arXiv subjects
Publications and source records attributed to Yao-Zhong Zhang.
It is shown that the (driven) quantum Rabi model and its 2-photon and 2-mode generalizations possess a hidden $sl(2)$-algebraic structure which explains the origin of the quasi-exact solvability of these models. It manifests the first appearance of a hidden algebraic structure in quantum spin-boson systems without $U(1)$ symmetry.
We construct the integrals of motion for the 5D deformed Kepler system with non-central potentials in $su(2)$ Yang-Coulomb monopole field. We show that these integrals form a higher rank quadratic algebra $Q(3; L^{so(4)}, T^{su(2)})\oplus so(4)$, with structure constants involving the Casimir operators of $so(4)$ and $su(2)$ Lie algebras. We realize the quadratic algebra in terms of the deformed oscillator and construct its finite-dimensional unitary representations. This enable us to derive the energy spectrum of the system algebraically. Furthermore we show that the model is multiseparable and allows separation of variables in the hyperspherical and parabolic coordinates. We also show the separability of its 8D dual system (i.e. the 8D singular harmonic oscillator) in the Euler and cylindrical coordinates.
We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in generalized Taub-NUT space. The Schrödinger equation of this model is solved in spherical coordinates in the framework of Stäckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.
By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the $N$-dimensional superintegrable Kepler-Coulomb model with non-central terms and the double singular oscillators of type ($n, N-n$). We show how the integrals of motion generate higher rank cubic algebra $C(3)\oplus L_1\oplus L_2$ with structure constants involving Casimir operators of the Lie algebras $L_1$ and $L_2$. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.
Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in pure and applied mathematics. We overview two new families of superintegrable Kepler-Coulomb systems with non-central terms and superintegrable Hamiltonians with double singular oscillators of type $(n,N-n)$ in $N$-dimensional Euclidean space. We present their quadratic and polynomial algebras involving Casimir operators of $so(N+1)$ Lie algebras that exhibit very interesting decompositions $Q(3)\oplus so(N-1)$, $Q(3)\oplus so(n)\oplus so(N-n)$ and the cubic Casimir operators. The realization of these algebras in terms of deformed oscillator enables the determination of a finite dimensional unitary representation. We present algebraic derivations of the degenerate energy spectra of these systems and relate them with the physical spectra obtained from the separation of variables.
We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak B(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus \mathfrak L(V);$ and {\rm (iii)} if $Δ(\mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} \not= 1$ for any $u \in D(V)$, then $\dim (\mathfrak B(V) ) = \infty$ if and only if there exists $V'$, which is twisting equivalent to $V$, such that $ \dim (\mathfrak L^ - (V')) = \infty.$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.
We introduce a new family of $N$-dimensional quantum superintegrable model consisting of double singular oscillators of type $(n,N-n)$. The special cases $(2,2)$ and $(4,4)$ were previously identified as the duals of 3- and 5-dimensional deformed Kepler-Coulomb systems with $u(1)$ and $su(2)$ monopoles respectively. The models are multiseparable and their wave functions are obtained in $(n,N-n)$ double-hyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra $Q(3)$ involving three generators, $so(n)$, $so(N-n)$ (i.e. $Q(3)\oplus so(n) \oplus so(N-n)$ ). The structure constants of the quadratic algebra themselves involve the Casimir operators of the two Lie algebras $so(n)$ and $so(N-n)$. Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.
We explore the entropic uncertainty relation in the curved background outside a Schwarzschild black hole, and find that Hawking radiation introduces a nontrivial modification on the uncertainty bound for particular observer, therefore it could be witnessed by proper uncertainty game experimentally. We first investigate an uncertainty game between a free falling observer and his static partner holding a quantum memory initially entangled with the quantum system to be measured. Due to the information loss from Hawking decoherence, we find an inevitable increase of the uncertainty on the outcome of measurements in the view of static observer, which is dependent on the mass of the black hole, the distance of observer from event horizon, and the mode frequency of quantum memory. To illustrate the generality of this paradigm, we relate the entropic uncertainty bound with other uncertainty probe, e.g., time-energy uncertainty. In an alternative game between two static players, we show that quantum information of qubit can be transferred to quantum memory through a bath of fluctuating quantum fields outside the black hole. For a particular choice of initial state, we show that the Hawking decoherence cannot counteract entanglement generation after the dynamical evolution of system, which triggers an effectively reduced uncertainty bound that violates the intrinsic limit $-\log_2c$. Numerically estimation for a proper choice of initial state shows that our result is comparable with possible real experiments. Finally, a discussion on the black hole firewall paradox in the context of entropic uncertainty relation is given.
