Also at ]Gauss-Olbers Space Technology Transfer Center, c/o ZARM, University
of Bremen, 28359 Bremen, Germany

# A gravitational metrological triangle

Claus Lämmerzahl1<EMAIL_ADDRESS>[  Sebastian
Ulbricht2,3<EMAIL_ADDRESS>1Center of Applied Space Technology and
Microgravity ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany
2Fundamentale Physik für Metrologie FPM, Physikalisch-Technische Bundesanstalt
PTB, Bundesallee 100, 38116 Braunschweig, Germany
3Institut für Mathematische Physik, Technische Universität Braunschweig,
Mendelssohnstraße 3, 38106 Braunschweig, Germany

(February 5, 2024)

###### Abstract

Motivated by the similarity of the mathematical structure of Einstein’s
General Relativity in its weak field limit and of Maxwell’s theory of
electrodynamics it is shown that there are gravitational analogues of the
Josephson effect and the quantum Hall effect. These effects can be combined to
derive a gravitational analogue of the quantum/electric metrological triangle.
The gravitational metrological triangle may have applications in metrology and
could be used to investigate the relation of the Planck constant to
fundamental particle masses. This allows for quantum tests of the Weak
Equivalence Principle. Moreover, the similarity of the gravitational and the
quantum/electrical metrological triangle can be used to test the universality
of quantum mechanics.

##  I Metrology, the new SI, and metrological triangles

A major aspect of metrology is the definition and dissemination of physical
units such as the SI base units second, meter, kilogram, coulomb, kelvin, mol,
and candela [1]. While once these units were based on specific realizations
and artefacts, they are now defined through the fixed numerical values of
seven fundamental constants. Besides the second as a given number of
oscillations of light, emitted from a certain atomic transition in Caesium,
those constants are the velocity of light $c$, the Planck constant $h$, the
electric charge $e$, the Boltzmann constant $k_{B}$, the Avogadro constant
$N_{A}$, and the luminous efficacy $K_{\mathrm{cd}}$ [2], each playing central
roles in various branches of physics. It is clear that, by this choice, the
Meter Convention established a deep relation between metrology and the
structure of physics and Special and General Relativity, in particular [3].

Before the redefinition of the SI was put into practice in 2019, it was needed
to measure the defining constants to highest precision [4, 5, 6]. In the case
of the electron charge $e$ and the Planck constant $h$ this only became
possible thanks to the quantum revolution in metrology, heralded by the
discovery of the Josephson effect and the quantum Hall effect [7, 8]. Each of
these effects gave experimental access to a fundamental measurand, the
Josephson constant $K_{\text{J}}=2e/h$ and the von Klitzing constant
$R_{\text{K}}=h/e^{2}$, respectively. Together with the electron charge $e$,
these constants form the _quantum/electrical metrological triangle_ [9, 10],
see Fig. 1, where the value of $e$ can be obtained from electron counting,
e.g. using a single electron pump. This framework allowed for different
schemes to measure $e$ and $h$, and thus offered substantial options for
consistency checks improving our understanding of the underlying physical
processes. Also today, after the SI redefinition, the quantum/electrical
metrological triangle remains a powerful tool for metrology and fundamental
physics.

Figure 1: The metrological triangle for $e$ as well as for $h$.

Nowadays, the constituting effects of the triangle are used to _realize_
electric standards, such as the units ampere, ohm, and volt which complement
the realizations of the SI base units. For instance, an array of Josephson
junctions is used to realize a voltage of $\pm 10\,\mathrm{V}$ with only
$\mathrm{nV}$ uncertainty [11, 12]. That voltage could be used further to
obtain a precise current from a quantum Hall device. The same device also can
be used to compare resistors with the exact value of the von Klitzing constant
$R_{\text{K}}=25.8128...\,\mathrm{k}\Omega$. Thus, the individual effects, but
also their interplay can be exploited for scientific and technical
application. Based on these unit realizations the effects can be exploited for
the investigation of electromagnetic fields, for instance by measuring the
induced voltage in a conductor or the magnetic flux in a SQUID. This also
allows for a precise and independent determination and detailed study of
additional fundamental constants, such as the vacuum permittivity
$\epsilon_{0}$ and vacuum permeability $\mu_{0}$, which should be related by
$\epsilon_{0}\mu_{0}=1/c^{2}$. From these, further quantities, e.g., the wave
impedance of the vacuum $Z_{0}=\sqrt{\mu_{0}/\epsilon_{0}}\approx 377\,\Omega$
and the fine structure constant $\alpha=e^{2}(4\pi\epsilon_{0}\hbar
c)^{-1}=Z_{0}(2R_{\text{K}})^{-1}\approx 1/137$ can be derived and measured in
optical and atomic physics experiments.

In this letter we will propose to follow the same line of thought to introduce
a _quantum/gravitational_ metrological triangle. For that we use the fact that
Einstein’s General Relativity in its weak field limit resembles the
mathematical structure of electrodynamics. We deduce the gravitational
analogues of the Josephson and the quantum Hall effect, which together with
the counting of elementary particle masses, form a triangle that provides a
great potential to investigate the interaction of gravity and quantum
particles in various experiments.

