Simulating general relativity and non-commutative geometry by non-paraxial quantum fluids

Quantum simulations of fundamental theories realize novel tests and investigations of inaccessible physical regimes. The phenomenology of the fundamental laws of physics modified at the Planck scale attracts a large community of scientists [1–7]. The challenge is to identify feasible experiments to test the apparently inaccessible limits of quantum mechanics and general relativity [8]. The difficulties in realizing such large-scale experiments trigger studying quantum simulations. The simulations are realizable in Earth-based laboratories in a human-life timescale. Researchers aim at realizing analog experimental models of black holes [9–11], Hawking radiation [12–15], inflation and universe expansion [16, 17], dark-matter models [18], and related phenomena [19–21]. The analogy is a fundamental tool in physics, and experimental and theoretical analogs may deepen our understanding of quantum gravity theories [22], and of other challenging proposals as time-asymmetric quantum mechanics [23–25]. All the considered quantum simulations in optics are limited to the paraxial regimes. Despite this enables to simulate interesting physics, the use of non-paraxial effects is so far unexplored.

In this manuscript, we study the orbital precession of a quantum fluid (figure 1(a)) due to the perturbation to quantum mechanics analog models induced by non-paraxial terms Quantum fluids are studied in the vast literature concerning Bose–Einstein condensates (BECs), classical and quantum nonlinear optics and polaritonics [26–30]. We consider a quantum fluid wave-packet with a non-vanishing angular momentum in a trapping potential. We show that the wave-packet elliptical orbit is perturbed by the new terms, and we find a direct analog of the orbital precession induced by post-Newtonian gravitation. The dynamics is equivalent to the known treatment of the precession of Mercury in general relativity [31–33].

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We also find a very interesting connection with the study related to the Generalized Uncertainty Principle (GUP), considered in sub-Planckian physics. Surprisingly enough, the orbital precession may also be explained in terms of non-commuting variables, as those that arise in studies in quantum gravity [34]. A large amount of literature deals with modifications of the uncertainty principle and related non-commutative geometry [35]. In standard quantum mechanics, there is no minimal value for the position uncertainty ΔX. However, theories in quantum gravity imply the existence of a minimal length scale, commonly identified with the Planck length ℓ_P [1]. Hence a lower bound for ΔX must be included in quantum mechanics, and the Heisenberg relation ΔXΔP ≥ ℏ/2, with the momentum uncertainty ΔP, must be generalized—also in the non-relativistic limit considered hereafter. The most accepted formulation of the so-called modified or GUP reads [36, 37]

$$\begin{eqnarray}&&{\rm{\Delta }}X{\rm{\Delta }}P\gt \displaystyle \frac{{\hslash }}{2}\left(1+\beta {\rm{\Delta }}{P}^{2}\right).\end{eqnarray} \tag{ 1 }$$

Equation (1) implies ${\rm{\Delta }}X\gt {\hslash }\sqrt{\beta }$ with β a unknown constant, which is eventually related to ℓ_P. We need experimental evidence to assess the validity of equation (1) and fix bounds for the value of β. The GUP in equation (1) arises in a theory-independent way. Many theories attempting to unify gravity and quantum mechanics—including string theory [38]—predict modifications of the Heisenberg principle as in equation (1). The GUP is also related to modified commutation relations and to non-commutative theories [34, 39–42]. The GUP has many implications as, for example, in cosmological dynamics, black-body radiation, wave-packet localization and related investigations reported by several authors [42–46]. Quantitative bounds for the β parameter in the modified Heisenberg principle in equation (1) were also reported [45]. Other authors discussed human-scale laboratory tests with optomechanical and orbiting classical objects [47–50]. However, the application of generalized quantum mechanics to the macroscopic world is questionable [40].

Analogs, i.e. physical systems governed by laws mathematically identical to those of the generalized quantum mechanics, were considered in the fields of optics and relativistic BECs [51–55]. The fact that non-paraxial quantum fluids enable the simulations of non-commutative quantum mechanics, and general relativistic effects, opens several possibilities for studying analogs of unexplored phenomena in the laboratory.

We study the generalized SE for quantum fluids with trapping potential V in two or three spatial dimensions

$$\begin{eqnarray}&&{\imath }{\hslash }\displaystyle \frac{\partial \psi }{\partial t}=-\displaystyle \frac{{{\hslash }}^{2}}{2m}{{\rm{\nabla }}}^{2}\psi +\displaystyle \frac{\beta {{\hslash }}^{4}}{m}{{\rm{\nabla }}}^{4}\psi +V({\bf{r}})\psi =\hat{H}\psi .\end{eqnarray} \tag{ 2 }$$

Equation (2) describes a non-interacting atomic BEC, realized by employing the Feshbach resonance [56]. Equation (2) also describes a polaritonic or photonic condensate. In the latter case, as extensively discussed in the literature, t corresponds to the propagation direction, and ℏ must be replaced by the reduced light wavelength λ/2π [57] equation (2) contains a kinetic term weighted by β. As considered by various authors [45, 52, 58], the additional kinetic term is the simplest modification to the SE that implies the GUP in equation (1) and arises from fundamental modifications to the geometry of the space-time at the Planck scale, which change the dispersion relation of free particles, as photons. In the optical case, corrections to paraxial approximation introduce the ∇⁴ψ term, and the ratio between λ and the beam waist determines β in a optical analog to the generalized quantum mechanics [52]. For cold atoms, higher order derivatives may arise from relativistic effects (not considered here), when the fluid velocity is comparable with the velocity of light c [59]. Another mechanism is the modification of the dispersion relation in models as doubly-special-relativity [60].

