Focus on Shortcuts to Adiabaticity

Shortcuts to Adiabaticity (STA) constitute driving schemes that provide an alternative to adiabatic protocols to control and guide the dynamics of classical and quantum systems without the requirement of slow driving. Research on STA advances swiftly with theoretical progress being accompanied by experiments on a wide variety of platforms. We summarize recent developments emphasizing advances reported in this focus issue while providing an outlook with open problems and prospects for future research.

We propose and analyze, focusing on non-adiabatic effects, a technique of manipulating quantum spin systems based on local 'cutting' and 'stitching' of the Heisenberg exchange coupling between the spins. This first operation is cutting of a bond separating a single spin from a linear chain, or of two neighboring bonds for a ring-shaped array of spins. We show that the disconnected spin can be in the ground state with a high-fidelity even after a non-adiabatic process. Next, we consider inverse operation of stitching these bonds to increase the system size. We show that the optimal control algorithm can be found by using common numerical procedures with a simple two-parametric control function able to produce a high-fidelity cutting and stitching. These results can be applied for manipulating ensembles of quantum dots, considered as prospective elements for quantum information technologies, and for design of machines based on quantum thermodynamics.

Shortcuts to adiabaticity (STA) provide an alternative to adiabatic protocols to guide the dynamics of the system of interest without the requirement of slow driving. We report the controlled speedup via STA of the nonadiabatic dynamics of a Fermi gas, both in the noninteracting and strongly coupled, unitary regimes. Friction-free superadiabatic expansion strokes, with no residual excitations in the final state, are demonstrated in the unitary regime by engineering the modulation of the frequencies and aspect ratio of the harmonic trap. STA are also analyzed and implemented in the high-temperature regime, where the shear viscosity plays a pivotal role and the Fermi gas is described by viscous hydrodynamics.

We introduce a phenomenological theory for many-body control of critical phenomena by engineering causally-induced gaps for quantum Hamiltonian systems. The core mechanisms are controlling information flow within and/or between clusters that are created near a quantum critical point. To this end, we construct inhomogeneous quantum phase transitions via designing spatiotemporal quantum fluctuations. We show how non-equilibrium evolution of disordered quantum systems can create new effective correlation length scales and effective dynamical critical exponents. In particular, we construct a class of causally-induced non-adiabatic quantum annealing transitions for strongly disordered quantum Ising chains leading to exponential suppression of topological defects beyond standard Kibble–Zurek predictions. Using exact numerical techniques for 1D quantum Hamiltonian systems, we demonstrate that our approach exponentially outperforms adiabatic quantum computing. Using strong-disorder renormalization group (SDRG), we demonstrate the universality of inhomogeneous quantum critical dynamics and exhibit the reconstructions of causal zones during SDRG flow. We derive a scaling relation for minimal causal gaps showing they narrow more slowly than any polynomial with increasing size of system, in contrast to stretched exponential scaling in standard adiabatic evolution. Furthermore, we demonstrate similar scaling behavior for random cluster-Ising Hamiltonians with higher order interactions.

We propose speeding up a single ion heat pump based on a tapered ion trap. If a trapped ion is excited in an oscillatory motion axially the radial degrees of freedom are cyclically expanded and compressed such that heat can be pumped between two reservoirs coupled to the ion at the turning points of oscillation. Through the use of invariant-based inverse engineering we can speed up the process without sacrificing the efficiency of each heat pump cycle. This additional control can be supplied with additional control electrodes or it can be encoded into the geometry of the radial trapping electrodes. We present a novel insight into how speed up can be achieved through the use of inverted harmonic potentials and verify the stability of such trapping conditions.

