Research

[ Japanese | English ]

Quantum annealing

Quantum annealing is known as a quantum-mechanical approach for optimization problems. In quantum annealing approaches, the total Hamiltonian consists of driving and problem Hamiltonians. If we choose a driving Hamiltonian so that constraints are naturally satisfied without penalty terms, the dimension of the state space is considerably reduced. Especially, for graph coloring, whose ground state is degenerate, a quantum annealing approach based on real-time quantum dynamics has some advantages.

Fig

Reference

[ Back to TopBack to Home ]

Many-body localization

In many-body quantum systems with disorder, the dynamical transition from the thermal to many-body localization (MBL) phase occurs as disorder strength increases. On the other hand, dependence on interaction coupling is nontrivial. The MBL transition is characterized by the ratio of disorder strength to the power of interaction coupling with respect to an extra critical exponent.

Fig

Reference

[ Back to TopBack to Home ]

Quantum dynamics under periodic driving

We can observe a wide variety of quantum dynamics in one-dimensional quantum systems under periodic driving.

Fig

Cold atoms in an optical lattice in the presence of both static and oscillating linear fields, under some conditions, show large oscillations termed Super-Bloch Oscillations (SBOs). The tunneling amplitudes between adjacent lattices are renormalized, and the effective tunneling depends on the amplitude and frequency of the driving field. For full-understanding of features of SBOs, an important phase correction, which depends on the phase of the driving field as well as its frequency and amplitude, must be included.

Quantum transport in one-dimensional quantum systems that are described by, e.g., Heisenberg model and Hubbard model can be controlled by means of periodic driving. Some of the eigenstates of the undriven Hamiltonian are bound states, and others are unpaired states. The effective tunneling amplitudes (exchange interactions) are renormalized under an oscillating field, and the renormalized amplitudes are different between bound-pair and unpaired states. Thus, one can control the relative direction and speed of transport for bound-pair and unpaired states.

The one-magnon spin dynamics in a quantum spin system can give remarkable transfer of quantum information: a wavepacket travels for a quite long time without broadening under some conditions. The left figure below shows a one-magnon wavepacket dynamics in the quantum Heisenberg chain under an oscillating field. The non-diffusive transport is explained by the corresponding classical phase space. The quantum spin model is equivalent to the driven Harper model, which has the classical phase space, under a certain condition, shown in the right figure below. The colored region, which is on a shearless torus, corresponds to the initial position of the quantum wavepacket. The white channels in the chaotic sea include shearless tori, which are the key to make the quantum transport non-dispersive.

Fig

Frustrated quantum XXZ spin chains with next-nearest-neighbor couplings show Gaussian orthogonal ensemble (GOE) spectral statistics. The energy spectral flow for such a spin chain under an oscillating magnetic field has many avoided crossings which lead to energy diffusion. Energy diffusion for the system depends on both the field strength and driving frequency. The diffusion coefficients obey the power law, whose power varies depending on the linear response and nonperturbative regimes, with respect to the strength and frequency of the field. The widths of the linear response and the nonperturbative regimes depend on the strength of frustrations.

References

[ Back to TopBack to Home ]

Level statistics

Level statistics of quantum systems generally show the Poisson distribution in a level-spacing distribution when the corresponding classical system is integrable. By contrast, the typical level-spacing distribution for a non-integrable system is the Wigner distribution. A quantum XXZ spin chain (anisotropic Heisenberg model) is integrable by the Bethe ansatz, although it does not have a corresponding classical system. Its level-spacing distribution is the Poisson distribution, and that of the XXZ spin chain with next-nearest-neighbor couplings, which is not solvable by the Bethe ansatz, is the Wigner distribution. This implies that the level statistics can be used to check the Bethe ansatz solvability. However, non-integrable quantum systems show non-Wigner distributions in some cases due to imperfect desymmetrization or finite size effects.

Fig

It was known that the level statistics of a standard Dicke model changes under quantum phase transition (QPT). We have investigated the level statistics of a pseudo-Hermitian Dicke Hamiltonian, which has real-number eigenvalues though it is not Hermitian. Its level-spacing distribution near the integrable limit is close to Poisson distribution, while it is Wigner distribution for the ranges of the parameters for which the Hamiltonian is non-integrable. The assertion that QPT is a precursor to a change in the level statistics, which is true in the standard Dicke model, is valid for the pseudo-Hermitian Dicke model only for a limited range of the parameter space because of its larger parameter space.

References

[ Back to TopBack to Home ]

Wrinkle formation on coatings

Wrinkles often emerge when double coating is applied. Considering wrinkle formation as buckling of an elastic thin film, we proposed a simple mathematical model. The mathematical model explains phenomena at early stage of wrinkle formation. Numerical simulations demonstrate reproduction of wrinkle formation.

