#  A note on the class number of certain real cyclotomic fields

Om Prakash  Kerala School of Mathematics, Kozhikode - 673571, Kerala, India.
<EMAIL_ADDRESS>
###### Abstract.

We construct an infinite family of real cyclotomic fields with non-trivial
class group. This result generalizes the result in [1] in the sense that our
family includes theirs.

###### Key words and phrases:

Diophantine equation, Continued fraction, Class number, Quadratic field, Real
cyclotomic field.

###### 2010 Mathematics Subject Classification:

Primary: 11D09, 11R29 Secondary: 11R11, 11R18

##  1\. Introduction

The ring of rational integers $\mathbb{Z}$ has unique factorization of
integers into primes. But $\mathcal{O}_{K}$, the ring of integers for any
number field $K$ does not necessarily have the property of unique
factorization. The extent of failure of unique factorization is measured by
the ideal class group which is a finite abelian group and its cardinality is
known as the class number of $K$. Although class number is the most
fundamental invariant of a number field, there is still a lot of mystery
around class numbers. For instance, a well known conjecture states that there
exist infinitely many number fields with class number $1$, or even with
bounded class number. The ‘class number one problem’ for real quadratic fields
is a ‘folklore conjecture’.

The class number of cyclotomic fields have been studied extensively since the
time mathematicians established linkage between Fermat’s last theorem and the
unique factorization properties of cyclotomic integers more than a century
ago. The class number $h_{n}$ of cyclotomic field $\mathbb{Q}({\zeta}_{n})$
can be written as the product

$h_{n}={h_{n}}^{+}{h_{n}}^{-}$

where ${h_{n}}^{+}$ denotes the class number of real cyclotomic field
$\mathbb{Q}({\zeta}_{n})^{+}$ ($:=\mathbb{Q}({\zeta}_{n}+{{\zeta}_{n}}^{-1})$)
of cyclotomic field $\mathbb{Q}({\zeta}_{n})$.

The class number of real cyclotomic field (also known as the ‘plus part’)
remains one of the most enigmatic object to study.

The ‘minus part’ ${h_{n}}^{-}$, which is also known as the relative class
number, is defined by the quotient

${h_{n}}^{-}=\frac{h_{n}}{{h_{n}}^{+}}.$

This is a tautology up to this point, but it is not tautological to note that
${h_{n}}^{-}$ can be calculated quite explicitly by using the class number
formula. On applying the class number formula to $\mathbb{Q}({\zeta}_{n})$ and
$\mathbb{Q}({\zeta}_{n})^{+}$ and than taking the quotient, leaves us with the
expression,

${h_{n}}^{-}=\mathcal{Q}m\prod_{\chi\hskip
8.5359pt\text{odd}}{(\frac{-1}{2}B_{1,\chi})}.$

Here the product runs over odd Dirichlet character $\chi$; $B_{1,\chi}$
denotes the generalised Bernoulli numbers and

$\mathcal{Q}=\begin{cases}1&{\rm~{}if~{}}n\hskip 2.84544pt\text{is a power of
prime}\\\ 2&{\rm~{}otherwise.}\end{cases}$

Despite the fact that ${h_{n}}^{-}$ grows exponentially with $n$;
${h_{n}}^{-}$ can be directly computed.

On the other hand the ‘plus part’ is still a mystery. Their Minkowski bounds
are too large to be effective for the fields with large $n$, and their
discriminant are also far to large for Odlyzko’s discriminant bound to treat
their class number.

Ankeny et al. [7] showed that, if $p=(2nq)^{2}+1$ is a prime where $q$ is also
a prime and $n>1$ is an integer then $h_{p}^{+}>1$. Lang [5] has also proved
the same result for $p=\\{(2n+1)q\\}^{2}+4$ where $q$ is a prime and $n\geq 1$
is an integer. In 1987, Osada [6] generalised [7] and [5].

In a recent work Mishra et al. [4] showed the existence of infinitely many
real cyclotomic fields with large class number. Their proof is existential. On
the other hand Chakraborty and Hoque [1] has given explicit families of real
cyclotomic fields having class number strictly bigger than $1$. More precisely
they proved the following:

######  Theorem 1.1.

(K.Chakraborty and A. Hoque)

  * 1.

Let $p\equiv 1\pmod{4}$ and $n\geq 1$ an integer. If $d=(2np)^{2}-1$, then
$h_{4d}^{+}>1$.

  * 2.

Let $p\equiv\pm 1\pmod{4}$ be a prime and $n\geq 1$ an integer. If
$d=(2np)^{2}+3$ and $n$ a multiple of $3$, then $h_{4d}^{+}>1$.

  * 3.

Let $p\equiv\pm 1\pmod{8}$ be a prime and $n\geq 1$ an integer. If
$d=((2n+1)p)^{2}+2$, then $h_{4d}^{+}>1$.

  * 4.

Let $p\equiv 1,3\pmod{8}$ be a prime not equal to 3 and $n\geq 1$ an integer.
If $d=((2n+1)p)^{2}-2$, then $h_{4d}^{+}>1$.

Their idea was to first prove that the equation $x^{2}-dy^{2}=\pm p$ has no
solution for $d$ of the form

$d=\begin{cases}(2np)^{2}-1\hskip 28.16838pt{\rm~{}with~{}}\hskip
5.69046ptp\equiv 1\pmod{4}\\\ (2np)^{2}+3\hskip
28.16838pt{\rm~{}with~{}}\hskip 5.69046ptp\equiv\pm 1\pmod{4}\\\
((2n+1)p)^{2}+2{\rm~{}with~{}}\hskip 5.69046ptp\equiv\pm 1\pmod{8}\\\
((2n+1)p)^{2}-2{\rm~{}with~{}}\hskip 5.69046ptp\equiv 1,3\pmod{8}\\\
\end{cases}$

where $p$ is a prime and $n$ is a positive integer. Then they lift the class
group of the quadratic field to the real cyclotomic field using the following
result of Yamaguchi [2].

