Commutation Relations in Mesoscopic Electric Circuits

In the talk, I briefly demonstrate the quantum theory for mesoscopic electric circuits and its applications. In the theory, the importance of the charge discreteness in a mesoscopic electric circuit is addressed. As a result, a new kind of commutation relation for electric charge and current occurred inevitably. The charge representation, canonical current representation and pseudo-current representation are discussed extensively. It not only provides a concrete realization of mathematical models which discuss the space quantization in high energy physics and quantum gravity but also presents a sequence of applications in condensed matter physics from a different point of view. A possible generalization to coupled circuits is also proposed. INTRODUCTION The dramatic achievement in nanotechnology has aroused tremendous developments in experimental physics in mesoscopic scale. Miniaturization of integrated circuits is undoubtedly a persistent trend for electronic device community. A theory for mesoscopic circuits was proposed by Li and Chen, in which the charge discreteness is first introduced in the quantization of electric circuits [1]. The possibility of space-time discreteness was early considered by Snyder [2] who indicated that the Lorentz invariance do not exclude quantized (discrete) space-time, it was also argued by Li [3] from the finiteness of the observed universe. S. Mantecinos, I. Saavedra, and O. Kunstmann [4] discussed the commutation relations of [2] arguing that it may be related to physics in high energy scale (10−10eV). Numerous attempts to the argument of exiting minimal position uncertainty were made [5] on the basis of various considerations in string theory as well as in quantum gravity. Actually, the approach of [1] not only provides a concrete realization of mathematical models for exploring the space quantization in high energy physics and quantum gravity but also presents a sequence of applications in condensed matter physics from a different point of view. For example, the persistent current is solved by regarding the mesoscopic metal ring as the circuit of a pure L-design. Application of the theory to a pure C-design gives rise to the Coulomb blockade solution [6]. BASIC DEFINITIONS AND COMMUTATION RELATIONS Let q̂ denote for the charge operator, and p̂ for the canonical conjugation of the charge satisfying [q̂, p̂] = ih̄. We call p̂ the canonical current operator since it is not only the canonical conjugation of charge but also the current operator in the quantization approach [7] where the charge was considered as a continuous variable. Taking into account of the discreteness of electronic charge in quantization procedure, we must impose that the eigenvalues of the self-adjoint operator q̂ take discrete values [1], i.e. q̂|n〉 = nqe|n〉 where n ∈ Z (set of integers) and qe = 1.602 × 10 coulomb, the elementary electric charge. We therefore introduce a minimum ‘shift operator’ Q̂ := ee in charge space, which satisfies [1] [q̂, Q̂] = −qeQ̂, Q̂ −1 = Q̂. (1) These relations determine the structure of the whole Fock space, accordingly, Q̂|n〉 = en+1 |n+1〉, Q̂|n〉 = en |n− 1〉 where αn’s being undetermined phases. The Fock space for the algebra (1) differs from the well known Fock space for the Heisenberg-Weyl algebra because the spectrum of the former is isomorphic to the set of integers Z but that of the later is isomorphic to the set of non-negative integers Z +{0}. Since {|n〉|n ∈ Z } spans a Hilbert space and q̂ is self-adjoint, both the completeness ∑ n∈Z |n〉〈n| = 1 and the orthogonality 〈n|m〉 = δnm faithful. The quasi-current Ĵ for a mesoscopic circuit is defined by Ĵ = −ih̄(Q̂ − Q̂)/qe which reduces to the canonical current in the limit qe → 0. The Hamiltonian of a mesoscopic LC-circuit is given by [1] Ĥ = − h̄ 2L Ĵ + 1 2C q̂ + εq̂, (2) where ε stands for the voltage source, L for inductance, and C for capacity of the circuit. Using eq.(1) we easily obtain the new commutation relations for the quasi-current operator, [q̂, Ĵ] = i h̄ 2 K̂, [q̂, K̂] = −i q e 2h̄ Ĵ , (3) where an auxiliary operator K̂ = Q̂ + Q̂ is introduced. Obviously eq.(3) obeys the SU(2) algebra after rescaling the operators. In terms of K̂ and Ĵ we can define a useful operator P̂ = ĴK̂ which we call the pseudo-current operator. Obviously the pseudo-current also reduces to canonical current in the limit qe → 0. With the help of (3), we obtain the following commutation relations, [q̂, P̂ ] = ih̄ ( 1 + ( qe 2h̄ )P̂ 2 ) . (4) Similar kind of commutation relation was considered earlier in [2] in searching the possibility of space-time discreteness. From the commutation relation (4) one will have a uncertainty relation [4,8] for the charge and pseudo-current [1], which is different from the conventional Heisenberg uncertainty relation. The definition of physical current Î arises from the Heisenberg equation Î = dq̂/dt = (1/ih̄)[q̂, Ĥ]. For the LC-design circuit, one can immediately obtain [9], Î = −i h̄ 2qeL (Q̂− Q̂). (5) PSEUDO-CURRENT REPRESENTATION We consider the pseudo-current representation P̂ |η〉 = η|η〉. The differential realization of commutation relation (4) is given by [2]