure of these models. The paper explores how this technique lets you pass through different symmetry regimes dynamically.

Lev: And they also showed that even when there's a non-zero homogeneous on-site interaction, the final state at the end of a loop stays close to the least decaying state of the broken PT-symmetry phase, even for large evolution times. That suggests some stability in certain dynamic regions.

Kai: So this work with "Exceptional point structure and chiral state transfer in a rate-operator-engineered PT-symmetric Bose-Hubbard dimer" shows that ROQJ isn't just a tool for unraveling equations, but a way to actively engineer the system's evolution by changing how we observe it.

Mira: It really pushes the idea that the control mechanism can be internal to the description of the dynamics itself, using these rate operator transformations instead of just tweaking external potentials.

Conclusion: Kai: So, we’re wrapping up this deep dive into the paper by Ashida, Gong, and Ueda about "Exceptional point structure and chiral state transfer in a rate-operator-engineered PT-symmetric Bose-Hubbard dimer."

Mira: Right. It all boils down to how you use these rate operator quantum jumps to actively design the non-Hermitian dynamics of a three-particle Bose-Hubbard model under environmental noise.

Lev: The authors constructed a specific PT symmetric non-Hermitian Hamiltonian with that two parameter imaginary diagonal part, which they call general. I'm interested in how robust that structure is when you actually try to run it on hardware.

Kai: Well, the paper shows how you can traverse different regions of the parameter space by switching the measurement scheme itself rather than just changing the physical knobs on your system or environment.

Mira: That’s a big shift because it means we can engineer vastly different jump trajectories and non-Hermitian dynamics simply by choosing a different way to measure what’s happening.

Lev: If you're trying to build an error correction scheme, knowing that you can dynamically move between unbroken and broken PT symmetry phases just by changing the measurement scheme is pretty powerful, but I need to know the overhead for that control.

Kai: The results show that even with a non-zero homogeneous on-site interaction, if you follow a certain path in parameter space—say clockwise or counter-clockwise—you get different final states.

Mira: And those final states are superpositions of eigenstates because the loops around certain order four exceptional points cause chiral state transfer.

Lev: So the numbers they find, like those order two exceptional lines and four order four points, they suggest a very specific topology in that parameter space we need to be aware of when designing our error correction protocols.

Kai: It means if we want to control these dynamics in a physical system, we don't have to rely only on tuning the external potentials; we can use measurement scheme engineering as a control mechanism.

Mira: And they also found that sometimes you can get direction dependent state transfer just by using inhomogeneous on-site interactions without ever having to cross or loop around those exceptional points.

Lev: That last part is interesting because it suggests a way to achieve state steering without the complexity of circling those difficult points, which would drastically reduce the error correction overhead if we could replicate that in a real system.

Kai: So what this paper really says is that the ROQJ formalism gives us a direct way to sculpt non-Hermitian dynamics, allowing us to navigate complex parameter spaces and achieve chiral state transfer through measurement control.

Mira: It moves the focus from just describing the master equation to actively engineering the system's evolution by changing how we observe it.

Lev: For anyone building real hardware, this suggests that measurement choice is a primary lever for controlling dynamics in these open systems, which is something we need to seriously consider for our error correction roadmap.

Jaakko Luomala, Federico Settimo, Kimmo Luoma, Iiro Vilja, Jyrki Piilo

University of Turku

quant-ph

Submitted: 2026-10-22

Updated: 2026-10-22

Comments: 16 + 1 pages, 11 + 1 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist The rate operator quantum jumps (ROQJ) allow for versatile modification of this non-Hermitian evolution, with each non-Hermitian Hamiltonian corresponding to a different continuous

Key concepts

Rate Operator Quantum Jumps (ROQJ): ROQJ is a method that transforms the master equation describing an open quantum system into stochastic pure state trajectories. By post-selecting on no jumps occurring, one obtains a trajectory governed by a specific non-Hermitian Hamiltonian, enabling versatile modification of the evolution based on different measurement schemes.
$PT$-Symmetric Non-Hermitian Hamiltonian: This is a specific type of non-Hermitian Hamiltonian designed to exhibit $PT$-symmetry. The study constructs such Hamiltonians with a general two-parameter imaginary diagonal part, generalizing the standard gain-loss Bose-Hubbard dimer model. This structure allows for exploring different symmetry regimes in the system's energy spectrum.
Exceptional Points (EPs): Exceptional points are specific parameter values where the eigenvalues and their corresponding eigenvectors coalesce. The study identifies four order four exceptional points, two of which are new compared to the usual gain-loss model. These EPs govern complex phenomena like chiral state transfer during time evolution in parameter space.