Letting $\hat{p}=-{\imath }{\hslash }{\rm{\nabla }}$, the Hamiltonian in equation (2) is

$$\begin{eqnarray}&&\hat{H}=\displaystyle \frac{{\hat{p}}^{2}}{2m}+\displaystyle \frac{\beta }{m}{\hat{p}}^{4}+V({\bf{r}})\end{eqnarray} \tag{ 3 }$$

with the position vector ${\bf{r}}=\left(x,y,z\right)=\left({x}^{1},{x}^{2},{x}^{3}\right)$. One introduces non-commutative coordinates by a new set of 'high-energy' variables (X^μ, P^μ), which, in the simplest case considered here, read (μ, ν = 1, 2, 3)

$$\begin{eqnarray}\begin{array}{rcl}{\hat{X}}^{\mu } & = & {\hat{x}}^{\mu };\\ {\hat{P}}^{\nu } & = & {\hat{p}}^{\nu }(1+\beta {\hat{p}}^{2}).\end{array}\end{eqnarray} \tag{ 4 }$$

By $[{\hat{x}}^{\mu },{\hat{x}}^{\nu }]=0$, $[{\hat{p}}^{\mu },{\hat{p}}^{\nu }]=0$, $[{\hat{x}}^{\mu },{\hat{p}}^{\nu }]={\imath }{\hslash }{\delta }^{\mu ,\nu }$, one has $\left[{\hat{P}}^{\mu },{\hat{P}}^{\nu }\right]=\left[{\hat{X}}^{\mu },{\hat{X}}^{\nu }\right]=0$ and [45, 50, 61]

$$\begin{eqnarray}&&\left[{\hat{X}}^{\mu },{\hat{P}}^{\nu }\right]={\imath }{\hslash }({\delta }^{\mu \nu }+\beta {\delta }^{\mu \nu }{\hat{P}}^{2}+2\beta {\hat{P}}^{\mu }{\hat{P}}^{\nu }).\end{eqnarray} \tag{ 5 }$$

This represents a weak version of non-commutative phase-space coordinates: at variance with standard quantum mechanics, the momentum and the position in different directions do not commute [34]. The standard case involves also commutation relations between position coordinates.

By ${\hat{P}}^{\mu }$, equation (2) has the traditional form

$$\begin{eqnarray}&&{\imath }{\hslash }{\psi }_{t}=\displaystyle \frac{{\hat{P}}^{2}}{2m}\psi +V({\bf{r}})\psi ,\end{eqnarray} \tag{ 6 }$$

but the commutation relations are modified and equation (1) holds true. The simplest effect of the additional kinetic term are shifts ΔE_k in the energetic levels of the eigenstates [62], ${\rm{\Delta }}{E}_{k}=\tfrac{\beta }{m}\langle k| {\hat{p}}^{4}| k\rangle =4m\beta \langle k| {V}^{2}| k\rangle$. Such perturbations may eventually occur in optical and atomic quantum fluids, however they are very difficult to observe [45]. We consider a more accessible phenomenology related to the dynamics of a wave-packet orbiting in the potential.

We study a wave-packet initially located at $r=\sqrt{{x}^{2}+{y}^{2}}={r}_{0}$ that rotates with a non-vanishing angular momentum (figure 1(a)). The trajectory is found in the hydrodynamic approximation [63, 64] by letting $\psi =A\exp ({\imath }{ \mathcal S }/{\hslash })$ with ${\hslash }\to 0$ [65]. The Hamilton–Jacobi equation for ${ \mathcal S }$ is [66, 67]

$$\begin{eqnarray}&&\displaystyle \frac{\partial { \mathcal S }}{\partial t}+\displaystyle \frac{{\left({\rm{\nabla }}{ \mathcal S }\right)}^{2}}{2m}+\displaystyle \frac{\beta }{m}{({\rm{\nabla }}{ \mathcal S })}^{4}+V=0.\end{eqnarray} \tag{ 7 }$$

By the classical Hamiltonian H(q, p) = p²/2m + (β/m)p⁴ + V, we have ${\partial }_{t}{ \mathcal S }+H({\bf{q}},{\rm{\nabla }}{ \mathcal S })=0$. Equation (7) is solved by the Hamilton system with ${\bf{p}}={\rm{\nabla }}{ \mathcal S }$ [67]

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\,{\rm{d}}{\bf{q}}}{\,{\rm{d}}t}={{\rm{\nabla }}}_{p}H,\\ \displaystyle \frac{\,{\rm{d}}{\bf{p}}}{\,{\rm{d}}t}=-{{\rm{\nabla }}}_{q}H.\end{array}\end{eqnarray} \tag{ 8 }$$

