# A proposed test of quantum mechanics with three connected atomic clock
transitions

Mark G. Raizen  Department of Physics, The University of Texas at Austin,
Austin, Texas, 78712  Gerald Gilbert  MITRE-Princeton, 200 Forrestal Road,
Princeton, NJ 08540  Dmitry Budker  Johannes Gutenberg-Universität Mainz,
Helmholtz-Institut Mainz, GSI Helmholtzzentrum für Schwerionenforschung, 55128
Mainz, Germany  Department of Physics, University of California, Berkeley,
California 94720, USA

###### Abstract

We consider possible extensions to quantum mechanics proposed by Steven
Weinberg, and re-analyze his prediction of a new test based upon three atomic
clocks in the same atom. We propose realistic experimental systems where this
hypothesis can be tested. Two systems already set limits on deviations from
quantum mechanics, while with another system, one would be able to search for
new physics at the limit of sensitivity of the best atomic clocks.

###### pacs:

Valid PACS appear here

##  I Introduction

Quantum mechanics is one of the fundamental pillars of modern physics. It has
remained essentially unchanged since it was developed almost 100 years ago
Weinberg (2015). Although in continuous practical use since its discovery,
there is general agreement that quantum mechanics is an incomplete theory as
it does not incorporate interactions with the environment which are mostly
unavoidable. For example, even atoms in isolation in a laboratory vacuum are
immersed in black-body radiation or other external electromagnetic fields. The
Schrödinger equation is time-symmetric, with no past and no future, just state
vectors, eigenvalues, and Hermitian operators. This creates a tension with
what is referred to as the macroscopic world, where there is a clear arrow of
time. Moreover, the underlying, dynamical assumptions of quantum mechanics do
not specify precisely how to account for a measurement apparatus, requiring
additional prescription from outside those assumptions. In an attempt to
resolve the situation, the concept of measurement-induced collapse of the
wave-function was invoked, known as the Copenhagen interpretation. An
alternative approach is the many-worlds interpretation, where endless branches
of parallel histories are created. Neither are appealing options, and do not
fit into a consistent framework that extends quantum mechanics. There are also
no experimental tests that can point to one or the other, a very
unsatisfactory state of affairs. This led Steven Weinberg to propose in 2014
that quantum mechanics can be formulated without the need for state vectors
Weinberg (2014). In 2016 Weinberg further proposed the possibility of
observing small departures from ordinary quantum mechanics by studying
implications of the Lindblad equation that might be detected by utilizing the
great precision of atomic clocks Weinberg (2016). Weinberg proposed that this
would have experimentally measurable consequences, specifically with atomic
clocks which have now reached an absolute accuracy of a part in 1018. He
envisioned an atom with three clock transitions, where each one could be
measured independently of the other two. These transitions would be from state
$\left|A\right\rangle$ to $\left|B\right\rangle$ denoted 1,
$\left|B\right\rangle$ to $\left|C\right\rangle$ denoted 2, and
$\left|A\right\rangle$ to $\left|C\right\rangle$, denoted 3, illustrated
schematically in Fig. 1. The manifestation of the extension of quantum
mechanics would be that the sum of the first two clock frequencies would not
be equal to the third, i.e., 1+2 $\neq$ 3\. Weinberg did not suggest in
Weinberg (2016) any concrete examples where this hypothesis could be tested.

Figure 1: The general idea of testing for a possible nonlinearity effect as
proposed by Steven Weinberg.

In this paper, we make the case for experimental systems to search for new
physics beyond standard quantum mechanics.

Two systems are a trapped 226Ra ion, and a trapped 40Ca ion where three such
transitions have, in fact, been measured, though they are not all atomic clock
transitions thereby precluding high accuracy. The newly proposed system is
neutral 171Yb atoms which offer the potential for extremely high accuracy
typical of the best atomic clocks.

While Weinberg’s extension of quantum mechanics has been a hitherto untested
hypothesis, his theory does not provide an exact quantitative prediction of
the size of the effect. Thus, tests in different systems are particularly
useful for probing the idea.

