Papers by Alexander Karabanov
Известия Коми научного центра Уральского отделения Российской академии наук, Sep 20, 2023
The Lax equations dL/dt = [M, L] play an important role in the integrability theory of nonlinear ... more The Lax equations dL/dt = [M, L] play an important role in the integrability theory of nonlinear evolution equations and quantum dynamics. In this work, tensor extensions of the Lax equations are suggested with M : V → V and L : T k (V) → V , k = 1, 2,. . ., on a complex vector space V. These extensions belong to the generalised class of Lax equations (introduced earlier by Bordemann) dL/dt = ρ k (M)L where ρ k is a representation of a Lie algebra. The case k = 1, ρ 1 = ad corresponds to the usual Lax equations. The extended Lax pairs are studied from the point of view of isomorphic deformations of multilinear structures, conservation laws, exterior algebras and cochain symmetries. Аннотация Уравнения Лакса dL/dt = [M, L] играют важную роль в теории интегрируемости нелинейных эволюционных уравнений и квантовой динамике. В данной работе предлагаются тензорные расширения уравнений Лакса с M : V → V и L : T k (V) → V , k = 1, 2,. .. на комплексном векторном пространстве V. Эти расширения относятся к обобщенному классу уравнений Лакса (введенному ранее Бордеманном) dL/dt = ρ k (M)L, где ρ kпредставление алгебры Ли. Случай k = 1, ρ 1 = ad соответствует обычным уравнениям Лакса. Расширенные пары Лакса изучаются с точки зрения изоморфных деформаций полилинейных структур, законов сохранения, внешних алгебр и коцепных симметрий.
Post-acquisition correction of NMR spectra distorted by dynamic and static field inhomogeneity of cryogen-free magnets
Journal of Magnetic Resonance
Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences
Lie algebras a with a complex underlying vector space V are studied that are automorphic with res... more Lie algebras a with a complex underlying vector space V are studied that are automorphic with respect to a given linear dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂ Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a Lie algebraic structure to the vector space of trajectories of the group Gt. The basics of the general structure of automorphic algebras a are described in terms of the eigenspace decomposition of the operatorM ∈ der(a) that determines the dynamics. Symmetries encoded by the presence of nonabelian automorphic algebras are pointed out connected to conservation laws, spectral relations and root systems. It is shown that, for a given dynamics Gt, automorphic algebras can be found via a limit transition in the space of Lie algebras on V along the trajectories of the group Gt itself. This procedure generalises the well-known Inönü-Wigner contraction and links adjoint representations of automorphic algebras to isospectral Lax representations on gl(V ). These resu...
Physical Review Letters, 2004
A method is presented for the coherent control of two-level systems when T 2 relaxation is signif... more A method is presented for the coherent control of two-level systems when T 2 relaxation is significant. The Bloch equations are rewritten as an equation of motion of the stereographic projection, ÿ, of the spin vector. This allows a Schur-type iteration used for the design of shaped pulses in magnetic resonance and coherent optics to be extended to include the effect of T 2. In general, the effect of T 2 on ÿ cannot be completely compensated for, although in practice it can be to a high degree. An example is presented of a driving field that produces a coherent superposition (no on-diagonal elements of the density matrix) over a chosen band of frequencies, in the presence of relaxation.
Cryogen-Free 400 Mhz (9.4 T) Solid State Mas Nmr System with Liquid State Nmr Potential
SSRN Electronic Journal
Physical Review A
A method based on spectral Green's functions is presented for the simulation of driven open quant... more A method based on spectral Green's functions is presented for the simulation of driven open quantum dynamics that can be described by the Lindblad master equation in Liouville density operator space. The method extends the Hilbert space formalism and provides simple algebraic connections between the driven and nondriven dynamics in the spectral frequency domain. The formalism shows remarkable analogies to the use of Green's functions in quantum field theory, such as the elementary excitation energies and the Dyson self-energy equation. To demonstrate its potential, we apply the method to a coherently driven dissipative ensemble of two-level systems comprising a single "active" subsystem interacting with N "passive" subsystems-a generic model with important applications in quantum optics and dynamic nuclear polarization. The method dramatically reduces the computational cost compared with simulations based on solving the full master equation, thus making it possible to study and optimize many-body correlated states up to the physically realistic limit of an arbitrarily large N.
