A compact derivation is given for propagators of a wide class of time-dependent Hamiltonians that describe coupled spin (``qubit'') pairs. As many as eight of the 16 operators in the system, each with its own time-dependent coefficient, can be handled with no more difficulty than a single spin in a magnetic field. A single first-order (in time), quadratically nonlinear differential equation for a classical function is the only nontrivial ingredient in constructing the full quantum propagator for a wide variety of magnetic coherences and qubit interactions.
A compact derivation is given for propagators of a wide class of time-dependent Hamiltonians that describe coupled spin (``qubit'') pairs. As many as eight of the 16 operators in the system, each with its own time-dependent coefficient, can be handled with no more difficulty than a single spin in a magnetic field. A single first-order (in time), quadratically nonlinear differential equation for a classical function is the only nontrivial ingredient in constructing the full quantum propagator for a wide variety of magnetic coherences and qubit interactions.
For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
Four-level systems in quantum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependent Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical, first-order differential equations. In particular, by exploiting $\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{su}(2)$ and $\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{u}(1)$ subalgebras of the full SU(4) dynamical group of the system, the nontrivial part of the final calculation is reduced to a single Riccati (first-order, quadratically nonlinear) equation, itself simply solved. Examples are provided of two-qubit problems from the recent literature, including implementation of two-qubit gates with Josephson junctions.
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.
Abstract In this paper, we find the quantum propagator for a general time-dependent quadratic Hamiltonian. The method is based on the properties of the propagator and the fact that the quantum propagator fulfills two independent partial differential equations originating from Heisenberg equations for positions and momenta. As an application of the method, we find the quantum propagator for a linear chain of interacting oscillators for both periodic and Dirichlet boundary conditions. The state and excitation propagation along the harmonic chain in the absence and presence of an external classical source is studied and discussed. The location of the first maxima of the probability amplitude P(n, τ) is a straight line in the (n, τ)-plane, indicating a constant speed of excitation propagation along the chain.