Multistate Landau-Zener Models Research Articles

Solution to a class of multistate Landau-Zener model beyond integrability conditions

We study a class of multistate Landau-Zener model which cannot be solved by integrability conditions or other standard techniques. By analyzing analytical constraints on its scattering matrix and performing fitting to results from numerical simulations of the Schrödinger equation, we find nearly exact analytical expressions of all its transition probabilities for specific parameter choices. We also determine the transition probabilities up to leading orders of series expansions in terms of the inverse sweep rate (namely, in the diabatic limit) for general parameter choices. We further show that this model can describe a Su-Schrieffer-Heeger chain with couplings changing linearly in time. Our work presents a new route, i.e., analytical constraint plus fitting, to analyze those multistate Landau-Zener models which are beyond the applicability of conventional solving methods.

Quantum emulation of the transient dynamics in the multistate Landau-Zener model

Quantum simulation is one of the most promising near term applications of quantum computing. Especially, systems with a large Hilbert space are hard to solve for classical computers and thus ideal targets for a simulation with quantum hardware. In this work, we study experimentally the transient dynamics in the multistate Landau-Zener model as a function of the Landau-Zener velocity. The underlying Hamiltonian is emulated by superconducting quantum circuit, where a tunable transmon qubit is coupled to a bosonic mode ensemble comprising four lumped element microwave resonators. We investigate the model for different initial states: Due to our circuit design, we are not limited to merely exciting the qubit, but can also pump the harmonic modes via a dedicated drive line. Here, the nature of the transient dynamics depends on the average photon number in the excited resonator. The greater effective coupling strength between qubit and higher Fock states results in a quasi-adiabatic transition, where coherent quantum oscillations are suppressed without the introduction of additional loss channels. Our experiments pave the way for more complex simulations with qubits coupled to an engineered bosonic mode spectrum.

Generic theory of characterizing topological phases under quantum slow dynamics

Dynamical characterization of equilibrium topological phases has attracted considerable attention in recent years. In this paper, we make a thorough exploration of the non-adiabatic characterization of topological phases under slow quench protocol. We first propose an exactly solvable multi-state Landau-Zener model that can be directly applied to the non-adiabatic slow quench dynamics of topological systems. Then we present two different schemes to characterize the bulk topology of the system based on the so called spin inversion surface. The first one needs least number of quenching processes, but requires to measure the gradients of time-averaged spin-polarization on the SIS. The second one only needs to measure the value of time-averaged spin-polarization on the SIS, thus makes it possible to directly characterize the topological phases by introducing an extra quenching process. Moreover, high-order SIS or band inversion surface (BIS) relying on the dimension reduction approach, is also generalized to the above two different characterization schemes. One can extract the topological invariant from pairs of points with opposite signs both on the 0D highest order BIS and on the 0D highest order SIS, which greatly simplifies the measurement strategy and characterization process. In a word, superior to the sudden quench protocol, we demonstrate that the topological invariant can be captured not only by the topological information on BIS, but also on the SIS. In particular, direct characterization of topological phases based on BIS and SIS can be realized.

Integrability in the multistate Landau-Zener model with time-quadratic commuting operators

All currently known exactly solvable multistate Landau–Zener (MLZ) models are associated with families of operators that commute with the MLZ Hamiltonians and depend on time linearly. There can also be operators that satisfy the integrability conditions with the MLZ Hamiltonians but depend on time quadratically. We show that, among the MLZ systems, such time-quadratic operators are much more common. We demonstrate then that such operators generally lead to constraints on the independent variables that parametrize the scattering matrix. Such constraints lead to asymptotically exact expressions for the transition probabilities in the adiabatic limit of a three-level MLZ model. New more complex fully solvable MLZ systems are also found.

Integrable multistate Landau–Zener models with parallel energy levels

We discuss solvable multistate Landau–Zener (MLZ) models whose Hamiltonians have commuting partner operators with ∼1/τ-time-dependent parameters. Many already known solvable MLZ models belong precisely to this class. We derive the integrability conditions on the parameters of such commuting operators, and demonstrate how to use such conditions in order to derive new solvable cases. We show that MLZ models from this class must contain bands of parallel diabatic energy levels. The structure of the scattering matrix and other properties are found to be the same as in the previously discussed completely solvable MLZ Hamiltonians.

A large class of solvable multistate Landau–Zener models and quantum integrability

The concept of quantum integrability has been introduced recently for quantum systems with explicitly time-dependent Hamiltonians (Sinitsyn et al 2018 Phys. Rev. Lett. 120 190402). Within the multistate Landau–Zener (MLZ) theory, however, there has been a successful alternative approach to identify and solve complex time-dependent models (Sinitsyn and Chernyak 2017 J. Phys. A: Math. Theor. 50 255203). Here we compare both methods by applying them to a new class of exactly solvable MLZ models. This class contains systems with an arbitrary number of interacting states and shows quick growth with N number of exact adiabatic energy crossing points, which appear at different moments of time. At each N, transition probabilities in these systems can be found analytically and exactly but complexity and variety of solutions in this class also grow with N quickly. We illustrate how common features of solvable MLZ systems appear from quantum integrability and develop an approach to further classification of solvable MLZ problems.