A novel method based on spectral Green functions is presented for the simulation of driven open q... more A novel method based on spectral Green functions is presented for the simulation of driven open quantum dynamics that can be described by the Lindblad master equation in Liouville density operator space. The method extends the Hilbert space formalism and provides simple algebraic connections between the driven and non-driven dynamics in the spectral frequency domain. The formalism shows remarkable analogies to the use of Green functions in quantum field theory such as the elementary excitation energies and the Dyson self-energy equation. To demonstrate its potential, we apply the novel method to a coherently driven dissipative ensemble of 2-level systems comprising a single “active” subsystem interacting with N identical “passive” subsystems — a generic model with important applications in quantum optics and dynamic nuclear polarization. The novel method dramatically reduces computational cost compared with simulations based on solving the full master equation, thus making it possib...
Theory of solid effect dynamic nuclear polarization based on adiabatic elimination in Liouville space
Physical Chemistry Chemical Physics, 2020
The study reveals heteronuclear-thermal mixing – a novel mechanism of dynamic nuclear polarizatio... more The study reveals heteronuclear-thermal mixing – a novel mechanism of dynamic nuclear polarization in a system with 1H and 19F nuclei.
Regular and Chaotic Dynamics, 2019
We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in exten... more We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.
Physical Review A, 2018
Dynamic nuclear polarization (DNP) is an out-of-equilibrium method for generating non-thermal spi... more Dynamic nuclear polarization (DNP) is an out-of-equilibrium method for generating non-thermal spin polarization which provides large signal enhancements in modern diagnostic methods based on nuclear magnetic resonance. A particular instance is cross effect DNP, which involves the interaction of two coupled electrons with the nuclear spin ensemble. Here we develop a theory for this important DNP mechanism and show that the non-equilibrium nuclear polarization build-up is effectively driven by three-body incoherent Markovian dissipative processes involving simultaneous state changes of two electrons and one nucleus. Our theoretical approach allows for the first time simulations of the polarization dynamics on an individual spin level for ensembles consisting of hundreds of nuclear spins. The insight obtained by these simulations can be used to find optimal experimental conditions for cross effect DNP and to design tailored radical systems that provide optimal DNP efficiency.
Physical Review Letters, 2017
We study an ensemble of strongly coupled electrons under continuous microwave irradiation interac... more We study an ensemble of strongly coupled electrons under continuous microwave irradiation interacting with a dissipative environment, a problem of relevance to the creation of highly polarized non-equilibrium states in nuclear magnetic resonance. We analyse the stationary states of the dynamics, described within a Lindblad master equation framework, at the mean-field approximation level. This approach allows us to identify steady state phase transitions between phases of high and low polarization controlled by the distribution of disordered electronic interactions. We compare the mean-field predictions to numerically exact simulations of small systems and find good agreement. Our study highlights the possibility of observing collective phenomena, such as metastable states, phase transitions and critical behaviour in appropriately designed paramagnetic systems. These phenomena occur in a low-temperature regime which is not theoretically tractable by conventional methods, e.g., the spin-temperature approach.
Physical Chemistry Chemical Physics, 2016
A quantum theory of thermal mixing dynamic nuclear polarisation in solids is developed displaying... more A quantum theory of thermal mixing dynamic nuclear polarisation in solids is developed displaying a good agreement with experiments.
Journal of magnetic resonance (San Diego, Calif. : 1997), 2016
We discuss the polarization dynamics during solid effect dynamic nuclear polarization (DNP) in a ... more We discuss the polarization dynamics during solid effect dynamic nuclear polarization (DNP) in a central spin model that consists of an electron surrounded by many nuclei. To this end we use a recently developed formalism and validate first its performance by comparing its predictions to results obtained by solving the Liouville von Neumann master equation. The use of a Monte Carlo method in our formalism makes it possible to significantly increase the number of spins considered in the model system. We then analyse the dependence of the nuclear bulk polarization on the presence of nuclei in the vicinity of the electron and demonstrate that increasing the minimal distance between nuclei and electrons leads to a rise of the nuclear bulk polarization. These observations have implications for the design of radicals that can lead to improved values of nuclear spin polarization. Furthermore, we discuss the potential to extend our formalism to more complex spin systems such as cross effect...
Physical Review Letters, 2015
Dynamic nuclear polarization (DNP) is a promising strategy for generating a significantly increas... more Dynamic nuclear polarization (DNP) is a promising strategy for generating a significantly increased non-thermal spin polarization in nuclear magnetic resonance (NMR) applications thereby circumventing the need for strong magnetic fields. Although much explored in recent experiments, a detailed theoretical understanding of the precise mechanism behind DNP is so far lacking. We address this issue by theoretically investigating solid effect DNP in a system where a single electron is coupled to an ensemble of interacting nuclei and which can be microscopically modelled by a quantum master equation. By deriving effective equations of motion that govern the polarization dynamics we show analytically that DNP can be understood as kinetically constrained spin diffusion. On the one hand this approach provides analytical insights into the mechanism and timescales underlying DNP. On the other hand it permits the numerical study of large ensembles which are typically intractable from the perspective of a quantum master equation. This paves the way for a detailed exploration of DNP dynamics which might form the basis for future NMR applications.