##  II The structure of weak field Einstein gravity

In what follows, we will motivate that gravity, in its weak field limit, is
described by similar equations as electrodynamics while both are fundamentally
different in nature. Einstein’s General Relativity is based on the Einstein
field equation

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G}{c^{4}}T_{\mu\nu}\,.$ (1)

Here, $R_{\mu\nu}$ and $R$ are the Ricci tensor and Ricci scalar,
respectively, and $g_{\mu\nu}$ is the space-time metric. $T_{\mu\nu}$ is the
energy momentum tensor and $G$ the Newton gravitational constant. A standard
weak field approximation uses a decomposition of the metric into a Minkowskian
background $\eta_{\mu\nu}$ and a small variation $h_{\mu\nu}\ll 1$

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\,.$ (2)

If we define the mass density $\rho:=\frac{1}{c^{2}}T^{tt}$, the mass flux
density $j^{i}:=\frac{1}{c^{2}}T^{ti}$, and the gravitational scalar and
vector potentials

$\phi:=\bar{h}^{tt}c^{2}\,,\qquad a^{i}:=\bar{h}^{ti}\,,$ (3)

where
$\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\eta^{\rho\sigma}h_{\rho\sigma}$
[13], then we obtain the linearized Einstein equations in harmonic gauge

$\displaystyle\boldsymbol{\nabla}\cdot{\boldsymbol{B}}_{\text{g}}$
$\displaystyle=0$
$\displaystyle\quad\boldsymbol{\nabla}\times{\boldsymbol{E}}_{\text{g}}+{\dot{\boldsymbol{B}}}_{\text{g}}$
$\displaystyle=0$ (4)
$\displaystyle\boldsymbol{\nabla}\cdot{\boldsymbol{E}}_{\text{g}}$
$\displaystyle=4\pi G\rho$
$\displaystyle\quad\boldsymbol{\nabla}\times{\boldsymbol{B}}_{\text{g}}-\frac{1}{c^{2}}{\dot{\boldsymbol{E}}}_{\text{g}}$
$\displaystyle=\frac{4\pi G}{c^{2}}\boldsymbol{j}$ (5)

where we defined the weak field gravitational field strengths, that is, the
gravitoelectric and the gravitomagnetic field strengths

${\bm{E}}_{\text{g}}=-\bm{\nabla}\phi-\dot{\bm{a}}\,,\quad{\bm{B}}_{\text{g}}=\bm{\nabla}\times\bm{a}\,.$
(6)

Boldface symbols are 3-vectors and the dot denotes the time derivative. With
that, (4) and (5) are the weak field Einstein equations, see e.g. [13, 14].

We may push even further the analogy to the Maxwell equations by eliminating
the velocity of light: Introducing another gravitational constant
$\mu_{\text{g}}:=c^{2}/G$ allows us to define the gravitoelectric and
gravitomagnetic excitations
${\bm{D}}_{\text{g}}:=\frac{1}{G}{\bm{E}}_{\text{g}}$ and
${\bm{H}}_{\text{g}}:=\mu_{\text{g}}{\bm{B}}_{\text{g}}$ and to rewrite the
inhomogeneous equations (5) as

$\displaystyle\bm{\nabla}\cdot{\bm{D}}_{\text{g}}$ $\displaystyle=4\pi\rho$
$\displaystyle\quad\bm{\nabla}\times{\bm{H}}_{\text{g}}-{\dot{\bm{D}}}_{\text{g}}$
$\displaystyle=4\pi\bm{j}\,.$ (7)

Having formulated the weak field Einstein equations (4) and (7) without the
speed of light we obtained on this level a pre-metric version of these
equations [15]. The constant $1/G$ now plays the role of a gravitoelectric
vacuum permittivity, and $1/\mu_{\text{g}}$ the role of a gravitomagnetic
vacuum permeability. $G$ gives the strength of the gravitoelectic attraction
between two masses and $\mu_{\text{g}}$ gives the strength of the
gravitomagnetic interaction between spinning masses. Accordingly, one can use
test particles to measure the gravitoelectric attraction generated by given
masses as well as the gravitomagnetic effects generated by rotating masses
such as the Lense-Thirring or the Schiff effect, from which one can determine
the two constants $G$ and $\mu_{\text{g}}$. Both constants together give a
velocity which usually is interpreted as the velocity of light $c$, that is,
$G\mu_{\text{g}}=c^{2}$. However, in principle we should interpret this $c$ as
the speed of gravity, $G\mu_{\text{g}}=c_{\text{g}}^{2}$. In this way, that
is, by measuring $\epsilon_{0}$ and $\mu_{0}$ from the motion of charged
particles and $G$ and $\mu_{\text{g}}$ from the motion of massive particles,
we have formulated a genuine test of whether the speed of gravity coincides
with the speed of light. – As in the electromagnetic case we also may define a
gravitational wave impedance of vacuum
$Z_{0}^{(\text{g})}:=\sqrt{G/\mu_{\text{g}}}$ which is $\sim 2.34\cdot
10^{-19}\;\text{m}^{2}/\text{kg}\,\text{s}$. The unit
$\text{m}^{2}/\text{kg}\,\text{s}$ is the mechanical analogy to the unit
$\Omega$ of the electrical resistance in the relation $I=U/R$, if $I$
describes a mass current in kg/s and $U$ a gravitational potential difference
in $\text{m}^{2}/\text{s}^{2}$.

