Non-Hermitian dynamics and PT-symmetry breaking in interacting mesoscopic Rydberg platforms

arXiv:2111.04378 · cond-mat.quant-gas, quant-ph · Submitted 2021-11-08 · 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: Today's paper: "Non-Hermitian dynamics and PT-symmetry breaking in interacting mesoscopic Rydberg platforms".

Mira: The gist The authors simulate dissipative dynamics of a mesoscopic system of long-range interacting particles mapped into non-Hermitian spin models with PT symmetry,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we're looking at this paper titled "Non-Hermitian dynamics and PT-symmetry breaking in interacting mesoscopic Rydberg platforms". What's the core idea here?

Mira: Basically, they take a system of long-range interacting particles, these mesoscopic systems, and map them onto non-Hermitian spin models that have PT symmetry. They find rich phase diagrams with both PT-symmetric and PT-broken phases.

Lev: So the thesis is that you can look at these dissipative dynamics and see how they behave in terms of this symmetry, right?

Kai: Yeah, and what makes it interesting is that they show how tuning some parameters can further enrich these dynamical regimes. They outline how we can probe this PT symmetry just by studying the system's dynamics over time.

Mira: And they point out that systems of Rydberg atoms or Rydberg ions with strong dipolar interactions are especially good for testing this because of how easily you can control those interactions and coupling strengths.

Lev: From an engineering standpoint, they show that for realistic parameters, these long-range interactions actually allow for the emergence of new PT-symmetric regions, which means they generate new PT-phase transitions.

Kai: It’s not just about finding some existing phases; it’s about finding entirely new ones when you consider how the interactions scale up with more particles. They also found these PT symmetry phase transitions by simply changing the configuration of the Rydberg atoms in their setup.

Mira: So, we're looking at how those configurations affect where the symmetry breaks and where it stays intact across these phase diagrams for this non-Hermitian model.

Lev: And they monitor these transitions by checking the populations of the Rydberg states; you see oscillatory or exponential dependence in each of those phases.

Kai: That's what’s cool—you can actually observe the symmetry breaking happening in real-time by watching how fast those state populations decay or oscillate.

Mira: The model they use, which is described by the effective non-Hermitian Hamiltonian PT = X sum i=one N one over two i x - i i z + X N i>j V ij i x j x + i y j y, which is what's shown on page two right <ref:2111.04378#pg1>?

Lev: Yeah, that Hamiltonian has the PT symmetry because the eigenvalues are real when the symmetry is preserved >

Kai: The critical values that limit those PT-unbroken and PT-broken phases are what they call exceptional points, and regions with a continuous set of these points are called exceptional lines.

Mira: So, determining these phase diagrams involves probing the system’s dynamics and reconstructing the eigenenergies to see where those limits lie.

Lev: The PT-symmetry phase diagrams get much richer as you increase N, for example, in the linear chain configuration, they show more regions when you look at N=three and N=four particles <ref:2111.04378#pg3,N=3$ and $N=4$ particles>.

Kai: For single particle systems alone, PT-symmetry is preserved when the ratio of coupling strength to decay rate is greater than one, but it breaks when that ratio is imaginary.

Paper summary: Mira: The paper shows that the PT-phase transitions occur right along these exceptional lines we just mentioned.

Lev: They also look at zigzag configurations, where the phase diagrams depend on an angle alpha because of how those interaction strengths V ij depend on it, as shown in Figure three <ref:2111.04378#pg3>.

Kai: And when they look at those zigzag configurations, for instance in the N=three and N=four cases with / = ten the zigzag configuration becomes an equilateral triangle when alpha equals pi over three, which increases the symmetry of the system > <ref:2111.04378#pg3,the zigzag configuration becomes an equilateral triangle>

Mira: That's a nice detail because it shows how geometric arrangement directly influences whether you have preserved or broken symmetry in this context.

Lev: The way they modulate parameters like, V, or periodically with some frequency omega F can make things even more interesting for the phase diagrams.

Kai: When you modulate the parameters, the eigenvalues of PT become time dependent, and you check if the eigenvalues of the time-evolution operator G PT(t) have a unitary modulus at two pi omega F >

Mira: Modulation enriches these phase diagrams because new PT-broken regions actually appear when the unmodulated model is sitting in a PT-symmetric phase.

Lev: So, if you're building this on real hardware, you need to consider that running this dynamics requires handling all those loss channels and decay rates accurately, which adds overhead to any error correction scheme >

Kai: Exactly. And the paper points out a limitation: they show these results for realistic parameters but they also note that the method itself is an effective model, so it's not a perfect description of every single thing happening in a real physical setup >

Mira: They mention that systems like Rydberg atoms are particularly well suited because you can tune the coupling between 1i and 2i by changing microwave field intensity, or change the interaction strength V by adjusting the principal quantum number of the Rydberg states or even just moving the ions > <ref:2111.04378#pg1>

Lev: And they show how you can tune that interaction strength V, for example, between zero and approximately two pi times ten MHz by changing those factors >

Kai: So, this means experimentalists have concrete ways to dial in those parameters to explore these phase diagrams directly on platforms like Rydberg atoms or ions.