##  II Existing limits

In searching for possible experimental systems where there are three
transitions that are tied together, we found existing data for a single
trapped 226Ra ion. While radium does not have any stable isotopes, the half-
life of 226Ra is about 1585.5 y. It is a daughter in the decay chain of 238U,
so can be found in nature as first discovered by Marie and Pierre Curie in
1898. Singly-ionized radium was confined in a RF Paul trap and spectroscopic
measurements were reported in a series of papers Holliman  _et al._ (2019,
2020, 2022). These papers did not cite the 2016 Weinberg paper, so, evidently,
the authors were not aware of that work. A schematic of the energy levels is
shown in Fig. 2. The ground state is $7s~{}^{2}S_{1/2}$, and two electric
quadrupole transitions, $7s~{}^{2}S_{1/2}\rightarrow 6d~{}^{2}D_{5/2}$ at 728
nm and $7s~{}^{2}S_{1/2}\rightarrow 6d~{}^{2}D_{3/2}$ at 828 nm. In addition,
there are two electric dipole transitions $6d~{}^{2}D_{3/2}\rightarrow
7p~{}^{2}P^{o}_{3/2}$ at 708 nm and $6d~{}^{2}D_{5/2}\rightarrow
7p~{}^{2}P^{o}_{3/2}$ at 802 nm. In a subsequent paper, the electric dipole
transition $7s~{}^{2}S_{1/2}\rightarrow 7p~{}^{2}P^{o}_{3/2}$ at 382 nm was
reported. To within the accuracy of these measurements, the results do not
exhibit any additional frequency shifts, but the precision was limited by the
fact that only one of the three transitions was an electric quadrupole, while
the other two were electric dipole. The quoted accuracy is around 30 MHz,
though could presumably be improved considerably with longer interrogation
times. Nevertheless, the 226Ra system does not satisfy the criterion proposed
by Weinberg of having three connected atomic clocks. The same data provide
other combinations of transitions, for example 828 nm and 708 nm reach the
same level as 728 and 802 nm. Again, only two of these transitions (828 nm and
728 nm) are electric quadrupole, while the other two are electric dipole.

A second case where there are existing data is a single trapped 40Ca ion (Ca
II). The atomic structure is similar to that of the radium ion, though the
transition frequencies and lifetimes are different. Transitions in Ca II were
investigated in the context of precision determination of isotope shifts
Kramida (2020) and “transition closures” were studied at the 100 kHz level
Müller  _et al._ (2020), although precise calculation (rather than direct
measurement) was used for some of the transitions. A fully experimental test
is possible using the electric quadrupole transition,
$4s~{}^{2}S_{1/2}\rightarrow 3d~{}^{2}D_{5/2}$ at 729 nm, which was accurately
measured Huang  _et al._ (2016) combining them with the electric dipole
transitions $4s~{}^{2}S_{1/2}\rightarrow 4p~{}^{2}P^{o}_{3/2}$ at 393 nm and
$4s~{}^{2}D_{5/2}\rightarrow 4p~{}^{2}P_{3/2}$ at 854 nm reported Müller  _et
al._ (2020). Currently, the accuracy of the limits in Ca II is around 100 kHz,
considerably better than in Ra II, and also may be improved in the future.

We now propose another atomic system where there are three connected clock
transitions.

Figure 2: Selected low-lying energy levels of Ra II Kramida  _et al._ (2021)
(not to scale).

##  III Proposed Test: 1-2-3 171Yb ensemble clocks

Various options for testing for Weinberg’s nonlinearity are offered by atomic
system with two electrons above closed shells, for example, neutral ytterbium
(see Fig. 3). Such atoms have an additional experimental advantage in that
there are well-developed techniques for their trapping and cooling and that
lattice clocks based on such atoms have been demonstrated Ludlow  _et al._
(2015), The natural choice of state 1 is the ground 4f146s2 state, while state
3 could be chosen from one of the $J=1,2$ states of the same (even) parity.
The even-parity $J=2$ states are connected to the ground state via electric
quadrupole (E2) transitions as well as by degenerate two-photon transitions
(i.e., transitions that can be driven by photons of the same wavelength). The
${}^{1}S_{0}\rightarrow$ ${}^{1}D_{2}$ transition was recently used for a
precision (uncertainty of $\approx 300$ Hz) determination of isotope shifts
Figueroa  _et al._ (2022). As for the transitions between the ground state and
$J=1$ even-parity states, they are forbidden by E2 selection rules, as well as
by selection rules for degenerate two-photon transitions, The way to drive
these transitions is to use nondegenerate photons or, alternatively, to rely
on mixing of $J$ states by hyperfine interactions or magnetic fields, see
Bowers  _et al._ (1996); Kozlov  _et al._ (2009) and references therein. In
order to ultimately achieve high spectroscopic resolution in the 1-3
transition, it is prudent to choose the upper level with the longest possible
radiative lifetime. The 4f145d6s 1D2 state used for the the isotope-shift
measurement has a lifetime of $\approx 6.7\,\mu$s Bowers  _et al._ (1996). The
E2 transition from the ground state is at 361 nm, while the degenerate two-
photon transition occurs at twice that wavelength (i.e., around 723 nm).