######  Lemma 1.1.

Let $h$ be the class number of quadratic field $\mathbb{Q}(\sqrt{d})$, if
$d>0$ is an integer with $\phi(d)>4$, where $\phi$ is Euler’s totient
function, then $h|{h_{4d}}^{+}$.

In this article, we obtain an infinite family of $d$’s with non-trivial class
number for real cyclotomic fields (which contains [1]).The main result of this
note is:

######  Theorem 1.2.

If $d=\alpha n^{2}+\beta n+\gamma$ is a square-free positive integer, then
${h_{4d}}^{+}>1$.

Here $\alpha,\beta$ and $\gamma$ are as in Proposition 1.

##  2\. Preliminary

Let us recall that the continued fraction expression for $\sqrt{d}$ where $d$
is a square-free integer has the form

$\sqrt{d}=[a_{0},\overline{a_{1},a_{2},...,a_{t-1},a_{t}=2a_{0}}]$

where $a_{1},a_{2},...,a_{t-1}$ is a symmetric sequence.

Moreover, the converse of the above fact is also true. The following
proposition describes the converse.

######  Proposition 1.

[3] Let $a_{1},a_{2},...,a_{t-1}$ ($t\geq 1$) be any symmetric sequence of
positive integers. Define a sequence $q_{i}$ for $-1\leq i\leq t$ by the
recurrence relation $q_{i+1}=a_{i+1}q_{i}+q_{i-1}$, $q_{-1}=0$, $q_{0}=1$.
Then the equation

$\sqrt{d}=[z,\overline{a_{1},a_{2},...,a_{t-1},2z]},z=\lfloor d\rfloor$ (2.1)

has infinitely many square-free positive solutions $d\equiv 2,3\pmod{4}$ if
and only if $q_{t-1}$ is odd or $q_{t-1}$ and $q_{t-2}q_{t-3}$ are both even.
When either of the condition is satisfied then all the solutions of (2.1) are
given by

$d=d(n)=\alpha n^{2}+\beta n+\gamma;~{}~{}z(n)=\eta n+\mu,$

for integers $n\geq 1$. Here $\alpha,\beta,\gamma,\eta,\mu$ depend on
$a_{1},a_{2},...,a_{t-1}$ with $\alpha\neq 0$ is a square; $\eta\neq 0$ and
the discriminant is

${\beta}^{2}-4\alpha\gamma=(-1)^{t}~{}\text{or}~{}4(-1)^{t}~{}\text{or}\hskip
5.69046pt16(-1)^{t}.$

Moreover, if $d(n)$ is square-free and $d\equiv 2,3\pmod{4}$, the fundamental
unit of $\mathbb{Q}(\sqrt{d})$ has the form

$\epsilon=p(n)+q\sqrt{d}$ (2.2)

where $p$ is a linear polynomial, $q=q_{t-1}$ and both $p$ and $q$ depend on
the sequence $a_{1},a_{2},...,a_{t-1}.$

In our case, we consider $t$ to be even so that the norm of the fundamental
unit, $Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}(\epsilon)=1$. Also
$a_{1},a_{2},...,a_{s-1}$ be any symmetric sequence of positive integers such
that $q_{t-1}$ is odd. Note that such class of symmetric sequences will be
non-empty because if we take $t=2$ and consider symmetric sequence $a_{1}=1$,
in this case $q_{1}=1$ is odd. Hence by Proposition 1, the equation

$\sqrt{d}=[z,\overline{a_{1},a_{2},...,a_{s-1},2z]},z=\lfloor d\rfloor$

has infinitely many square-free positive integer solutions $d\equiv
3\pmod{4}$. For example, if we take the same sequence $a_{1}=1$ as above then
we can see that there are infinitely many $d$’s with $d\equiv 3\pmod{4}$. We
aren’t interested in $d\equiv 2\pmod{4}$ for reasons which would be clear when
we prove non-triviality of the class number. Therefore for us $d$ has the
form:

$d=\alpha n^{2}+\beta n+\gamma.$

If $d$ is square-free, the fundamental unit of $\mathbb{Q}(\sqrt{d})$ is of
the form:

$\epsilon=p(n)+q\sqrt{d}.$

##  3\. Solvability of a Diophantine equation

######  Lemma 3.1.

The Diophantine equation

$x^{2}-dy^{2}=\pm p$

has no integer solutions for $d=\alpha n^{2}+\beta n+\gamma$.

###### Proof.

First we consider

$x^{2}-dy^{2}=p.$ (3.1)

If possible let us assume that (3.1) has an integer solution $(x_{0},y_{0})$
and without loss of generality we assume that it is the smallest solution with
$y_{0}\geq 1$. Then,

$x_{0}^{2}-dy_{0}^{2}=p.$ (3.2)

We re-write it as

$Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}(x_{0}-y_{0}\sqrt{d})=p$ (3.3)

and multiply the resulting equation (3.3) by $\epsilon$ given by (2.2). Thus

$p=Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}((x_{0}-y_{0}\sqrt{d})\epsilon).$

Which is,

$\displaystyle p$ $\displaystyle=$ $\displaystyle
Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}((x_{0}-y_{0}\sqrt{d})(p(n)+q\sqrt{d})).$
$\displaystyle=$ $\displaystyle
Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}((p(n)x_{0}-dqy_{0})+(x_{0}q-p(n)y_{0})\sqrt{d}).$
$\displaystyle=$
$\displaystyle(p(n)x_{0}-dqy_{0})^{2}-(x_{0}q-p(n)y_{0})^{2}d.$

Now by the minimality of $y_{0}$, we have,

$y_{0}\leq|x_{0}q-p(n)y_{0}|.$

######  Case 1.

If $y_{0}\leq x_{0}q-p(n)y_{0}$.