We consider here a z-independent radial potential with polar symmetry: V = V(r). Two-dimensional condensates are routinely considered in the literature [27, 30, 65]. In polar coordinates (r, θ, z), with conjugate momenta (p_r, p_θ, p_z), we have

$$\begin{eqnarray}&&H=\displaystyle \frac{1}{2m}\left({p}_{r}^{2}+\displaystyle \frac{{p}_{\theta }^{2}}{{r}^{2}}\right)+\displaystyle \frac{\beta }{m}{\left({p}_{r}^{2}+\displaystyle \frac{{p}_{\theta }^{2}}{{r}^{2}}\right)}^{2}+V(r).\end{eqnarray} \tag{ 9 }$$

The corresponding Lagrangian does not depend explicitly on θ, hence the conjugate momentum ${p}_{\theta }=l$ is conserved, and the motion occurs in the z plane, with p_z = 0. By the conserved l, equation (8) are written as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\,{\rm{d}}\theta }{\,{\rm{d}}t}=\displaystyle \frac{l}{{{mr}}^{2}}\left[1+4\beta \left({p}_{r}^{2}+\displaystyle \frac{{l}^{2}}{{r}^{2}}\right)\right],\displaystyle \frac{\,{\rm{d}}{p}_{\theta }}{\,{\rm{d}}t}=0,\\ \displaystyle \frac{\,{\rm{d}}r}{\,{\rm{d}}t}=\displaystyle \frac{{p}_{r}}{m}\left[1+4\beta \left({p}_{r}^{2}+\displaystyle \frac{{l}^{2}}{{r}^{2}}\right)\right],\displaystyle \frac{\,{\rm{d}}{p}_{r}}{\,{\rm{d}}t}=-\displaystyle \frac{\partial H}{\partial r}=-V^{\prime} (r)+\displaystyle \frac{{l}^{2}}{{{mr}}^{3}}\left[1+4\beta \left({p}_{r}^{2}+\displaystyle \frac{{l}^{2}}{{r}^{2}}\right)\right],\end{array}\end{eqnarray} \tag{ 10 }$$

with $V^{\prime} =\,{\rm{d}}V/\,{\rm{d}}r$ and, for a parabolic potential,

$$\begin{eqnarray}&&V=\displaystyle \frac{1}{2}m{{\rm{\Omega }}}^{2}{r}^{2}.\end{eqnarray} \tag{ 11 }$$

Figure 1(b) shows the numerical solutions of equation (10) with r(0) = r₀ and p_r(0) = 0. When β = 0 and $l\ne 0$, the orbit is elliptical (dashed line in figure 1(b)). When β > 0—continuous line in figure 1(a)—the orbit exhibits a precession (clockwise for l < 0, and counter-clockwise for l > 0). Figure 1(c) shows the precession angle Δ with the particle at t = 0, x = r₀ and y = 0.

As the orbit rotates, the maximal position in the y coordinate is amplified as shown in figure 1(d). This orbital enhancement of the quantum gravity effect resembles the known gravity slingshot assist adopted to alter the speed of a spacecraft in orbital mechanics [68]: the radial acceleration at any turn emphasizes the phenomenology.

We remark that the precession can be related to non-commutative coordinates in the phase-space that arise because of the quantum gravity terms If we introduce the classical counter-part of the generalized momenta (4),

$$\begin{eqnarray}&&{P}_{r,\theta }={p}_{r,\theta }(1+\beta {p}^{2})\end{eqnarray} \tag{ 12 }$$

the Hamiltonian H is written as in the case $\beta =0$:

$$\begin{eqnarray}&&H=\displaystyle \frac{1}{2m}\left({P}_{r}^{2}+\displaystyle \frac{{P}_{\theta }^{2}}{{r}^{2}}\right)+V(r).\end{eqnarray} \tag{ 13 }$$

However, while the Poisson brackets $\{r,{p}_{\theta }\}=\{\theta ,{p}_{r}\}=0$ vanish [67], the corresponding quantities for the generalized momenta are

$$\begin{eqnarray}&&\{r,{P}_{\theta }\}={r}^{2}\{\theta ,{P}_{r}\}=2\beta {P}_{r}{P}_{\theta }.\end{eqnarray} \tag{ 14 }$$

Therefore, one has modified mechanics with non-commuting coordinates. In the following, we show that the precession is of the order of magnitude of the brackets in equation (14), revealing the link between the non-commutative geometry and GUP phenomenology.

Equation (10) give the precession angle Δ by $\,{\rm{d}}\theta /\,{\rm{d}}r=l/[{p}_{r}(r){r}^{2}]$ with p_r(r) expressed in terms of the conserved quantities H = E₀ and p_θ = l. However, no closed form can be found, and figure 2 shows numerical results. We obtain estimates by considering a nearly circular orbit with r ≃ r₀. If β = 0, then ${p}_{\theta }=l={{mr}}^{2}\dot{\theta }$ is constant, and the period is ${T}_{0}=2\pi {{mr}}_{0}^{2}/l$. For β > 0 and r ≃ r₀

$$\begin{eqnarray}&&{{mr}}_{0}^{2}\dot{\theta }\simeq l\left[1+4\beta \left({p}_{r}^{2}+\displaystyle \frac{{l}^{2}}{{r}_{0}^{2}}\right)\right]\geqslant l\left(1+4\beta {p}_{0}^{2}\right),\end{eqnarray} \tag{ 15 }$$

with ${p}_{0}^{2}=\tfrac{{l}^{2}}{{r}_{0}^{2}}$. In a time interval Δt, ${\rm{\Delta }}\theta \gtrsim \tfrac{2\pi {\rm{\Delta }}t}{{T}_{0}}(1+4\beta {p}_{0}^{2})$, and, when Δt = T₀, we have the lower bound

$$\begin{eqnarray}&&{\rm{\Delta }}={\rm{\Delta }}\theta -2\pi \simeq 8\pi \beta {p}_{0}^{2}.\end{eqnarray} \tag{ 16 }$$

