arXiv:1305.6403 · Physical Review Letters

Driving at the Quantum Speed Limit: Optimal Control of a Two-Level System

Gerhard C. Hegerfeldt

Published Dec 27, 2013• 232 citations

Abstract

A remarkably simple result is derived for the minimal time Tmin required to drive a general initial state to a final target state by a Landau-Zener-type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is bounded by a constant c. Then, for large c, the optimal driving is of type bang-off-bang and for increasing c one recovers the unconstrained result. However, for smaller c the optimal driving can suddenly switch to bang-bang type. We discuss the notion of quantum speed limit time.

DOI: 10.1103/physrevlett.111.260501

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arXiv:1305.6403 · Physical Review Letters

Driving at the Quantum Speed Limit: Optimal Control of a Two-Level System

Gerhard C. Hegerfeldt

Published Dec 27, 2013• 232 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.111.260501

Related papers

High-speed driving of a two-level system

Gerhard C. Hegerfeldt

A simple result is found for the optimal protocol of drivings for a general two-level Hamiltonian, which transports a given initial state to a given final state in minimal time, under additional conditions on the drivings. If one of the three possible drivings is unconstrained in strength the problem is analytically completely solvable. A surprise arises for a class of states when one driving is bounded by a constant $c$ and the other drivings are constant. Then, for large $c$ , the optimal driving is of type bang-off-bang and for increasing $c$ one recovers the unconstrained result. However, for smaller $c$ the optimal driving can suddenly switch to bang-bang type. It is also shown that for general states one may have a multistep protocol. The present paper explicitly proves and considerably extends the author's results contained in Phys. Rev. Lett. 111, 260501 (2013).A simple result is found for the optimal protocol of drivings for a general two-level Hamiltonian, which transports a given initial state to a given final state in minimal time, under additional conditions on the drivings. If one of the three possible drivings is unconstrained in strength the problem is analytically completely solvable. A surprise arises for a class of states when one driving is bounded by a constant $c$ and the other drivings are constant. Then, for large $c$, the optimal driving is of type bang-off-bang and for increasing $c$ one recovers the unconstrained result. However, for smaller $c$ the optimal driving can suddenly switch to bang-bang type. It is also shown that for general states one may have a multistep protocol. The present paper explicitly proves and considerably extends the author's results contained in Phys. Rev. Lett. 111, 260501 (2013).

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c

and the other drivings are constant. Then, for large

c

, the optimal driving is of type bang-off-bang and for increasing

c

one recovers the unconstrained result. However, for smaller

c

the optimal driving can suddenly switch to bang-bang type. It is also shown that for general states one may have a multistep protocol. The present paper explicitly proves and considerably extends the author's results contained in Phys. Rev. Lett. 111, 260501 (2013).

Nonlinear two-level Landau-Zener type equations for systems with relevance for Bose-Einstein condensates and nonlinear optics are considered and the minimal time

{T}_{\mathrm{min}}

to drive an initial state to a given target state is investigated. Surprisingly, the nonlinearity may be canceled by a time-optimal unconstrained driving and

{T}_{\mathrm{min}}

becomes independent of the nonlinearity. For constrained and unconstrained driving explicit expressions are derived for

{T}_{\mathrm{min}}

, the optimal driving, and the protocol.

{u}_{\text{max}}

. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small

{u}_{\text{max}}

, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.