Then,

$(1+p(n))y_{0}\leq x_{0}q.$

We use (3.2) and set $p(n)=x_{1}n+x_{2}$. This leads to

$\displaystyle\\{(1+x_{1}n+x_{2})^{2}-q^{2}d\\}y_{0}^{2}$ $\displaystyle\leq$
$\displaystyle q^{2}p.$ $\displaystyle\\{(1+x_{1}n+x_{2})^{2}-q^{2}(\alpha
n^{2}+\beta n+\gamma)\\}y_{0}^{2}$ $\displaystyle\leq$ $\displaystyle q^{2}p.$

Also

$(x_{1}^{2}-q^{2}\alpha)n^{2}+(2x_{1}+2x_{1}x_{2}-q^{2}\beta)n+(x_{2}^{2}+2x_{2}+1-q^{2}\gamma)\leq\frac{q^{2}p}{y_{0}^{2}}.$
(3.4)

Since $x_{1},x_{2},p,q,\alpha,\beta,\gamma,y_{0}$ are fixed integers,
therefore, (3.4) implies that there are finitely many $n^{\prime}s$, which
further implies that there are finitely many $d^{\prime}s$. This leads to a
contradiction.

######  Case 2.

If $y_{0}\leq p(n)y_{0}-x_{0}q$.

Then,

$x_{0}q\leq(p(n)-1)y_{0}.$

Again, using (3.2) and setting $p(n)=x_{1}n+x_{2}$, we get,

$q^{2}p\leq\\{(x_{1}n+x_{2}-1)^{2}-q^{2}d\\}y_{0}^{2}.$

This leads to

$q^{2}p\leq\\{(x_{1}n+x_{2}-1)^{2}-q^{2}(\alpha n^{2}+\beta
n+\gamma)\\}y_{0}^{2}$

and

$q^{2}(\alpha n^{2}+\beta
n+\gamma)-(x_{1}n+x_{2}-1)^{2}\leq\frac{-q^{2}p}{y_{0}^{2}}.$

Thus

$(q^{2}\alpha-x_{1}^{2})n^{2}+(q^{2}\beta-2x_{1}-2x_{1}x_{2})n+(q^{2}\gamma-
x_{2}^{2}-2x_{2}-1)\leq\frac{-q^{2}p}{y_{0}^{2}}.$ (3.5)

By the same argument as in case(1), we have a contradiction to the fact that
there are infinitely many $d^{\prime}s$.

Next, we consider

$x^{2}-dy^{2}=-p.$ (3.6)

Suppose that (3.6) has an integer solution. Without loss of generality assume
that $(x_{0},y_{0})$ is the smallest solution with $y_{0}\geq 1$.

Then,

$x_{0}^{2}-y_{0}^{2}=-p.$ (3.7)

We re-write

$Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}(x_{0}-y_{0}\sqrt{d})=-p.$ (3.8)

and multiply the resulting equation (3.8) by the fundamental unit $\epsilon$.

Thus,

$-p=Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}((x_{0}-y_{0}\sqrt{d})\epsilon)$

Which is,

$\displaystyle-p$ $\displaystyle=$ $\displaystyle
Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}((x_{0}-y_{0}\sqrt{d})(p(n)+q\sqrt{d})).$
$\displaystyle=$ $\displaystyle
Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}((p(n)x_{0}-dqy_{0})+(x_{0}q-p(n)y_{0})\sqrt{d}).$
$\displaystyle=$
$\displaystyle(p(n)x_{0}-dqy_{0})^{2}-(x_{0}q-p(n)y_{0})^{2}d.$

Now by the minimality of $y_{0}$, we have,

$y_{0}\leq|x_{0}q-p(n)y_{0}|.$

######  Case 3.

If $y_{0}\leq x_{0}q-p(n)y_{0}$. Then,

$(1+p(n))y_{0}\leq x_{0}q.$

We use (3.7) and set $p(n)=x_{1}n+x_{2}$. This leads to

$\displaystyle-q^{2}p$ $\displaystyle\geq$
$\displaystyle\\{(1+x_{1}n+x_{2})^{2}-q^{2}d\\}y_{0}^{2}$ $\displaystyle\geq$
$\displaystyle\\{(1+x_{1}n+x_{2})^{2}-q^{2}(\alpha n^{2}+\beta
n+\gamma)\\}y_{0}^{2}.$

This implies

$\frac{-q^{2}p}{y_{0}^{2}}\geq(x_{1}^{2}-q^{2}\alpha)n^{2}+(2x_{1}+2x_{1}x_{2}-q^{2}\beta)n+(x_{2}^{2}+2x_{2}+1-q^{2}\gamma),$

which contradicts the fact that there are infinitely many $d^{\prime}s$.

######  Case 4.

If $y_{0}\leq p(n)y_{0}-x_{0}q$.

