Jumping up and down arbitrary-order excited Schrödinger’s cats formally satisfying the Pauli exclusion principle

Abstract

We present a formal extension of the Pauli exclusion principle to the arbitrary-order jumping up and down excited Schrödinger’s cats. We show that the excited symmetric Schrödinger’s cat of the n-order will occupy the same n-order excited eigenstate of the harmonic oscillator when the “alive” and “dead” cat’s constituents closely approach in space and time each other. The antisymmetric n-order cat states, vice versa, will never occupy the same excited n-order eigenstate. For example, the respective probabilities for antisymmetric cat states of n = 1, 2, and 3 orders to jump up onto the corresponding energy levels with n = 2, 3, and 4, and, at the same time, jump down onto the levels with n = 0, 1, and 2, are found to be in the ratios: 2/3:1/3, 3/5:2/5, and 4/7:3/7, with the asymptotic saturated ratio of 1/2:1/2 for the cat states distributed over the higher-energy levels. Moreover, we demonstrate the remarkable analogies between linear and nonlinear dynamics of the higher-order Schrödinger’s cats revealed in the framework of the nonlinear Schrödinger equation model with harmonic oscillator potential.

Introduction

Many aspects of the Schrödinger’s cats behavior in the harmonic oscillator potential are still not fully understood. In particular, it was demonstrated that the interchange of the “alive” and “dead” constituents in the whole Schrödinger cat wave function opens the possibility to reveal one interesting, but obviously, only formal analogy with the Pauli exclusion principle [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. The simplest Schrödinger’s cat state is composed by a superposition of two canonical coherent states, which are initially localized in different regions of space at the same time and illustrated by the quantum superposition of “alive” and “dead” cat states. This state represents the well-known Schrödinger’s cat paradox – the Schrödinger’s “gedaken cat” – a thought experiment, in which “a cat” is prepared in a quantum superposition of “alive” and “dead” states, i.e., “a cat” is neither “alive” nor “dead”, but rather a linear combination of the two, being simultaneously “alive” and “dead” until the measurement occurs  [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45]. Although this so-called “Schrödinger’s cat” was for decades a matter of academic curiosity and heated epistemological debates, these states have found many fundamental applications in modern physics, applied mathematics, and quantum computations (see, for example, the works [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45] and references therein).

Recently, the profound physical linkage between the Pauli exclusion principle and the symmetries of the Schrödinger’s cats has been revealed for the simplest model of the Schrödinger’s cat represented by a quantum superposition of two harmonic oscillator ground-state Gaussian wave functions that are localized in different regions of space at the same time  [11]. Naturally, the subsequent general question arises as to whether the formal analogy with the Pauli exclusion principle demonstrates itself in the dynamics of the so-called higher-order excited Schrödinger’s cats composed by two initially translated $n$-order Hermitian$-$Gaussian eigenfunctions belonging to the higher-energy $n$-excited eigenstates of the quantum harmonic oscillator. We show that the symmetric $n$-order cat states will occupy the same $n$-order excited state of the harmonic oscillator when these cat’s constituents closely approach in space and time each other, whereas the antisymmetric strongly overlapping $n$-order cat states, vice versa, will never occupy the same harmonic oscillator state. It turns out that the antisymmetric strongly overlapping in space and time $n$-order “alive” and “dead” Schrödinger’s cat states will be distributed over the neighboring energy states of the quantum harmonic oscillator, namely, with certain probability one part of the cat jumps up onto the $(n+1)$ excited energy level and the another part jumps down onto $(n-1)$ energy level. This effect could be considered as the extension of the formal analogy of the Pauli exclusion principle to the arbitrary-order jumping up and down excited Schrödinger’s cats. By way of illustration we study in details how the first $n=1$ order antisymmetric excited Schrödinger’s cat jumps up onto the $n=2$ excited energy level with the probability equal to 2/3, and, at the same time, jumps down onto the $n=0$ ground state energy level with the probability equal to 1/3, so that the complete probability remains to be conserved. The probabilities of the antisymmetric cat states of order $n$ to be distributed over the neighboring upper $(n+1)$ and the lower $(n-1)$ energy states depend strongly on the cat’s order $n$. For example, the respective probabilities for antisymmetric cat states of orders $n=1,2$, and 3, to jump up onto the corresponding excited energy levels with $n=2,3$, and 4, and, at the same time, jump down onto the levels with $n=0,1$, and 2, are found to be in the ratios: $2∕3:1∕3$, $3∕5:2∕5$, and $4∕7:3∕7$, with the asymptotic saturated ratio $1∕2:1∕2$ for the cat states distributed over the higher-energy levels $n\gg 1$. Moreover, we demonstrate the existence of the remarkable analogies between linear and nonlinear dynamics of the higher-order Schrödinger’s cats revealed in the framework of the nonlinear Schrödinger equation model with harmonic oscillator potential. We conclude that the wave and particle-like analogies in the dynamics of the symmetric and antisymmetric Schrödinger’s cat states confirm at least formally the Pauli statement that “a symmetric solution can never develop into an antisymmetric one, and vice versa”, and that “for particles with antisymmetric states it can, therefore, never happen that two particles find themselves in the same state” [1], [2], [3], [4], [5], [6]. The formal extension of the Pauli exclusion principle to the arbitrary-order jumping up and down excited Schrödinger’s cats is essentially the content of our study.