arXiv subjects

Alexandre Faribault

Publications and source records attributed to Alexandre Faribault.

At least 19 recordsLinked to original sources

arXiv2024

Integrability and dark states of the XX spin-1 central spin model in a transverse field

It was recently shown that, for central spin-1/2 and central spin-1, the XX central spin model is integrable in the presence of a magnetic field oriented perpendicular to the XY plane in which the coupling exists. In the spin-1/2 case, it was also shown, through an appropriate limit of the non-skew symmetric XXZ Richardson-Gaudin models, that it remained integrable even when the magnetic field is tilted to contain an in-plane component. Although the model has not yet been shown to explicitly belong to a known class of Richardson- Gaudin models, we show, in this work, that the spin-1 case also remains integrable in a titled magnetic field. We do so by writing explicitly the complete set of conserved charges, then showing that these operators obey polynomial relations. It is finally demonstrated numerically that dark states, for which the central spin is completely unentangled with the bath, can emerge at strong enough coupling just as they do in the central spin-1/2 model in an arbitrarily oriented magnetic field.

cond-mat.mes-hall↗

arXiv2022

Reduced Density Matrices / Static Correlation Functions of Richardson-Gaudin States Without Rapidities

Seniority-zero geminal wavefunctions are known to capture bond-breaking correlation. Among this class of wavefunctions, Richardson-Gaudin states stand out as they are eigenvectors of a model Hamiltonian. This provides a clear physical picture, clean expressions for reduced density matrix (RDM) elements, and systematic improvement (with a complete set of eigenvectors). Known expressions for the RDM elements require the computation of rapidities, which are obtained by first solving for the so-called eigenvalue based variables (EBV) then root-finding of a Lagrange interpolation polynomial. In this manuscript we obtain expressions for the RDM elements directly in terms of the EBV. The final expressions can be computed with the same cost as the rapidity expressions. Therefore, except in particular circumstances, it is entirely unnecessary to compute rapidities at all. The RDM elements require numerically inverting a matrix and while this is usually undesirable we demonstrate that it is stable, except when there is degeneracy in the single-particle energies. In such cases a different construction would be required.

physics.chem-ph↗

arXiv2022

Gradient corrections to the local density approximation in the one-dimensional Bose gas

The local density approximation (LDA) is the central technical tool in the modeling of quantum gases in trapping potentials. It consists in treating the gas as an assembly of independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it is justified when the correlation length is larger than the size of the cells. The LDA is often regarded as a crude approximation, particularly in the ground state of the one-dimensional (1D) Bose gas, { where the correlation length is "therefore said to be" infinite (in the sense that correlation functions decay as a power law).} Here we take another look at the LDA. The local density $\rho(x)$ is viewed as a functional of the trapping potential $V(x)$, to which one applies a gradient expansion. The zeroth order in that expansion is the LDA. The first-order correction in the gradient expansion vanishes due to reflection symmetry. At second order, there are two corrections proportional to $d^2V/dx^2$ and $(dV/dx)^2$, and we propose a method to determine the corresponding coefficients by a perturbative calculation in the Lieb-Liniger model. This leads to an expression for the coefficients in terms of matrix elements of the density operator, which can in principle be evaluated numerically for an arbitrary coupling constant; here we show how to efficiently evaluate the coefficient associated to the curvature of the potential $d^2V/dx^2$, which dominates the deviation to LDA near local minima or maxima of the trapping potential. Both coefficients are evaluated analytically in the limits of infinite repulsion (hard-core bosons) and small repulsion (quasi-condensate).} The corrected LDA density profiles are compared to DMRG calculations, with significant improvement compared to zeroth-order LDA.

cond-mat.quant-gas↗

arXiv2022

Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field

Eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian, Richardson-Gaudin (RG) states, are used as a variational wavefunction Ansatz for strongly-correlated electronic systems. These states are geminal products whose coefficients are solutions of non-linear equations. Previous results showed un-physical behaviour but in this contribution it is shown that with only the variational solution for the ground state, all the seniority-zero states are quite well approximated. The difficulty is in choosing the correct RG state. The systems studied showed a clear choice and we expect it should always be possible to reason physically which state to choose.

physics.chem-ph↗

arXiv2021

Strong-coupling emergence of dark states in XX central spin models

It was recently shown that the XX central spin model is integrable in the presence of a magnetic field perpendicular to the plane in which the coupling exists. A large number of its eigenstates are such that the central spin is not correlated to the environmental spins it is coupled to. In this work, we first demonstrate that the XX-central spin model remains integrable in the presence of an arbitrarily oriented magnetic field. We then show that, provided the coupling is strong enough, dark states can actually be found even in the presence of an in-plane magnetic field. We finally provide a simple explanation of this result and demonstrate its universality for a variety of distinct distributions of the coupling of the central spin to the various bath spins.

