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The experimental optimization of a two-qubit controlled-Z (CZ) gate is realized following two different data-driven gradient ascent pulse engineering (GRAPE) protocols in the aim of optimizing the gate operator and the output quantum state, respectively. For both GRAPE protocols, the key computation of gradients utilizes mixed information of the input Z-control pulse and the experimental measurement. With an imperfect initial pulse in a flattop waveform, our experimental implementation shows that the CZ gate is quickly improved and the gate fidelities subject to the two optimized pulses are around 99%. Our experimental study confirms the applicability of the data-driven GRAPE protocols in the problem of the gate optimization.

A combination of digitized shortcut to adiabaticity (STA) and sequential digitized adiabaticity is implemented in a superconducting quantum device to determine the electronic states in two example systems, the ${\mathrm{H}}_{2}$ molecule and the topological Bernevig-Hughes-Zhang (BHZ) model. For ${\mathrm{H}}_{2}$, a short internuclear distance is chosen as a starting point, at which the ground and excited states are obtained via digitized STA. From this starting point, a sequence of internuclear distances is built. The eigenstates at each distance are sequentially determined from those at the previous distance via the digitized adiabaticity, leading to the potential energy landscapes of ${\mathrm{H}}_{2}$. The same approach is applied to the BHZ model and the valence and conduction bands are obtained with a high degree of accuracy along the $X$-$\mathrm{\ensuremath{\Gamma}}$-$X$ line cut of the first Brillouin zone. Furthermore, a numerical simulation of this method is performed to successfully extract the ground states of hydrogen chains with lengths of between three and six atoms.

Measurement-based feedback control is central in quantum computing and precise quantum control. Here we realize a fast and flexible field-programmable-gate-array-based feedback control in a superconducting Xmon qubit system. The latency of room-temperature electronics is custom optimized to be as short as 140 ns. Projective measurement of a signal qubit produces a feedback tag to actuate a conditional pulse gate to the qubit. In a feed-forward process, the measurement-based feedback tag is brought to a different target qubit for a conditional control. In a two-qubit experiment, the feedback and feed-forward controls are simultaneously actuated in consecutive steps. A quantum number is then generated by the signal qubit, and a random walk of the target qubit is correspondingly triggered and realized on the Bloch sphere. Our experiment provides a conceptually simple and intuitive benchmark for the feedback control in a multi-qubit system. The feedback system can be further scaled up for more complex feedback control experiments.

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., 21 pages, 5 figures

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., 18 pages, 6 figures

The significance of topological phases has been widely recognized in the community of condensed matter physics. The well controllable quantum systems provide an artificial platform to probe and engineer various topological phases. The adiabatic trajectory of a quantum state describes the change of the bulk Bloch eigenstates with the momentum, and this adiabatic simulation method is however practically limited due to quantum dissipation. Here we apply the `shortcut to adiabaticity' (STA) protocol to realize fast adiabatic evolutions in the system of a superconducting phase qubit. The resulting fast adiabatic trajectories illustrate the change of the bulk Bloch eigenstates in the Su-Schrieffer-Heeger (SSH) model. A sharp transition is experimentally determined for the topological invariant of a winding number. Our experiment helps identify the topological Chern number of a two-dimensional toy model, suggesting the applicability of the fast adiabatic simulation method for topological systems., Comment: 37 pages, 8 figures including Supplementary

For a frequency-tunable two-qubit system, a controlled-Z (CZ) gate can be realized by adiabatically driving the qubit system through an avoided level crossing between an auxiliary state and computational levels. Here, we theoretically propose a fast CZ gate using a shortcut-to-adiabaticity (STA). Experimentally, the STA CZ gate is implemented with a 52 ns control pulse for two coupled superconducting Xmon qubits. Measured fidelity of the STA CZ gate is higher than 96.0%, in both quantum process tomography and randomized benchmarking. The protocol allows a flexible design of the evolution time and control waveform. We suggest that this `fast adiabatic' CZ gate can be directly applied to other multi-qubit quantum systems., Comment: 7 pages, 4 figures

Following a recently proposed decomposition technique [C. R. Duan et al., Phys. Rev. B 95, 214308 (2017)], we inspect the zero-temperature spin-boson model for five different cutoff functions of the spectral density. With oscillatory and non-oscillatory exponentially decaying functions to decompose the bath correlation function, the hierarchy equation of motion is reliably extended to each spectral density under our investigation. The predicted spin dynamics is gradually converged with the increase of the hierarchic expansion order and the number of decomposing basis functions. Our systematic study of different cutoff functions expands previous results of the delocalized-localized phase transition with the exponential and sudden cutoffs in the spectral density.