Having discussed the strong analogy between electrodynamics and weak field
gravity regarding the field equations as well as the equations of motion of
charged and massive particles, we now ask for implications of this analogy for
massive quantum particles.

##  III A gravitational Josephson effect

The analogy between electrostatic and Newtonian gravitational forces gives a
motivation to formulate a gravitational analogue of the electric Josephson
effect. For a gravitational Josephson effect we first need a gravitational
Josephson junction. This can be realized using quantum condensates on a
horizontal table [16, 17] with a step of height $h_{0}$ and a barrier, giving
a non-symmetric double well potential (DWP). In the following we consider
neutral quantum particles.

The non-symmetric DWP we are considering is a box of width $2b$, a step of
height $h_{0}$ at $x=0$ and a barrier of finite height and of width $2a$, see
Fig 2. The whole DWP is in a constant gravitational field. The step provides a
constant potential difference $gh_{0}$ between two regions where $g$ is the
constant gravitational acceleration. The total Hamiltonian of the system is
given by $H=H_{0}+V(x)$ with

$H_{0}=-\frac{\hbar^{2}}{2m}\Delta+mgz+\begin{cases}V_{0}&\text{for
$|x|<a$}\\\ 0&\text{elsewhere}\end{cases}\,,$ (8)

describing a symmetric DWP with a tunnel barrier of height $V_{0}$, and a
small potential step $h_{0}$ making the DWP non-symmetric

$V(x)=\begin{cases}mgh_{0}&\text{for $x<-a$}\\\
0&\text{elsewhere}\end{cases}\,.$ (9)

We will solve this problem whereby the non-symmetric part will be treated as
perturbation.

We have a stationary situation so that $H\psi=E\psi$ with $E>0$. Since in the
considered problem the $y$ dependence is trivial we will omit this degree of
freedom in the following. Then the above problem is a problem in $x$ and $z$.
We assume $\psi(x,z)=\psi_{x}(x)\psi_{z}(z)$ and take $\psi_{x}(x)=e^{\pm
ikx}$ where the $k$ is allowed to differ in the regions I,II and in the
barrier region m. Inserting this separation ansatz in the Schrödinger equation
we obtain the following three equations

$\displaystyle\left(E-mgh_{0}-\frac{\hbar^{2}k_{\text{I}}^{2}}{2m}\right)\psi_{z\text{I}}$
$\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\psi_{z\text{I}}^{\prime\prime}+mgz\,\psi_{z\text{I}}$
$\displaystyle\left(E-V_{0}-\frac{\hbar^{2}k_{\text{m}}^{2}}{2m}\right)\psi_{z\text{m}}$
$\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\psi_{z\text{m}}^{\prime\prime}+mgz\,\psi_{z\text{m}}$
(10)
$\displaystyle\left(E-\frac{\hbar^{2}k_{\text{II}}^{2}}{2m}\right)\psi_{z\text{II}}$
$\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\psi_{z\text{II}}^{\prime\prime}+mgz\,\psi_{z\text{II}}\,,$

where the prime denotes a derivative with respect to $z$. See also Fig. 2. A
global solution to these equations is given by

$\psi_{z}(z)\sim\text{Ai}\left(\frac{z}{\ell}-\epsilon_{i}\right)\,$ (11)

that reduces to $\psi_{z\text{I}}$, $\psi_{z\text{m}}$ and $\psi_{z\text{II}}$
in the corresponding domains. Here
$\ell=\left(2m^{2}g/\hbar^{2}\right)^{-1/3}$ is a characteristic length scale
and $\epsilon_{i}$ is a zero of the Airy function, see e.g. [16]. The energy
and the momenta take the values
$E=\frac{\hbar^{2}k^{2}}{2m}+mg\ell\epsilon_{i}$, $k_{\text{II}}\equiv k$,
$k_{\text{I}}=\sqrt{k^{2}-2m^{2}gh_{0}/\hbar^{2}}$ and
$k_{\text{m}}=\sqrt{2mV_{0}/\hbar^{2}-k^{2}}\equiv\kappa$.

Figure 2: The choice of the height $h_{0}$. The potential is shown in gray,
the Airy function is given in terms of their position probability. The
horizontal dashed line indicates the node of the Airy function which coincides
with the height $h_{0}$ of the left region. $2a$ is the width of the tunnel
barrier $V_{0}$ and $2b$ is the size of the box. (The $z$-axis shows the
density of the wave function depending on $z$ but also shows the height of the
potential $V_{0}$.)

