Hamiltonian Description Research Articles

Automated Discovery of Autonomous Quantum Error Correction Schemes

We can encode a qubit in the energy levels of a quantum system. Relaxation and other dissipation processes lead to decay of the fidelity of this stored information. Is it possible to preserve the quantum information for a longer time by introducing additional drives and dissipation? The existence of autonomous quantum error correcting codes answers this question in the positive. Nonetheless, discovering these codes for a real physical system, i.e., finding the encoding and the associated driving fields and bath couplings, remains a challenge that has required intuition and inspiration to overcome. In this work, we develop and demonstrate a computational approach based on adjoint optimization for discovering autonomous quantum error correcting codes given a Hamiltonian description of a physical system. We implement an optimizer that searches for a logical subspace and control parameters to better preserve quantum information. We demonstrate our method on a system of a harmonic oscillator coupled to a lossy qubit, and find that varying the Hamiltonian distance in Fock space—a proxy for the control hardware complexity—leads to discovery of different and new error correcting schemes. We discover what we call the 3 code, realizable with a Hamiltonian distance d=2, and propose a hardware-efficient implementation based on superconducting circuits.1 MoreReceived 11 August 2021Revised 4 December 2021Accepted 7 March 2022DOI:https://doi.org/10.1103/PRXQuantum.3.020302Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAdiabatic quantum optimizationMachine learningQuantum error correctionQuantum information processing with continuous variablesPhysical SystemsSuperconducting qubitsQuantum Information

Spontaneous radiation of black holes

We provide an explicitly hermitian hamiltonian description for the spontaneous radiation of black holes, which is a many-level, multiple-degeneracy generalization of the usual Janeys-Cummings model for two-level atoms. We show that under single-particle radiation and standard Wigner-Wiesskopf approximation, our model yields exactly thermal type power spectrum as hawking radiation requires. While in the many-particle radiation cases, numeric methods allow us to follow the evolution of microscopic state of a black hole exactly, from which we can get the firstly increasing then decreasing entropy variation trend for the radiation particles just as the Page-curve exhibited. Basing on this model analysis, we claim that two ingredients are necessary for resolutions of the information missing puzzle, a spontaneous radiation like mechanism for the production of hawking particles and proper account of the macroscopic superposition happening in the full quantum description of a black hole radiation evolution and, the working logic of replica wormholes is an effect account of this latter ingredient.As the basis for our interpretation of black hole Hawking radiation as their spontaneous radiation, we also provide a fully atomic like inner structure models for their microscopic states definition and origins of their Bekenstein-Hawking entropy, that is, exact solution families to the Einstein equation sourced by matter constituents oscillating across the central point and their quantization. Such a first quantization model for black holes' microscopic state is non necessary for our spontaneous radiation description, but has advantages comparing with other alternatives, such as string theory fuzzball or brick wall models.

The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space

Helicity, a topological degree that measures the winding and linking of vortex lines, is preserved by ideal (barotropic) fluid dynamics. In the context of the Hamiltonian description, the helicity is a Casimir invariant characterizing a foliation of the associated Poisson manifold. Casimir invariants are special invariants that depend on the Poisson bracket, not on the particular choice of the Hamiltonian. The total mass (or particle number) is another Casimir invariant, whose invariance guarantees the mass (particle) conservation (independent of any specific choice of the Hamiltonian). In a kinetic description (e.g., that of the Vlasov equation), the helicity is no longer an invariant (although the total mass remains a Casimir of the Vlasov’s Poisson algebra). The implication is that some “kinetic effect” can violate the constancy of the helicity. To elucidate how the helicity constraint emerges or submerges, we examine the fluid reduction of the Vlasov system; the fluid (macroscopic) system is a “sub-algebra” of the kinetic (microscopic) Vlasov system. In the Vlasov system, the helicity can be conserved if a special helicity symmetry condition holds. To put it another way, breaking helicity symmetry induces a change in the helicity. We delineate the geometrical meaning of helicity symmetry and show that for a special class of flows (the so-called epi-two-dimensional flows), the helicity symmetry is written as ∂γ = 0 for a coordinate γ of the configuration space.

