Corrigendum: Stern-Gerlach interferometry for tests of quantum gravity and general applications

A Corrigendum on
Stern-Gerlach interferometry for tests of quantum gravity and general applications

by Lokare Y (2022). Front. Phys. 10:785125. doi: 10.3389/fphy.2022.785125

In the original article, two references were not cited. These are as follows:

20. Marshman RJ, Mazumdar A, Bose S. Locality and entanglement in table-top testing of the quantum nature of linearized gravity. Phys Rev A (Coll Park) (2020) 101:052110. doi:10.1103/physreva.101.052110

21. Bose S, Mazumdar A, Schut M, Toros M. Mechanism for the quantum natured gravitons to entangle masses. Phys Rev D (2022) 105:106028. doi:10.1103/physrevd.105.106028

The citations appearing as Ref. [20] and Ref. [21] in the reference list, respectively) have now been inserted in Section 4: Entanglement dynamics in the density matrix formalism.

Additionally, there was a mistake in the original article relating to a reference (i.e., Ref. [18] in the original article) which has now been withdrawn from arXiv. Previously this was listed as:

18. Lokare Y. A complete analysis of spin coherence in the full-loop stern-gerlach interferometer using non-squeezed and squeezed coherent states of the quantum harmonic oscillator (2021).

The original article referenced and discussed results appearing in Ref. [18]. Section 5, Paragraphs 1-6 and equations 22-26 have now been removed and the section rewritten. This section now reads:

“In principle, it is possible to realize full-loop Stern-Gerlach interferometers using pure Bose-Einstein condensates (BECs). A possible arrangement is to have a BEC initially confined within a harmonic trap, which is then released and probed in a Stern-Gerlach interferometric setup. To this end, we consider one such case, as proposed and/or analyzed in Ref. [22]. We work under the crude assumption that interactions between atoms in the BEC are effectively negligible, in light of which the analysis will simply reduce to the single-particle limit. Note here that a rapid free expansion of the BEC post-release is being assumed. The wave-packet of a BEC satisfies the Gross-Pitaevskii equation, as follows

$\begin{array}{c}i\hslash \frac{\partial \psi \left(t\right)}{\partial t}=H_{\text{MF}}\left(t\right)\psi \left(t\right),\hfill \end{array}(23)$

where the Hamiltonian H_MF(t) in the mean-field approximation assumes the form $H_{\text{MF}}(t)=-\frac{{\hslash }^2}{2m}{\nabla }^2+U(r,t)+g|\psi {|}^2$. Here, note that U(r, t) denotes the time-dependent potential that arises from external effects such as gravity, etc. (g here denotes the mean-field coupling parameter which is characterized in terms of the s-wave scattering length (denoted as a) of the BEC in question, as g = 4πℏ²a/m). Given Eq. 23, it is possible to derive a closed-form expression for the evolution of a BEC wave-packet in the center-of-mass frame [22]. This can be done so as long as the time-dependent potential U(r, t) assumes a quadratic profile around the BEC center-of-mass. The scaling ansatz (see Ref. [22]) is shown to be exact if one approximates the wave function at the time of release by a Gaussian wave-packet that evolves in this quadratic potential (interatomic interactions have been ignored). Thus, we consider the form of the wave function before release from the harmonic trap (approximated by a Gaussian and assumed to be stationary initially) as follows

$\begin{array}{c}{\psi }_i\left(z,t=0\right)={\left(2\pi \right)}^{-1/4}{\left(\delta z\right)}^{-1/2}\exp \left(-\frac{z^2}{4{\left(\delta z\right)}^2}\right),\hfill \end{array}(24)$

where δz denotes the width of the initially prepared Gaussian wave-packet. The one-dimensional scaling factor assumes the form $\gamma \equiv \sqrt{1+{\omega }^2{t}^2}$ and the scaled spatial and momentum splitting between the wave-packets is given as [22]

$\begin{array}{c}\mathrm{\Delta }\bar{z}\left(t\right)=\mathrm{\Delta }z\left(t\right)/\gamma ,\hfill \end{array}(25)$

and

$\begin{array}{c}\mathrm{\Delta }{\bar{p}}_z\left(t\right)=\gamma \mathrm{\Delta }{p}_z\left(t\right)-m\frac{d\gamma }{dt}\mathrm{\Delta }z\left(t\right),\hfill \end{array}(26)$

where m is the mass of the atoms in the BEC, $\mathrm{\Delta }\bar{z}(t)$ is the macroscopic splitting between the wave-packets in position space, and $\mathrm{\Delta }{\bar{p}}_z(t)$ denotes the macroscopic splitting between the wave-packets in momentum space. From Eq. 11 and Eq. 24, we get for the interference signal strength ϕ_BEC [computed over the total time-of-flight τ]

