اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAlgebraic Bethe Ansatze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models
235
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the last decades. In general, non-Hermitian Hamiltonians are constructed by a textit{ad hoc} manner. Here, we study the (2+1) Dirac
oscillator and show that in the context of the $kappa$--deformed Poincare-Hopf algebra its Hamiltonian is non-Hermitian but having real eigenvalues. The non-Hermiticity steams from the $kappa$-deformed algebra. From the mapping in [Bermudez textit{et al.}, Phys. Rev. A textbf{76}, 041801(R) 2007], we propose the $kappa$-JC and $kappa$--AJC models, which describe an interaction between a two-level system with a quantized mode of an optical cavity in the $kappa$--deformed context. We find that the $kappa$--deformation modifies the textit{Zitterbewegung} frequencies and the collapse and revival of quantum oscillations. In particular, the total angular momentum in the $z$--direction is not conserved anymore, as a direct consequence of the deformation.
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 fo
r 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.
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 whic
h 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.
The Jaynes-Cummings-Gaudin model describes a collection of $n$ spins coupled to an harmonic oscillator. It is known to be integrable, so one can define a moment map which associates to each point in phase-space the list of values of the $n+1$ conserv
ed Hamiltonians. We identify all the critical points of this map and we compute the corresponding quadratic normal forms, using the Lax matrix representation of the model. The normal coordinates are constructed by a procedure which appears as a classical version of the Bethe Ansatz used to solve the quantum model. We show that only elliptic or focus-focus singularities are present in this model, which provides an interesting example of a symplectic toric action with singularities. To explore these, we study in detail the degeneracies of the spectral curves for the $n=1$ and $n=2$ cases. This gives a complete picture for the image of the momentum map (IMM) and the associated bifurcation diagram. For $n=2$ we found in particular some lines of rank 1 which lie, for one part, on the boundary of the IMM, where they behave like an edge separating two faces, and which go, for another part, inside the IMM.
Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal bas
is of the Hilbert space. With the XXZ spin torus model and the open XXX spin-1/2 chain as examples, we show that for a given inhomogeneous T-Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