Beyond the Tavis-Cummings model: Revisiting cavity QED with ensembles of quantum emitters

The interaction of an ensemble of $N$ two-level atoms with a single mode electromagnetic field is described by the Tavis-Cummings model. There, the collectively enhanced light-matter coupling strength is given by $g_N = \sqrt{N} \bar{g}_1$, where $\bar{g}_1$ is the ensemble-averaged single-atom coupling strength. Formerly, this model has been employed to describe and to analyze numerous cavity-based experiments. Here, we show that this is only justified if the effective scattering rate into non-cavity modes is negligible compared to the cavity's free-spectral range. In terms of experimental parameters, this requires that the optical depth of the ensemble is low, a condition that is violated in several state-of-the-art experiments. We give quantitative conditions for the validity of the Tavis-Cummings model and derive a more general Hamiltonian description that takes into account the cascaded interaction of the photons with all consecutive atoms. We show that the predictions of our model can differ quantitatively and even qualitatively from those obtained with the Tavis-Cummings model. Finally, we present experimental data, for which the deviation from the predictions of the Tavis-Cummings model is apparent. Our findings are relevant for all experiments in which optically dense ensembles of quantum emitters are coupled to an optical resonator.

Model-based two-layer control design for optimal power management in wind-battery microgrids

In this paper, a comprehensive model in Hamiltonian form of a Microgrid (MG) composed of heterogeneous components, i.e. wind turbine generator, battery storage and local loads together with their power conversion units, is developed. The proposed model analytically captures the energy conversion capabilities of different sustainable energy sources. Based on this model description, novel primary (nonlinear PI) and secondary controllers (receding horizon) are proposed that ensure boundedness of the currents injected by each energy source and optimal power management operation of the entire MG. Furthermore, closed-loop stability analysis is rigorously proven for both primary and secondary control loops taking into account the accurate Hamiltonian description of the whole MG that includes the energy conversion characteristics. Detailed simulation results of the entire MG connected to a weak grid and operating in islanded mode are provided to validate the proposed model, the control design and the stability analysis under various scenarios.

Distribution of energy in propagation for ocean extreme wave generation in hydrodynamics laboratory

BACKGROUND AND OBJECTIVES: The hydrodynamic uncertainty of the ocean is the reason for testing marine structures as an initial consideration. This uncertainty has an impact on the natural structure of the topography as well as marine habitats. In the hydrodynamics laboratory, ships and offshore structures are tested using mathematical models as input to the wave marker. For large wavenumbers, Benjamin Bona Mahony's equation has a stable direction and position in the wave tank. During their propagation, the generated waves exhibit modulation instability and phase singularity phenomena. These two factors refer to Benjamin Bona Mahony as a promising candidate for generating extreme waves in the laboratory. The aim of this research is to investigate the distribution of energy in each modulation frequency change. The Hamiltonian formula that describes the phenomenon of phase singularity is used to observe energy. This data is critical in determining the parameters used to generate extreme waves.METHODS: The envelope of the Benjamin Bona Mahony wave group can be used to study the Benjamin Bona Mahony wave. The Benjamin Bona Mahony wave group is known to evolve according to the Nonlinear Schrodinger equation. The Hamiltonian governs the dynamics of the phase amplitude and proves the Nonlinear Schrodinger equation's singularity for finite time. The Hamiltonian is derived from the appropriate Lagrangian for Nonlinear Schrodinger and then transformed into the Hamiltonian with the displaced phase-amplitude variable.FINDINGS: Potential energy is related to wave amplitude and kinetic energy is related to wave steepness in the study of surface water waves. When , the maximum wave amplitude and steepness are obtained. When , extreme waves cannot be formed due to steepness. This is due to the possibility of breaking waves into smaller waves on the shore. In terms of position, the energy curve is symmetrical.CONCLUSION: According to Hamiltonian's description of the energy distribution, the smaller the modulation frequency, the greater the potential and kinetic energy involved in wave propagation, and vice versa. While the wave's amplitude and steepness will be greatest for a low modulation frequency, and vice versa. The modulation frequency considered as an extreme wave generator is , because the resulting amplitude is quite high and the energy in the envelope is also quite large.