The linear absorption spectrum of a multisite system can be written as a weighted accumulation of elements of an absorption matrix. In the framework of the quantum-classical Liouville equation (QCLE), a mean-field approximation is introduced to simplify the calculation of the absorption matrix. The classical bath oscillators are propagated under partially shifted potentials, which reflects an averaged behavior after considering quantum jumps of the system states. For a specified initial condition, the time-dependent shifting possibility of each bath potential is given by the time evolution of site population estimated by the Redfield equation or the noninteracting blip approximation. The two hybrid QCLE approaches are tested in various models, including biased and unbiased two-site models, a subnetwork and the whole monomer of Fenna-Matthews-Olson, and harmonic and anharmonic baths. With numerically excellent results, the numerical studies show reliability and flexibility of the hybrid QCLE in calculating the absorption matrix and spectrum.

In the framework of the quantum-classical Liouville equation (QCLE), the quantum kinetic expansion (QKE) of the spin-boson model is extended to an arbitrary combination of the bath potential and the system-bath interaction. The mixed quantum-classical estimation of the QKE rate kernels and modification functions are transformed into averages of deterministic classical trajectories over the Wigner initial distribution. For the standard spin-boson model, the QCLE-QKE method produces exactly the same result as that from full quantum dynamics and the numerical applicability of the approximate action-angle initial distribution is verified. For an anharmonic bath with the quartic potential, the QCLE-QKE calculation under the action-angle initial distribution illustrates the influence of this specific anharmonicity. With the increase of the quartic parameter, the fourth order QKE corrections are suppressed and the short-time population transfer is accelerated together with an enhanced quantum oscillation.

An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model. In this approach, a complete set of orthonormal functions are used to expand an arbitrary bath correlation function. As a result, a complete dynamic basis set is constructed by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element. The reliability of the extended HEOM is demonstrated by comparison with the stochastic Hamiltonian approach under room-temperature classical ohmic and sub-ohmic noises and the multilayer multiconfiguration time-dependent Hartree theory under zero-temperature quantum ohmic noise. Upon increasing the order in the hierarchical expansion, the result obtained from the extended HOEM systematically converges to the numerically exact answer.

For a general two-cluster network, a new methodology of the cluster-based generalized quantum kinetic expansion (GQKE) is developed in the matrix formalism under two initial conditions: the local cluster equilibrium and system-bath factorized states. For each initial condition, the site population evolution follows exactly a distinct closed equation, where all the four terms involved are systematically expanded over inter-cluster couplings. For the system-bath factorized initial state, the numerical investigation of the two models, a biased (2, 1)-site system and an unbiased (2, 2)-site system, verifies the reliability of the GQKE and the relevance of higher-order corrections. The time-integrated site-to-site rates and the time evolution of site population reveal the time scale separation between intra-cluster and inter-cluster kinetics. The population evolution of aggregated clusters can be quantitatively described by the approximate cluster Markovian kinetics.

For a general two-cluster energy transfer network, a new methodology of the generalized quantum kinetic expansion (GQKE) method is developed, which predicts an exact time-convolution equation for the cluster population evolution under the initial condition of the local cluster equilibrium state. The cluster-to-cluster rate kernel is expanded over the inter-cluster couplings. The lowest second-order GQKE rate recovers the multichromophoric Forster theory (MCFT) rate. The higher-order corrections to the MCFT rate are systematically included using the continued fraction resummation form, resulting in the resummed GQKE method. The reliability of the GQKE methodology is verified in two model systems, revealing the relevance of higher-order corrections.

In the spin-boson model, a continued fraction form is proposed to systematically resum high-order quantum kinetic expansion (QKE) rate kernels, accounting for the bath relaxation effect beyond the second-order perturbation. In particular, the analytical expression of the sixth-order QKE rate kernel is derived for resummation. With higher-order correction terms systematically extracted from higher-order rate kernels, the resummed quantum kinetic expansion (RQKE) approach in the continued fraction form extends the Pade approximation and can fully recover the exact quantum dynamics as the expansion order increases., accepted by J. Chem. Phys

The polarization dependence and temporal profile of the fifth-order Raman response function and corresponding correlation function in liquid Xe are studied both analytically and numerically. Based on the symmetry of an isotropic sample, the fifth-order Raman response function has twelve distinct tensor elements, ten of which are independent, and the corresponding correlation function has twelve distinct tensor elements, seven of which are independent. The coefficients for decomposition into independent components are calculated explicitly based on the tensor property of an isotropic sample and are used to identify different coupling mechanisms in liquid Xe. The two-dimensional profile of the fifth-order Raman response function is evaluated by a simple hydrodynamic expression derived using the Gaussian factorization scheme. An alternative approach reduces the fifth-order Raman response function to time correlation functions that are easy to compute.