The orbital period T is also altered by GUP. For a Δθ = 2π, the relative variation

$$\begin{eqnarray}&&\delta T=T/{T}_{0}-1\simeq -4\beta {p}_{0}^{2}\end{eqnarray} \tag{ 17 }$$

is compared with the numerical calculation in figure 2(b).

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To outline the link with general relativity, we compare our result with the analysis of the precession of the perihelion of Mercury, as originally considered by Einstein [31–33]. We consider a nearly circular orbit and write the orbit equation starting from the conservation of energy in equation (9). By using $\,{\rm{d}}r/\,{\rm{d}}\theta ={p}_{r}{r}^{2}/l$ from equation (10), and u = 1/r we have

$$\begin{eqnarray}&&H=\displaystyle \frac{{l}^{2}}{2m}\left(\mu +2\beta {l}^{2}{\mu }^{2}\right)+\tilde{V}(u),\end{eqnarray} \tag{ 18 }$$

with $\mu \equiv {\left({\rm{d}}u/{\rm{d}}\theta \right)}^{2}+{u}^{2}$ and $\tilde{V}(u)=V(1/u)$. By deriving equation (18) one obtains, at the lowest order in β

$$\begin{eqnarray}&&\displaystyle \frac{\,{{\rm{d}}}^{2}u}{\,{\rm{d}}{\theta }^{2}}+u=-\displaystyle \frac{m}{{l}^{2}}\tilde{V}^{\prime} (u)(1-4\beta {l}^{2}\mu )\end{eqnarray} \tag{ 19 }$$

with $\tilde{V}^{\prime} (u)=\,{\rm{d}}\tilde{V}/\,{\rm{d}}u$. Equation (19) for β = 0 furnishes the orbit equation [67]; for β > 0, an additional contribution to the effective potential perturbs the orbit. Equation (19) is written as

$$\begin{eqnarray}&&\displaystyle \frac{\,{{\rm{d}}}^{2}u}{\,{\rm{d}}{\theta }^{2}}+u=-\displaystyle \frac{m}{{l}^{2}}\tilde{V}^{\prime} (u)+\beta { \mathcal F }\left(\displaystyle \frac{\,{\rm{d}}u}{\,{\rm{d}}\theta },u\right)\end{eqnarray} \tag{ 20 }$$

with

$$\begin{eqnarray}&&{ \mathcal F }=4m\mu \tilde{V}^{\prime} (u)=4m\left[{\left(\displaystyle \frac{{\rm{d}}u}{{\rm{d}}\theta }\right)}^{2}+{u}^{2}\right]\tilde{V}^{\prime} (u)\end{eqnarray} \tag{ 21 }$$

representing the perturbation due to the modified uncertainty principle, which disappears for β = 0.

For a nearly circular orbit $u\simeq (1/{r}_{0})$, and a parabolic potential $\tilde{V}(u)=m{{\rm{\Omega }}}^{2}/(2{u}^{2})$, the first term in the right hand side of (20) $-m\tilde{V}^{\prime} (u)/{l}^{2}={m}^{2}{{\rm{\Omega }}}^{2}/({l}^{2}{u}^{3})$ is nearly a constant, and plays the role of the gravitational field. This shows the way the kinetic term induces an effective force, as general relativity induces a correction to the Newton force.

In a perturbative expansion in β, ${ \mathcal F }$ in (20) produces a driving force term, as detailed in the following; such a term is in perfect analogy with the Einstein analysis of the Mercury orbit. We consider the solution for β = 0 representing an elliptical orbit with eccentricity e:

$$\begin{eqnarray}&&{u}_{0}=\sqrt{\displaystyle \frac{m{\rm{\Omega }}}{l}}\left[1+e\,\cos (2\theta )\right].\end{eqnarray} \tag{ 22 }$$

Equation (22) is valid for the parabolic potential, similar results are obtained for other potentials as, for example, the Newtonian gravitational potential. The perihelion corresponds to maximal u for θ_n = nπ with n = 0, 1, 2, .....