Shortcuts to adiabaticity are techniques allowing rapid variation of the system Hamiltonian without inducing excess heating. Fast optical transfer of atoms between different locations is an important application of shortcuts to adiabaticity. We show that the common boundary conditions on the atomic position, which are imposed to find the driving trajectory, lead to highly non-practical boundary conditions for the optical trap. Our experimental results demonstrate that, as a result, previously suggested trajectories are likely to fall short of the expectation. We develop two complementary methods that solve this boundary conditions problem by adding more degrees of freedom to the trajectory parameter space. In the first method, this is achieved by the addition of a spectral component at the trapping frequency, while in the second we use a polynomial trajectory of an order high enough to account for the new boundary conditions. We experimentally demonstrate that this approach allows us to construct highly non-adiabatic movements with no residual sloshing. Our techniques can also account for non-harmonic terms in the confining potential.

Shortcuts to adiabaticity (STA) provide control protocols to guide the dynamics of a quantum system through an adiabatic reference trajectory in an arbitrary prescheduled time. Designing STA proves challenging in complex quantum systems when the dynamics of the degrees of freedom span different time scales. We introduce counterdiabatic Born–Oppenheimer dynamics (CBOD) as a framework to design STA in systems with a large separation of energy scales. CBOD exploits the Born–Oppenheimer approximation to separate the Hamiltonian into effective fast and slow degrees of freedom and calculate the corresponding counterdiabatic drivings for each sub-system. We show the validity of the CBOD technique via an example of coupled harmonic oscillators, which can be solved exactly for comparison, and further apply it to a system of two-charged particles.

In a 'shortcut to adiabaticity' (STA) protocol, the counter-diabatic Hamiltonian, which suppresses the non-adiabatic transition of a reference 'adiabatic' trajectory, induces a quantum uncertainty of the work cost in the framework of quantum thermodynamics. Following a theory derived recently (Funo et al 2017 Phys. Rev. Lett. 118 100602), we perform an experimental measurement of the STA work statistics in a high-quality superconducting Xmon qubit. Through the frozen-Hamiltonian and frozen-population techniques, we experimentally realize the two-point measurement of the work distribution for given initial eigenstates. Our experimental statistics verify (i) the conservation of the average STA work and (ii) the equality between the STA excess of work fluctuations and the quantum geometric tensor.

Motivated by recent results using shaken optical lattices to perform atom interferometry, we explore the splitting of an atom cloud trapped in a phase-modulated ('shaken') optical lattice. Using a simple analytic model we are able to show that we can obtain the simplest case of ±2ℏk_L splitting via single-frequency shaking. This is confirmed both via simulation and experiment. Furthermore, we are able to split with a relative phase θ between the two split arms of 0 or π depending on our shaking frequency. Addressing higher-order splitting, we determine that ±6ℏk_L splitting is sufficient to be able to accelerate the atoms in counterpropagating lattices. Finally, we show that we can use a genetic algorithm to optimize ±4ℏk_L and ±6ℏk_L splitting to within ≈0.1% by restricting our optimization to the resonance frequencies corresponding to single- and two-photon transitions between Bloch bands. As a proof-of-principle, an experimental demonstration of simplified optimization of 4ℏk_L splitting is presented.

We propose an enlarged framework to study transformations that drive an underdamped Brownian particle in contact with a thermal bath from an equilibrium state to a new one in an arbitrarily short time. To this end, we make use of a time and space-dependent potential, that plays a dual role: confine the particle, and manipulate the system. In the special case of an isothermal compression or decompression of a harmonically trapped particle, we derive explicit protocols that perform this quick transformation, following an inverse engineering method. We focus on the properties of these protocols, which crucially depend on two key dimensionless numbers that characterize the relative values of the three timescales of the problem, associated with friction, oscillations in the confinement and duration of the protocol. In particular, we show that our protocols encompass the known overdamped version of this problem and extend it to any friction for decompression and to a large range of frictions for compression.