Within the framework of the hierarchy equation of motion (HEOM), the quantum kinetic expansion (QKE) method of the spin-boson model is reformulated in the matrix representation. The equivalence between the two formulations (HEOM matrices and quantum operators) is numerically verified from the calculation of the time-integrated QKE rates. The matrix formulation of the QKE is extended to the system-bath factorized initial state. Following a one-to-one mapping between HEOM matrices and quantum operators, a quantum kinetic equation is rederived. The rate kernel is modified by an extra term following a systematic expansion over the site-site coupling. This modified QKE is numerically tested for its reliability by calculating the time-integrated rate and non-Markovian population kinetics. For an intermediate-to-strong dissipation strength and a large site-site coupling, the population transfer is found to be significantly different when the initial condition is changed from the local equilibrium to system-bath factorized state.

The algebraic structure of the quantum Morse oscillator is explored to formulate the coherent state, the phase-space representations of the annihilation and creation operators, and their classical limits. The formulation allows us to calculate the linear and nonlinear quantum response functions for microcanonical Morse systems and to demonstrate the linear divergence in the corresponding classical response function. On the basis of the uncertainty principle, the classical divergence is removed by phase-space averaging around the microcanonical energy surface. For the Morse oscillator, the classical response function averaged over quantized phase space agrees exactly with the quantum response function for a given eigenstate. Thus, phase-space averaging and quantization provide a useful way to establish the classical-quantum correspondence of anharmonic systems.

An asymptotic scaling theory is presented using the conceptual basis of trapping-free subspace (i.e., orthogonal subspace) to establish the generic mechanism of optimal efficiency of excitation energy transfer in light-harvesting systems. A quantum state orthogonal to the trap will exhibit noise-assisted transfer, clarifying the significance of initial preparation. For such an initial state, the efficiency is enhanced in the weak damping limit ($⟨t⟩\ensuremath{\sim}1/\ensuremath{\Gamma}$), and suppressed in the strong damping limit ($⟨t⟩\ensuremath{\sim}\ensuremath{\Gamma}$), analogous to Kramers turnover in classical rate theory. An interpolating expression $⟨t⟩=A/\ensuremath{\Gamma}+B+C\ensuremath{\Gamma}$ quantitatively describes the trapping time over the entire range of the dissipation strength, and predicts the optimal efficiency at ${\ensuremath{\Gamma}}_{\mathrm{opt}}\ensuremath{\sim}J$ for homogenous systems. In the presence of static disorder, the scaling law of transfer time with respect to dephasing rate changes from linear to square root, suggesting a weaker dependence on the environment. The prediction of the scaling theory is verified in a symmetric dendrimer system by numerically exact quantum calculations. Though formulated in the context of excitation energy transfer, the analysis and conclusions apply in general to open quantum processes, including electron transfer, fluorescence emission, and heat conduction.

With sulfur partially substituted by 3d transition metals, magnetism in a molybdenum disulphide monolayer is investigated comprehensively by the calculation of a spin-polarized density functional theory. The magnetic moment induced by a single defect is found to be equal to the absolute value of the magnetic moment of this defect atom subtracted by two Bohr magnetons. A detailed analysis on the locally projected density of states demonstrates that the underlying mechanism can be qualitatively interpreted in a simple ionic scenario. Subsequently, quasi one-dimensional superlattices of defects are built for the exploration of long range magnetic orders. Among four candidates of chromium and copper superlattices for high temperature dilute magnetic semiconductors (DMSs), a particular chromium superlattice holds the promise of a room temperature DMS against the change of the on-site Coulomb interaction.

Following the calculation of optimal energy transfer in thermal environment in our first paper (Wu et al., New J. Phys., 2010, 12, 105012), full quantum dynamics and leading-order `classical' hopping kinetics are compared in the seven-site Fenna-Matthews-Olson (FMO) protein complex. The difference between these two dynamic descriptions is due to higher-order quantum corrections. Two thermal bath models, classical white noise (the Haken-Strobl-Reineker model) and quantum Debye model, are considered. In the seven-site FMO model, we observe that higher-order corrections lead to negligible changes in the trapping time or in energy transfer efficiency around the optimal and physiological conditions (2% in the HSR model and 0.1% in the quantum Debye model for the initial site at BChl 1). However, using the concept of integrated flux, we can identify significant differences in branching probabilities of the energy transfer network between hopping kinetics and quantum dynamics (26% in the HSR model and 32% in the quantum Debye model for the initial site at BChl 1). This observation indicates that the quantum coherence can significantly change the distribution of energy transfer pathways in the flux network with the efficiency nearly the same. The quantum-classical comparison of the average trapping time with the removal of the bottleneck site, BChl 4, demonstrates the robustness of the efficient energy transfer by the mechanism of multi-site quantum coherence. To reconcile with the latest eight-site FMO model, the quantum-classical comparison with the flux network analysis is summarized in the appendix. The eight-site FMO model yields similar trapping time and network structure as the seven-site FMO model but leads to a more disperse distribution of energy transfer pathways., Comment: submitted to Journal of Chemical Physics