For β > 0, one adopts a perturbative expansion at the lowest order in the eccentricity e and β. The perturbation force ${ \mathcal F }$ reads

$$\begin{eqnarray}&&{ \mathcal F }\,={ \mathcal F }\left(\displaystyle \frac{\,{\rm{d}}{u}_{0}}{\,{\rm{d}}\theta },{u}_{0}\right)\simeq -4{m}^{3/2}{{\rm{\Omega }}}^{3/2}{l}^{1/2}\left[1-e\cos (2\theta )\right].\end{eqnarray} \tag{ 23 }$$

The perturbed orbit equation (20) is

$$\begin{eqnarray}&&\displaystyle \frac{\,{{\rm{d}}}^{2}u}{\,{\rm{d}}{\theta }^{2}}+u=\displaystyle \frac{{m}^{2}{{\rm{\Omega }}}^{2}}{{l}^{2}{u}^{3}}+4\beta {{\rm{\Omega }}}^{3/2}{m}^{3/2}{l}^{1/2}\left[1-e\cos (2\theta )\right].\end{eqnarray} \tag{ 24 }$$

The forcing term in (24) is directly corresponding to the term obtained by Einstein representing the correction to the orbit due to general relativity. One can solve equation (24) at the lowest order in β and e as

$$\begin{eqnarray}&&u=\sqrt{\displaystyle \frac{m{\rm{\Omega }}}{l}}\left\{1-\beta m{\rm{\Omega }}l+e\left[\cos (2\theta )+\beta {\rm{\Omega }}{ml}\theta \sin (2\theta )\right]\right\}.\end{eqnarray} \tag{ 25 }$$

By writing equation (25) as

$$\begin{eqnarray}&&u\simeq \sqrt{\displaystyle \frac{m{\rm{\Omega }}}{l}}\left\{1-\beta m{\rm{\Omega }}l+e\cos \left[2\theta \left(1-\displaystyle \frac{\beta m{\rm{\Omega }}l}{2}\right)\right]\right\},\end{eqnarray} \tag{ 26 }$$

one sees that the maximum of u in equation (26) occurs for

$$\begin{eqnarray}&&{\theta }_{n}=n\pi \left(1+\displaystyle \frac{\beta m{\rm{\Omega }}l}{2}\right).\end{eqnarray} \tag{ 27 }$$

Hence the perihelion shifts for each half-orbit by an amount πβmΩl/2. Being $l\cong m{\rm{\Omega }}{r}_{0}^{2}$ for nearly circular orbits, this result is consistent with the estimate above within numerical factors due to the definition of the precession angle. Equation (27) shows that the effect accumulates as the precession angle grows with n.

It is instructive to write the previous results in a geodesic formulation [69]. We give in the following the expression of Φ and τ in our case.

We introduce in equation (10) the proper time τ by

$$\begin{eqnarray}&&\displaystyle \frac{\,{\rm{d}}t}{\,{\rm{d}}\tau }=\displaystyle \frac{1}{1+4\beta \left({p}_{r}^{2}+\tfrac{{l}^{2}}{{r}^{2}}\right)}.\end{eqnarray} \tag{ 28 }$$

In terms of τ, equation (10) read

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\,{\rm{d}}\theta }{\,{\rm{d}}\tau } & = & \displaystyle \frac{l}{{{mr}}^{2}},\displaystyle \frac{\,{\rm{d}}{p}_{\theta }}{\,{\rm{d}}\tau }=0,\\ \displaystyle \frac{\,{\rm{d}}r}{\,{\rm{d}}\tau } & = & \displaystyle \frac{{p}_{r}}{m},\displaystyle \frac{\,{\rm{d}}{p}_{r}}{\,{\rm{d}}\tau }=-\displaystyle \frac{\partial }{\partial r}\left[{\rm{\Phi }}(r)+\displaystyle \frac{{l}^{2}}{2{{mr}}^{2}}\right],\end{array}\end{eqnarray} \tag{ 29 }$$

where Φ is a modified potential given by the solution of the equation

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r}=\displaystyle \frac{1}{1+4\beta \left({p}_{r}^{2}+\tfrac{{l}^{2}}{{r}^{2}}\right)}\displaystyle \frac{\partial V}{\partial r}.\end{eqnarray} \tag{ 30 }$$

The expression for Φ is found at the lowest order in β by using the energy conservation H = E. After equation (9) we have

$$\begin{eqnarray}&&\beta \left({p}_{r}^{2}+\displaystyle \frac{{p}_{\theta }^{2}}{{r}^{2}}\right)\simeq 2m\beta E-2\,m\beta V+o(\beta ),\end{eqnarray} \tag{ 31 }$$

which, used in (30), gives at the lowest order in β

$$\begin{eqnarray}&&{\rm{\Phi }}=V(1-8m\beta E)+4m\beta {V}^{2}\simeq V+4m\beta {V}^{2},\end{eqnarray} \tag{ 32 }$$

where we have used the term mβE ≪ 1, as we are studying the perturbation to Newtonian dynamics (with generic potential) and the energy is also a small quantity. In the analogy with general relativity, this corresponds to consider a particle with energy much smaller than the rest energy mc², with c the vacuum light velocity.

For the proper time in (33), we have

$$\begin{eqnarray}&&\displaystyle \frac{\,{\rm{d}}t}{\,{\rm{d}}\tau }\simeq 1+8m\beta V,\end{eqnarray} \tag{ 33 }$$

which shows how the time speeds up at large r for a parabolic potential, hence the local velocity decreases, and this causes the precession of the orbit.