Adiabatic braiding of Majorana zero modes can be used for topologically protected quantum information processing. While extremely robust to many environmental perturbations, these systems are vulnerable to noise with high-frequency components. Ironically, slower processes needed for adiabaticity allow more noise-induced excitations to accumulate, resulting in an antiadiabatic behavior that limits the precision of Majorana gates if some noise is present. In a recent publication (2017 Phys. Rev. B 96 075158), fast optimal protocols were proposed as a shortcut for implementing the same unitary operation as the adiabatic braiding in a low-energy effective model. These shortcuts sacrifice topological protection in the absence of noise but provide performance gains and remarkable robustness to noise due to the shorter evolution time. Nevertheless, gates optimized for vanishing noise are suboptimal in the presence of noise. If we know the noise strength beforehand, can we design protocols optimized for the existing unavoidable noise, which will effectively correct the noise-induced errors? We address this question in the present paper, focusing on the same low-energy effective model. We find such optimal protocols using simulated-annealing Monte Carlo simulations. The numerically found pulse shapes, which we fully characterize, are in agreement with Pontryagin's minimum principle, which allows us to arbitrarily improve the approximate numerical results (due to discretization and imperfect convergence) and obtain numerically exact optimal protocols. The protocols are bang–bang (sequence of sudden quenches) for vanishing noise, but contain continuous segments in the presence of multiplicative noise due to the nonlinearity of the master equation governing the evolution. We find that such noise-optimized protocols completely eliminate the above-mentioned antiadiabatic behavior. The final error corresponding to these optimal protocols monotonically decreases with the total time (in three different regimes). A liner fit to 1/τ indicates extrapolation of the cost function to finite value in the $\tau \to \infty$ limit. However, quadratic and cubic fits are more suggestive of the cost function extrapolating to zero in the limit of infinite time. Our results set the precision limit of the device as a function of the noise strength and total time.

Based on a 'shortcut-to-adiabaticity' (STA) scheme, we theoretically design and experimentally realize a set of high-fidelity single-qubit quantum gates in a superconducting Xmon qubit system. Through a precise microwave control, the qubit is driven to follow a fast 'adiabatic' trajectory with the assistance of a counter-diabatic field and the correction of derivative removal by adiabatic gates. The experimental measurements of quantum process tomography and interleaved randomized benchmarking show that the process fidelities of our STA quantum gates are higher than 94.9% and the gate fidelities are higher than 99.8%, very close to the state-of-art gate fidelity of 99.9%. An alternate of high-fidelity quantum gates is successfully achieved under the STA protocol.

There is recent interest in determining energy costs of shortcuts to adiabaticity (STA), but different definitions of 'cost' have been used. We demonstrate the importance of taking into account the control system (CS) for a fair assessment of energy flows and consumptions. We model the energy consumption and power to transport an ion by a STA protocol in a multisegmented Paul trap. The ion is driven by an externally controlled, moving harmonic oscillator. Even if no net ion-energy is gained at destination, setting the time-dependent control parameters is a macroscopic operation that costs energy and results in energy dissipation for the short time scales implied by the intrinsically fast STA processes. The potential minimum is displaced by modulating the voltages on control (dc) electrodes. A secondary effect of the modulation, usually ignored as it does not affect the ion dynamics, is the time-dependent energy shift of the potential minimum. The non trivial part of the energy consumption is due to the electromotive forces to set the electrode voltages through the low-pass filters required to preserve the electronic noise from decohering the ion's motion. The results for the macroscopic CS (the Paul trap) are compared to the microscopic power and energy of the ion alone. Similarities are found—and may be used quantitatively to minimize costs—only when the CS-dependent energy shift of the harmonic oscillator is included in the ion-energy.

We introduce a bang-bang shortcut to adiabaticity for the Dicke model, which we implement via a two-dimensional array of trapped ions in a Penning trap with a spin-dependent force detuned close to the center-of-mass drumhead mode. Our focus is on employing this shortcut to create highly entangled states that can be used in high-precision metrology. We highlight that the performance of the bang-bang approach is comparable to standard preparation methods, but can be applied over a much shorter time frame. We compare these theoretical ideas with experimental data which serve as a first step towards realizing this theoretical procedure for generating multi-partite entanglement.