Then,

$x_{0}q\leq(p(n)-1)y_{0}.$

We use (3.7) and set $p(n)=x_{1}n+x_{2}$. This leads to

$\displaystyle-q^{2}p$ $\displaystyle\leq$
$\displaystyle\\{(x_{1}n+x_{2}-1)^{2}-q^{2}d\\}y_{0}^{2}.$
$\displaystyle-q^{2}p$ $\displaystyle\leq$
$\displaystyle\\{(x_{1}n+x_{2}-1)^{2}-q^{2}(\alpha n^{2}+\beta
n+\gamma)\\}y_{0}^{2}.$ $\displaystyle\frac{q^{2}p}{y_{0}^{2}}$
$\displaystyle\geq$ $\displaystyle q^{2}(\alpha n^{2}+\beta
n+\gamma)-(x_{1}n+x_{2}-1)^{2}.$

This implies

$(q^{2}\alpha-x_{1}^{2})n^{2}+(q^{2}\beta-2x_{1}-2x_{1}x_{2})n+(q^{2}\gamma-
x_{2}^{2}-2x_{2}-1)\leq\frac{q^{2}p}{y_{0}^{2}},$

which contradicts the fact that there are infinitely many $d^{\prime}s$.

This completes the proof of Lemma 3.1 ∎

##  4\.  proof of the Theorem (1.2)

Let $p\equiv 1\pmod{4}$ be a prime. Since $d\equiv 3\pmod{4}$, the Legendre
symbol

$\left(\frac{d}{p}\right)=1.$

This implies, $p$ splits completely in $\mathbb{Q}(\sqrt{d})$, i.e.
$p=\mathfrak{P}\mathfrak{P}^{{}^{\prime}}$, where $\mathfrak{P}$ and
$\mathfrak{P}^{{}^{\prime}}$ are prime ideals. We also have
$Nm(\mathfrak{P})=Nm(\mathfrak{P}^{{}^{\prime}})=p$.

Suppose if possible the class number $h$ of $\mathbb{Q}(\sqrt{d})$ is $1$.
Then $\mathfrak{P}$ is a principal ideal in
$\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$, say $\mathfrak{P}=\langle
a+b\sqrt{d}\rangle$. Thus, we have

$p=Nm(\mathfrak{P})=Nm_{{\mathbb{Q}(\sqrt{d})}/\mathbb{Q}}(a+b\sqrt{d})=|a^{2}-db^{2}|.$

This implies

$a^{2}-db^{2}=\pm p,$ (4.1)

which contradicts the lemma(3.1). Hence $h\neq 1$. Except for finitely many
$d^{\prime}s$, we have $\phi(d)>4$, therefore $h|{h_{4d}}^{+}$ (by Lemma 1.1).
Since $h>1$, we have ${h_{4d}}^{+}>1$.

##  5\. Concluding Remarks

As mentioned earlier, if we consider the sequence $a_{1}=1$ for $t=2$, then in
this case $d=(z+1)^{2}-1$. Now for $d$ to be square-free $z$ has to odd. Hence
$d$ has to be of the form $(2np)^{2}-1$, which is precisely the family
considered in [1]. Thus here we get a more generalized and larger family.

## Acknowledgement

The author would like to thank Dr. Mohit Mishra for going through the article
and his continuous support in the completion of this article. The scenic
ambience of KSoM and support of the colleagues played an important role in
completing the work.

## References

  * [1] K.Chakraborty and A. Hoque, Pell-type equations and class number of the maximal real subfield of a cyclotomic field Ramanujan J., 46 (2018), 727–742.
  * [2] I. Yamaguchi, On the class number of real subfield of cyclotomic field, J. Reine Angrew. Math., 272 , (1975), 217–220.
  * [3] A. Dahl and V. Kala, Distribution of class numbers in continued fraction families of real quadratic fields Proc. Edinb. Math. Soc. (2), 61 (2018), 1193–-1212.
  * [4] M. Mishra, R. Schoof and L. Washington, Class groups of real cyclotomic fields, Monatsh. Math., 195 (2021), 489–-496.
  * [5] S. D. Lang, Note on the class number of the maximal real subfield of a cyclotomic field, J. Reine Angrew. Math., 290 (1977), 70–72.
  * [6] H. Osada, Note on class number of maximal real subfield of cyclotomic field, Manuscripta Math., 58 (1987), 215–227.
  * [7] N. C. Ankeny, S. Chowla and H. Hasse, On the class number of the maximal real subfield of a cyclotomic field, J. Reine Angrew. Math., 217 (1965), 217–220.


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: Current address: Department
of Chemistry, Dalhousie University, Halifax, NS B3J 4H9, Canada