Hamiltonian description of internal ocean waves with Coriolis force

<p style="text-indent:20px;">The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a <em>nearly</em> Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.</p>

Pressure wave characteristics in a bubble-liquid mixture via Kudryashov–Sinelshchikov equation

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system.

Small oscillations of a vortex ring: Hamiltonian formalism and quantization

This article investigates small oscillations of a vortex ring with zero thickness that evolves under the Local Induction Equation (LIE). We deduce the differential equation that describes the dynamics of these oscillations. We suggest the new approach to the Hamiltonian description of this dynamic system. This approach is based on the extension of the set of dynamical variables by adding the circulation Γ as a dynamical variable. The constructed theory is invariant under the transformations of the Galilei group. The appearance of this group allows for a new viewpoint on the energy of a vortex filament with zero thickness. We quantize this dynamical system and calculate the spectrum of the energy and acceptable circulation values. The physical states of the theory are constructed with help of coherent states for the Heisenberg–Weyl group.

Detecting two photons with one molecule

We apply input-output theory with quantum pulses [AH Kiilerich, K M\o lmer, Phys. Rev. Lett. {\bf 123}, 123604 (2019)] to a model of a new type of two-photon detector consisting of one molecule that can detect two photons arriving sequentially in time. The underlying process is distinct from the usual two-photon absorption process where two photons arriving simultaneously and with frequencies adding up to the resonance frequency are absorbed by a single molecule in one quantum jump. Our detector model includes a Hamiltonian description of the amplification process necessary to convert the microscopic change in the single molecule to a macroscopic signal.

Contact geometry and quantum thermodynamics of nanoscale steady states

We develop a geometric formalism suited for describing the quantum thermodynamics of a certain class of nanoscale systems (whose density matrix is expressible in the McLennan–Zubarev form) at any arbitrary non-equilibrium steady state. It is shown that the non-equilibrium steady states are points on control parameter spaces which are in a sense generated by the steady state Massieu–Planck function. By suitably altering the system’s boundary conditions, it is possible to take the system from one steady state to another. We provide a contact Hamiltonian description of such transformations and show that moving along the geodesics of the friction tensor results in a minimum increase of the free entropy along the transformation. The control parameter space is shown to be equipped with a natural Riemannian metric that is compatible with the contact structure of the quantum thermodynamic phase space which when expressed in a local coordinate chart, coincides with the Schlögl metric. Finally, we show that this metric is conformally related to other thermodynamic Hessian metrics which might be written on control parameter spaces. This provides various alternate ways of computing the Schlögl metric which is known to be equivalent to the Fisher information matrix.

Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime.

Three Roads to the Geometric Constraint Formulation of Gravitational Theories with Boundaries

The Hamiltonian description of mechanical or field models defined by singular Lagrangians plays a central role in physics. A number of methods are known for this purpose, the most popular of them being the one developed by Dirac. Here, we discuss other approaches to this problem that rely on the direct use of the equations of motion (and the tangency requirements characteristic of the Gotay, Nester and Hinds method), or are formulated in the tangent bundle of the configuration space. Owing to its interesting relation with general relativity we use a concrete example as a test bed: an extension of the Pontryagin and Husain–Kuchař actions to four dimensional manifolds with boundary.

Hamiltonian and exclusion statistics approach to discrete forward-moving paths.

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an alternative, simpler form for its logarithm that makes its polynomial structure explicit.

Contact Hamiltonian Description of 1D Frictional Systems

In this paper, we consider contact Hamiltonian description of 1D frictional dynamics with no conserved force. Friction forces that are monomials of velocity, and sum of two monomials are considered. For that purpose, quite general forms of contact Hamiltonians are taken into account. We conjecture that it is impossible to give a contact Hamiltonian description dissipative systems where the friction force is not in the form considered in this paper.