Integrable Time-Dependent Quantum Hamiltonians.

We formulate a set of conditions under which the nonstationary Schrödinger equation with a time-dependent Hamiltonian is exactly solvable analytically. The main requirement is the existence of a non-Abelian gauge field with zero curvature in the space of system parameters. Known solvable multistate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time dependence into various quantum integrable models while maintaining their integrability. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.

Loss of adiabaticity with increasing tunneling gap in nonintegrable multistate Landau-Zener models

We consider the simplest non-integrable model of multistate Landau-Zener transition. In this model two pairs of levels in two tunnel coupled quantum dots are swept passed each other by the gate voltage. Although this 2 * 2 model is non-integrable, it can be solved analytically in the limit when the inter-level energy distance is much smaller than their tunnel splitting. The result is contrasted to the similar 2 * 1 model, in which one of the dots contains only one level. The latter model does not allow interference of the virtual transition amplitudes, and it is exactly solvable. In 2 * 1 model, the probability for a particle, residing at time t -> -\infty in one dot, to remain in the same dot at t -> \infty falls off exponentially with tunnel coupling. By contrast, in 2 * 2 model, this probability grows exponentially with tunnel coupling. The physical origin of this growth is the formation of the tunneling-induced collective states in the system of two dots. This can be viewed as manifestation of the Dicke effect.

Multistate Landau-Zener models with all levels crossing at one point

We discuss common properties and reasons for integrability in the class of multistate Landau-Zener (MLZ) models with all diabatic levels crossing at one point. Exploring the Stokes phenomenon, we show that each previously solved model has a dual one, whose scattering matrix can be also obtained analytically. For applications, we demonstrate how our results can be used to study conversion of molecular into atomic Bose condensates during passage through the Feshbach resonance, and provide purely algebraic solutions of the bowtie and special cases of the driven Tavis-Cummings model (DTCM).

The quest for solvable multistate Landau-Zener models

Recently, integrability conditions (ICs) in mutistate Landau-Zener (MLZ) theory were proposed []. They describe common properties of all known solved systems with linearly time-dependent Hamiltonians. Here we show that ICs enable efficient computer assisted search for new solvable MLZ models that span complexity range from several interacting states to mesoscopic systems with many-body dynamics and combinatorially large phase space. This diversity suggests that nontrivial solvable MLZ models are numerous. In addition, we refine the formulation of ICs and extend the class of solvable systems to models with points of multiple diabatic level crossing.

Constraints on scattering amplitudes in multistate Landau-Zener theory

We derive a set of constraints, which we will call hierarchy constraints (HCs), on scattering amplitudes of an arbitrary multistate Landau-Zener model (MLZM). The presence of additional symmetries can transform such constraints into nontrivial relations between elements of the transition probability matrix. This observation can be used to derive complete solutions of some MLZMs or, for models that cannot be solved completely, to reduce the number of independent elements of the transition probability matrix.

Differential and total cross sections of mutual neutralization in low-energy collisions of isotopes ofH++H−

Mutual neutralization in the collisions of ${\mathrm{H}}^{+}$ and ${\mathrm{H}}^{\ensuremath{-}}$ is studied both theoretically and experimentally. The quantum-mechanical ab initio model includes covalent states associated with the $\mathrm{H}(1)+\mathrm{H}(n\ensuremath{\le}3)$ limits and the collision energy ranges from 1 meV to 100 eV. The reaction is theoretically studied for collisions between different isotopes of the hydrogen ions. From the partial wave scattering amplitude, the differential and total cross sections are computed. The differential cross section is analyzed in terms of forward- and backward-scattering events, showing a dominance of backward scattering which can be understood by examining the phase of the scattering amplitudes for the gerade and ungerade set of states. The isotope dependence of the total cross section is compared with the one obtained using a semiclassical multistate Landau-Zener model. The final state distribution analysis emphasizes the dominance of the $n=3$ channel for collisions below 10 eV, while at higher collision energies, the $n=2$ channel starts to become important. For collisions of ions forming a molecular system with a larger reduced mass, the $n=2$ channel starts to dominate at lower energies. Using a merged ion-beam apparatus, the branching ratios for mutual neutralization in ${\mathrm{H}}^{+}$ and ${\mathrm{H}}^{\ensuremath{-}}$ collisions in the energy range from 11 to 185 eV are measured with position- and time-sensitive particle detectors. The measured and calculated branching ratios satisfactorily agree with respect to state contributions.