Now we have to incorporate the boundary conditions regarding the coordinate
$z$: $\psi_{z}(h_{0})=0$, and $\psi_{z}(0)=0$. One way to achieve this is to
choose $h_{0}=(\epsilon_{i}-\epsilon_{j})\ell$ for any two zeros $i>j$ of the
Airy function. The boundary and jump conditions regarding the coordinate $x$
leads to the quantization of the energy. The corresponding energies and wave
functions can be determined along the usual procedure, see the Supplement
Materials. The non-symmetric DWP potential then is implemented as
perturbation. This way we obtain the Josephson equations

$\displaystyle i\hbar\dot{w}_{\text{L}}$ $\displaystyle=$ $\displaystyle
E_{\text{L}}w_{\text{L}}+Kw_{\text{R}}$ (12) $\displaystyle
i\hbar\dot{w}_{\text{R}}$ $\displaystyle=$ $\displaystyle
E_{\text{R}}w_{\text{R}}+Kw_{\text{L}}\,,$ (13)

characterizing the tunneling dynamics in a non-symmetric DWP. Here, the
$w_{\text{L,R}}$ are the probabilities to find the particle in the region I or
II. $E_{\text{L,R}}$ is the energy of the corresponding states, and
$K=\frac{1}{2}\Delta E$ with $\Delta E=mgh_{0}$.

We obtained equations which have exactly the same structure as for the
description of the Josephson effect in superconductivity. Thus, we obtain the
same solutions, that is, for a constant $h_{0}\neq 0$ we get the AC Josephson
effect with a current $I=I_{0}\sin(\nu t)$ with $\nu=\frac{m}{h}v_{0}$ where
$v_{0}=gh_{0}$. For a mass of 1 amu and $h=1\;\text{cm}$ we get
$\nu=10^{4}\;\text{Hz}$ and for $h\sim 10\;\mu\text{m}$ we have $\nu\sim
10\;\text{Hz}$. Based on that we can define a gravitational Josephson constant

$\displaystyle
K_{\text{J}}^{\text{(g)}}:=\frac{m}{h}\qquad\text{in}\qquad\frac{\text{s}}{{\text{m}}^{2}}=\frac{\text{Hz}}{\frac{\text{m}^{2}}{\text{s}^{2}}}\,.$
(14)

For neutrons, for instance, we get $K_{\text{J}}^{\text{(g)}}=1.2667\cdot
10^{6}\frac{\text{s}}{{\text{m}}^{2}}$. In order to measure this effect one
needs a collective quantum system as, e.g., a Bose-Einstein condensate. For a
system of incoherent particles the phases are different and average out so
that no net current will be observable. In fact, the Josephson constant for
BECs in a non-symmetric DWP has been measured in [18, 19].

This Josephson junction may be used to build up a gravitational SQUID, see
Fig. 3(a). For any ring quantum particle current in a rotating frame using
$\bm{k}=\bm{v}+\bm{\omega}\times\bm{x}$ we have

$2\pi n=\oint\bm{k}\cdot d\bm{x}=\frac{m}{\hbar}\oint\boldsymbol{v}\cdot
d\boldsymbol{x}+\frac{m}{\hbar}\Phi^{(g)}\approx\frac{m}{\hbar}\Phi^{(g)}$
(15)

for small velocities. Based on that we can define a flux quantum
$\Phi_{0}^{(g)}:=\frac{h}{m}$ so that

$\Phi^{(g)}=n\Phi_{0}^{(g)}\,.$ (16)

This is deeply related to the Sagnac effect. The rotation $\bm{\omega}$ may
consist of a kinematical rotation and of the gravitomagnetic rotation.

In analogy to the electromagnetic case the current in Fig. 3(b) is given by

$I_{\text{tot}}=2I_{c}\sin\delta\cos\left(\pi\frac{\Phi^{(g)}}{\Phi_{0}^{(g)}}\right)$
(17)

with the critical current $I_{c}$ and with

$\delta=\gamma_{a}+\pi\frac{\Phi^{(g)}}{\Phi_{0}^{(g)}}$ (18)

where $\gamma_{a}$ is the phase shift at the Josephson junction $a$, Fig.
3(b). The typical resolution of the flux is $\sim 10^{-6}\Phi_{0}^{(g)}$. For
an area of $\Sigma\sim 10^{-2}\;{\text{m}}^{2}$ this implies a sensitivity of
$\omega\sim 10^{-10}\;\text{Hz}$ for measuring a rotation. This is a gyroscope
based on the Sagnac effect for matter waves realized through a gravitational
SQUID for BECs, see also [19].

Figure 3: Quantum particle SQUID for measuring rotations and gravitomagnetism.