The geodesic formulation of the (29) is found by defining x⁰ = t, x¹ = r and x² = θ, so we have

$$\begin{eqnarray}&&\displaystyle \frac{\,{{\rm{d}}}^{2}{x}^{\alpha }}{\,{\rm{d}}{\tau }^{2}}+{{\rm{\Gamma }}}_{\,\beta \gamma }^{\alpha }\displaystyle \frac{\,{\rm{d}}{x}^{\beta }}{\,{\rm{d}}\tau }\displaystyle \frac{\,{\rm{d}}{x}^{\gamma }}{\,{\rm{d}}\tau }=0,\end{eqnarray} \tag{ 34 }$$

where Greek indices span 0, 1, 2 = t, r, θ. The non-vanishing Christoffel symbols are

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\,00}^{1}={{\rm{\Gamma }}}_{\,{tt}}^{r}=\displaystyle \frac{1}{m}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r},{{\rm{\Gamma }}}_{\,22}^{1}={{\rm{\Gamma }}}_{\,\theta \theta }^{r}=-r,{{\rm{\Gamma }}}_{\,12}^{2}={{\rm{\Gamma }}}_{\,21}^{2}={{\rm{\Gamma }}}_{\,r\theta }^{\theta }=\displaystyle \frac{1}{r},\end{eqnarray} \tag{ 35 }$$

and the (equation (34)) read explicitly as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\,{{\rm{d}}}^{2}{x}^{0}}{\,{\rm{d}}{\tau }^{2}}=0\displaystyle \frac{\,{{\rm{d}}}^{2}{x}^{1}}{\,{\rm{d}}{\tau }^{2}}+{{\rm{\Gamma }}}_{\,00}^{1}{\left(\displaystyle \frac{{\rm{d}}{x}^{0}}{{\rm{d}}\tau }\right)}^{2}+{{\rm{\Gamma }}}_{\,22}^{1}{\left(\displaystyle \frac{{\rm{d}}{x}^{2}}{{\rm{d}}\tau }\right)}^{2}=0\\ \displaystyle \frac{\,{{\rm{d}}}^{2}{x}^{2}}{\,{\rm{d}}{\tau }^{2}}+2\,{{\rm{\Gamma }}}_{\,12}^{2}\displaystyle \frac{\,{\rm{d}}{x}^{1}}{\,{\rm{d}}\tau }\displaystyle \frac{\,{\rm{d}}{x}^{2}}{\,{\rm{d}}\tau }=0.\end{array}\end{eqnarray} \tag{ 36 }$$

Equation (29) with τ and Φ are the Newtonian limit of the relativistic particle with metric [69].

$$\begin{eqnarray}&&\,{\rm{d}}{s}^{2}=-(1+2{\rm{\Phi }})\,{\rm{d}}{\tau }^{2}+\displaystyle \frac{1}{1+2{\rm{\Phi }}}\,{\rm{d}}{r}^{2}+{r}^{2}\,{\rm{d}}{\theta }^{2}.\end{eqnarray} \tag{ 37 }$$

A similar approach, but in a quantum framework, is developed in [70], and allows to find the link between equation (37) and the SE in (6). Other solutions to the same problem are given in [71, 72] for cosmological descriptions of a homogeneous, isotropic universe. We describe the free particle, whose motion is described by geodesic equation, through a field $\psi (t,{\bf{r}})$ such that $| \psi (t,{\bf{r}}){| }^{2}$ is the probability to find the particle in the position r at time t. It must be governed by

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{g}^{2}\psi =-{k}_{\mu }{k}^{\mu }\psi ,\end{eqnarray} \tag{ 38 }$$

where ${{\rm{\nabla }}}_{g}^{2}=\tfrac{1}{\sqrt{g}}\tfrac{\partial }{\partial {x}^{\mu }}\left(\sqrt{g}{g}^{\mu \nu }\tfrac{\partial }{\partial {x}^{\nu }}\right)$ is the Laplacian operator in a curved spacetime. Specifically, equation (38) rules the motion of a particle with kinetic energy $K=\tfrac{{{\hslash }}^{2}}{2{m}_{0}}{k}_{j}{k}^{j}$, j = 1, 2, 3, mass at rest m₀, and relativistic momenta ${p}_{\mu }={\hslash }{k}_{\mu }=\left(-\tfrac{E}{c},{\bf{p}}\right)$, ${p}^{\mu }={\hslash }{k}^{\mu }=\left(\tfrac{E}{c},{\bf{p}}\right)$. It turns out that ${k}_{\mu }{k}^{\nu }=-\tfrac{{E}^{2}}{{{\hslash }}^{2}{c}^{2}}+\tfrac{2{m}_{0}K}{{{\hslash }}^{2}}=-{\left(\tfrac{{m}_{0}c}{{\hslash }}\right)}^{2}$. The next step is to find a metric tensor ${g}_{\mu \nu }$ such that, through a covariant transformation, equation (38) is equivalent to equation (6) in a flat space, that is, the particle moves under the effect of a gravitational parabolic potential. This has been already done for a spherical potential [70], and in the Newtonian limit K ≪ m₀c², the metric in (37) is recovered. Extensions to the parabolic potential are possible and already explored via another scheme of derivation [71, 72], and will be treated in a general context in future works.