Mira: The conclusion of the paper is that platforms with a few particles and long-range binary interactions are excellent places to study these PT-symmetry phase transitions >

Lev: It’s a good playground for studying effective single-particle models in a many-body environment, and it has direct applications for quantum simulations and even quantum information processing >

Kai: So, what this means for us is that we have a clear path to test these theoretical concepts using these controllable physical systems to see how the dynamics manifest across different interaction geometries.

Conclusion: Kai: So, we're wrapping up this look at "Non-Hermitian dynamics and PT-symmetry breaking in interacting mesoscopic Rydberg platforms" by those authors.

Mira: They essentially show how you can map these complex particle interactions onto a non-Hermitian spin model that has PT symmetry, and they found rich phase diagrams with both symmetric and broken phases.

Lev: It’s interesting because they aren't just doing pure theory; they're showing how this maps onto systems we can actually build, like Rydberg atoms or ions.

Kai: Exactly. The implication here is that PT symmetry isn't just some abstract math thing; it's something you can observe in a real experimental setup where you control the interactions directly.

Mira: They highlight how tuning parameters like the coupling strength and the interaction V lets you navigate these phase transitions, finding new regions where symmetry breaks when it wasn't there before.

Lev: From what I see, if we were to try and run this on actual hardware, we’d need to worry about those decay rates one and two which the model has built in.

Kai: Right. So the authors suggest that studying the dynamics—how fast those populations decay or oscillate—is a direct way to probe if you're in a symmetric or broken phase.

Mira: And they point out that this setup, with long-range interactions, allows for new PT-symmetric regions to pop up when you increase the number of particles N.

Lev: That tells us that adding more interacting components changes the physics significantly and creates these new transition points along those exceptional lines.

Kai: So, these findings suggest that using controllable few-particle systems with long-range forces is a really good place to explore how non-Hermitian physics manifests in many-body environments.

Mira: And they confirm that this framework has direct applications for simulating complex quantum processes and even for designing quantum information protocols.

Lev: It opens the door for us to think about using these non-Hermitian dynamics not just as a curiosity, but as a way to model real lossy or dissipative systems.

Kai: We'll be looking at how this connects to our next segment on how we actually build and measure these specific Rydberg platforms.

José A. S. LourenÃo, Gerard Higgins, Chi Zhang, Markus Hennrich, Tommaso Macrǭ

Departamento de Física Teƙrica e Experimental, Universidade Federal do Rio Grande do Norte · Department of Physics, Stockholm University

cond-mat.quant-gas, quant-ph

Submitted: 2021-11-08

Updated: 2021-11-09

Journal ref: Phys. Rev. A 106, 023309 (2022)

DOI: 10.1103/PhysRevA.106.023309

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

Importance score: 80/100

The gist: The gist The authors simulate dissipative dynamics of a mesoscopic system of long-range interacting particles mapped into non-Hermitian spin models with PT symmetry, finding rich PT-phase diagrams

Key concepts

$\\mathcal{PT}$ Symmetry: This is a specific type of symmetry in physics where a mathematical transformation (parity ($\\mathcal{P}$) and time reversal ($\\mathcal{T}$)) leaves the physical laws unchanged. In this context, it means the system's energy eigenvalues remain real when this symmetry is present, which is crucial for defining the stable 'symmetric' phase.
Non-Hermitian Hamiltonian: This is a mathematical description of a quantum system where the energy operator (Hamiltonian) does not have to be Hermitian. Unlike standard quantum mechanics where observables must be real, non-Hermitian systems can still exhibit real eigenvalues under specific symmetry conditions, allowing for the study of $\mathcal{PT}$ symmetry.
Exceptional Points: These are special parameter values in a system where both the eigenvalues and their corresponding eigenvectors coalesce. In $\mathcal{PT}$-symmetric systems, these points mark the boundaries between different phases—specifically, where the system transitions from having real eigenvalues (stable phase) to complex ones (broken phase).
Rydberg Atoms/Ions: These are highly excited atoms or ions used as physical platforms for quantum simulation. They are ideal because they can be precisely controlled using microwave fields to tune coupling strengths and exhibit strong, tunable long-range dipolar interactions, making them perfect testbeds for these complex many-body models.

Terminology

Phase Transition — 2× in this paper · explained in 2 episodes
Phase Transitions — 2× in this paper · explained in 2 episodes
Dynamical regimes — 1× in this paper · explained in 2 episodes
Symmetry Breaking — 1× in this paper · explained in 3 episodes

Summary

The gist The authors simulate dissipative dynamics of a mesoscopic system of long-range interacting particles mapped into non-Hermitian spin models with PT symmetry, finding rich PT-phase diagrams with PT-symmetric and PT-broken phases.

