The early quantum theory of electrical conductivity in crystal lattices by Bloch and Zener [1,2] led to the striking prediction that a homogeneous static electric field induces an oscillatory rather than uniform motion of the electrons. These so-called Bloch oscillations have never been observed in natural crystals because the scattering time of the electrons by the lattice defects is much shorter than the Bloch period. In semiconductor superlattices the larger spatial period leads to a much shorter Bloch period (,600 fs) and Bloch oscillations have recently been observed through the emission of THz radiation by the electrons [3]. Here we present Bloch oscillations of atoms in the fundamental energy band of a periodic optical potential. We directly measure the atomic momentum distribution evolving in time under the influence of a constant inertial force for various potential depths. We experimentally observe oscillation periods in the millisecond range as well as positive and negative effective masses. Bloch oscillations are a pure quantum effect which can be explained in a simple one-dimensional model. The periodicity of the lattice (period d) leads to a band structure (Fig. 1) of the energy spectrum of the particle and the corresponding eigenenergies Ensqd and eigenstates jn, ql (Bloch states) are labeled by the discrete band index n and the continuous quasimomentum q; Ensqd and jn, ql are periodic functions of q with period 2pyd and q is conventionally restricted to the first Brillouin zone g2pyd, pydg [4]. Under the influence of a constant external force F, weak enough not to induce interband transitions, a given Bloch state jn, qs0dl evolves (up to a phase factor) into the state jn, qstdl according to qstd qs0d 1 Ftyh . (1) This evolution is periodic with a period tB hyjFjd corresponding to the time required for the quasimomentum to scan a full Brillouin zone. The mean velocity in jn, qstdl
A general Hamiltonian H of electrons in finite concentration, interacting via any two-body coupling inside a crystal of arbitrary dimension, is considered. For simplicity and without loss of generality, a one-band model is used to account for the electron-crystal interaction. The electron motion is described in the Hilbert space ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin \ensuremath{\zeta}=0,\ifmmode\pm\else\textpm\fi{}1 are considered in this work. The Hamiltonian then reads H=
HD+${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where
HD consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that (
HD+${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills (
HD-${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=0. We prove also that the eigenvectors of class 1 have off-diagonal long-range order, whereas those of class 2 do not. Finally, our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity. \textcopyright{} 1996 The American Physical Society.
We provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of physical operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Sharpness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Sturdiness. Filters are non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilitieso follows. A more detailed abstract is provided in the paper.
We present an operator-based derivation of Bloch's theorem. In this approach, we first construct an explicit operator form of the Hamiltonian by writing the potential part in terms of an infinite sum of the momentum-translation operators. Next, we prove that it commutes with the position-translation operator corresponding to a lattice translation, using the exponential re-ordering identity. We then build on Merzbacher's treatment of simultaneous diagonalization of commuting operators to reproduce Kittel's central equation and also to derive a new formula for the simultaneous eigenket corresponding to a specific energy eigenvalue. Moreover, through the Born rule, we provide a precise account of the position and momentum probability distributions for Bloch states, and we examine the concept of crystal momentum. Finally, we apply our findings to the Kronig–Penney model, where we numerically depict the band structure and illustrate the delocalized nature of the single electron.
The orthogonality requirement usually specified for the eigenfunctions φ of a quantum-mechanical operator P which has a continuous spectrum of eigenvalues λ, is ∫φ*(λ)φ(λ′) dλ=δ(λ−λ′). The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound-state wavefunction ψ in the form ∫ a(λ)φ(λ) dλ. The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion ψ=∫ a(λ)φ(λ) dλ is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground-state hydrogen atom. When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single-valued eigenfunctions.