Dynamics of a composite quantum bouncing ball

TL;DR

This paper models a quantum particle with internal states bouncing on a reflector under gravity, predicting that the particle's position expectation can revive over time, revealing complex quantum dynamics.

Contribution

It introduces a novel quantum bouncing ball model incorporating internal degrees of freedom and predicts revival phenomena in its position expectation.

Findings

Revival of position expectation is possible in the quantum bouncing system.

The model combines internal degrees of freedom with gravitational bouncing dynamics.

Predictions suggest observable revival phenomena in such quantum systems.

Abstract

I construct the state of a quantum particle with internal degree of freedom bouncing on a perfect reflector in presence of gravity. I predict from the result that revival of position expectation is possible for such system.

Equations

$$\hat{H}=\hat{M}{c}^2+\frac{{\hat{P}}^2}{2\hat{M}}+\hat{M}g\hat{z}$$

$$\begin{split}\hat{H}_{0}=mc^{2}+\dfrac{\hat{P}^{2}}{2m}+mg\hat{z}+\hat{H}_{int}\\ \hat{H}_{1}=\dfrac{\hat{H}_{int}}{mc^{2}}\Big{(}-\dfrac{\hat{P}^{2}}{2m}+mg\hat{z}\Big{)}\end{split}$$

$${\hat{H}}_0|{\psi }_n,i⟩=(m{c}^2+E_n+E_i)|{\psi }_n,i⟩$$

$$E_{ni}^{(1)}=⟨{\psi }_n,i^{(0)}\mid {\hat{H}}_1\mid {\psi }_n,i^{(0)}⟩=\frac{E_i{E}_n}{m{c}^2}$$

$$\begin{array}{r}|{\psi }_n,i^{(1)}⟩=\sum _{\{m,j\}\ne \{n,i\}}\frac{⟨{\psi }_m,j^{(0)}\mid {\hat{H}}_1\mid {\psi }_n,i^{(0)}⟩}{(E_{ni}^{(0)})-E_{mj}^{(0)}}|{\psi }_m,j^{(0)}⟩\\ =\frac{E_i}{m{c}^2}\sum _{m(\ne n)}^{\infty }\frac{(-{)}^{m-n+2}2}{(E_m-E_n{)}^3}|{\psi }_m,i^{(0)}⟩\end{array}$$

$$\rho ={\rho }_{cm}(0)\otimes {\rho }_{int}$$

$${\rho }_t=e^{-\frac{it\hat{H}}{\hslash }}\cdot \rho \cdot e^{\frac{it\hat{H}}{\hslash }}$$

$${\rho }_{red}=\sum _i\frac{e^{-\beta E_i}}{Z}\cdot e^{-\frac{itH(E_i)}{\hslash }}\cdot {\rho }_{cm}(0)\cdot e^{\frac{itH(E_i)}{\hslash }}$$

$$\rho_{red}=\sum_{m,n,i}\dfrac{e^{-\beta E_{i}}}{Z}\cdot e^{-\Big{(}\dfrac{igt(\alpha_{m}-\alpha_{n})(m+\dfrac{E_{i}}{c^{2}})}{k_{i}\hbar}\Big{)}}c_{m}c_{n}^{*}\ket{\psi_{m}}\bra{\psi_{n}}$$

$$\rho_{red}=\sum_{m,n,i}\dfrac{e^{-\beta E_{i}}}{Z}\cdot e^{-\Big{(}\dfrac{igt(\alpha_{m}-\alpha_{n})(m+\dfrac{E_{i}}{3c^{2}})}{k\hbar}\Big{)}}c_{m}c_{n}^{*}\ket{\psi_{m}}\bra{\psi_{n}}$$

$$\rho_{cm}=\sum_{m,n}c_{m}c_{n}^{*}e^{-\Big{(}\dfrac{igt(\alpha_{m}-\alpha_{n})(m+\dfrac{\bar{E}}{3c^{2}})}{k\hbar}\Big{)}}e^{-\delta}\ket{\psi_{m}}\bra{\psi_{n}}$$

Full text

1 Introduction

Atomic cloud bouncing on a perfectly reflecting surface(i.e mirror) was first experimentally demonstrated by Aminoff et al. [1]. Quantum bouncing ball was analytically studied by J. Gea-Banacloche [2]. In the same time and afterwards it has been practically realised using Bose Einstein condensates [3], ultracold neutrons [4] and found huge application especially in the field of non Newtonian interaction, gravimetry, dark energy [5] and many more.

It has been shown by Brukner et. al. Centre of mass motion of a composite particle (particle with internal degree of freedom) is influenced by its internal degree of freedom(IDOF) due to contribution of IDOF into mass energy of the system. As a result mass becomes a dynamical variable resulting superposition of mass at low energy regime. These formulations lead to interesting phenomena such as decoherence due to gravitational time dilation and quantum violation of Einstein’s equivalence principle [6]. Though these formulations has resulted some controversy and alternative interpretations [7, 8, 9].