In this article we consider a bosonic Josephson junction, a model system composed by two coupled nonlinear quantum oscillators which can be implemented in various physical contexts, initially prepared in a product of weakly populated coherent states. We quantify the maximum achievable entanglement between the modes of the junction and then use shortcuts to adiabaticity, a method developed to speed up adiabatic quantum dynamics, as well as numerical optimization, to find time-dependent controls (the nonlinearity and the coupling of the junction) which bring the system to a maximally entangled state.

We present an effective and fast (few microseconds) procedure for transferring a Bose–Einstein condensate from the ground state in a harmonic trap into the desired bands of an optical lattice. Our shortcut method is a designed pulse sequence where the time duration and the interval in each step are fully optimized in order to maximize robustness and fidelity of the final state with respect to the target state. The atoms can be prepared in a single band with even or odd parity, and superposition states of different bands can be prepared and manipulated. Furthermore, we extend this idea to the case of two-dimensional or three-dimensional optical lattices where the energies of excited states are degenerate. We experimentally demonstrate various examples and show very good agreement with the theoretical model. Efficient shortcut methods will find applications in the preparation of quantum systems, in quantum information processing, in precise measurement and as a starting point to investigate dynamics in excited bands.

We present a detailed theoretical analysis of the implementation of shortcut-to-adiabaticity protocols for the fast transport of neutral atoms with atom chips. The objective is to engineer transport ramps with durations not exceeding a few hundred milliseconds to provide metrologically relevant input states for an atomic sensor. Aided by numerical simulations of the classical and quantum dynamics, we study the behavior of a Bose–Einstein condensate in an atom chip setup with realistic anharmonic trapping. We detail the implementation of fast and controlled transports over large distances of several millimeters, i.e. distances 1000 times larger than the size of the atomic cloud. A subsequent optimized release and collimation step demonstrates the capability of our transport method to generate ensembles of quantum gases with expansion speeds in the picokelvin regime. The performance of this procedure is analyzed in terms of collective excitations reflected in residual center of mass and size oscillations of the condensate. We further evaluate the robustness of the protocol against experimental imperfections.

We propose a mean to obtain computationally useful resource states also known as cluster states, for measurement-based quantum computation, via transitionless quantum driving algorithm. The idea is to cool the system to its unique ground state and tune some control parameters to arrive at computationally useful resource state, which is in one of the degenerate ground states. Even though there is set of conserved quantities already present in the model Hamiltonian, which prevents the instantaneous state to go to any other eigenstate subspaces, one cannot quench the control parameters to get the desired state. In that case, the state will not evolve. With involvement of the shortcut Hamiltonian, we obtain cluster states in fast-forward manner. We elaborate our proposal in the one-dimensional Kitaev honeycomb model, and show that the auxiliary Hamiltonian needed for the counterdiabatic driving is of M-body interaction.

We propose an efficient strategy to find optimal control functions for state-to-state quantum control problems. Our procedure first chooses an input state trajectory, that can realize the desired transformation by adiabatic variation of the system Hamiltonian. The shortcut-to-adiabaticity formalism then provides a control Hamiltonian that realizes the reference trajectory exactly but on a finite time scale. As the final state is achieved with certainty, we define a cost functional that incorporates the resource requirements and a perturbative expression for robustness. We optimize this functional by systematically varying the reference trajectory. We demonstrate the method by application to population transfer in a laser driven three-level Λ-system, where we find solutions that are fast and robust against perturbations while maintaining a low peak laser power.

