Exact scattering matrix of graphs in magnetic field and quantum noise

We consider arbitrary quantum wire networks modelled by finite, noncompact, connected quantum graphs in the presence of an external magnetic field. We find a general formula for the total scattering matrix of the network in terms of its local scattering properties and its metric structure. This is applied to a quantum ring with $N$ external edges. Connecting the external edges of the ring to heat reservoirs, we study the quantum transport on the graph in ambient magnetic field. We consider two types of dynamics on the ring: the free Schr\"odinger and the free massless Dirac equations. For each case, a detailed study of the thermal noise is performed analytically. Interestingly enough, in presence of a magnetic field, the standard linear Johnson-Nyquist law for the low temperature behaviour of the thermal noise becomes nonlinear. The precise regime of validity of this effect is discussed and a typical signature of the underlying dynamics is observed

Reflection-transmission quantum Yang-Baxter equations

We explore the reflection–transmission quantum Yang–Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow us to establish, for the first time, a direct relationship with the different previous works on the subject and make evident the advantages of the reflection–transmission algebra as a universal approach to integrable systems with impurities

Spontaneous symmetry breaking in the non-linear Schrodinger hierarchy with defect

We introduce and solve the one-dimensional quantum non-linear Schrodinger (NLS) equation for an N-component field defined on the real line with a defect sitting at the origin. The quantum solution is constructed using the quantum inverse scattering method based on the concept of Reflection-Transmission (RT) algebras recently introduced. The symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra. We classify all the corresponding reflection and transmission matrices. This provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly. Finally, we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry.Comment: discussion on symmetry breaking has been improved and examples adde

The quantum non-linear Schrodinger model with point-like defect

We establish a family of point-like impurities which preserve the quantum integrability of the non-linear Schrodinger model in 1+1 space-time dimensions. We briefly describe the construction of the exact second quantized solution of this model in terms of an appropriate reflection-transmission algebra. The basic physical properties of the solution, including the space-time symmetry of the bulk scattering matrix, are also discussed.Comment: Comments on the integrability and the impurity free limit adde

Quantum field theory on quantum graphs and application to their conductance

We construct a bosonic quantum field on a general quantum graph. Consistency of the construction leads to the calculation of the total scattering matrix of the graph. This matrix is equivalent to the one already proposed using generalized star product approach. We give several examples and show how they generalize some of the scattering matrices computed in the mathematical or condensed matter physics litterature. Then, we apply the construction for the calculation of the conductance of graphs, within a small distance approximation. The consistency of the approximation is proved by direct comparison with the exact calculation for the `tadpole' graph.Comment: 32 pages; misprints in tree graph corrected; proofs of consistency and unitarity adde

Quantum Wire Networks with Magnetic Field

International audienceThe charge transport and the noise of a quantum wire network, made of threesemi-infinite external leads attached to a ring crossed by a magnetic flux, areinvestigated. The system is driven away from equilibrium by connecting theexternal leads to heat reservoirs with different temperatures and/or chemicalpotentials. The properties of the exact scattering matrix of this configurationas a function of the momentum, the magnetic flux and the transmission along thering are explored. We derive the conductance and the noise, describing indetail the role of the magnetic flux. In the case of weak coupling between thering and the reservoirs, a resonant tunneling effect is observed. We alsodiscover that a non-zero magnetic flux has a strong impact on the usualJohnson-Nyquist law for the pure thermal noise at small temperatures

LAPTH

Solving the quantum non-linear Schrödinger equation with δ-type impurit

Solving the quantum non-linear Schrodinger equation with delta-type impurity.

We establish the exact solution of the nonlinear Schrödinger equation with a delta-function impurity, representing a pointlike defect which reflects and transmits. We solve the problem both at the classical and the second quantized levels. In the quantum case the Zamolodchikov–Faddeev algebra, familiar from the case without impurities, is substituted by the recently discovered reflection-transmission (RT) algebra, which captures both particle–particle and particle–impurity interactions. The off-shell quantum solution is expressed in terms of the generators of the RT algebra and the exact scattering matrix of the theory is derived