In this second part about dynamics of atomic system we revisit the logic application of
SU(2) dynamics. We reiterate that solution of quantum dynamics systems can be represented geometrically. Such geometric representations of solutions provides intuitive physical insights. To which end studying dynamics of Quantum systems via
su(2) Lie algebra proves to be convenient way of obtaining geometric solution for two level systems. In this paper link is established between two formalisms that made use of Lie algebra to describe equation of motion for quantum system. In both approaches the Hamiltonian and the density matrix are expressed as a linear combination of the Lie group. To exemplify the approach we consider a very well studied two level system whose coupling laser pulse have area of
2π. Beyond establishing link between these two formalism we provide analytical solutions using Magnus expansion by exploiting the Sylvester formula where knowledge of eigenvalues suffice to obtain solution. Furthermore we also obtained two constants of motion by assuming time dependent detuning whose time. Consequently we have shown how one can have two disjoint subspaces whose evolution vector is independent of each other. As an application we present two classical binary logic machines, namely CNOT- gate and parity checker, that the two level system's dynamics mimics.
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
The ultrafast (picosecond) coherent dynamics of exciton systems in semiconductors can be approximately described by bosonic mean-field equations. These equations are nonlinear and therefore difficult to solve analytically. It is thus important to study the general dynamical properties of these equations, such as the underlying symmetry and corresponding conservation laws. It is shown in this paper that, for an
N-species exciton system (e.g., heavy-hole and light-hole excitons), a mean-field Hamiltonian (including the coupling to external fields and fermionic corrections) can be formulated which is a member of the
su(N,N) algebra. As a consequence, the equations of motion for the center-of-mass momentum dependent exciton distribution and the coherent biexciton amplitude can be cast into a form similar to that of the optical Bloch vector in two-level atoms that belong to the algebra
su(2) [or, more generally,
N-level atoms with algebra
su(N)]. It is shown that the analog to the Bloch sphere in
N-level atoms is an unbounded hypersurface (generalized hyperboloid) that constrains the motion of the exciton distribution and coherent biexciton amplitude. Further constants of motions that constrain the motion on the hypersurface are found from an
su(N,N) generalization to the Hioe-Eberly method in
su(N) systems (
N-level atoms) [F. Hioe and J. Eberly, Phys. Rev. Lett. 47, 838 (1981)].
Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group S N and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.
The quantum dynamics of arbitrary
N-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary
N-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension
is general for any
N. The time evolution of an
N-level system [generated by a complex element of the SU(
N) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in
n Euclidean dimensions [the rotation group SO(
n)] is then mapped directly to
n oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent
N-level systems scales as
N2 for the density matrix formalism but increases only linearly with
N for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.
A set of time-dependent basis states is obtained with a group of unitary transformations generated by a Lie algebra. Applying the time-dependent variational principle to the trial function subspace constructed from the linear combination of the time-dependent basis states gives rise to a set of ‘‘classical’’ equations of motion for the group parameters and the expansion coefficients from which the time evolution of the system state can be determined. The formulation is developed for a general Lie algebra as well as for the commonly encountered algebra containing homogeneous polynominal products of the coordinate Q and momentum P operators (or equivalently the boson creation a° and annihilation a operators) of order 0, 1, and 2. Explicit expressions for the transition amplitudes are derived by virtue of the cannonical transformation properties of the unitary transformation. The applicability of the present formalism in a variety of problems is implied by two illustrative examples: (a) a parametric amplifier; (b) the collinear collision of an atom with a Morse oscillator.