To the theory of coupled oscillations passing through the resonance
Regular and Chaotic Dynamics, 2006
ABSTRACT The problem of coupled oscillations is considered in the case when a stable equilibrium ... more ABSTRACT The problem of coupled oscillations is considered in the case when a stable equilibrium of a globally averaged system passes through a resonance curve. Questions of persistence of invariant tori and transition to a self-synchronization are particularly discussed.
Physical Chemistry Chemical Physics, 2012
A strategy is described for simulations of solid effect dynamic nuclear polarisation that reduces... more A strategy is described for simulations of solid effect dynamic nuclear polarisation that reduces substantially the dimension of the quantum mechanical problem. Averaging the Hamiltonian in the doubly rotating frame is used to confine the active space to the zero quantum coherence subspace. A further restriction of the Liouville space is made by truncating higher spin order states, which are weakly populated due to the presence of relaxation processes. Based on a dissipative transport equation, which is used to estimate the transport of the magnetisation starting from single spin order to higher spin order states, a minimal spin order for the states is calculated that needs to be taken into account for the spin dynamics simulation. The strategy accelerates individual spin calculations by orders of magnitude, thus making it possible to simulate the polarisation dynamics of systems with up to 25 nuclear spins.
Molecular Physics, 2014
Relaxation plays a crucial role in the spin dynamics of dynamic nuclear polarisation. We review h... more Relaxation plays a crucial role in the spin dynamics of dynamic nuclear polarisation. We review here two different strategies that have recently been used to incorporate relaxation in models to predict the spin dynamics of solid effect dynamic nuclear polarisation. A detailed explanation is provided how the Lindblad-Kossakowski form of the master equation can be used to describe relaxation in a spin system. Fluctuations of the spin interactions with the environment as a cause of relaxation are discussed and it is demonstrated how the relaxation superoperator acting in Liouville space on the density operator can be derived in the Lindblad-Kossakowski form by averaging out non-secular terms in an appropriate interaction frame. Furthermore we provide a formalism for the derivation of the relaxation superoperator starting with a choice of a basis set in Hilbert space. We show that the differences in the prediction of the nuclear polarisation dynamics that are found for certain parameter choices arise from the use of different interaction frames in the two different strategies. In addition we provide a summary of different relaxation mechanism that need to be considered to obtain more realistic spin dynamic simulations of solid effect dynamic nuclear polarisation.
Spectral Resolution and Inversion of the Bloch Equations with Relaxation
Letters in Mathematical Physics, 2007
ABSTRACT It is demonstrated that the linear Bloch equations, describing near-resonant excitation ... more ABSTRACT It is demonstrated that the linear Bloch equations, describing near-resonant excitation of two-level media with relaxation, can be resolved into a 3n-dimensional nonlinear system associated with a special spectral problem, generalizing the classical Zakharov–Shabat spectral problem. Remarkably, for n = 1 it is the well-known Lorenz system, and for n > 1 several such systems coupled with each other in a manner dependant on the excitation pulse. The unstable manifold of a saddle equilibrium point in this ensemble characterizes possible excitations of the spins from the initial equilibrium state. This enables us to get a straightforward geometric extension of the inverse scattering method to the damped Bloch equations and hence invert them, i.e., design frequency selective pulses automatically compensated for the effect of relaxation. The latter are essential, for example, in nuclear magnetic resonance and extreme nonlinear optics.
On the accuracy of the state space restriction approximation for spin dynamics simulations
The Journal of Chemical Physics, 2011
We present an algebraic foundation for the state space restriction approximation in spin dynamics... more We present an algebraic foundation for the state space restriction approximation in spin dynamics simulations and derive applicability criteria as well as minimal basis set requirements for practically encountered simulation tasks. The results are illustrated with nuclear magnetic resonance (NMR), electron spin resonance (ESR), dynamic nuclear polarization (DNP), and spin chemistry simulations. It is demonstrated that state space restriction yields accurate results in systems where the time scale of spin relaxation processes approximately matches the time scale of the experiment. Rigorous error bounds and basis set requirements are derived.
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Papers by Alexander Karabanov