Summary

The gist The rate operator quantum jumps (ROQJ) allow for versatile modification of this non-Hermitian evolution, with each non-Hermitian Hamiltonian corresponding to a different continuous measurement scheme > page 1.

Methodology

The research applies the ROQJ formalism to modify the non-Hermitian dynamics of the open three-particle Bose-Hubbard dimer model under environmental non-local dephasing to construct a PT-symmetric non-Hermitian Bose-Hubbard Hamiltonian with a general two parameter PT-symmetric imaginary diagonal part > page 1. The ROQJ allows for versatile modification of this non-Hermitian evolution, with each non-Hermitian Hamiltonian corresponding to a different continuous measurement scheme > page 1. The formalism unravels the GKSL ME into stochastic pure state trajectories, and by post-selecting on no jumps occurring one obtains a trajectory generated by a non-Hermitian Hamiltonian > page 1. The deterministic evolution is governed by the effective Hamiltonian K = H − i/2 sum i=1 γi L†iLi > page 3.

Symmetry Structure of the Spectrum

The study characterizes the symmetry structure of the spectrum for different on-site interaction strengths, finding an unbroken PT-symmetry region enclosed by order two exceptional lines of different character and four order four exceptional points, two of which are not found in the usual gain-loss Bose-Hubbard dimer > page 1. In this parameter space, they also find regions with degenerate imaginary parts of the eigenvalues, also not found in the gain-loss Bose-Hubbard dimer > page 1. The effect of the interaction term U on the spectrum and PT-symmetry breaking of the effective Hamiltonian (10) in parameter space (l, k) can be described by presenting phase diagrams of broken/unbroken PT-symmetry for different interaction strengths U > page 5. For vanishing interaction strength U = 0, branching points related to PT-symmetry breaking can be found to form two types of exceptional lines: the first type consists of two order two exceptional points (4PTS to DIP in Fig. 1(a) and first branching point in Fig. 3(b)), and the second type consists of order two exceptional points (4PTS to 2PTS and 2PTS to 0PTS in Fig. 1(a) and both branching points in Fig. 3(c)) > page 6.

Chiral State Transfer

Time-dependent rate operator transformations are used to investigate state transfer under slowly varying parameter loops, where loops encircling an order four exceptional point exhibit chiral state transfer > page 1. Chiral state transfer is also found with non-zero homogeneous on-site interaction when the trajectory crosses over broken PT-symmetry regions of different character > page 1. Using inhomogeneous on-site interactions, they find direction dependent state transfer between eigenstates without crossing or encircling exceptional points > page 2. The final states after performing a slow loop in parameter space depends on the direction of the traversal around the loop but not, usually, on the initial state > page 3.

Dynamics under Different Interaction Regimes

The study investigates three cases for non-zero homogeneous on-site interaction: zero on-site interaction, non-zero homogeneous on-site interaction, and non-zero inhomogeneous on-site interaction > page 2. For vanishing interaction U = 0, the final state is in most cases independent of the initial state because the state has sufficient time to reach the least decaying state at late times before re-entering the phase of unbroken PT-symmetry > page 10. For non-zero, homogeneous on-site interaction, CCW and CW trajectories have different least decaying states in the long time limit, exhibiting chiral behavior in the final state > page 12. In the case of inhomogeneous on-site interactions U1 = −U2, analogous behavior to chiral state transfer when traversing the trajectory in different directions was found in the model studied here > page 13.

Control via Measurement Scheme

The paper demonstrates that instead of varying physical parameters of system or the environment, one can instead vary the measurement scheme to form time-dependent non-Hermitian Hamiltonians using ROQJ > page 14. This opens options for future theoretical and experimental research in control of non-Hermitian dynamics using different measurement schemes instead of varying physical parameters > page 14. The versatility of the ROQJ method allows one to engineer vastly different jump trajectories and non-Hermitian dynamics by changing the continuous measurement scheme > page 1.

