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We propose a simple method of synchronous combined modulations to generatethe exact analytic solutions for a spin-orbit (SO) coupled ultracold atom heldin a non-Hermitian double-well potential. Based on the obtained analyticalsolutions, we mainly study the parity-time ($\mathcal{PT}$) symmetry of thissystem and the system stability for both balanced and unbalanced gain-lossbetween two wells. Under balanced gain and loss, the effect of the proportionalconstants between synchronous combined modulations and the SO-coupling strengthon the $\mathcal{PT}$-symmetry breaking is revealed analytically. Surprisingly,we find when the Zeeman field is present, the stable spin-flipping tunnelingbetween two wells can not occur in the non-Hermitian SO-coupled ultracoldatomic system, but the stable spin-conserving tunneling can be performed. Underunbalanced gain and loss, the unique set of parameter conditions that can causethe system to stabilize is found. The results may provide a possibility for theexact control of $\mathcal{PT}$-symmetry breaking and quantum spin dynamics ina non-Hermitian SO-coupled system.

Analytical results for a spin-orbit coupled atom held in a non-Hermitian double well under synchronous combined modulation