The ability to faithfully and rapidly transport quantum particles by means of a moving trap has been a long-standing theoretical and experimental challenge, especially when the surrounding medium is randomly fluctuating (dissipative). It has preoccupied all of us, particularly Arnab Chakrabarti and Biswarup Ash, for years because of the importance of faithful (high-fidelity) and fast transfer of particles such as cold atoms via moving traps between nodes of quantum information processing setups. Our intuition and analysis have suggested that the well-established approach to such problems by exerting compensating forces on the trap, known as "shortcuts to adiabaticity", may not always be a good choice, especially for quantum particles that tend to leak out of their shallow traps while they move. But what could be the alternative?
One day, Gershon Kurizki, while watching "Modern Times" with his granddaughters, was inspired by Charlie Chaplin's evasive maneuvers as a waiter trying to avoid random collisions with the crowd (see Fig. 1 of the article, skillfully illustrated by Mayank Shreshtha). He thought it would be advisable to adopt the following analogous strategy in case the trapped substance was drink in a glass, filled to the brim ( which is quite unstable): accelerate the trap as much as possible, so that it traverses the medium within the time it still “remembers” previous collisions (“memory time”) and the spilling of the quantum “drink” by its movement.
Control of quantum processes that is faster than the memory time of simple quantum systems that are affected by the surrounding medium was introduced by Gershon Kurizki's group nearly thirty years ago, but would it be adequate for trapped quantum-particle motion? Arnab and Biswarup showed that it would indeed, by developing an in-depth comprehensive theory of these complex and subtle processes, with help and guidance from Igor Mazets, Xi Chen and Gershon. The results have surpassed our expectations: they provide a practical, feasible recipe for delivering trapped quantum particles to their target site with high fidelity and speed, which can be markedly superior to "shortcuts to adiabaticity". Charlie Chaplin might have used our suggestions to improve his moving through the rowdy crowd even if he had been ordered to deliver a vessel filled with quantum liquid to his customer. We hope that many researchers would benefit from this recipe.
The dilemma: Minimize medium effects or non-adiabaticity ?
A quantum particle in a shallow trap is an ``unstable" wave-packet, prone to leakage out of the trap due to any small perturbation, even in the absence of an environment. This leakage can be minimized by demanding very slow (adiabatic) trap motion. However, in experimental setups, where the environment almost invariably causes friction or heating – slow motion of the trap increases the probability of environment-induced leakage, reducing the chances of keeping the particle in the trap. Moreover, slow trap motion is incompatible with the strive for high-speed quantum computing operations with atoms. We thus need an optimal trade-off between trap motion speed and the achievable transfer fidelity.
In a simple but realistic scenario involving one-dimensional shallow-trap motion we found that the trap acceleration is the sole parameter that controls both environment - induced and motion-induced leakage, when all other parameters are fixed. Hence, we formulate the problem as an optimal control task of finding the trap acceleration time-dependence that results in the highest survival probability (fidelity) of the particle.
Mathematical Challenge and an engineering solution:
The problem of determining the optimal trap acceleration that maximizes the transport fidelity, has been reduced by us to an Euler-Lagrange (EL) equation that should be solved to find the optimal time-dependent trajectory of the trap. However, the resulting equation has turned out to be non-linear and integro-differential, for which there is no standard analytical solution. Even a brute-force numerical solution may be unfeasible in certain cases.
Instead, we have found an analytical method to solve the problem in a suitable non-linear regime of trap-speeds, by resorting to a major generalization of the Kofman-Kurizki (KK) universal formula for optimal control of quantum systems coupled to a thermal environment (a ``bath"). The KK formula prescribes the time-variation of the system that minimizes the ``bath" effect, but thus far it has been applied to systems with few discrete levels, whereas the moving wave-packet problem is associated with a continuum of states. Our generalized KK approach seeks the optimal time-variation of the system-bath coupling caused by the trap acceleration, for this continuum, in order to maximize transport fidelity in a non-perturbative approximation. Intriguingly, it treats nonadiabatic leakage as an additional "bath". The optimal solution method is motivated by the FM demodulation technique used in frequency-discriminator circuits. The result is a linearized, but higher-order, version of the integro-differential EL equation, which we have solved as a Neumann series using a suitable Green’s function. Our method has been benchmarked against the brute-force successive approximation protocol.
Comparison to standard Hamiltonian protocols like "shortcuts-to-adiabaticity" (STA) based on compensating "counterdiabatic" forces, shows that our approach can yield much higher transport fidelities, all parameters being equal. Futhermore, STA methods using invariant-based reverse-engineering techniques may only be applicable to specific trapping potentials, whereas our method has no such limitations.
Applications:
Our method of finding optimal trap acceleration to achieve the highest fidelity is versatile, in that it may account for multiple mechanisms of wavepacket leakage from the trap and is applicable in diverse models: trapped atoms transported by optical tweezers in quantum computers or simulators; multi-atom impurities moving through an ultra-coldatom condensate as part of a quantum refrigeration cycle; ions transported in a trap or fast charged particles channelled through solids, provide typical examples. Analogous problems that can be addressed by this method include the control of molecular dissociation or collisions,
Take-away:
Control of non-adiabatic leakage of an unstable wave-packet from a shallow trap moving through a thermal, frictional (dissipative) medium, has been reduced to the problem of stabilizing the wave-packet against coupling to multiple "baths". Trap acceleration has been deemed a sole control parameter for this complex problem, capable of providing optimal tradeoff of transport speed and fidelity.