# Direct observation of geometric-phase interference in dynamics around a
conical intersection

C. H. Valahu  School of Physics, University of Sydney, NSW 2006, Australia
ARC Centre of Excellence for Engineered Quantum Systems, University of Sydney,
NSW 2006, Australia  V. C. Olaya-Agudelo  ARC Centre of Excellence for
Engineered Quantum Systems, University of Sydney, NSW 2006, Australia  School
of Chemistry, University of Sydney, NSW 2006, Australia  R. J. MacDonell  ARC
Centre of Excellence for Engineered Quantum Systems, University of Sydney, NSW
2006, Australia  School of Chemistry, University of Sydney, NSW 2006,
Australia  University of Sydney Nano Institute, University of Sydney, NSW
2006, Australia  T. Navickas  School of Physics, University of Sydney, NSW
2006, Australia  ARC Centre of Excellence for Engineered Quantum Systems,
University of Sydney, NSW 2006, Australia  A. D. Rao  School of Physics,
University of Sydney, NSW 2006, Australia  ARC Centre of Excellence for
Engineered Quantum Systems, University of Sydney, NSW 2006, Australia  M. J.
Millican  School of Physics, University of Sydney, NSW 2006, Australia  ARC
Centre of Excellence for Engineered Quantum Systems, University of Sydney, NSW
2006, Australia  J. B. Pérez-Sánchez  Department of Chemistry and
Biochemistry, University of California San Diego, La Jolla CA, 92093, USA  J.
Yuen-Zhou  Department of Chemistry and Biochemistry, University of California
San Diego, La Jolla CA, 92093, USA  M. J. Biercuk  School of Physics,
University of Sydney, NSW 2006, Australia  ARC Centre of Excellence for
Engineered Quantum Systems, University of Sydney, NSW 2006, Australia  C.
Hempel  School of Physics, University of Sydney, NSW 2006, Australia  ARC
Centre of Excellence for Engineered Quantum Systems, University of Sydney, NSW
2006, Australia  Quantum Center, ETH Zurich, CH-8093 Zurich, Switzerland  ETH
Zurich-PSI Quantum Computing Hub, Paul Scherrer Institut, 5232 Villigen,
Switzerland  T. R. Tan<EMAIL_ADDRESS>School of Physics,
University of Sydney, NSW 2006, Australia  ARC Centre of Excellence for
Engineered Quantum Systems, University of Sydney, NSW 2006, Australia  I.
Kassal<EMAIL_ADDRESS>ARC Centre of Excellence for Engineered
Quantum Systems, University of Sydney, NSW 2006, Australia  School of
Chemistry, University of Sydney, NSW 2006, Australia  University of Sydney
Nano Institute, University of Sydney, NSW 2006, Australia

###### Abstract

Conical intersections are ubiquitous in chemistry and physics, often governing
processes such as light harvesting, vision, photocatalysis, and chemical
reactivity. They act as funnels between electronic states of molecules,
allowing rapid and efficient relaxation during chemical dynamics. In addition,
when a reaction path encircles a conical intersection, the molecular
wavefunction experiences a geometric phase, which can affect the outcome of
the reaction through quantum-mechanical interference. Past experiments have
measured indirect signatures of geometric phases in scattering patterns and
spectroscopic observables, but there has been no direct observation of the
underlying wavepacket interference. Here, we experimentally observe geometric-
phase interference in the dynamics of a wavepacket travelling around an
engineered conical intersection in a programmable trapped-ion quantum
simulator. To achieve this, we develop a technique to reconstruct the two-
dimensional wavepacket densities of a trapped ion. Experiments agree with the
theoretical model, demonstrating the ability of analog quantum simulators—such
as those realised using trapped ions—to accurately describe nuclear quantum
effects.

Light drives molecular processes as important as photosynthesis,
photocatalysis, and vision. Absorbing a photon promotes a molecule to an
excited electronic state, triggering chemical dynamics and reactivity. The
molecule will eventually return to the ground state; often, this relaxation
happens on ultrafast (fs–ps) timescales at molecular geometries where two
electronic energy surfaces have the same energy, known as conical
intersections [1, 2, 3]. By acting as funnels between electronic states for
the molecular wavefunction, conical intersections enable rapid non-radiative
electronic transitions and have a decisive role in chemical dynamics, from
charge-transfer processes to photochemical reactions [4].

The path taken during molecular dynamics involving conical intersections can
profoundly alter chemical reaction outcomes. In particular, a geometric phase
[5] causes quantum interference of wavepackets encircling a conical
intersection [6, 7, 8, 9]. Accounting for geometric phase is necessary in
quantum chemistry calculations because the resulting interference changes the
ratio of reactive and non-reactive outcomes in scattering cross-sections [10,
11, 12, 13] and alters vibrational spectra [14, 15, 16]. Indeed, recent
experiments have detected indirect signatures of geometric phase in reactive
scattering [17, 18]. An elegant proposal for revealing spectroscopic
signatures of geometric phase involves interference signals from pairs of
excitation pulses [19, 20, 21], but it remains unimplemented due to
challenging state preparation.

Conical intersections and the associated geometric phase are general phenomena
that also appear in other branches of physics [3]. In general, a conical
intersection can form in any parameter-dependent quantum system where two
energy surfaces cross. In molecules, the parameters are usually the normal
modes of nuclear motion, but, in condensed-phase systems, conical
intersections commonly arise as Dirac cones in reciprocal (momentum) space
[3]. These include the Dirac cones in graphene [22], in superconductors [23],
and in the Rashba [24] and Dresselhaus [25] treatments of spin-orbit coupling.

An unambiguous observation of geometric-phase interference in wavepacket
dynamics around a conical intersection remains an outstanding challenge. In a
molecular or solid-state system, it would require a full reconstruction of the
wavepacket dynamics on ultrafast timescales, which is possible in small
molecules [26], but has never been used to characterise geometric phase.

Analog quantum simulators present a new opportunity to access quantum dynamics
on laboratory-accessible timescales [27, 28, 29, 30, 31]. In such systems, a
one-to-one correspondence between the degrees of freedom of the chemical or
physical system and those of the simulator makes it possible to replicate the
target dynamics in a controllable and measurable manner, as well as explore
new parameter regimes in a controllable fashion.

Several controllable quantum systems have been proposed to engineer conical
intersections and study signatures of geometric phase. Most of these quantum
simulations have been performed in reciprocal space to simulate solid-state
systems, including geometric phases around Dirac points [32, 33, 3].
Theoretical proposals for simulating molecular conical intersections have
included using trapped Rydberg ions to simulate electronic populations [34],
circuit quantum electrodynamics to simulate emission spectra [35], and cavity
quantum electrodynamics to simulate collapse-revival characteristics of a
spreading wavepacket [36]. To date, the only experimental quantum simulation
of a chemical conical intersection demonstrated branching between different
photochemical reaction products with strong dissipation [37].