Conclusion

The rate operator quantum jumps formalism has been applied to engineer the non-Hermitian dynamics of the open Bose-Hubbard model with three particles subject to non-local dephasing > page 14. The ROQJ formalism can be used to design time-dependent PT -symmetric non-Hermitian Hamiltonians that can freely traverse part of the parameter space of the Hamiltonian, passing between unbroken and broken PT -symmetry phases and circling exceptional points over time > page 14. The research shows that diverse modifications of non-Hermitian dynamics can be accomplished through switching of the measurement scheme defined by an unraveling of the master equation while keeping the physical parameters fixed > page 14. The final state at the end of a loop stayed close to the least decaying state of the broken PT-symmetry phase even for large total evolution times when non-zero homogeneous on-site interaction was present > page 12.

References

[1] Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics 69, 249 (2020) > page 15

[2] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2007) > page 15

[3] B. Vacchini, Open Quantum Systems (Springer, 2024) > page 15

[4] J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Physical Review Letters 68, 580 (1992) > page 15

[5] C. W. Gardiner, A. S. Parkins, and P. Zoller, Wave-function quantum stochastic differential equations and quantum-jump simulation methods, Physical Review A 46, 4363 (1992) > page 15

[6] K. Mølmer, Y. Castin, and J. Dalibard, Monte Carlo wave-function method in quantum optics, Journal of the Optical Society of America B 10, 524 (1993) > page 15

[7] M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Reviews of Modern Physics 70, 101 (1998) > page 15

[8] F. Settimo and J. Piilo, Quantum jump unravelings for non-Markovian open system dynamics: a review (2026), arXiv: 2605.07797 > page 15

[9] V. Gorini, A. Kossakowski, and E. C. Sudarshan, Completely positive dynamical semigroups of n-level systems, J. Math. Phys. 17, 821 (1976) > page 15

[10] G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976) > page 2

[11] A. Smirne, M. Caiaffa, and J. Piilo, Rate operator unraveling for open quantum system dynamics, Phys. Rev. Lett. 124, 190402 (2020) > page 3

[12] D. Chru´sci´nski, K. Luoma, J. Piilo, and A. Smirne, How to design quantum-jump trajectories via distinct master equation representations, Quantum 6, 835 (2022) > page 3

[13] F. Settimo, K. Luoma, D. Chru´sci´nski, B. Vacchini, A. Smirne, and J. Piilo, Generalized-rate-operator quantum jumps via realization-dependent transformations, Phys. Rev. A 109, 062201 (2024) > page 3

[14] J. Luomala, K. Luoma, I. Vilja, and J. Piilo, Engineering of non-hermitian trajectories and phase structure in an open bose-hubbard model via rate operator transformations (2026), arXiv:2609.02490 > page 15

[15] R. Uzdin, A. Mailybaev, and N. Moiseyev, On the observability and asymmetry of adiabatic state flips generated by exceptional points, Journal of Physics A: Mathematical and Theoretical 44, 435302 (2011) > page 6

[16] S. Khandelwal, W. Chen, K. W. Murch, and G. Haack, Chiral bell-state transfer via dissipative liouvillian dynamics, Phys. Rev. Lett.

Improvements for AI systems

Increase robustness in quantum state engineering by designing measurement schemes instead of relying solely on physical parameter variation. This allows AI systems to generate time-dependent non-Hermitian Hamiltonians by varying the measurement scheme to form time-dependent non-Hermitian Hamiltonians.
Develop algorithms for identifying chiral state transfer pathways within parameter space, specifically those involving parameter loops encircling an order four exceptional point. This enables AI to predict and exploit directional dependence in quantum state evolution under controlled measurement protocols.
Create predictive models for the symmetry structure of open quantum systems, such as the regions of unbroken PT-symmetry (green), partially unbroken PT-symmetry (blue), white and red based on interaction strengths. This allows AI to determine when a system is most likely to exhibit specific dynamical behaviors like chiral state transfer or unique steady states.
Implement control strategies for non-Hermitian dynamics that utilize the least decaying eigenstate as an asymptotic target. By understanding how populations tend toward these states, AI can design measurements or parameter sweeps that drive the system into a desired final state quickly.
Design adaptive measurement protocols for systems with inhomogeneous on-site interactions to achieve direction dependent state transfer without crossing or encircling exceptional points. This capability allows AI to navigate complex interaction landscapes efficiently for specific quantum control goals.