In this article we will construct the state of a composite quantum bouncing ball.

2 State of a composite quantum bouncing ball

Hamiltonian of a free particle with internal degree of freedom in presence of gravtational field in laboratory reference frame is given by [6]

[TABLE]

where $\hat{M}=m\hat{\mathbb{I}}_{int}+\hat{H}_{int}/c^{2}$, $\hat{\mathbb{I}}_{int}$ is the identity operator in the internal state space of the particle, $\hat{H}_{int}$ is internal Hamiltonian of the particle. Assuming lowest eigenvalue of $\hat{H}_{int}$ to be zero, $mc^{2}$ is the mass energy of the particle at ground state. $\hat{H}$ is valid upto first order correction of $\hat{H}_{int}/mc^{2}$ and hence the Hamiltonian can be written as $\hat{H}=\hat{H}_{0}+\hat{H}_{1}$ where,

[TABLE]

To find the state of quantum bouncing ball we need to solve above Hamiltonian with boundary condition – state $\psi=0$ at $z=0$ (because of the mirror). Whole problem is treated as one dimensional.

2.1 Solution of $H_{0}$:

Solution of Schrödinger equation for Hamiltonian \Big{(}\dfrac{\hat{P}^{2}}{2m}+mg\hat{z}\Big{)} with above mentioned boundary condition is [2] \psi_{n}=N_{n}Ai\Big{(}k\Big{(}z-\dfrac{E_{n}}{mg}\Big{)}\Big{)}=N_{n}Ai(kz-\alpha_{n}). Here $n$ corresponds $n^{th}$ eigenstate, $\{\alpha_{n}\}$ are zeroes of airy function, $k=(2m^{2}g/\hbar^{2})^{1/3}$, $N_{n}$ is the normalisation constant. Eigenvalue $E_{n}=(mg\alpha_{n}/k)$.

So, if $\{\ket{i}\}$ is the eigenbasis for $\hat{H}_{int}$ with eigenvalues $\{E_{i}\}$ then eigenvalue equation for $\hat{H}_{0}$ is

[TABLE]

2.2 Correction for $H_{1}$:

Here first order perturbation theory is applied. $\hat{H}=\hat{H}_{0}+\hat{H}_{1}=\hat{H}_{0}+\lambda\hat{V}$ where $\lambda=\dfrac{1}{mc^{2}}\ll 1$ and \hat{V}=\hat{H}_{int}\Big{(}-\dfrac{\hat{P}^{2}}{2m}+mg\hat{z}\Big{)}. Straightforward calculation gives –

Correction in energy:

[TABLE]

Correction in eigenstates:

[TABLE]

3 Time development:

Let initially particle is in a superposition state $\ket{\phi}=\sum_{n}c_{n}\ket{\psi_{n}}\otimes\sum_{i}a_{i}\ket{i}$. Density matrix of the particle after time $t$

[TABLE]

After time t,

[TABLE]

Tracing out internal degree of freedom we get the reduced density matrix. If we assume IDOF(s) are in a thermal state with temperature $T$

[TABLE]

where $H(E_{i})=\dfrac{\hat{P}^{2}}{2(m+\dfrac{E_{i}}{c^{2}})}+(m+\dfrac{E_{i}}{c^{2}})g\hat{z}$. $Z$ is the partition function, $\beta=(k_{B}T)^{-1}$, $k_{B}$ is Boltzmann constant. Evaluating equation ((8)) we get

[TABLE]

Where k_{i}=\Big{(}\dfrac{2(m+\dfrac{E_{i}}{c^{2}})^{2}g}{\hbar}\Big{)}^{1/3}. Since $1/(mc^{2})\ll 1$, we can write above expression upto first order in $1/(mc^{2})$, as–

[TABLE]

Expanding around average energy $\bar{E}$ upto second order we get

[TABLE]

where $\delta=\dfrac{\left\langle\Delta E^{2}\right\rangle t^{2}g^{2}(\alpha_{m}-\alpha_{n})^{2}}{9\hbar^{2}c^{4}k^{2}}$, $\Delta E$ is the fluctuation. Now $\left\langle\Delta E^{2}\right\rangle\propto T^{2}$ where $T$ is temperature associated to IDOF. Usually the factor $e^{-\delta}$ will suppress the off diagonal term of reduced density matrix. At low temperature $e^{-\delta}\rightarrow 1$. So, quantum state at low temperature is basically like a particle without any degree of freedom with an additional factor $\bar{E}/3c^{2}$ added with mass in the numerator.

4 Conclusion:

So, at low temperature we will observe usual collapse and revival of position expectation. Next we will study application of above formulation.