Cooper-pair number–phase Wigner function for the bosonic operator Josephson model

2006 ◽  
Vol 359 (6) ◽  
pp. 580-586 ◽  
Author(s):  
Hong-Yi Fan ◽  
Ji-Suo Wang ◽  
Shu-guang Liu
Keyword(s):  
Wigner Function ◽  
Cooper Pair ◽  
Bosonic Operator ◽  
Pair Number

QUANTUM STATE OF JOSEPHSON JUNCTION AS COOPER PAIR NUMBER-PHASE ENTANGLED STATE IN THE BOSONIC OPERATOR JOSEPHSON MODEL

2007 ◽  
Vol 21 (21) ◽  
pp. 3697-3706 ◽  
Author(s):  
HONG-YI FAN ◽  
JI-SUO WANG ◽  
XIANG-GUO MENG
Keyword(s):  
Josephson Junction ◽  
Quantum State ◽  
Cooper Pair ◽  
Geometric Distribution ◽  
Entangled State ◽  
Phase Operator ◽  
Schmidt Decomposition ◽  
Bosonic Operator ◽  
Pair Number

Based on Feynman's explanation about Cooper pair that "a bound pair acts as a Bose particle" and the bosonic operator Hamiltonian of the Josephson junction, we realize that the quantum state of the Josephson junction is a Cooper pair number-phase entangled state constructed by the phase operator across the junction. Its Schmidt decomposition is derived. The Cooper pair number-phase squeezed state's projection onto this entangled state leads to a geometric distribution.

COOPER-PAIR-NUMBER–PHASE SQUEEZING FOR THE BOSONIC HAMILTONIAN MODEL OF JOSEPHSON JUNCTION

2006 ◽  
Vol 20 (17) ◽  
pp. 1041-1047 ◽  
Author(s):  
HONG-YI FAN ◽  
JI-SUO WANG ◽  
YUE FAN
Keyword(s):  
Josephson Junction ◽  
Cooper Pair ◽  
Entangled State ◽  
Entangled State Representation ◽  
State Representation ◽  
Extra Energy ◽  
Bosonic Operator ◽  
Hamiltonian Model ◽  
Pair Number ◽  
Two Sides

Based on Feynman's explanation about the Cooper pair that "a bound pairs act as a Bose particle" and the bosonic operator Hamiltonian of Josephson junction (H.-Y. Fan, Int. J. Mod. Phys. B17 (2003) 2599) as well as the entangled state representation, we establish a possible number–phase squeezing mechanism for the Cooper-pair and the phase difference between the two sides of the junction. We find that when an extra energy (e.g. microwave radiation) is provided to the junction, then this squeezing mechanism can happen.

Phase State as a Cooper-Pair Number: Phase Minimum Uncertainty State for Josephson Junction

2003 ◽  
Vol 17 (13) ◽  
pp. 2599-2608 ◽  
Author(s):  
Hong-Yi Fan
Keyword(s):  
Josephson Junction ◽  
Cooper Pair ◽  
Uncertainty Relation ◽  
Phase State ◽  
Phase Operator ◽  
Operator Formalism ◽  
Hamiltonian Model ◽  
Pair Number ◽  
Minimum Uncertainty ◽  
Phase Uncertainty

Based on Feynman's explanation that a Cooper pair is "a bound pair act as a Bose particle", we propose a bosonic phase operator formalism and a bosonic Hamiltonian model for Josephson junction. The Cooper pair number — phase uncertainty relation is thus established. The corresponding minimum uncertainty state is derived which turns out to be a phase state.

The inner bound of quantum spacetime

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Wigner Function ◽  
Uncertainty Principle ◽  
Quantum Spacetime

This article addresses the connection of the UNCERTAINTY PRINCIPLE with the WIGNER FUNCTION.

Superfluidity and Superconductivity

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Magnetic Field ◽  
Wave Function ◽  
Wave Packet ◽  
Long Range ◽  
Cooper Pair ◽  
Bose Condensation ◽  
Range Order ◽  
Condensed State ◽  
Long Range Order ◽  
Coherent Wave

Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ‎ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.

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