Multi-particle and multi-state Landau-Zener model: Dynamic matrix approach

The paper presents the multi-particle and multi-state Landau-Zener problem and focuses on indistinguishable particles with degenerate states applying the Dynamics matrix approach. It is observed that the probabilities are described by the Binomial law with the limiting values that achieved exact results for spin $ S=1/2$ and $ S=1$ Landau-Zener problems. The derivation of the generalized multi-particle probability function is observed to be equivalent to solving a Landau-Zener problem for particle number equal to twice the spin.

Analytical results for state-to-state transition probabilities in the multistate Landau-Zener model by nonstationary perturbation theory

Multistate generalizations of Landau-Zener model are studied by summing entire series of perturbation theory. A new technique for analysis of the series is developed. Analytical expressions for probabilities of survival at the diabatic potential curves with extreme slope are proved. Degenerate situations are considered when there are several potential curves with extreme slope. New expressions for some state-to-state transition probabilities are derived in degenerate cases.

Reply to comment on ‘Exact results for survival probability in the multistate Landau–Zener model’

After using a corrected change of variables, Dobrescu and Sinitsyn (2006 J. Phys. B: At. Mol. Opt. Phys. 39 1253) subsequently circumvent the direct calculation of the perturbation theory expansion by comparing the problem to the bow-tie model that was solved exactly previously (Ostrovsky and Nakamura 1997 J. Phys. B: At. Mol. Opt. Phys. 30 6939). We argue that the direct calculations are feasible and interesting: for instance, they are able to enlighten the specific structure of the survival probability in the bow-tie model, as illustrated by some examples.

Comment on ‘Exact results for survival probability in the multistate Landau–Zener model’

We correct the proof of the Brundobler–Elser formula (BEF) provided in Volkov and Ostrovsky (2004 J. Phys. B: At. Mol. Opt. Phys. 37 4069) and continued in the appendix of Volkov and Ostrovsky (2005 J. Phys. B: At. Mol. Opt. Phys. 38 907). After showing that some changes of variables employed in these articles are used erroneously, we propose an alternative change of variables which solves the problem. In our proof, we reveal the connection between the BEF for a general N-level Landau–Zener system and the exactly solvable bow-tie model. The special importance of the diabatic levels with maximum/minimum slope is emphasized throughout.

No-go theorem for bands of potential curves in multistate Landau–Zener model

In the multistate Landau–Zener model all diabatic potential curves are linear functions of time. We consider the case where there is a band of parallel potential curves with slope larger (smaller) than any of the other slopes in the system. In such a situation transitions from a lower (higher) lying state within the band to any upper (lower) state are counterintuitive, since in the simple semiclassical picture they are possible only via propagation backwards in time. We rigorously prove that the probabilities of such transitions are exactly zero. In other words the energy of states within the band can only decrease (increase). The theoretical method employed is based on analysis of perturbation theory series to arbitrary order.

Exact results for survival probability in the multistate Landau–Zener model

An exact formula is derived for survival probability in the multistate Landau–Zener model in the special case where the initially populated state corresponds to the extremal (maximum or minimum) slope of a linear diabatic potential curve. The formula was originally guessed by S Brundobler and V Elzer (1993 J. Phys. A: Math. Gen. 26 1211) based on numerical calculations. It is a simple generalization of the expression for the probability of diabatic passage in the famous two-state Landau–Zener model. Our result is obtained via analysis and summation of the entire perturbation theory series.

Multiparticle Landau-Zener problem: Application to quantum dots

We propose a simple ansatz that allows to generate new exactly solvable multi-state Landau-Zener models. It is based on a system of two decoupled two-level atoms whose levels vary with time and cross at some moments. Then we consider multiparticle systems with Heisenberg equations for annihilation operators having similar structure with Shr\"odinger equation for amplitudes in multistate Landau-Zener models and show that the corresponding Shr\"odinger equation in multiparticle sector belongs to the multistate Landau-Zener class. This observation allows to generate new exactly solvable models from already known ones. We discuss possible applications of the new solutions in the problem of the driven charge transport in quantum dots.

Nonadiabatic dynamics in evaporative cooling of trapped atoms by a radio-frequency field

Magnetically trapped neutral atoms can be cooled with the evaporation technique. This is typically done by using a radio-frequency (rf) field that adiabatically couples trapped and untrapped internal atomic states for atoms with kinetic energies above a value set by the field frequency. The rf field can also induce nonadiabatic changes of internal atomic spin states $(F,M)$ that lead to heating and enhanced loss of atoms. In this paper we use wave-packet simulations to show that the evaporation process can induce these nonadiabatic transitions, which change the internal spin state of doubly spin-polarized (2,2) trapped atoms. We also verify the validity of a multistate Landau-Zener model in describing the nonadiabatic dynamics. In addition, we calculate exchange relaxation rate coefficients for collisions between atoms in the $(2,M)$ states of ${}^{23}\mathrm{Na}$ atoms. Large exchange relaxation coefficients for ${}^{23}\mathrm{Na}$ as compared to ${}^{87}\mathrm{Rb} F=2$ suggest that evaporative cooling of (2,2) Na will be more difficult than for the corresponding state of Rb.