From g_μν, two important results are soon attained: the link between Einstein field equations and the SE [72], and the GUP effects in the quantization of the curved space [73]. Einstein equations are ${G}_{\mu \nu }=\tfrac{8\pi G}{{c}^{4}}{T}_{\mu \nu }$, with stress-energy tensor ${t}_{\mu \nu }=(\rho +p){U}_{\mu }{U}_{\nu }-{{pg}}_{\mu \nu }$ of a perfect fluid of density ρ at pressure p. U_μ is the four velocity. The quantization of the curved space follows from the canonical commutation relations $\left[{x}^{\mu },{p}^{\nu }\right]={\imath }{\hslash }{g}^{\mu \nu }$ [73]. These, when transformed in the previous variables, must give equation (5), and thus the dependence of the metric tensor on β.

We validate our theoretical analysis on the generalized SE equation (2) in two dimensions. We adopt dimensionless coordinates s = t/t_N, u = x/w_N, v = y/w_N, with ${w}_{N}^{2}={\hslash }{t}_{N}/m$, and t_N = Ω⁻¹ and we have from equation (2)

$$\begin{eqnarray}&&{\imath }{\partial }_{s}\varphi =\displaystyle \frac{1}{2}{{\rm{\nabla }}}_{{uv}}^{2}\varphi +\displaystyle \frac{\varepsilon }{8}{{\rm{\nabla }}}_{{uv}}^{4}\varphi +\displaystyle \frac{1}{2}({u}^{2}+{v}^{2})\varphi ,\end{eqnarray} \tag{ 39 }$$

with ${{\rm{\nabla }}}_{u,v}^{2}={\partial }_{u}^{2}+{\partial }_{v}^{2}$, and $\varepsilon =8m{\rm{\Omega }}\beta {\hslash }$. ε = 0 corresponds to the conventional paraxial regime. At t = s = 0, a Gaussian wave-packet with waist w_o is the initial condition:

$$\begin{eqnarray}&&\varphi (u,v,s=0)={\varphi }_{o}{{\rm{e}}}^{-{\left(\tfrac{u-{u}_{o}}{\sqrt{2}{w}_{o}}\right)}^{2}-{\left(\displaystyle \frac{v}{\sqrt{2}{w}_{o}}\right)}^{2}}{{\rm{e}}}^{{\imath }{k}_{l}v}.\end{eqnarray} \tag{ 40 }$$

In (40), the angular momentum is l = ℏk_l u₀. If ε = 0 and k_l = 0, the wave-packet oscillates without orbiting in the y-direction (not reported). Figure 3(a) shows the evolution, for ε = 0 and k_l = 20, by various snapshots of $| \varphi {| }^{2}$: the bottom panel reveals the elliptical orbit. When ε = 0.01, as in figure 3(b), we have evidence of the predicted precession. Figure 3 shows representative simulations, similar results occur for all the considered cases. Figure 4(a) shows a volumetric visualization of the solution of equation (39) for a longer timescale with respect to figure 3, for β = ε = 0. Panels in figure 4 give the trajectories of the wave-packet center of mass, which, for ε = 0 do not reveal any precession. Figure 4(b) for ε > 0 demonstrates the precession and the reduction of the orbital period.

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To analyze possible experimental tests, we recall that β is typically expressed in terms of the dimensionless

$$\begin{eqnarray}&&{\beta }_{0}=\beta \displaystyle \frac{{{\hslash }}^{2}}{{{\ell }}_{p}^{2}}=\beta {M}_{{\rm{P}}}^{2}{c}^{2},\end{eqnarray} \tag{ 41 }$$

with M_P the Planck mass, and c the vacuum light velocity. According to some authors, β₀ < 10³⁴ that we adopt as an optimistic upper bound for a real test of GUP phenomenology [45, 74]. For emulations, e.g. by paraxial light, we have β₀ = 10⁵⁵ [52, 54].

We consider a wave-packet at initial distance r₀ from the center of the potential with tangential velocity v₀ and $| l| ={{mv}}_{0}{r}_{0}={p}_{0}{r}_{0}$. For a harmonic trap, the quantum fluid size is ${{\ell }}_{B}^{2}\cong {\hslash }/m{\rm{\Omega }}$ [65], and we take ${r}_{0}\cong {{\ell }}_{B}$. A key point here is that the precession angle and the delay with respect to the unperturbed case (β = 0) increase at any orbit. Hence, the longer the observation time, the more accessible the measurement of the predicted perturbation. For a 1-degree precession, the number of orbits is ${{ \mathcal N }}_{1}=\pi /(180{\rm{\Delta }})={\left(\tfrac{{M}_{{\rm{P}}}c}{{{mv}}_{0}}\right)}^{2}/(1440{\beta }_{0})$. With β₀ = 10³⁴, ℓ_B = 100 μm, and ${v}_{0}={10}^{-10}c$, for ⁸⁷Rb BEC, we have ${{ \mathcal N }}_{1}\simeq {10}^{18}$ orbits occurring in a time interval ${{ \mathcal N }}_{1}{T}_{0}\simeq {10}^{15}$ s, which is experimentally inaccessible. On the contrary, if we consider a photonic condensate simulation, with β₀ = 10⁵⁵, m = 10⁻³⁶ kg, and v₀ = c [27], we have ${{ \mathcal N }}_{1}=0.03$: an experimentally testable 30-degrees precession in T₀ = 2 ps.

