Speeding up and slowing down the relaxation of a qubit by optimal control

Abstract

We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As byproducts of our analysis we find that: (i) if the qubit is initially in a thermal state hotter than the environmental bath, quantum control cannot speed up its natural cooling rate; (ii) if the qubit is initially in a thermal state colder than the bath, it can reach the fixed point of the dynamics in finite time if a strong control field is applied; (iii) in the presence of unconstrained quantum control it is possible to keep the evolved state indefinitely and arbitrarily close to special initial states which are far away from the fixed points of the dynamics.

Optimizing for an arbitrary perfect entangler. II. Application

“…Eqs. (21) and (22). First, we consider the case E J J (t) = E J (t), i.e., using only a single control pulse.…”

“…In the present context, it is mathematically infeasible to calculate gradients of F LEC and F PE as given in Eqs. (21) and (22) with respect to the states (as needed for the Krotov update formula in Sec. IV below), since the functionals depend on the states in a highly nontrivial way.…”

“…For example, it is possible to obtain controls that are robust with respect to experimentally unavoidable fluctuations by accounting for these fluctuations in the optimization [15][16][17]. Similarly, when treating the quantum system as open, one can explore the limits on fidelity imposed by decoherence [6,[18][19][20][21] or even identify control mechanisms that rely on the explicit form of coupling to the environment [22][23][24]. Quantum optimal control also allows us to develop time-optimal strategies, resulting in protocols that perform transformations at the fastest possible pace compatible with energy and information constraints-the so-called quantum speed limit (QSL) [25][26][27][28][29][30].…”

Controlling open quantum systems: tools, achievements, and limitations

“…The example of cooling [85,86,88,148,149,155,156,181,182] was already taken as reference to introduce optimal control of open quantum systems in Sec. IV A.…”

“…For a few exceptional cases, for example one or two spins (or qubits) [144][145][146][147][148][149], a harmonic oscillator [150,151], or a sequence of Λ-systems subject to decay [152,153], the external controls can be determined using geometric techniques based on Pontryagin's maximum principle [4]. Typically, however, the control problem cannot be solved in closed form, and one needs to resort to numerical optimization.…”