Fig

Reference

[ Back to TopBack to Home ]

Magnetic domain pattern on a Co/Ni multilayer

The experiment by means of spin-polarized low energy electron microscopy (SPLEEM) has shown that the magnetic anisotropy, and hence magnetic domain patterns, in a Co/Ni multilayer changes depending on the number of layers. In the experiment, pairs of 2-ML-thick Ni and 1-ML-thick Co layers were deposited on W(110). Perpendicular magnetization appears after the deposition of Ni layers. In contrast, in-plane magnetization appears after Co deposition for the first and second pairs. However, after the deposition of the third Co/Ni pair, magnetization remains perpendicular to the surface even after Co deposition. When domain patterns appear in the perpendicular magnetization images, the in-plane magnetization images show clear domain-wall structures. The numerical simulation, which is based on the Landau-Lifshitz-Gilbert (LLG) equation, well reproduce magnetic domain patterns with clear domain walls and the change in magnetic anisotropy.

Fig

References

[ Back to TopBack to Home ]

Magnetic domain pattern in a spinor BEC

When a Bose-Einstein condensation (BEC) has spin degrees of freedom, it is called a spinor BEC. In spinor BECs, spin and gauge degrees of freedom couple in various manners, leading to nontrivial properties of spin waves and topological excitations such as spin textures and vortices. The figures show examples of magnetic domain patterns (left: transverse magnetization, right: longitudinal magnetization) that are simulated by means of the Gross-Pitaevskii (GP) equation.

Fig

If we focus on spin degrees of freedom, we can derive a hydrodynamic equation from the GP equation under some assumptions. The hydrodynamic description extracts the dynamics of spin wave and affords a straightforward approach by which to investigate the spin dynamics of the condensate.

The hydrodynamic equation with dissipation has the same form as the equation of the spin dynamics in a conducting ferromagnet. Thus, the domain pattern which is similar to that of a ferromagnetic thin film can be reproduced by the hydrodynamic equation simulation.

References

[ Back to TopBack to Home ]

Domain pattern in a ferromagnetic thin film

Ferromagnetic garnet thin films show domain patterns, which consist of up-spin and down-spin domains. In order to observe the magnetic domain patterns under a zero field in experiments, usually an external magnetic field is applied once up to the saturation field, and then it is removed. The characteristics of domain patterns depend on the decreasing rate of the external field. When the field decreases rapidly, a sea-island structure (right figure) appears, while a labyrinth structure (left figure) is observed for a slowly decreasing field.

Fig

Under an oscillating field, we can see some interesting patterns such as lattice patterns, concentric-ring patterns and traveling patterns moving very slowly compared with the frequency of the field. The key idea to analyze the effect of the oscillating field on these patterns is to separate the dynamics into a rapidly oscillating part and a slowly varying part.

The Landau-Lifshitz-Gilbert (LLG) equation is often used for simulation of magnetic domain patterns. If the Gilbert damping is removed from the LLG equation, it merely describes the precession of magnetization with a ferromagnetic interaction. However, even without the damping, domains that look similar to those of scalar fields are formed.

References

[ Back to TopBack to Home ]

Shear-induced onion phase

Surfactants in water can form various kinds of assemblies. Multi-lamellar vesicles, which are also called onions, are one of the interesting morphologies induced by shear flow. Onions form a two-dimensional close-packed honeycomb structure, and their centers are supposed to be on a plane (say the x-y plane). Focusing on the surface height (z(x,y)) of onions, we can make a phenomenological model to simulate the top view of the onion phase.

Fig

Reference

[ Back to TopBack to Home ]

Semi-automatic timetabling

Scheduling parallel sessions of an academic meeting is a complicated task. Organizers have to consider a lot of constraints. For example, sessions with the same topic and those with largely overlapped audiences should not be in the same timeslot; some sessions often need to avoid particular timeslots; and the number of meeting rooms is often limited. If each presentation is assigned to an appropriate session, an antiferromagnetic Potts model can be used for semi-automatic timetabling.

Fig

Reference

[ Back to TopBack to Home ]

Mathematical model of dengue fever

Dengue fever is a mosquito-borne viral disease. In non-tropical urban cities such as Tokyo, mosquitoes live mainly in parks (not business districts or residential areas), and they are active only in summer. We proposed a mathematical model that consists of high-risk-area (e.g., parks) and low-risk-area (e.g., residential areas) parts. The result calculated from the model implies that the restriction of the travel between the two areas can promote the epidemic in some situations.

Fig

Reference

[ Back to TopBack to Home ]