In this paper, we study how to apply a periodic driving field to control stable spin tunneling in a non-Hermitian spin–orbit (SO) coupled bosonic double-well system. By means of a high-frequency approximation, we obtain the analytical Floquet solutions and their associated quasienergies and thus construct the general non-Floquet solutions of the dissipative SO coupled bosonic system. Based on detailed analysis of the Floquet quasienergy spectrum, the profound effect of system parameters and the periodic driving field on the stability of spin-dependent tunneling is investigated analytically and numerically for both balanced and unbalanced gain–loss between two wells. Under balanced gain and loss, we find that the stable spin-flipping tunneling is preferentially suppressed with the increase of gain–loss strength. When the ratio of Zeeman field strength to periodic driving frequency Ω/ω is even, there is a possibility that continuous stable parameter regions will exist. When Ω/ω is odd, nevertheless, only discrete stable parameter regions are found. Under unbalanced gain and loss, whether Ω/ω is even or odd, we can get parametric equilibrium conditions for the existence of stable spin tunneling. The results could be useful for the experiments of controlling stable spin transportation in a non-Hermitian SO coupled system.

We study the Hankel determinants associated with the weight w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n→∞ and t→1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.

We apply the uniform asymptotics method proposed by Bassom, Clarkson, Law and McLeod [4] to a special Painlevé V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation, which have been established earlier by McCoy and Tang via the isomonodromy and WKB methods.

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In this paper, we investigate the spin and tunneling dynamics of a spin-orbit-coupled noninteracting Bose-Einstein condensate in a periodically driven non-Hermitian double-well potential. Under high-frequency driving, we obtain the effective time-averaged Hamiltonian by using the standard time-averaging method, and analytically calculate the Floquet quasienergies, revealing that the parity-time (PT)-breaking phase transition appears even for arbitrarily small non-Hermitian parameters when the spin-orbit coupling strength takes half-integer value, irrespective of the values of other parameters used. When the system is PT-symmetric with balanced gain and loss, we find numerically and analytically that in the broken PT-symmetric regions, there will exist the net spin current together with a vanishing atomic current, if we drop the contribution of the exponential growth of the norm to the current behaviors. When the system is non-PT-symmetric, though the quasienergies are partial complex, a stable net spin current can be generated by controlling the periodic driving field, which is accompanied by a spatial localization of the condensate in the well with gain. The results deepen the understanding of non-Hermitian physics and could be useful for engineering a variety of devices for spintronics.

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As a new application of the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod, we provide a simpler and more rigorous proof of the connection formulas of some special solutions of the fifth Painlevé equation, which have been established earlier by Andreev and Kitaev.

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