This study shows that the usual time evolution operator can act as a U(1) gauge transformation on the initial wavefuntion. When considering the spin freedom in the system, the time evolution operator corresponds to a non-Abelian SU(2) gauge transformation for spin-1/2 particles and a SU(3) gauge transformation for spin-1 particles. If we adopt the adiabatic approximation in the magnetization dynamics of a ferromagnetic system driven by a spin-polarized current, a temperature-dependent non-Abelian thermal gauge potential will appear. We can employ a temperature-dependent Landau-Lifshitz-Gilbert (LLG) equation to describe the magnetization dynamics in the semi-classical limit, a linear approximated solution for this equation is given analytically.
This study shows that the usual time evolution operator can act as a U(1) gauge transformation on the initial wavefuntion. When considering the spin freedom in the system, the time evolution operator corresponds to a non-Abelian SU(2) gauge transformation for spin-1/2 particles and a SU(3) gauge transformation for spin-1 particles. If we adopt the adiabatic approximation in the magnetization dynamics of a ferromagnetic system driven by a spin-polarized current, a temperature-dependent non-Abelian thermal gauge potential will appear. We can employ a temperature-dependent Landau-Lifshitz-Gilbert (LLG) equation to describe the magnetization dynamics in the semi-classical limit, a linear approximated solution for this equation is given analytically.
We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-21 particles in a magnetic field Bi(t) can be formulated in a natural way as an SU(2) gauge theory, with the magnetic field Bi(t) playing the role of the gauge potential Ai. The present approach can also be applied to systems of n levels with time-dependent potentials, U(n) being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schr\"odinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed.
We explore chaotic dynamics in SU(2) gauge theory by extracting the Lyapunov exponent under both in and out-of-equilibrium conditions to understand how chaos develops and influences the approach to thermalization. In thermal equilibrium at very high temperatures, the hard, electric, and magnetic scales are well separated, and one can write an effective theory for the soft gauge fields (magnetic modes) which are classical due to their large occupation number. In the vicinity of the critical temperature Tc, SU(2) gauge theory falls within the same universality class as Z2 scalar field theory. Exploiting this correspondence, we determine the Lyapunov exponent in the scalar theory through out-of-time-ordered correlators (OTOC). To probe non-equilibrium dynamics, we prepare the gauge fields in a highly occupied, non-thermal initial state. Under classical Yang–Mills evolution, the system evolves toward a self-similar scaling regime characterized by a scale separation similar to that of the thermal case. Within this regime, we again compute the Lyapunov exponent and compare it to its equilibrium counterpart at similar energy densities. This comparison allows us to quantify the timescale over which the system approaches thermal equilibrium.
The time-evolution operator is in general gauge dependent. Its gauge transformation property follows from the gauge transformation of the wave function and ensures gauge-invariant matrix elements. The same transformation property is shown here to follow from the formal solution of the Schr\"odinger equation for the time-evolution operator, which is a time-ordered exponential of the time integral of the Hamiltonian. The gauge transformation property of the time-evolution operator in the interaction picture is also obtained. The perturbation expansion of the time-evolution operator in one gauge can be transformed to give the perturbation expansion for the time-evolution operator in another gauge. The A\ensuremath{\cdot}p versus E\ensuremath{\cdot}r controversy in the electric dipole approximation is resolved by specifying the correct initial and final states.
We derive the quantum Langevin equation (QLE) for a harmonically single trapped cold atom subjected to artificial non-Abelian gauge potentials and linearly coupled to a heat bath. The independent-oscillator (IO) and the momentum-momenta coupling models are studied. In each case, the non-Abelian effect on the QLE is pointed out for a U(2) gauge transformation. For the IO model, only the generalized Lorentz force is modified by the appearance of an additive term. For the momentum-momenta coupling model, the generalized Lorentz force as well as the friction force are subjected to modifications. The dependence of the system on the magnetic field is explicit even if the gauge potential is uniform in space.
Given a time-independent Hamiltonian H , one can construct a time-dependent Hamiltonian Ht by means of the gauge transformation Ht=UtHUt†−iUt∂tUt† . Here Ut is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on Ut ensuring that Ht is physical as long as H is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians Ht and H such that the driven many-body dynamics governed by Ht can be reduced to the quench dynamics due to the time-independent H . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H=Ht is time-dependent but dynamically integrable.
In this work, we study the effect of non-Abelian gauge potential on the magnetic moment of a cold atom which is linearly coupled to a quantum heat bath through two different coupling schemes: (i) coordinate-coordinate coupling and (ii) momentum-momentum coupling. The bath is modelled as a collection of independent quantum harmonic oscillators. In each coupling scheme, the effect of the non-Abelian magnetic field on magnetic moment are explicitly demonstrated for a U(2) gauge transformation. For both schemes, we show that the magnetic moment has a different expression than that for the Abelian case. We consider two illustrative heat bath spectrum (Ohmic bath and Drude model) to evaluate the magnetic moment in the low temperature limit.