We study the quantum correlation and quantum communication channel of both free scalar and fermionic fields in de Sitter space, while the Planckian modification presented by the choice of a particular $α$-vacuum has been considered. We show that the occurrence of degradation of quantum entanglement between field modes for an inertial observer in curved space, due to the radiation associated with its cosmological horizon. Comparing with standard Bunch-Davies choice, the possible Planckian physics causes some extra decrement on the quantum correlation, which may provide the means to detect quantum gravitational effects via quantum information methodology in future. Beyond single-mode approximation, we construct proper Unruh modes admitting general $α-$vacua, and find a convergent feature of both bosonic and fermionic entanglements. In particular, we show that the convergent points of fermionic entanglement negativity are dependent on the choice of $α$. Moreover, an one-to-one correspondence between convergent points $H_c$ of negativity and zeros of quantum capacity of quantum channels in de Sitter space has been proved.
We introduce a new superintegrable Kepler-Coulomb system with non-central terms in $N$-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give a algebraic derivation of spectrum of the superintegrable system. We show how the $so(N+1)$ symmetry algebra of the $N$-dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure constants involving a Casimir operator of $so(N-1)$ Lie algebra. We construct the quadratic algebra and the Casimir operator. We show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum.
We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $\mathfrak B(V)$; (ii) Nichols Lie algebra $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.
We present a new tractable quantum Rabi model for $N$-level atoms by extending the $\mathbb Z_2$ symmetry of the two-state Rabi model. The Hamiltonian is $\mathbb Z_N$ symmetric and allows the parameters in the level separation terms to be complex while remaining hermitian. This latter property means that the new model is {\em chiral}, which makes it differ from any existing $N$-state Rabi models in the literature. The $\mathbb Z_N$ symmetry provides partial diagonalization of the general Hamiltonian. The exact isolated (i.e. exceptional) energies of the model have the Rabi-like form but are $N$-fold degenerate. For the three-state case ($N=3$), we obtain three transcendental functions whose zeros give the regular (i.e. non-exceptional) energies of the model.
We introduce a new family of quasi-exactly solvable generalized isotonic oscillators which are based on the pseudo-Hermite exceptional orthogonal polynomials. We obtain exact closed-form expressions for the energies and wavefunctions as well as the allowed potential parameters for the first two members of the family using the Bethe ansatz method. Numerical calculations of the energies reveal that member potentials have multiple quasi-exactly solvable eigenstates and the number of states for higher members are parameter dependent.
By considering a unified treatment, we present quasi exact polynomial solutions to both the Klein-Gordon and Dirac equations with the family of soft-core Coulomb potentials $V_q(r)=-Z/\left(r^q+β^q\right)^{1/q}$, $Z>0$, $β>0$, $q\geq 1$. We consider cases $q=1$ and $q=2$ and show that both cases are reducible to the same basic ordinary differential equation. A systematic and closed form solution to the basic equation is obtain using the Bethe ansatz method. For each case, the expressions for the energies and the allowed parameters are obtained analytically and the wavefunctions are derive in terms of the roots of a set of Bethe ansatz equations.
We present (exact) solutions of the Dirac equation with equally mixed interactions for a single fermion bounded by the family of fractional power singular potentials. Closed-form expressions as well as numerical values for the energies were obtained. The wave functions and the allowed values of the potential parameters for the first two members of the family are obtained in terms of a set of algebraic equations. The non relativistic limit is also discussed and using the Hellmann-Feynmann theorem, some useful expectation values are obtained.
We study the solvability of the time-independent matrix Schrödinger differential equations of the quantum Rabi model and its 2-photon and two-mode generalizations in Bargmann Hilbert spaces of entire functions. We show that the Rabi model and its 2-photon and two-mode analogs are quasi-exactly solvable. We derive the exact, closed-form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces. Up to a normalization factor, the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations.
The uncertainty principle bounds our ability to simultaneously predict two incompatible observables of a quantum particle. Assisted by a quantum memory to store the particle, this uncertainty could be reduced and quantified by a new Entropic Uncertainty Relation (EUR). In this Letter, we explore how the relativistic motion of the system would affect the EUR in two sample scenarios. First, we show that the Unruh effect of an accelerating particle would surely increase the uncertainty if the system and particle entangled initially. On the other hand, the entanglement could be generated from nonuniform motion once the Unruh decoherence is prevented by utilizing the cavity. We show that, in a uncertainty game between an inertial cavity and a nonuniformly accelerated one, the uncertainty evolves periodically with respect to the duration of acceleration segment. Therefore, with properly chosen cavity parameters, the uncertainty bound could be protected. Implications of our results for gravitation are also discussed.
We analyze the two-mode squeezed harmonic oscillator and the $k$th-order harmonic generation within the framework of Bargmann-Hilbert spaces of entire functions. For the displaced, single-mode squeezed and two-mode squeezed harmonic oscillators, we derive the exact, closed-form expressions for their energies and wave functions. For the $k$th-order harmonic generation with $k\geq 3$, our result indicates that it does not have eigenfunctions and is thus ill-defined in the Bargmann-Hilbert space.