##  IV A gravitational quantum Hall effect

With mass currents, e.g. neutrons, atoms, or BECs, on a rotating surface a
gravitational analogue of the quantum Hall effect can be realized. According
to Fig. 4, in thin rotating plate the incoming current (red) will be deviated
to the right. This can be compensated by a small gravitational acceleration in
$y$-direction induced by a slight tilting of the $x-y$-plane in $y$-direction.

The Hamilton operator of a point mass in a rotating frame [20] is given by

$H=\frac{1}{2m}{\bm{p}}^{2}+\bm{\omega}\cdot\bm{L}\,,$ (19)

where $\bm{L}$ is the angular momentum of the particle. If the rotation is
around the $z$-axis and the particles move only in $x$-$y$-plane, see Fig. 4,
then $p_{z}=0$ and the Hamilton operator reads

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2m}{\boldsymbol{p}}^{2}-\omega_{z}yp_{x}$ (20)
$\displaystyle\approx$
$\displaystyle\frac{1}{2m}\left(p_{x}-m\omega_{z}y\right)^{2}+\frac{1}{2m}p_{y}^{2}\,,$
(21)

where we neglected terms quadratic in the rotation. This Hamiltonian has
exactly the same structure as the Hamilton operator for the quantum Hall
effect in the Landau gauge. One can calculate the gravitational analogue of
the Landau levels and the mass current for the case that the Landau levels are
filled up to some level $s$. By tilting the plane one may also induce a small
gravitational acceleration needed to compensate the rotation induced
acceleration, see Fig. 4. This then gives a gravitational quantum Hall effect
described by the transversal mass current resistivity

$\rho_{xy}=\frac{h}{m^{2}}\frac{1}{s}\,.$ (22)

This leads to the gravitational von Klitzing constant

$R_{\text{K}}^{(\text{g})}:=\frac{h}{m^{2}}\qquad\text{in}\qquad\frac{{\text{m}}^{2}}{\text{kg}\;\text{s}}$
(23)

for the used masses. If we take for instance neutrons or hydrogen atoms, then
$R_{\text{K}}^{(\text{g})}=2.481\cdot
10^{18}\;\frac{{\text{m}}^{2}}{\text{kg}\;\text{s}}$.

Figure 4: Setup for a gravitational quantum Hall effect. Neutrons or atoms are
moving on a thin rotating plate with a current $I$. The rotation induces an
acceleration field orthogonal to the motion of the particles. A compensation
of the transversal acceleration can be achieved by a small tilt of the plane
around the $x$-axis what induces a small acceleration $\bm{a}$.

The classical gravitational Hall effect adds the Coriolis and gravitational
acceleration $\bm{a}=\bm{g}+\bm{v}\times\bm{\Omega}$ with $\bm{v}=\bm{p}/m$,
and requires $\bm{a}=0$, $\bm{\Omega}=\Omega_{z}{\bm{e}}_{z}$ and
$\bm{v}\bot\bm{\Omega}$. In equilibrium we have $v_{y}=0$ so that
$\boldsymbol{g}=v_{x}\Omega_{z}{\boldsymbol{e}}_{y}$. Accordingly, the induced
acceleration can be compensated by the plane being slightly tilted in
$y$-direction (for $\Omega=0.1\;\text{Hz}$ and $v_{x}=10\;\text{cm/s}$ we need
$g=1\;\text{cm/s}^{2}=10^{-3}g_{\text{Earth}}$ what requires a tilt angle of
$\sim 1^{\prime}$).

This can also be described within the Drude model. The equation of motion
including a friction/damping term for reaching an equilibrium ($\tau$ is a
damping time) then is
$\bm{a}=\bm{g}+\bm{v}\times\bm{\Omega}-\frac{\bm{v}}{\tau}$. In equilibrium
$\bm{a}=0$ and with $\bm{j}=nm\bm{v}$ we get Ohm’s law $\bm{g}=\rho\bm{j}$
with the resistivity

$\rho=\begin{pmatrix}\frac{1}{nm\tau}&-\frac{\Omega_{z}}{nm}\\\
\frac{\Omega_{z}}{nm}&\frac{1}{nm\tau}\end{pmatrix}\,,$ (24)

from which we can read off the longitudinal and transversal (Hall) resistivity

$\rho_{xx}=\frac{1}{nm\tau}\qquad\rho_{xy}=\frac{\Omega_{z}}{nm}\,.$ (25)

##  V A gravitational metrological triangle

Having discussed the gravitational Josephson effect and the gravitational
quantum Hall effect, we can use these results to construct the gravitational
analogue of the quantum/electrical metrological triangle. In total analogy to
the electric constants $e$, $K_{\text{J}}$, and $R_{\text{K}}$, which obey the
relations [9, 10]

