Bardeen-Cooper-Schrieffer State Representation and Pairing in the Fermionic Tonks-Girardeau Gas

TL;DR

This paper presents a new wave function representation for the fermionic Tonks-Girardeau gas, demonstrating it as a number-conserving BCS paired state and providing methods for experimental detection of pairing.

Contribution

It introduces a novel wave function based on occupation numbers and natural orbitals, establishing the fermionic Tonks-Girardeau gas as a paired BCS state with explicit expressions and detection strategies.

Findings

The ground state is a number-conserving BCS state.

Explicit coefficients for the BCS state are derived.

Methods for experimental detection of pairing are proposed.

Abstract

We introduce an alternative expression for the ground state wave function of the fermionic Tonks-Girardeau gas. Our wave function is constructed based on the occupation numbers and natural orbitals of the one-body density matrix. We demonstrate that the newly found wave function describes the ground state of the fermionic Tonks-Girardeau gas under any external potential. By expressing the proposed wave function in the framework of second quantization, we show that the ground state of the fermionic Tonks-Girardeau gas is a number-conserving Bardeen-Cooper-Schrieffer (BCS) state. We provide explicit expressions for the corresponding coefficients that describe the fermionic Tonks-Girardeau gas as a number-conserving BCS state. Additionally, the suitable form of the proposed wave function in second quantization allows us to derive the necessary expectation values to experimentally detect pairing in the fermionic Tonks-Girardeau gas. With this, we prove and show how to detect that the fermionic Tonks-Girardeau gas not only exhibits non-trivial quantum correlations but is also a paired state.