Very interesting is the measurement of the time delay. For example, one can measure the delay of the orbital oscillation with respect to a metrological reference clock. The period shrinks at any orbit by a relative amount $\delta T=-4{\beta }_{0}{\left(\tfrac{{{mv}}_{0}}{{{\rm{M}}}_{P}c}\right)}^{2}$. For ⁸⁷Rb BEC, we have δT ≃ −10⁻²⁰: after 10⁶ orbits (15 h of observation), one has a delay of 1fs. The bounds for the modifications of the uncertainty principle become more precise when increasing the observation time. In the photonic simulation, one has 0.1 ps delay for 1 orbit.

We remark that in order to observe the precession, one has to generate states with an initial angular momentum in a trapping potential, as in equation (40). Considering the fact that our model applies to many physical systems, as in photonics, polaritonics, and BECs of atoms and photons, we remark that different approaches to realize experiments may be taken into account.

The literature and the experimental realizations of beams and condensates with angular momentum is so vast that cannot be reviewed here (see for example [75]). We will discuss in the following some representative cases.

For optical propagation, a trapping transverse parabolic potential is realized by graded index systems, as lenses or optical fibers. A further possibility is to consider an array of optical lenses, which, as shown in [76], may also emulate a parabolic medium. A further possible framework is given by highly nonlocal nonlinear optical systems, as thermal media, where related studies have been reported [77, 78].

In these devices, the initial state with angular momentum in equation (40) is excited by a beam spatially displaced with respect to the center of the trapping potential with an initial phase tilt (a sketch is given in figure 5). The angular momentum is varied by the incidence angle with respect to the input plane. The beam follows a motion described by an optical orbit as represented in our theoretical analysis and in the simulations in figure 4. As a representative case, one can consider a beam with a transverse size of the order of few microns, with 1 μm wavelength. The propagation length in fibers can be kilometers, this may enable a precise monitoring of the spiraling trajectories, and the deviation from an elliptical orbit due to the precession is here predicted.

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For atomic BEC, states with angular momentum have been reported in a large number of articles (the interested readers may consider the references in [75]). The preparation of the initial state is conceptually similar to the optical case discussed above. In a wide trapping potential, a fraction of condensed atoms is launched with an angular momentum parallel to the trap axis. This results into a spiraling motion of the atoms. An interesting possibility for putting atom into rotation is using transfer of orbital angular momentum from photons by stimulated Raman processes with Laguerre–Gauss optical beams, as, e.g. analyzed in [79]. This approach creates persistent currents in superfluid Bose gases, and the observation of precession dynamics in these systems may provide evidence of the effects discussed in this manuscript.

The case of polaritonic condensates is particularly relevant, as the generation of states with angular momentum has been actively investigated in recent years. For example, authors in [80] demonstrated that angular momentum can be transferred to an exciton-polariton BEC by an external incoherent pump. In a parabolic potential, which is always present in this class of condensates, one can generate a rotating motion, and correspondingly observe the predicted precession. Recent developments [81] show that it is possible to precisely control the amount of optical angular momentum, and the generation of various spinning states. These results show that the observation of the precession here predicted is within current technologies for polariton superfluids.

As further possible experimental framework, we mention the case of photonic BECs. [82] We are not aware of the experimental generation of photonic BEC with angular momentum. One can however figure out approaches similar to those discussed above for polaritonic BEC. Angular momentum may be transferred by an external pump to the condensate. A further possibility is adopting symmetry breaking microcavities (e.g. by axicon mirrors) to generate spinning photonic BECs. The detailed analysis of these possibilities is beyond the scope of this manuscript.

The longtime observation of an orbiting quantum fluid is a feasible experimental road to realize analogs of orbital precession effects in general relativity and post-Newtonian corrections to the gravity law. We predict that the orbit of a wave-packet in a trapping potential exhibits a precession with an intriguing connection with the well-known anomalous precession of the perihelion of Mercury, the first experimental test of general relativity. We show that the precession is also linked to the modification of the uncertainty principle predicted by the most studied theories of quantum gravity. The resulting non-commutative geometry, for which the momentum and the position in different directions do not commute, is derived from general relativity and interpreted in the quantum framework of a massive particle under a parabolic potential. The link between its motion equations and the corresponding geodesic equations for a free particle in a bonding gravitational field connects the quantum theory to the general relativity. Moreover, the analogy with the original Einstein's solutions addresses the existence of additional effective quantum forces occurring at the Planck scale.

With reference to feasible real laboratory tests of quantum gravity theories, the perturbation to the orbital period duration may become accessible after a large number of orbits because of the cumulative amplification during time. In photonic analogs, one can have β₀ ≃ 10⁵⁵, and quantum simulations and experiments with non-paraxial light, or polaritonic condensates, are well within current experimental possibilities. By quantum simulations, one may test the mathematical models, and also conceive improved frameworks for real probes of Planckian physics. This is specifically relevant for theories based on non-commutative coordinates [34], for which experiments are lacking.

This publication was made possible through the support of a grant from the John Templeton Foundation (grant number 58277). We also acknowledge support the H2020 QuantERA ERA-NET cofund Quomplex (grant number 731473) and H2020 PhoQus project (grant number 820392), Sapienza Ateneo (2016 and 2017), PRIN 2017 PELM (Grant number 20177PSCKT).