quant-ph↗

arXiv2019

Read-Green points and level crossings in XXZ central spin models and $p_x+ip_y$ topological superconductors

In this work, we study the full set of eigenstates of a $p_x+ip_y$ topological superconductor coupled to a particle bath which can be described in terms of an integrable Hamiltonian of the Richardson-Gaudin class. The results derived in this work also characterise the behaviour of an anisotropic XXZ central spin model in a external magnetic field since both types of Hamiltonian are know to share the exact same conserved quantities making them formally equivalent. We show how by ramping the coupling strength (or equivalently the magnetic field acting in the z-direction on the central spin), each individual eigenstate undergoes a sequence of gain/loss of excitations when crossing the specific values known as Read-Green points. These features are shown to be completely predictable, for every one of the $2^N$ eigenstates, using only two integers obtainable easily from the zero-coupling configuration which defines the eigenstate in question. These results provide a complete map of the particle-number sectors (superconductor) or magnetisation sectors (central spin) involved in the large number of level-crossings which occur in these systems at the Read-Green points. It further allows us to define quenching protocols which could create states with remarkably large excitation-number fluctuations.

cond-mat.mes-hall↗

arXiv2018

"Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models

In this work we construct the eigenstates of the most general spin-1/2 Richardson-Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the $S^x_iS^x_j$, $S^y_iS^y_j$ and $S^z_iS^z_j$ terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r-matrices. The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U(1) so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided.

math-ph↗

arXiv2018

Integrable spin-1/2 Richardson-Gaudin XYZ models in an arbitrary magnetic field

We establish the most general class of spin-1/2 integrable Richardson-Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin-1/2 relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson-Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this approach does not provide expressions for the exact eigenstates, it allows their eigenvalues to be obtained, and expectation values of local observables can then be calculated from the Hellmann-Feynman theorem.

math-ph↗

arXiv2018

Steady-states of out-of-equlibrium inhomogeneous Richardson-Gaudin quantum integrable models in quantum optics

In this work we present numerical results for physical quantities in the steady-state obtained after a variety of product-states initial conditions are evolved unitarily, driven by the dynamics of quantum integrable models of the rational (XXX) Richardson-Gaudin family, which includes notably Tavis-Cummings models. The problem of interest here is one where a completely inhomogeneous ensemble of two-level systems (spins-1/2) are coupled to a single bosonic mode. The long-time averaged magnetisation along the z-axis as well as the bosonic occupation are evaluated in the diagonal ensemble by performing the complete sum over the full Hilbert space for small system sizes. These numerically exact results are independent of any particular choice of Hamiltonian and therefore describe general results valid for any member of this class of quantum integrable models built out of the same underlying conserved quantities. The collection of numerical results obtained can be qualitatively understood by a relaxation process for which, at infinitely strong coupling, every initial state will relax to a common state where each spin is in a maximally coherent superposition of its $\left|\uparrow\right>$ and $\left|\downarrow\right>$ states, i.e. they are in-plane polarised, and consequently the bosonic mode is also in a maximally coherent superposition of different occupation number states. This bosonic coherence being a feature of a superradiant state, we shall loosely use the term superradiant steady-state to describe it. A finite value of the coupling between the spins and the bosonic mode then leads to a long-time limit steady-state whose properties are qualitatively captured by a simple "dynamical" vision in which the coupling strength $V$ plays the role of a time $t_V$ at which this "relaxation process" towards the common strong coupling superradiant steady-state is interrupted.

cond-mat.mes-hall↗

arXiv2018

Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models

In this work we demonstrate how one can, in a generic approach, derive a set of $N$ simple quadratic Bethe equations for integrable Richardson-Gaudin (RG) models built out of $N$ spins-1/2. These equations depend only on the $N$ eigenvalues of the various conserved charges so that any solution of these equations defines, indirectly through the corresponding set of eigenvalues, one particular eigenstate. The proposed construction covers the full class of integrable RG models of the XYZ (including the subclasses of XXZ and XXX models) type realised in terms of spins-1/2, coupled with one another through $σ_i^x σ_j^x $, $σ_i^y σ_j^y $, $σ_i^z σ_j^z $ terms, including, as well, magnetic field-like terms linear in the Pauli matrices. The approach exclusively requires integrability, defined here only by the requirement that $N$ conserved charges $R_i$ (with $i = 1,2 \dots N$) such that $\left[R_i,R_j\right] =0 \ (\forall \ i,j)$ exist . The result is therefore valid, and equally simple, for models with or without $U(1)$ symmetry, with or without a properly defined pseudo-vacuum as well as for models with non-skew symmetric couplings.