Figure 1: Directly detecting a geometric phase through wavepacket
interference. a, A motional wavepacket is initially displaced to the minimum
of the potential energy surface, after which it begins to encircle the conical
intersection, denoted CI. b, Initial wavepacket density in 2D (left), and
integrated over $Q_{1}$ (right). c, After sufficient time evolution, the two
components of the wavepacket destructively interfere due to geometric phase,
giving a nodal line along $Q_{2}=0$ (dotted line). d, Motional wavepacket
density at the maximum interference time $T$. e, If the geometric phase were
neglected, the two wavepacket components would interfere constructively. f,
Density at $t=T$ with geometric phase neglected. Contours in b, d, and f
correspond to the potential energy surface $E_{-}$. g, The Jahn-Teller
Hamiltonian $H^{\mathrm{JT}}$ is engineered in an ion-trap quantum simulator
with a single ^171Yb+ ion. The ion (white sphere) is confined in a Paul trap
and $H^{\mathrm{JT}}$ is realised using two simultaneous laser-induced
interactions (purple and pink, corresponding to colour-coded terms in
$H^{\mathrm{JT}}$).

Here, we present the observation of the destructive interference caused by
geometric phase during dynamics of a wavepacket around a conical intersection.
We implement a controllable conical intersection by engineering a Jahn-Teller
Hamiltonian in a trapped-ion quantum simulator that employs a mixed-qudit-
boson (MQB) encoding in which both the ion’s electronic and motional degrees
of freedom are used [38]. This work is not merely a simulation of geometric
phase: the ion is a real, observable, and measurable quantum system that
undergoes conical-intersection dynamics, allowing us to directly observe the
geometric-phase interference of its motional wavepacket. To this end, our
experiment introduces a resource-efficient reconstruction method to image the
wavepacket’s probability density, directly showing the destructive self-
interference as the wavepacket encircles a conical intersection. Experimental
measurements match theoretical predictions, demonstrating the utility of
quantum simulators to give insights into properties that have otherwise been
impossible to measure directly for chemical systems.

## Results

In an MQB simulator [38], the electronic and vibrational degrees of freedom
that are to be simulated are represented in a qudit and a set of bosonic
modes. We realise a conical intersection using an ^171Yb+ ion confined in a
Paul trap, where two vibrations are encoded directly in the ion’s transverse
vibrational modes (B1 and B2), while two electronic states are encoded in the
ion’s qubit (qudit with $d=2$) states comprising the two hyperfine levels of
the ^2S_1/2 ground state (detailed in Methods). This approach has recently
been employed to predict molecular spectra using time-domain simulations [39],
and provides resource-scaling advantages relative to conventional methods of
quantum simulation [38].

Figure 2: Experimental protocol for geometric-phase dynamics simulation and
wavepacket reconstruction. (top) Quantum circuit diagram for a single-trapped-
ion simulator, consisting of a qubit (Q, solid line) and 2 bosonic modes (B1,
B2, wavy lines), and (bottom) corresponding experimental pulse sequence. (i)
Preparation of fiducial states by optical pumping and cooling. (ii)
Initialisation: B1 is displaced by $D_{1}(-\kappa/\sqrt{2}\omega)$,
implemented using an SDF pulse with surrounding qubit $\pi/2$ pulses to map to
the correct basis. (iii) Time evolution under the Jahn-Teller Hamiltonian for
duration $t$, implemented using two simultaneous SDF pulses in different
bases, acting on the two bosonic modes. (iv) Reconstruction measurement: a
state detection (SD) pulse collapses the qubit state, and the circuit proceeds
only if the measured state was $\ket{\downarrow}$ (rounded shape). An
additional single-qubit pulse (dashed) is introduced to retrieve information
entangled with the $\ket{\uparrow}$ state. The qubit probabilities
$p_{\downarrow}$ and $p_{\uparrow}$ are calculated from the mid-circuit
measurement outcomes. Controlled displacements acting in the $\sigma_{x}$
basis (denoted ) map the motional degrees of freedom onto the qubit, allowing
the real part of the characteristic function to be measured. Its imaginary
part is obtained using an additional $R_{x}(\pi/2)$ pulse prior to the
controlled displacements (dotted). Successive $R_{x}$ rotations shown
separately in the circuit are combined into a single pulse in the experiment.

To demonstrate geometric-phase interference, we implement the $E\otimes e$
Jahn-Teller model [6], a standard model of geometric-phase effects in
molecules [40, 8]. It consists of two electronic states coupled with two
vibrational modes, described by the potential energy

$V^{\mathrm{JT}}=\frac{\omega}{2}(Q_{1}^{2}+Q_{2}^{2})+\kappa(\sigma_{z}Q_{1}+\sigma_{x}Q_{2}),$
(1)

where $\sigma_{x}$ and $\sigma_{z}$ are the Pauli matrices acting on the
electronic states and $Q_{j}=(a_{j}^{{\dagger}}+a_{j})/\sqrt{2}$ is the
dimensionless position coordinate for the $j$th vibrational mode, with
creation and annihilation operators $a_{j}^{{\dagger}}$ and $a_{j}$. $\kappa$
is the vibronic coupling strength, and $\omega$ is the frequency of both
vibrational modes. The Jahn-Teller Hamiltonian is given by
$H^{\mathrm{JT}}=\omega(P_{1}^{2}+P_{2}^{2})/2+V^{JT}$, where $P_{j}$ is the
conjugate momentum of $Q_{j}$. We set $\hbar=1$ throughout.

