Controllability on relaxation-free subspaces: On the relationship between adiabatic population transfer and optimal control

Abstract

We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer. For a single intermediate decaying state, we recover the Stimulated Raman Adiabatic Passage (STIRAP) process which we identify as the global optimum in the limit of infinite control time. We also present analytical solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control under the assumption of finite power. If two or more consecutive states in an N -level chain are subject to decay, complete population transfer with finite-power controls is not possible.

Optimal efficiency of a noisy quantum heat engine

“…The situation is reminiscent of STIRAP in a Λ-type atom, where perfect population transfer is achieved between two ground states coupled through a lossy excited state, which is actually never populated in the adiabatic (long time) limit. The attainment of ideal performance in the presence of dephasing can be attributed to the existence of a path along the noise-free subspace (E, 0, 0) connecting the initial and final states [31]. Note that this result seems to be in contrast with the finite limiting value for δ obtained in [21], Eq.…”

“…Note that this control strategy, to make a transfer between two variables through an intermediate variable which is kept small, has been used for the spin-order transfer along an Ising spin chain [25]. The situation is reminiscent of STIRAP in a Λ-type atom, where perfect population transfer is achieved between two ground states coupled through a lossy excited state, which is actually never populated in the adiabatic (long time) limit [31,32]. From (25) we find that during the application of the feedback law it is C/E h = √ uǫ < ǫ, while for ǫ → 0 it is additionally L/E h ≈ 2γ p ǫx 2 /x 1 < 2γ p ǫ.…”

Optimization of STIRAP-based state transfer under dissipation

“…Past work on the optimization of STIRAP for atomic population transfer was performed in the absence of decoherence, with the objective of reducing the non-adiabatic leakage and minimizing the pulse area or duration [30]. Other studies considered the effect of dissipation by including decoherence of the intermediate level [31][32][33][34], thus the crucial trade-off between decoherence and adiabaticity was not taken into consideration.…”

Training Schrödinger’s cat: quantum optimal control

“…Typically, this requires understanding of the geometry of the control problem from which one can deduce the structure of the optimal solution, a proof of global optimality and physical limits, such as the minimal time to reach the target [5]. Mathematical tools that were developed recently [11][12][13][68][69][70] could tackle problems of increasing difficulty, including fundamental control problems for closed [14,[71][72][73][74][75] and open quantum systems [76][77][78][79][80][81]. This method is able to treat quantum control problems ranging from two and three level quantum systems or two and three coupled spins to two-level dissipative quantum systems with dynamics governed by the Lindblad equation.…”