Model Description

The system is modeled by two internal levels for each particle, labeled 1i and 2i>, which are Rabi-coupled with coupling strength omega and decay to a third state 0i with rates Γ1 and Γ2 respectively The particles interact via long-range exchange interactions, where the interaction strength Vij can be assumed to be a e.g. dipolar (∝ i − j −3) interaction The effective non-Hermitian Hamiltonian is given by Hˆeff = X N i=1 1/2 omegaˆσ i x − iΓˆσ i z − i Γ1 + Γ2 / 2 I i + X N i>j Vij 2 σˆi xσˆj x + σˆi yσhat j y The PT-symmetric non-Hermitian Hamiltonian is H PT = X N i=1 1/2 omegaˆσ i x − iΓˆσ i z + X N i>j Vij 2 σˆi xσhat j x + σhat i yσhat j y.

PT Symmetry Analysis

The PT symmetry of H PT is determined by whether it has real eigenvalues, as the PT symmetry is preserved when H PT has real eigenvalues The critical values that limit the PT-unbroken and PT-broken phases are the exceptional points Also, the existence of a continuous set of exceptional points that limit regions with PT-unbroken and PT-broken phases are called exceptional lines. The PT symmetry phase diagrams can be determined by probing the system’s dynamics and reconstructing the system’s eigenenergies.

PT Phase Diagrams

The PT-symmetry phase diagrams become richer with increasing N, as seen in the linear chain of particles where more regions are present in the N = 3 and N = 4 cases For single particle systems, PT-symmetry is preserved when omega/Γ > 1 and broken when it is imaginary. The PT-phase transitions occur along exceptional lines. In zigzag configurations, the PT-phase diagrams depend on the angle α as a result of the interaction strengths Vij dependence on α.

Dynamics and Experimental Probing

The PT symmetry of a quantum system can be probed by measuring the system’s dynamics The dynamics are understood through the density matrix ρ(t) = ψ0(t)ihψ0(t), where the state ψ0(t)i = exp−iHˆPT t1i. In the PT-symmetric phase, the eigenvalues are real and the elements of the density matrix represent physical populations. When the system parameters are set into the PT-broken region, physical populations rapidly decay depopulating the Rydberg states.

Experimental Implementation

Systems of Rydberg atoms and systems of Rydberg ions are well suited for implementing the Hamiltonian in Eq. 1 because they can be exquisitely controlled and exhibit strong dipolar interactions For a Rydberg system, the coupling strength omega between 1i and 2i can be tuned between 0 and 2π × 500 MHz by changing the intensity of the coupling microwave field. The interaction strength V can be tuned between 0 and ≈ 2π×10 MHz by changing either the principal quantum number of the Rydberg states, by changing the distance between the ions, else by changing the detuning of the microwave field from the S1/2 ↔ P1/2 resonance.

Modulation Effects

The PT-symmetry phase diagrams can be enriched further by periodically modulating parameters like omega, V or Γ with a characteristic frequency ωF. When the parameters are modulated, the eigenvalues of H PT become time dependent, and PT symmetry is assessed by checking if the eigenvalues of the time-evolution operator G PT (t) have unitary modulus at 2πωF. Modulation enriches phase diagrams because new PT-broken regions appear when the unmodulated model is in the PT-symmetric phase.

Conclusion

In conclusion, platforms with few particles with long-range binary interactions are an excellent playground for studying PT-symmetry phase transitions The investigation generalizes previous theoretical and experimental research on effective single-particle models to a many-body environment with direct applications to quantum simulations and quantum information processing.

Acknowledgments

The authors acknowledge stimulating discussions with Y. Pará This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), by the Knut & Alice Wallenberg Foundation (through the Wallenberg Centre for Quantum Technology [WACQT]) and by the CAPES-STINT project Strong correlations in Cavity and Ion Quantum Simulators.

Improvements for AI systems

A PT-symmetry probing method for quantum systems can be implemented by measuring the system’s dynamics to determine whether PT is preserved or broken based on whether the eigenvalues of HˆPT are real.
The improved AI system can accurately predict the dynamical regimes of mesoscopic systems by mapping them onto non-Hermitian spin models and analyzing their PT -phase diagrams, allowing for the identification of PT -symmetric and PT -broken phases.
The AI can determine the PT-symmetry phase based on experimental observables by calculating population dynamics, specifically accessing the populations of the Rydberg states to reconstruct the system’s eigenenergies.
The system can simulate complex many-body dynamics, including those with long-range interacting particles, enabling the study of how interactions and geometric arrangement of coupled spins enrich the PT phase diagrams.
The AI can predict phase transitions by modulating system parameters, as demonstrated by finding that the PT nonequilibrium dynamical phase diagrams can be further enriched when parameters like dissipation strength or coupling strength are varied.