Diagonalisation of $V^{\mathrm{JT}}$ in the electronic basis leads to
cylindrically symmetric potential energy surfaces along $Q_{1}$ and $Q_{2}$,
with energies
$E_{\pm}=\omega(Q_{1}^{2}+Q_{2}^{2})/2\pm\kappa\sqrt{Q_{1}^{2}+Q_{2}^{2}}$
(see fig. 1). The conical intersection is present at the point of highest
symmetry ($Q_{1}=Q_{2}=0$), where the two potential energy surfaces are
degenerate. The minimum of $E_{-}$ occurs where
$Q_{1}^{2}+Q_{2}^{2}=(\kappa/\omega)^{2}$.

The effects of geometric phase on dynamics around a conical intersection can
be directly observed from the motional probability density, fig. 1a–d. As the
initial wavepacket, we choose the ground state of the non-interacting
vibrational Hamiltonian,
$H_{0}=\omega(a^{\dagger}_{1}a_{1}+a^{\dagger}_{2}a_{2})$, displaced to the
potential-energy minimum at $Q_{1}=-\kappa/\omega$, $Q_{2}=0$ (fig. 1a–b).
During the time evolution, the wavepacket splits into two components evolving
in opposite directions around the conical intersection. The two components
overlap at $Q_{1}>0$, causing destructive interference at the nodal line
$Q_{2}=0$, where their equal and opposite geometric phases lead to a vanishing
density (fig. 1c–d). By contrast, if geometric phase were disregarded, the two
wavepacket fragments would interfere constructively, reaching maximum
amplitude at $Q_{2}=0$ (fig. 1e–f).

To map the Jahn-Teller model onto the MQB simulator, we rewrite
$H^{\textrm{JT}}$ in the interaction picture with respect to $H_{0}$,

$\displaystyle H_{\mathrm{I}}^{\mathrm{JT}}={}$
$\displaystyle\frac{\kappa}{\sqrt{2}}\sigma_{z}(a_{1}^{{\dagger}}e^{i\omega
t}+a_{1}e^{-i\omega t})+$
$\displaystyle\frac{\kappa}{\sqrt{2}}\sigma_{x}(a_{2}^{{\dagger}}e^{i\omega
t}+a_{2}e^{-i\omega t}),$ (2)

which can be implemented using tunable light-atom interactions to enact qubit-
boson couplings. We achieve this implementation using a coherent state-
dependent force (SDF) enacted by stimulated Raman transitions driven with a
$355\text{\,}\mathrm{n}\mathrm{m}$ pulsed laser [41, 42]. Driving transitions
near bosonic mode $j$ leads to the Hamiltonian

$H^{\mathrm{SDF}}_{j,\phi_{s}}(\delta,\Omega,\phi_{m})=\frac{\Omega}{2}\sigma_{\phi_{s}}(a_{j}^{\dagger}e^{i(\phi_{m}+\delta
t)}+a_{j}e^{-i(\phi_{m}+\delta t)}),$ (3)

where $\sigma_{\phi_{s}}=\sigma_{x}\cos\phi_{s}+\sigma_{y}\sin\phi_{s}$ and
$\phi_{s}$ and $\phi_{m}$ are the phases associated with the qubit and the
bosonic mode, respectively (see Methods). $\Omega$ and $\delta$ are the Rabi
frequency and detuning of the laser from the bosonic mode, respectively. We
use the notation $H^{\mathrm{SDF}}_{j,x}$ and $H^{\mathrm{SDF}}_{j,y}$ for SDF
interactions where $\phi_{s}=0$ and $\phi_{s}=\pi/2$, respectively.
Interactions in the $\sigma_{z}$ basis are obtained using a qubit basis
rotation,
$H^{\mathrm{SDF}}_{j,z}=R_{x}(\pi/2)H^{\mathrm{SDF}}_{j,y}R_{x}(-\pi/2)$,
where $R_{x}(\theta)$ are driven qubit rotations around the Bloch sphere.
$H^{\mathrm{JT}}$ can then be implemented in a programmable way using two
simultaneous SDFs (see fig. 1g),

$H^{\mathrm{JT}}_{\mathrm{I}}=H^{\mathrm{SDF}}_{1,z}(\omega,\sqrt{2}\kappa,0)+H^{\mathrm{SDF}}_{2,x}(\omega,\sqrt{2}\kappa,0).$
(4)

The parameters $\kappa$ and $\omega$ are chosen to produce a clear wavepacket
interference. To achieve this, $\kappa/\omega$ should be large enough that the
wavepacket prepared at the minimum of the potential energy surface (at
$Q_{1}=-\kappa/\omega$) has negligible overlap with the conical intersection.
However, $\kappa/\omega$ should also be kept small enough to mitigate
vibrational decoherence that increases with larger vibrational excitations. To
balance these considerations, we choose $\kappa/\omega=1.5$, for which the
wavepacket has only 1.7% of the density at $Q_{1}\geq 0$. $\kappa$,
implemented by adjusting the Rabi frequency, is maximised to increase the
speed of the dynamics; its value is constrained by the available SDF-laser
power to $\kappa=2\pi\times$1.00\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
yielding $\omega=2\pi\times$667\text{\,}\mathrm{H}\mathrm{z}$$. With these
parameters, the wavepackets are expected to experience the greatest geometric-
phase interference at $T=$1.59\text{\,}\mathrm{m}\mathrm{s}$$, which was
computationally predicted as half the time at which the width of the
probability density is minimized.