Understanding the mechanisms of efficient and robust energy transfer in light-harvesting systems provides new insights for the optimal design of artificial systems. In this paper, we use the Fenna–Matthews–Olson (FMO) protein complex and phycocyanin 645 (PC 645) to explore the general dependence on physical parameters that help maximize the efficiency and maintain its stability. With the Haken–Strobl model, the maximal energy transfer efficiency (ETE) is achieved under an intermediate optimal value of dephasing rate. To avoid the infinite temperature assumption in the Haken–Strobl model and the failure of the Redfield equation in predicting the Forster rate behavior, we use the generalized Bloch–Redfield (GBR) equation approach to correctly describe dissipative exciton dynamics, and we find that maximal ETE can be achieved under various physical conditions, including temperature, reorganization energy and spatial–temporal correlations in noise. We also identify regimes of reorganization energy where the ETE changes monotonically with temperature or spatial correlation and therefore cannot be optimized with respect to these two variables., National Science Foundation (U.S.) (NSF 0806266), National Institutes of Health (U.S.) (Grant 0556268), Massachusetts Institute of Technology. Energy Initiative (Seed grant), United States. Dept. of Energy (DOE grant number DE-SC0001088)

A theoretical approach is developed to derive a hierarchy of mode-coupling equations for the dynamics of concentrated colloidal suspensions, which improves the prediction of the colloidal glass transition. Our derivation is based on a matrix formalism for stochastic dynamics and the resulting recursive expressions for irreducible memory functions. The 1st order truncation of the generalized mode-coupling closure recovers mode-coupling theory, whereas its 2nd and 3rd order truncations provide corrections. The predictions of the transition volume fraction and Debye-Waller parameter for the hard-sphere colloidal system improve with the increasing mode-coupling order and compare favorably with experimental measurements. The key predictions of mode-coupling theory (MCT) are the ergodic-to-nonergodic transition and associated dynamic scaling [1–3]. These predictions have been supported by numerical simulations of simple liquids and scattering measurements of colloidal suspensions, which are often described as a many-body Brownian system with pairwise interactions. Because of its simplicity and the wealth of experimental investigations, an assembly of spherical colloidal particles has been an ideal system for demonstrating dynamic slowdown and the glass transition both experimentally and theoretically [4]. In the standard projection operator formalism, the co

The reorientational relaxation of molecular systems is important in many phenomenon and applications. In this paper, we explore the reorientational relaxation of a model Brownian rotor lattice system with short range interactions in both the high and low temperature regimes. In this study, we use a basis set expansion to capture collective motions of the system. The single particle basis set is used in the high temperature regime, while the spin wave basis is used in the low temperature regime. The equations of motion derived in this approach are analogous to the generalized Langevin equation, but the equations render flexibility by allowing nonequilibrium initial conditions. This calculation shows that the choice of projection operators in the generalized Langevin equation (GLE) approach corresponds to defining a specific inner-product space, and this inner-product space should be chosen to reveal the important physics of the problem. The basis set approach corresponds to an inner-product and projection operator that maintain the orthogonality of the spherical harmonics and provide a convenient platform for analyzing GLE expansions. The results compare favorably with numerical simulations, and the formalism is easily extended to more complex systems.

We apply a new formalism to derive the higher-order quantum kinetic expansion (QKE) for studying dissipative dynamics in a general quantum network coupled with an arbitrary thermal bath. The dynamics of system population is described by a time-convoluted kinetic equation, where the time-nonlocal rate kernel is systematically expanded on the order of off-diagonal elements of the system Hamiltonian. In the second order, the rate kernel recovers the expression of the noninteracting-blip approximation (NIBA) method. The higher-order corrections in the rate kernel account for the effects of the multi-site quantum coherence and the bath relaxation. In a quantum harmonic bath, the rate kernels of different orders are analytically derived. As demonstrated by four examples, the higher-order QKE can reliably predict quantum dissipative dynamics, comparing well with the hierarchic equation approach. More importantly, the higher-order rate kernels can distinguish and quantify distinct nontrivial quantum coherent effects, such as long-range energy transfer from quantum tunneling and quantum interference arising from the phase accumulation of interactions.