$\begin{array}{ll}\hfill {\varphi }_{\text{BEC}}& =\frac{1}{\sqrt{2\pi }}\frac{1}{\delta z}{\int }_{-\infty }^{\infty }\exp \left(-{\left(\frac{z-\mathrm{\Delta }\bar{z}\left(t\right)}{2\delta z}\right)}^2-{\left(\frac{z+\mathrm{\Delta }\bar{z}\left(t\right)}{2\delta z}\right)}^2\right)\hfill \\ \hfill & \times \exp \left(-2i\frac{\mathrm{\Delta }{\bar{p}}_z\left(t\right)}{\hslash }z\right)dz.\hfill \end{array}(27)$

The integral in Eq. 27 is a standard Gaussian integral, simplifying which we get

$\begin{array}{c}{\varphi }_{\text{BEC}}=\exp \left(-\frac{1}{2}\left({\left(\frac{\mathrm{\Delta }\bar{z}\left(\tau \right)}{\delta z}\right)}^2+{\left(\frac{\mathrm{\Delta }{\bar{p}}_z\left(\tau \right)}{\delta p_z}\right)}^2\right)\right),\hfill \end{array}(28)$

where δp_z denotes the initial momentum uncertainty in the initially prepared BEC wave-packet (note: the Gaussian wave-packet is a minimum uncertainty state that saturates the uncertainty principle, for which δz δp_z = ℏ/2). From Eqs 25, 26, 28, one can deduce an approximate closed-form expression for ϕ_BEC.

To maximize the Stern-Gerlach interference signal strength, Eq. 28 suggests that the following conditions must hold¹

$\begin{array}{c}\mathrm{\Delta }\bar{z}\left(\tau \right)\ll \delta z,\hfill \end{array}(29)$

and

$\begin{array}{c}\mathrm{\Delta }{\bar{p}}_z\left(\tau \right)\ll \delta p_z.\hfill \end{array}(30)$

A more robust analysis is however in order, for the following reasons. In an experimental setup, it is quite possible that the initial BEC wave-packet might not assume a Gaussian profile, in light of which Eqs. 27, 28 would not be sufficient to quantify the interference signal strength. Moreover, analyzing the model in the single-particle limit yields only approximate results¹. It is in fact, necessary to consider a full quantum many-body treatment of a trapped impure BEC (i.e., by incorporating finite-temperature effects), which from an experimental point of view, seems more reasonable. Experimental realizations of the kind described here have already been reported in the literature. Ref. [23] for instance, reports the realization of a high stability Stern-Gerlach spatial fringe interferometer with pure BECs.

Using heavy neutral test masses instead of atomic clouds is another viable, yet formidable approach¹. It is worth bearing in mind however that such an experiment would demand a delicate balance between several experimental parameters. A more realistic implementation of such a setup (more specifically, the quantification of the Stern-Gerlach interference signal strength) will have to take into account the effects of the field gradient present in the x − y plane (per Maxwell’s equations), in addition to the one applied along z¹. This warrants a 2D analysis of such a setup [14]. In recent years, there have been attempts to address some of these issues. Marshman et al. not long ago analyzed a numerical model of a slightly modified version of an SG interferometric setup [24] by including the effects of the field gradient in the x − y plane (in addition to the one already existing along the z direction). Furthermore, they consider field gradients of intermediate strengths and the effect of the diamagnetic properties of the test mass in their analysis. Their results demonstrate that the introduction of a gradient-free region (see Ref. [24] for more details) along the wave-packet trajectories can facilitate the acceleration of micron-sized test masses in the interferometric setup, which in turn can help one realize more efficient splitting between the individual wave-packets.

Considering the above correction, Eq. 33 appearing in Section 6 of the original article is re-labelled to Eq. 31.

In addition, Refs. [20] and [97] appearing in the original article have now been published in peer-reviewed journals. Therefore, their journal references have been added and now read as follows:

20. Japha Y. Unified model of matter-wave-packet evolution and application to spatial coherence of atom interferometers. Phys Rev A (Coll Park) (2021) 104:053310. doi:10.1103/physreva.104.053310

24. Marshman RJ, Mazumdar A, Folman R, Bose S. Constructing nano-object quantum superpositions with a Stern-Gerlach interferometer. Phys Rev Res (2022) 4:023087. doi:10.1103/physrevresearch.4.023087

Finally, the Acknowledgments have been modified to better reflect the contributions of Prof. A. Mazumdar and Prof. S. Bose. The updated statement reads:

“The author YL would like to express his gratitude to his former mentors, A. Mazumdar and S. Bose, with whom he has had several useful and/or insightful discussions on matter-wave interferometry and related areas (in particular, realizing quantum gravity tests using Stern-Gerlach interferometry).”

The author apologizes for this error and states that this does not change the scientific conclusions of the article in any way. The original article has been updated.

Footnotes

¹Bose S, Mazumdar A. Private Communication. (2021).

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