$\displaystyle e=\frac{2}{K_{\text{J}}R_{\text{K}}}\quad$ ,
$\displaystyle\quad h=\frac{4}{K_{\text{J}}^{2}R_{\text{K}}}\,,$ (26)

we have found their gravitational counterparts, which satisfy

$\displaystyle
m=\frac{1}{K_{\text{J}}^{(\text{g})}R_{\text{K}}^{(\text{g})}}\quad$ ,
$\displaystyle\quad
h=\frac{1}{K_{\text{J}}^{(\text{g})2}R_{\text{K}}^{(\text{g})}}\,,$ (27)

see Fig. 5. Here, we find that the mass $m$ appears on the same level as $2e$
in (26), where it represents the charge of a Cooper pair. In (27), however,
all effects needed to realize the corresponding experiments are purely
gravitational and without electromagnetic interaction. At one hand, this could
open a new window to quantum based metrology. At the other hand, this
relations give raise to a number of possible consistency checks, to test our
understanding of quantum systems in gravity. Ignoring for a moment, that we
fixed $h$ in the current SI, we could assume experiments, where a fundamental
mass $m$ is precisely known. In this version of the gravitational metrological
triangle, we would have access to three measurands

$h\,,\qquad m/K_{\text{J}}^{(\text{g})}\,,\qquad
m^{2}/R_{\text{K}}^{(\text{g})}\,,$ (28)

which have to coincide. Within the modern SI, where the Planck constant $h$ is
exactly known we, in turn, have three methods to obtain the mass of a chosen
reference particle,

$m\,,\qquad
K_{\text{J}}^{(\text{g})}h\,,\qquad\sqrt{R_{\text{K}}^{(\text{g})}h}\,,$ (29)

which allows similar insights as in the previous case. In that scenario,
however, we have an additional option, since the fixed constant $h$ is the
same as for quantum/electrical metrological triangle. Assuming the Plank
constant to have the same value for the electromagnetic _and_ the
gravitational interaction of quantum particles, we can relate its identities
from Eqs. (26) and (27) to obtain

$K_{\text{J}}^{2}R_{\text{K}}=4K_{\text{J}}^{(\text{g})2}R_{\text{K}}^{(\text{g})}\,.$
(30)

Thus, we find both triangles to be not independent, what allows to check
whether the Planck constant $h$ is the same in the gravitational and in the
electrical context. This might be interpreted as a test for the universality
of quantum mechanics [21].

Figure 5: The metrological triangle for $m$ as well as for $h$.

##  VI Summary and discussion

A gravitational analogue of the quantum/electrical metrological triangle has
been introduced. The new triangle is based on quantum effects of neutrons or
atoms in a gravitational field – in complete analogy to the electromagnetic
interaction of charged quantum particles used in the quantum/electrical
triangle. This is possible since the mathematical structure of weak field
gravity resembles the governing equations of electromagnetism. We showed that
two fundamental quantum effects can also be realized with the gravitational
interaction: the gravitational Josephson effect and the gravitational quantum
Hall effect. Together with particle counting in analogy to the count of
electrons in a single electron pump those can be used to trace back for
instance mass measurements to the values of the Planck constant $h$ and a
chosen atomic or fundamental particle mass $m$. Fundamental masses can be
determined to tremendous precision, for instance in particle trap experiments
[6, 22]. With that a pure quantum realization of the kilogram – besides the
Avogadro experiment [23] and the Kibble balance [24] – may be possible, see
also [25].

In addition to their metrological aspects, the gravitational Josephson effect
and the gravitational quantum Hall effect can also provide fundamental test of
the behavior of quantum systems in gravity: They are suitable to construct
tests of the universality of quantum mechanics, i.e, whether quantum mechanics
is valid and the same for interactions of quantum particles with
electromagnetic and gravitational fields. According to [18, 19], the
gravitational Josephson constant might be measurable with an accuracy of
$10^{-3}$. The gravitational quantum Hall effect has not yet been
experimentally realized.

As in the electromagnetic case one may use devices based on the discussed
effects for a better exploration of the dynamics of matter and, thus, of the
gravitational fields. However, the new approach will not help in improving the
precision of measurements due to the weakness of gravity on the laboratory
scale. Owing to the Weak Equivalence Principle, the mass of a test particle
plays no role in the dynamics of test particles, and in the astrophysical
context a mass $M$ appears in the combination $GM$ only.

Nevertheless, from the motion of test objects it is possible to measure the
gravitational constants $G$ and $\mu_{\text{g}}$. The Newton gravitational
constant $G$ is given by the force between two massive bodies [6], and the
gravitomagnetic constant $\mu_{\text{g}}$ can be determined from the Lense-
Thirring effect [26]. $G$ can be measured with 0.1% precision, and for
$\mu_{\text{g}}$ one approaches 1%. In analogy to the electromagnetic case one
may – in principle – measure $G$ with clocks in a gravitational field given by
a laboratory mass or measure the gravitomagnetic field with gravitational
SQUIDs.