We study phase imprinting on Bose–Einstein condensates (BECs) with the fast-forward scaling theory revealing a nontrivial scaling property in quantum dynamics. We introduce a wave packet with uniform momentum density (WPUM) which has peculiar properties but is short-lived. The fast-forward scaling theory is applied to derive the driving potential for creation of the WPUMs in a predetermined time. Fast manipulation is essential for the creation of WPUMs because of the instability of the state. We also study loading of a BEC into a predetermined Bloch state in the lowest band from the ground state of a periodic potential. Controlled linear potential is not sufficient for creation of the Bloch state with large wavenumber because the change in the amplitude of the order parameter is not negligible. We derive the exact driving potential for creation of predetermined Bloch states using the obtained theory.

A systematic approach to design robust control protocols against the influence of different types of noise is introduced. We present control schemes which protect the decay of the populations avoiding dissipation in the adiabatic and nonadiabatic regimes and minimize the effect of dephasing. The effectiveness of the protocols is demonstrated in two different systems. Firstly, we present the case of population inversion of a two-level system in the presence of either one or two simultaneous noise sources. Secondly, we present an example of the expansion of coherent and thermal states in harmonic traps, subject to noise arising from monitoring and modulation of the control, respectively.

We propose a quantum speedup method for adiabatic generation of cat states in bosonic Josephson junctions via shortcuts to adiabaticity. We apply approximated counter-diabatic driving to a bosonic Josephson junction using the Holstein–Primakoff transformation. In order to avoid the problem of divergence in counter-diabatic driving, we take finite-size corrections into account. The resulting counter-diabatic driving is well-defined over whole processes. Schedules of the counter-diabatic driving consist of three steps; the counter-diabatic driving in the disordered phase, smoothly and slowly approaching the critical point, and the counter-diabatic driving in the ordered phase. Using the counter-diabatic driving, adiabatic generation of cat states is successfully accelerated. The enough large quantum Fisher information ensures that generated cat states are highly entangled.

We investigate a thermodynamic cycle using a Bose–Einstein condensate (BEC) with nonlinear interactions as the working medium. Exploiting Feshbach resonances to change the interaction strength of the BEC allows us to produce work by expanding and compressing the gas. To ensure a large power output from this engine these strokes must be performed on a short timescale, however such non-adiabatic strokes can create irreversible work which degrades the engine's efficiency. To combat this, we design a shortcut to adiabaticity which can achieve an adiabatic-like evolution within a finite time, therefore significantly reducing the out-of-equilibrium excitations in the BEC. We investigate the effect of the shortcut to adiabaticity on the efficiency and power output of the engine and show that the tunable nonlinearity strength, modulated by Feshbach resonances, serves as a useful tool to enhance the system's performance.

A shortcut to adiabaticity is a recipe for generating adiabatic evolution at an arbitrary pace. Shortcuts have been developed for quantum, classical and (most recently) stochastic dynamics. A shortcut might involve a counterdiabatic (CD) Hamiltonian that causes a system to follow the adiabatic evolution at all times, or it might utilize a fast-forward (FF) potential, which returns the system to the adiabatic path at the end of the process. We develop a general framework for constructing shortcuts to adiabaticity from flow fields that describe the desired adiabatic evolution. Our approach encompasses quantum, classical and stochastic dynamics, and provides surprisingly compact expressions for both CD Hamiltonians and FF potentials. We illustrate our method with numerical simulations of a model system, and we compare our shortcuts with previously obtained results. We also consider the semiclassical connections between our quantum and classical shortcuts. Our method, like the FF approach developed by previous authors, is susceptible to singularities when applied to excited states of quantum systems; we propose a simple, intuitive criterion for determining whether these singularities will arise, for a given excited state.

We apply the method of shortcuts to adiabaticity to nonequilibrium systems. For unitary dynamics, the system Hamiltonian is separated into two parts. One of them defines the adiabatic states for the state to follow and the nonadiabatic transitions are prevented by the other part. This property is implemented to the nonequilibrium entropy production and we find that the entropy is separated into two parts. The separation represents the Pythagorean theorem for the Kullback–Leibler divergence and an information-geometric interpretation is obtained. We also study a lower bound of the entropy, which is applied to derive a trade-off relation between time, entropy and state distance.