We introduce the new concept of "nodal phase correlation" (NPC) into many-particle quantum mechanics. NPC is a particular quantum correlation among particles, which is important for current-carrying states of isolated systems. Mathematically it is characterized by a definite relation in configuration space between the phase and the modulus of the many-particle wave function. We illustrate the significance of NPC by showing the crucial role it plays in magnetic flux quantization and in certain macroscopic interference effects in superconducting rings. To this end we partly rederive the theory of these phenomena. Our method is based on the BCS theory. We critically examine previous work, in particular the single-particle pairing theory, which is considered to be a standard approach to magnetic flux quantization. This theory does not give rise to NPC. We are able to show that the corresponding many-particle wave function is in fact unrealistic and not consistent with the BCS theory of superconductivity.
We introduce the new concept of "nodal phase correlation" (NPC) into many-particle quantum mechanics. NPC is a particular quantum correlation among particles, which is important for current-carrying states of isolated systems. Mathematically it is characterized by a definite relation in configuration space between the phase and the modulus of the many-particle wave function. We illustrate the significance of NPC by showing the crucial role it plays in magnetic flux quantization and in certain macroscopic interference effects in superconducting rings. To this end we partly rederive the theory of these phenomena. Our method is based on the BCS theory. We critically examine previous work, in particular the single-particle pairing theory, which is considered to be a standard approach to magnetic flux quantization. This theory does not give rise to NPC. We are able to show that the corresponding many-particle wave function is in fact unrealistic and not consistent with the BCS theory of superconductivity.
The nodal structure of the Schr\"odinger wave function is used to describe an N-particle system in a stationary quantum state. In terms of nodal lines (quantized flux lines) together with their circulation numbers, two fields are defined: the microscopic, singular nodal flux field n (r) and the macroscopic, regular nodal flux density field N (r). Their definition arises in a natural way from a quantum-mechanical vector potential defined by the gradient of the multivalued phase function associated with the pattern of quantized flux lines. The flux of the fields n (r) and N (r) is quantized and shown to be proportional to the circulation of the velocity in the system. The unit circulation for a system of bosons with a particle spin s=0 and mass m0 is equal to m0h. In the case of charged particles, the field (qc)N(r) resembles Jehle's formulation of the magnetic field of a lepton as a superposition of quantized flux lines. In our case, however, the presence of quantized flux lines, forming closed loops follows from the mere existence of a current density and does not have to be assumed. By a simple argument, using permutation symmetry, it is shown that a flux quantum of the size 2qhc is possible.
The requirement that the quantal wave function be single-valued is examined in the light of two recent developments: The effect of a magnetic vector potential on a particle moving in a multiply connected field-free region (Aharonov-Bohm effect), and the flux quantization inside a superconducting ring. Both of these effects are new applications of conventional elementary quantum mechanics. The single-valuedness problem is considered, with its historical background, for particles with no spin, particles with spin 12, and rigid bodies. It is shown that the single-valuedness condition, if properly adapted to the particular physical model, is deeply rooted in the foundations of quantum mechanics. However, wave functions which superficially appear to be double-valued are useful in an helicity representation of spinors and in the construction of nuclear wave functions.
We revisit the quantization of the order parameter, which we refer to as third quantization, from the perspective of the commutation relation between the phase operator of the order parameter and the particle-number operator. We show that this macroscopic commutation relation does not constitute an independent fundamental postulate added to quantum mechanics, but instead emerges naturally from second quantization in the thermodynamic limit for both bosonic and fermionic many-body systems. In this sense, both Bose-Einstein condensates (BECs) and Bardeen-Cooper-Schrieffer (BCS) states can be understood as macroscopic quantum states described by bosonic coherent states: in BEC, bosons condense into a single coherent mode with a well-defined phase, while in BCS systems, collective excitations of Cooper pairs can also acquire an effectively bosonic coherent description. On this basis, we propose a new macroscopic interpretation of the BCS-BEC crossover. To characterize this crossover, we model a conventional superconductor as an assembly of macroscopically separated superconducting segments. As the intra-segment coupling increases, the system evolves from a BCS-like regime toward a BEC-like regime, in which the segments collectively behave as macroscopic coherent states. Inter-segment tunneling then locks their phases, establishes global phase coherence, and gives rise to a bulk Bose-Einstein condensate. The phase diagram of the BCS-BEC crossover can thus be understood as a manifestation of a macroscopic quantum process governed by the coherent-state dynamics of the order parameter. Our results provide a unified perspective on BEC, BCS superconductivity, and the BCS-BEC crossover within the framework of third quantization.
We have analyzed the quantum interference effects in the macroscopic “superconducting molecule.” The composite system consists of two massive superconducting rings, each interrupted by a Josephson junction, which are at the same time weakly coupled with one another. The special case of coupling via the Josephson four-terminal junction is considered. The structure of the macroscopic quantum states in an applied magnetic field is calculated. It is shown that, depending on the values of the magnetic fluxes through each ring, the system displays two groups of states, the “orthostates” with both induced currents going in the same direction, and the “parastates” with the opposite currents and with the total induced flux locked to zero value. The transition to the flux locked state with changing of the total applied flux is sudden and is preserved in a certain interval which is determined by the difference of the fluxes applied through each ring. It makes the system sensitive to small gradients of the external magnetic field.
Recently, a new theory of superconductivity has been put forward that attributes the origin of superconductivity to the appearance of a non-trivial Berry connection from many-electron wave functions. This theory reproduces the major results of the BCS theory with conserving the particle number, and predicts the single-electron supercurrent tunneling across the Josephson junction with keeping the correct Josephson relation. We re-examine the quantization of superconducting qubit circuits by taking into account the above development, and show that the dynamical variables used in the standard theory, the flux nodes relating to the voltage, should be replaced by those relating to the electromagnetic vector potential. The fact that the Josephson junction tunneling allows the single-electron supercurrent tunneling is the reason for the existence of excited single electrons in superconducting qubits with Josephson junctions. We predict that it will be avoided by weakening the coupling between two superconductors in the Josephson junction.
In a ring of s-wave superconducting material the magnetic flux is quantized in units of Φ0=2eh. It is well known from the theory of Josephson junctions that if the ring is interrupted with a piece of d-wave material, then the flux is quantized in one-half of those units due to a additional phase shift of π. We reinterpret this phenomenon in terms of geometric phase. We consider an idealized hetero-junction superconductor with pure s-wave and pure d-wave electron pairs. We find, for this idealized configuration, that the phase shift of π follows from the discontinuity in the geometric phase and is thus a fundamental consequence of quantum mechanics.