A further issue is a hypothetical time dependence of the constants. While the
SI-defining constants are constant by definition, the measured constants like
$\epsilon_{0}$, $\mu_{0}$, or $G$ (or parameters relevant for the standard
model of elementary particles) may depend on time. While a time dependent $G$
will not interfere with the constancy of the defining constants a time-
dependent $\epsilon_{0}$, for example, will influence the energy levels of
atoms and, thus, the definition of the second. However, since the second is
based on one transition only, nothing will change and everything still is
consistent. One may detect a hypothetical time-dependence of $\epsilon_{0}$ –
what is equivalent to a time dependence of the fine structure constant –
through the comparison of different energy transitions within one atom but
this does not influence the particular transition chosen for the definition of
the second. In any case, any hypothetical time dependence of $G$ or the fine
structure constant has been experimentally excluded with highest precision
[27, 28].

From $G$ and $\mu_{\text{g}}$ further quantities can be derived, e.g., the
gravitational wave impedance $Z_{0}^{(\text{g})}:=\sqrt{G/\mu_{\text{g}}}\sim
2.34\cdot 10^{-19}\text{m}^{2}/\text{kg}\,\text{s}$, which governs the
propagation of gravitational waves in vacuum.

Having started our investigation at the level of Einstein equations, we also
have to take the Weak Equivalence Principle as granted. That means that one
has to be sure that the gravitational interaction acts universally, that is,
independent from the composition and weight of pointlike bodies. If we
distinguish between inertial and gravitational mass, then inspection of the
derivation of the gravitational Josephson and quantum Hall effects shows that
the gravitational Josephson constant depends on the gravitational mass
$K_{\text{J}}^{(g)}=\frac{m_{\text{g}}}{h}$ and the gravitational von Klitzing
constant on the product of inertial and gravitational mass
$R_{\text{K}}^{(g)}=\frac{h}{m_{\text{i}}m_{\text{g}}}$. Then in the relations
(27) the left hand sides will be replaced by $h\rightarrow
h\frac{m_{\text{i}}}{m_{\text{g}}}$ and $m\rightarrow m_{\text{i}}$,
respectively. This agrees with the general observation that in all equations
of motion of test matter the gravitational mass appears in the combination
$m_{\text{g}}/m_{\text{i}}$. Thus, in principle the MTs also provide a pure
quantum test of the Weak Equivalence Principle when realized with different
types of, e.g., atoms. The recently published final result of the MICROSCOPE
mission provided a classical test of the Weak Equivalence Principle confirming
it at the order $10^{-15}$ [29], see also [30].

###### Acknowledgements.

We thank H. Abele, G. Czycholl, F. W. Hehl, T. Mehlstäubler, A. Pelster, G.
Schäfer, A. Surzhykov, and J. Ullrich for fruitful discussions. We acknowledge
the support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy-EXC-2123 “QuantumFrontiers” –
Grant No. 390837967 and the CRC 1464 “Relativistic and Quantum-based Geodesy”
(TerraQ).

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##  VII Supplementary material

Here we derive the essential equations for the gravitational Josephson effect.
For that we first determine the energies and eigenstates for the DWP and then
the dynamics of symmetric and antisymmetric states in a non-symmetric DWP.

###  VII.1 The symmetric double well potential

We take the symmetric DWP as given by (8). We employ the boundary conditions
in $x$-direction, that is, $\psi_{x}(\pm b)=0$ and the smoothness of the wave
function at $x=\pm a$. In what follows, we first consider the symmetric
potential ($h_{0}=0$) and treat the asymmetry as a perturbation afterwards.
Then the wave functions are

$\psi_{x}(x)=\begin{cases}A\sin\left(k\left(x+b\right)\right)&\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{for
}-b<x<-a\\\ A\frac{e^{\kappa a}\sin\left(k(b-a)\right)}{se^{\kappa
a}+e^{-\kappa a}}\\\
\qquad\times\left(e^{\kappa(x-a)}+se^{-\kappa(x+a)}\right)&\text{for }|x|<a\\\
As\sin\left(k\left(-x+b\right)\right)&\text{for }a<x<b\,,\end{cases}$ (31)

where $s=\pm 1$ stands for the symmetric/antisymmetric solution, respectively.
$A$ is a normalization factor and $\kappa$ and $k$ have to obey the condition

$\kappa\tanh^{s}\left(\kappa a\right)=-k\cot\left(k(b-a)\right)\,,$ (32)

what leads to a discrete set of solutions for $k$ and $\kappa$. For
simplicity, we consider the case of weak coupling between I and II, that is,
$\kappa a\gg 1$, and of low energies of the particles, $\frac{k}{\kappa}\ll
1$. We then obtain the momentum and energy eigenvalues

$\displaystyle k_{n}$
$\displaystyle=\frac{1}{b-a}\left(n\pi-\frac{k_{n}^{0}}{\kappa_{n}^{0}}\left(1\pm
e^{-2\kappa_{n}^{0}a}\right)\right)$ (33) $\displaystyle E_{i,n,\pm}$
$\displaystyle=E_{n}^{0}\left(1-\frac{2}{(b-a)\kappa_{n}^{0}}\left(1\pm
2e^{-2\kappa_{n}^{0}a}\right)\right)+mg\ell\epsilon_{i}\,.$ (34)

with $\kappa_{n}^{0}=\frac{1}{\hbar}\sqrt{2m(V_{0}-E_{n}^{0})}$,
$k_{n}^{0}=\frac{n\pi}{b-a}$, and $E_{n}^{0}=(\hbar k_{n}^{0})^{2}/(2m)$ where
$n$ should be not too large. We also define the average energy and the energy
difference

