R. Uzdin, Dalla Torre, E.G. , Kosloff, R. , and Moiseyev, N. . 2013.
“Effects Of An Exceptional Point On The Dynamics Of A Single Particle In A Time-Dependent Harmonic Trap.”. Physical Review A - Atomic, Molecular, And Optical Physics, 88, Pp. 022505.
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Abstract The time evolution of a single particle in a harmonic trap with time-dependent frequency ω(t) has been well studied. Nevertheless, here we show that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter μ=[dω(t)/dt]/ω2(t) fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at μ=2. At this transition point, the time evolution has an exceptional point (EP) at all instants. This situation, where an EP of a time-dependent Hermitian Hamiltonian is obtained at any given time, is very different from other known cases. In the present case, we show that the order of the EP depends on the set of observables used to describe the dynamics. Our finding is relevant to the dynamics of a single ion in a magnetic, optical, or rf trap, and of diluted gases of ultracold atoms in optical traps.
Non-unitary operations generated by an effective non-Hermitian Hamiltonian can be used to create quantum state manipulations which are impossible in Hermitian quantum mechanics. These operations include state preparation (or cooling) and non-orthogonal state discrimination. In this work we put a lower bound on the resources needed for the construction of some given non-unitary evolution. Passive systems are studied in detail and a general feature of such systems is derived. After interpreting our results using the singular value decomposition, several examples are studied analytically. In particular, we put a lower bound on the resources needed for non-Hermitian state preparation and non-orthogonal state discrimination.
Unambiguous-nonorthogonal-state discrimination has a fundamental importance in quantum information. Moreover, it can be used for entanglement distillation and secure communication. The discrimination is carried out by a positive operator-valued measure (POVM) generalized measurement, which is typically implemented by coupling the system to an ancilla. We find a trade-off between the needed energy resources and the evolution time needed to implement the POVM and express it in terms of an actionlike cost inequality. We find the realization that minimizes this actionlike cost and show that, in this case, the cost is determined by the maximal population transfer from the system to the ancilla. We demonstrate our findings in an example of a three-level system coupled to a laser.