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Flux quantization and off-diagonal long-range order in thin cylinders containing a "weak link"

Michael Edward Melich

Published Jan 1, 1966• 0 citations
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Abstract

The equilibrium states of a thin-walled hollow cylinder containing a 'freak link", (e .g. a Josephson junction ) are examined in a Bose gas model calculation. It is found that these states may be such that the fluxoid is not an integer multiple of the flux quantum. The deviation from integer flux quantization depends upon the coupling strength of the "weak link" and the externally applied magnetic flux. The first order reduced density matrix (the 1-matrix) for the boson system is then investigated utilizing the general features of the model calculation. The fluxoid quantization and persistent currents are considered in terms of the modifications (due to the link) of the off-diagonal long- ra nge order of the 1-ma tr ix .

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Physical Review•Jun 14, 1965•0 citations
The equilibrium states of a thin-walled hollow cylinder containing a 'freak link", (e .g. a Josephson junction ) are examined in a Bose gas model calculation. It is found that these states may be such that the fluxoid is not an integer multiple of the flux quantum. The deviation from integer flux quantization depends upon the coupling strength of the "weak link" and the externally applied magnetic flux. The first order reduced density matrix (the 1-matrix) for the boson system is then investigated utilizing the general features of the model calculation. The fluxoid quantization and persistent currents are considered in terms of the modifications (due to the link) of the off-diagonal long- ra nge order of the 1-ma tr ix .
Some properties of the reduced density matrix are considered in their general relation to the current produced by a system of charged particles in a thin-walled long cylinder. The common connection of persistent currents and off-diagonal long-range order to the velocity distribution function will be particularly stressed. Essential features of this function, characteristic for the superconducting state, are discussed and exemplified by applying an analogous procedure to the case of boson and fermion systems.
Relations between the existence of ODLRO in the reduced density matrixρ n (for alln≧1) and that of superconducting currents in a hollow cylinder are discussed. Some concrete behaviours ofρ n , including the power law decaying behaviour, in the coordinate space representation are considered and the corresponding effective superconducting particle densities are given. Some restraint conditions caused by the existence of the ODLRO (including quasi-ODLRO) have been found and discussions on the minimum unit of flux quantization are also included.
It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.
The Hamiltonian of a superconducting thin loop interrupted by a double-barrier Josephson junction is derived. The parameter ε, which takes account of the difference in the Josephson coupling of the two single junctions in the SISIS structure, is seen to allow variation of the energy gap ΔE01 between the ground state and the first excited flux number state of the system. An additional parameter γ, describing the coupling between the outer electrodes of the double-barrier Josephson junction, is introduced. Persistent currents in the system are studied in terms of the usual parameter α, representing the ratio between the Josephson energy and the magnetic energy, and of the externally applied flux.
For two-component Fermi gases at zero temperature, through a general derivation based on the BCS wave function, our research shows that the nonuniform molecular Bose-Einstein condensate has off-diagonal long-range order and can be described well by an order parameter when the size of molecules is much smaller than the size of the whole system. We also give the equation of state and nonlinear evolution equation for the order parameter of nonuniform molecular condensates, which is similar to the Gross-Pitaevskii equation of atomic Bose condensate. The nonlinear evolution equation is applied to consider the Josephson effect for two weakly linked molecular condensates in the presence of nonuniform magnetic field. We find clear particle-number oscillation where spatially dependent binding energy of molecules plays an important role.
The thermodynamic properties of the hard-sphere Bose gas are calculated from the one- and two-particle density matrices. The one- and two-particle density matrices are evaluated in the Bogoliubov approximation. The off-diagonal elements of the one- and two-particle density matrices are studied for the presence or absence of off-diagonal long-range order in various temperature regions. Off-diagonal long-range order was found in the single-particle density matrix below the Bose-Einstein transition temperature. The two-particle density matrix also exhibited off-diagonal long-range order whenever the single-particle density matrix did.