$\displaystyle E_{i,n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(E_{i,n,+}+E_{i,n,-})$ (35) $\displaystyle\Delta
E_{n}$ $\displaystyle=$ $\displaystyle
E_{i,n,+}-E_{i,n,-}=\frac{8E_{n}^{0}}{(b-a)\kappa_{n}^{0}}e^{-2\kappa_{n}^{0}a}$
(36)

of symmetric and anti-symmetric states of given $n$ and $i$.

###  VII.2 The gravitational Josephson equations for the non-symmetric DWP

Having solved the problem for the symmetric DWP, we now investigate the
influence of the small potential step $gh_{0}$. Application of the standard
first order perturbation theory to the small potential step (9) yields the
corrected states

$\tilde{\psi}_{i,n,s}(x,z)=\psi_{i,n,s}(x,z)+\frac{s}{2}\frac{mgh_{0}}{\Delta
E_{n}}\,\psi_{i,n,-s}(x,z)$ (37)

obeying to first order approximation

$[H_{0}+V(x)]\tilde{\psi}_{i,n,s}(x,z)=\left(E_{i,n,s}+\frac{1}{2}mgh_{0}\right)\tilde{\psi}_{i,n,s}(x,z)\,.$
(38)

Thus, the small potential step $gh_{0}$ leads to a mixing between symmetric
$(s=+1)$ and anti-symmetric $(s=-1)$ wave functions, while mixing between
states of different $n$ is suppressed by a factor $\Delta E_{n}/E_{i,n,s}$. In
what follows we drop the indices $n$ and $i$ assuming them to be fixed and
show them only when needed.

We now want to define left and right bin states by a superposition of eigen
states of the non-symmetric DWP (37). With the ansatz

$\displaystyle\psi_{\text{L}}$ $\displaystyle=$
$\displaystyle\cos\theta\,\tilde{\psi}_{+}+\sin\theta\,\tilde{\psi}_{-}$ (39)
$\displaystyle\psi_{\text{R}}$ $\displaystyle=$
$\displaystyle\sin\theta\,\tilde{\psi}_{+}-\cos\theta\,\tilde{\psi}_{-}$ (40)

we minimize
$\int_{-b}^{0}|\psi_{\text{R}}|^{2}dx+\int_{0}^{b}|\psi_{\text{L}}|^{2}dx$
what is the probability to find a particle described by $\psi_{\text{R}}$ in
the left bin and vice versa. One finds, that the best fitting for left and
right states are given by

$\displaystyle\psi_{\text{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\psi_{+}+\psi_{-}\right)$ (41)
$\displaystyle\psi_{\text{R}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\psi_{+}-\psi_{-}\right)\,.$ (42)

These are the same states as the optimal left and right states for the
symmetric DWP. However, in this case $\theta$ reads

$\theta\approx-\frac{\pi}{4}+\frac{mgh_{0}}{2\Delta E_{n}}$ (43)

and the action of the Hamiltonian is given by

$\displaystyle(H_{0}+V)\psi_{\text{L}}$
$\displaystyle=E_{\text{L}}\psi_{\text{L}}+K\psi_{\text{R}}$ (44)
$\displaystyle(H_{0}+V)\psi_{\text{R}}$
$\displaystyle=E_{\text{R}}\psi_{\text{R}}+K\psi_{\text{L}}$ (45)

with the energies and the coupling constants

$\displaystyle E_{i,n,\text{R}/\text{L}}$
$\displaystyle=E_{i,n}\pm\frac{1}{2}\Delta E_{n}\cos(2\theta)\approx
E_{i,n}\pm\frac{1}{2}mgh_{0}$ (46) $\displaystyle K_{n}$
$\displaystyle=-\frac{1}{2}\Delta E_{n}\sin(2\theta)\approx\frac{1}{2}\Delta
E_{n}\,.$ (47)

We find in particular that the energy difference
$E_{i,n,\text{R}}-E_{i,n,\text{L}}=mgh_{0}$ is related to the potential step
(9) and does not depend on $i$ or $n$.

The wave functions $\psi_{\text{L/R}}(x)$ are no eigenstates of the Hamilton
operator. In order to check how the actual wave function decomposes into the
left and right wave functions we make the ansatz

$\Psi(t,x,z)=\frac{1}{\sqrt{2}}\left[w_{\text{L}}(t)\psi_{\text{L}}(x,z)+w_{\text{R}}(t)\psi_{\text{R}}(x,z)\right]$
(48)

and determine the equations of motion for the pobabilities $w_{\text{L/R}}(t)$
from $i\hbar\dot{\Psi}(x,t)=[H_{0}+V(x)]\Psi(x,t)$.