Response theory for nonequilibrium steady states of open quantum systems

We introduce a response theory for open quantum systems within nonequilibrium steady-states subject to a Hamiltonian perturbation. Working in the weak system-bath coupling regime, our results are derived within the Lindblad-Gorini-Kossakowski-Sudarshan formalism. We find that the response of the system to a small perturbation is not simply related to a correlation function within the system, unlike traditional linear response theory in closed systems or expectations from the fluctuation-dissipation theorem. In limiting cases, when the perturbation is small relative to the coupling to the surroundings or when it does not lead to a change of the eigenstructure of the system, a perturbative expansion exists where the response function is related to a sum of a system correlation functions and additional forces induced by the surroundings. Away from these limiting regimes however, the secular approximation results in a singular response that cannot be captured within the traditional approach, but can be described by reverting to a microscopic Hamiltonian description. These findings are illustrated by explicit calculations in coupled qubits and anharmonic oscillators in contact with bosonic baths at different temperatures.

Generalized Holographic Principle, Gauge Invariance and the Emergence of Gravity à la Wilczek

We show that a generalized version of the holographic principle can be derived from the Hamiltonian description of information flow within a quantum system that maintains a separable state. We then show that this generalized holographic principle entails a general principle of gauge invariance. When this is realized in an ambient Lorentzian space-time, gauge invariance under the Poincaré group is immediately achieved. We apply this pathway to retrieve the action of gravity. The latter is cast à la Wilczek through a similar formulation derived by MacDowell and Mansouri, which involves the representation theory of the Lie groups SO(3,2) and SO(4,1).

Density-functional-theory approach to the Hamiltonian adaptive resolution simulation method

In the Hamiltonian adaptive resolution simulation method (H–AdResS) it is possible to simulate coexisting atomistic (AT) and ideal gas representations of a physical system that belong to different subdomains within the simulation box. The Hamiltonian includes a field that bridges both models by smoothly switching on (off) the intermolecular potential as particles enter (leave) the AT region. In practice, external one-body forces are calculated and applied to enforce a reference density throughout the simulation box, and the resulting external potential adds up to the Hamiltonian. This procedure suggests an apparent dependence of the final Hamiltonian on the system’s thermodynamic state that challenges the method’s statistical mechanics consistency. In this paper, we explicitly include an external potential that depends on the switching function. Hence, we build a grand canonical potential for this inhomogeneous system to find the equivalence between H–AdResS and density functional theory (DFT). We thus verify that the external potential inducing a constant density profile is equal to the system’s excess chemical potential. Given DFT’s one-to-one correspondence between external potential and equilibrium density, we find that a Hamiltonian description of the system is compatible with the numerical implementation based on enforcing the reference density across the simulation box. In the second part of the manuscript, we focus on assessing our approach’s convergence and computing efficiency concerning various model parameters, including sample size and solute concentrations. To this aim, we compute the excess chemical potential of water, aqueous urea solutions and Lennard–Jones (LJ) mixtures. The results’ convergence and accuracy are convincing in all cases, thus emphasising the method’s robustness and capabilities.

Open systems in classical mechanics

Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories LagSy and HamSy that, respectively, provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of LagSy and HamSy correspond to such open systems, and the composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives rise to a functor from LagSy to HamSy that translates from the Lagrangian to the Hamiltonian perspective.

Understanding radiative transitions and relaxation pathways in plexcitons

Summary Molecular aggregates on plasmonic nanoparticles have emerged as attractive systems for the studies of polaritonic light-matter states, called plexcitons. Such systems are tunable, scalable, easy to synthesize, and offer sub-wavelength confinement, all while giving access to the ultrastrong light-matter coupling regime, promising a plethora of applications. However, the complexity of these materials prevented the understanding of their excitation and relaxation phenomena. Here, we follow the relaxation pathways in plexcitons and conclude that while the metal destroys the optical coherence, the molecular aggregate coupled to surface processes significantly contributes to the energy dissipation. We use two-dimensional electronic spectroscopy with theoretical modeling to assign the different relaxation processes to either molecules or metal nanoparticle. We show that the dynamics beyond a few femtoseconds has to be considered in the language of hot electron distributions instead of the accepted lower and upper polariton branches and establish the framework for further understanding.