Figure 3: Selected low-lying energy levels in Yb I Kramida  _et al._ (2021) (not to scale). Their properties relevant to the 1-2-3 test are listed it Table 1 State |  | Energy (cm-1) Kramida  _et al._ (2021) | Lifetime
---|---|---|---
4f146s2 |  1S0 | 0 |
4f146s6p |  3P0 | 17 288.439 |  $\approx$20 s Safronova  _et al._ (2018)
|  3P1 | 17 992.007 | 866 ns Beloy  _et al._ (2012)
|  3P2 | 19 710.388 |  $\approx$9 s Mishra and Balasubramanian (2001)
4f135d6s2 | J=2 | 23 188.518 |  $\approx$1 min Dzuba  _et al._ (2018); Safronova  _et al._ (2018)
4f145d6s |  3D1 | 24 489.102 | 329 ns Beloy  _et al._ (2012)
|  3D2 | 24 751.948 | 460 ns Bowers  _et al._ (1996)
|  1D2 | 27 677.665 | 6.7 $\mu$s Bowers  _et al._ (1996)
Table 1: Properties of Yb I states shown in Fig. 3. Lifetimes of metastable
states depend on hyperfine mixing and are given for 171Yb.

We now discuss the choice of level 2 and the 1-2 and 2-3 transitions. One
option for level 2 is the metastable 4f146s6p 3P0 state. Here, the 1-2
transition can occur due to hyperfine mixing with the ${}^{3}P_{1}$ state for
an isotope with nonzero nuclear spin (e.g., 171Yb with $I=1/2$. The advantage
here is that existing ytterbium lattice clocks already operate on this
transition. The 2-3 transition can proceed by a similar mechanism and the
clock operation on several transitions in the same atom can proceed as
proposed in Safronova  _et al._ (2018) by alternating measurements on
different transitions, with adiabatic switching of the lattice wavelength to
avoid systematics.

For an even narrower 1-3 transition, one may choose to work with the
metastable odd-parity $J=2$ state at 23188.518 cm-1, the lifetime of which is
predicted to be on the order of 1 min Dzuba  _et al._ (2018); Safronova  _et
al._ (2018). In fact, this is exactly the system considered in Dzuba  _et al._
(2018); Safronova  _et al._ (2018) for the dual atomic clock in one and the
same atom in the context of the search for possible variation of fundamental
constants Dzuba  _et al._ (2018); Safronova  _et al._ (2018). For the 1-3
transition, one would use either an E1-M1 two-photon transition or,
alternatively, a hyperfine-interaction induced two-photon transition at 852
nm.

Several specific schemes exist in Yb for the 1-2-3 test, and it appears that
such tests can be carried out with existing technologies, easily to sub-Hz
accuracies in the initial experiments with lattice clocks, and eventually down
to the full capacity of such clocks, i.e., another several orders of magnitude
better.

##  IV Outlook and Conclusions

The cases considered here are representative; other combinations of clock
transitions are possible. Interesting possibilities for 1-2-3 test exist with
molecules, for instance, the HD+ molecular ions Kortunov  _et al._ (2021),
transitions in highly charged relativistic ions Wojtsekhowski and Budker
(2021), and perhaps even with nuclear gamma transitions Budker  _et al._ .

The highest accuracy with neutral atoms to date has been reported with Yb and
Sr atoms trapped in optical lattices, and other promising clock transitions
are currently being investigated. Three clock transitions in an atom are the
minimum number for testing Weinberg’s hypothesis, though more transitions
could provide cross checks and higher sensitivity 111We note that four-
transition closures were investigated via collinear spectroscopy of Ca+ Müller
_et al._ (2020)..

Once the 1-2-3 tests start pushing the limits of sensitivity, one will need to
contend with subtle effects that determine the generally asymmetric lineshapes
observed in experiment, where the relation between the lineshape parameters
and the “clock readout” will need to be carefully understood. One such effect
is quantum interference Udem  _et al._ (2019), the treatment of which with
Lindblad formalism could perhaps be extended to include the Weinberg’s
extension of quantum mechanics.

In summary, we outline here realistic experimental tests of a possible
extension of quantum mechanics first proposed by Steven Weinberg in 2016. If a
violation is found, this would require a reformulation of quantum mechanics to
account for inevitable coupling to the environment.

## Acknowledgements

This paper is dedicated to the memory of our mentor and friend, Steven
Weinberg. We are grateful to Surjeet Rajendran, Nataniel L. Figueroa, Stephan
Schiller, Piet Schmidt, Thomas Udem and David Allcock for useful discussions.
The work of MGR was supported by the Sid W. Richardson Foundation. The work of
GG was supported by the MITRE Quantum Moonshot program. The work of DB was
supported by the Deutsche Forschungsgemeinschaft (DFG) - Project ID 423116110
and by the Cluster of Excellence Precision Physics, Fundamental Interactions,
and Structure of Matter (PRISMA+ EXC 2118/1) funded by the DFG within the
German Excellence Strategy (Project ID 39083149).

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