math-ph↗

arXiv2017

Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach.

math-ph↗

arXiv2016

Persisting correlations of a central spin coupled to large spin baths

The decohering environment of a quantum bit is often described by the coupling to a large bath of spins. The quantum bit itself can be seen as a spin $S=1/2$ which is commonly called the central spin. The resulting central spin model describes an important mechanism of decoherence. We provide mathematically rigorous bounds for a persisting magnetization of the central spin in this model with and without magnetic field. In particular, we show that there is a well defined limit of infinite number of bath spins. Only if the fraction of very weakly coupled bath spins tends to 100\% does no magnetization persist.

quant-ph↗

arXiv2015

Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin

In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson-Gaudin models which, in addition to $N-1$ spins $\frac{1}{2}$, contains one arbitrarily large spin $S$. The proposed determinant representation is written in terms of a set of variables which, from previous work, are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic Bethe equations. Such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the usual highly non-linear Bethe equations. It can therefore offer significant gains in stability and computation speed.

math-ph↗

arXiv2014

Algebraic Bethe Ansätze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models

In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled to a single bosonic mode. The ensemble of resulting models which we call Dicke-Jaynes-Cummings- Gaudin models are particularly relevant for the description of light-matter interaction in the context of quantum optics. Having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions. We also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges. Since these eigenvalues obey a much simpler set of quadratic Bethe equations, the resulting expressions could then offer important simplifications for the numerical treatment of these models.

math-ph↗

arXiv2013

Spin decoherence due to a randomly fluctuating spin bath

We study the decoherence of a spin in a quantum dot due to its hyperfine coupling to a randomly fluctuating bath of nuclear spins. The system is modelled by the central spin model with the spin bath initially being at infinite temperature. We calculate the spectrum and time evolution of the coherence factor using a Monte Carlo sampling of the exact eigenstates obtained via the algebraic Bethe ansatz. The exactness of the obtained eigenstates allows us to study the non-perturbative regime of weak magnetic fields in a full quantum mechanical treatment. In particular, we find a large non-decaying fraction in the zero-field limit. The crossover from strong to weak fields is similar to the decoherence starting from a pure initial bath state treated previously. We compare our results to a simple semiclassical picture [Merkulov et al., Phys. Rev. B 65, 205309 (2002)] and find surprisingly good agreement. Finally, we discuss the effect of weakly coupled spins and show that they will eventually lead to complete decoherence.

cond-mat.mes-hall↗

arXiv2013

Integrability-based analysis of the hyperfine-interaction -nduced decoherence in quantum dots

Using the Algebraic Bethe Ansatz in conjunction with a simple Monte Carlo sampling technique, we study the problem of the decoherence of a central spin coupled to a nuclear spin bath. We describe in detail the full crossover from strong to weak external magnetic field field, a limit where a large non-decaying coherence factor is found. This feature is explained by Bose-Einstein-condensate-like physics which also allows us to argue that the corresponding zero frequency peak would not be broadened by statistical or ensemble averaging.

cond-mat.mes-hall↗

arXiv2012

On the determinant representations of Gaudin models' scalar products and form factors

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnovs determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.

math-ph↗

arXiv2012

Non-equilibrum dynamics in the strongly excited inhomogeneous Dicke model

Using the exact eigenstates of the inhomogeneous Dicke model obtained by numerically solving the Bethe equations, we study the decay of bosonic excitations due to the coupling of the mode to an ensemble of two-level (spin 1/2) systems. We compare the quantum time-evolution of the bosonic mode population with the mean field description which, for a few bosons agree up to a relatively long Ehrenfest time. We demonstrate that additional excitations lead to a dramatic shortening of the period of validity of the mean field analysis. However, even in the limit where the number of bosons equal the number of spins, the initial instability remains adequately described by the mean-field approach leading to a finite, albeit short, Ehrenfest time. Through finite size analysis, we also present indications that the mean field approach could still provide an adequate description for thermodynamically large systems even at long times. However, for mesoscopic systems one cannot expect it to capture the behavior beyond the initial decay stage in the limit of an extremely large number of excitations.

cond-mat.mes-hall↗