Extension of the Morris-Shore transformation to multilevel ladders

Abstract

We describe situations in which chains of N degenerate quantum energy levels, coupled by timedependent external fields, can be replaced by independent sets of chains, of length N , N − 1, . . . , 2, and sets of uncoupled single states. The transformation is a generalization of the two-level MorrisShore transformation [J.R. Morris and B.W. Shore, Phys. Rev. A 27, 906 (1983)]. We illustrate the procedure with examples of three-level chains.

Scalable quantum search using trapped ions

“…The new states we shall call MS states, since they can be obtained by the multilevel Morris-Shore (MS) factorization [30].…”

Optimized control of multistate quantum systems by composite pulse sequences

“…In this section we demonstrate the potential applications of composite pulses in multistate quantum systems with MS symmetry, that is, systems amenable to the MS decomposition [24,31]. These systems obey three conditions: (i) Their states can be cast into two sets with the same RWA energies, 0 orh , the implication being that any two states of the same set are on two-photon resonance; (ii) couplings exist between states of different sets but not within the same set, that is, jk = 0 if states |j and |k are in the same set; and (iii) all couplings can be different but they must have the same time dependence f (t).…”

Generation of Dicke states using adiabatic passage

“…1(a)]. This is one of the simplest examples of a more general prescription known as the Morris-Shore transformation [14,15]. If the two ions have the Rabi frequencies of the same time dependence, the two-state system will consist of |↓↓ |1 and a Dicke state (or a Bell state) |D (1) 2 |0 = (1/ √ 2)(|↓↑ + |↑↓ )|0 , with the uncoupled state being equal to another Bell state with the same N a , (1/ √ 2)(|↓↑ − |↑↓ )|0 .…”

“…The latter extension requires extending the Morris-Shore transformation to multilevel ladders [15]; in this case, the initial state |↓ · · · ↓ |N e is coupled to the final state by a ladder with N e + 1 energy levels. With unitary operations induced by resonant optical pulses, the dynamics in this multilevel diagram never produce a unit population in the final state |D (N e ) N |0 [10,16], whereas when using RAP, such an objective can be accomplished, at least under ideal conditions [9].…”

“…The laser is assumed to illuminate both ions equally. In that case, the antisymmetric Bell state (1/ √ 2) (|↓↑ − |↑↓ ) becomes uncoupled with other internal states and can be excluded from consideration [14,15]. The existence of |↑ ↑ |1 can be neglected since it is only indirectly coupled with |↓↓ |1 via |D |1 through off-resonant carrier couplings.…”