Figure 3: Wavepacket dynamics around an engineered conical intersection. a-d
Reconstructed two-dimensional motional densities of the ^171Yb+ ion at times
$t=0$, $0.9T$, $T$, and $2T$, with theoretical predictions in the insets. The
nodal line at $Q_{2}=0$, $Q_{1}>0$ is a direct signature of geometric-phase
interference. e One-dimensional motional densities (with the $Q_{1}$
coordinate integrated out) at more values of $t$. Geometric-phase interference
causes the dip at $Q_{2}=0$ around $t=T$. The motional densities are obtained
using eq. 5 (and its one-dimensional equivalent) from the characteristic
functions, $\chi(i\beta_{1},i\beta_{2})$ and $\chi(i\beta_{2})$, measured
using the circuit in fig. 2 (see Methods for details).

We probe the dynamics of the geometric phase around the conical intersection
by reconstructing the ion’s motional probability densities at different
evolution times $t$. The experimental sequence consists of four stages, shown
in fig. 2. (i) Preparation of the qubit and cooling of the vibrational modes
to their ground states is achieved by optical pumping, Doppler cooling, and
sideband cooling. (ii) Initialisation consists of displacing B1 to
$Q_{1}=-\kappa/\omega$ by applying an SDF interaction
$H^{\mathrm{SDF}}_{1,z}(0,\Omega_{1},\pi/2)$ for a duration $\tau$. This
applies the displacement operator $D_{1}(-\Omega_{1}\tau/2)$, where
$\Omega_{1}$ and $\tau$ are chosen to implement
$D_{1}(-\kappa/\sqrt{2}\omega)$. (iii) Evolution of the system under
$H^{\mathrm{JT}}_{I}$ is achieved by applying the two simultaneous SDF
interactions of eq. 4 for an experimentally variable duration $t$. (iv)
Reconstruction of the joint densities of B1 and B2 is achieved by measuring
the characteristic function

$\chi(i\beta_{1},i\beta_{2})=\bra{\Psi}D_{1}(i\beta_{1})D_{2}(i\beta_{2})\ket{\Psi},$
(5)

where $\ket{\Psi}$ is the total wavefunction of the system, and $\beta_{1}$
and $\beta_{2}$ are real numbers. See Methods for details.

The joint probability densities are reconstructed using the circuit in fig. 2.
Two SDF pulses are sequentially applied on B1 and B2, and
$\chi(i\beta_{1},i\beta_{2})$ is scanned over $\beta_{1}$, $\beta_{2}$. These
measurements yield the joint probability density via the Fourier transform of
the measured characteristic function

$|\Psi(Q_{1},Q_{2})|^{2}=\\!\iint\frac{d\beta_{1}d\beta_{2}}{2\pi^{2}}e^{-i\sqrt{2}(Q_{1}\beta_{1}+Q_{2}\beta_{2})}\chi(i\beta_{1},i\beta_{2}).$
(6)

In further detail, we measure $\chi(i\beta_{1},i\beta_{2})$ by mapping
information from the multimode bosonic system onto the qubit using SDF pulses,
moving beyond previous works on direct single-mode [43, 44, 45, 46] and
indirect multimode reconstructions [47]. The reconstruction consists of
preparing the qubit in $\ket{\downarrow}$ and applying two successive SDF
interactions, $H^{\mathrm{SDF}}_{1,x}(0,\Omega_{2},0)$ and
$H^{\mathrm{SDF}}_{2,x}(0,\Omega_{2},0)$ with durations $\tau_{1}$ and
$\tau_{2}$. Doing so results in controlled displacements $D_{1}(i\beta_{1}/2)$
and $D_{2}(i\beta_{2}/2)$, where $\beta_{j}=\Omega_{2}\tau_{j}$.
$\chi(i\beta_{1},i\beta_{2})$ is measured for different values of $\beta_{1}$
and $\beta_{2}$ by varying $\tau_{1}$ and $\tau_{2}$. We reconstruct the
characteristic functions of the bosonic modes entangled with the
$\ket{\downarrow}$ and $\ket{\uparrow}$ qubit states independently.
Reconstructing the $\ket{\downarrow}$ component is achieved by adding a mid-
circuit measurement which projects out the $\ket{\uparrow}$ component (see
Methods). The experiment is repeated with an additional $R_{x}(\pi)$ pulse
prior to the mid-circuit measurement to reconstruct the $\ket{\uparrow}$
component. The qubit probabilities $p_{\downarrow}$ and $p_{\uparrow}$ are
calculated from the success rate of the mid-circuit measurement. After the
displacements, measuring the qubit in the $\sigma_{z}$ basis gives the real
part of the characteristic function,
$\langle\sigma_{z}\rangle=\operatorname{Re}\chi(i\beta_{1},i\beta_{2})$.
Repeating the experiment with an additional $R_{x}(\pi/2)$ pulse prior to the
displacements gives the imaginary part, after which the full
$\chi(i\beta_{1},i\beta_{2})$ is obtained by adding the real and imaginary
parts associated with both $\ket{\downarrow}$ and $\ket{\uparrow}$.

The reconstructed probability densities in fig. 3a–d demonstrate a direct
measurement of the wavepacket interference caused by the geometric phase. At
$t=0$, the initial wavepacket is prepared at $(Q_{1},Q_{2})=(-1.5,0)$. As the
wavepacket evolves around the conical intersection, the nodal line becomes
visible at $Q_{2}=0$ and is most pronounced at $t=T$; this is a direct
observation of destructive interference due to geometric phase. Finally, at
$t=2T$, the two wavepackets recombine close to their initial position. The
experimental results agree well with theoretical predictions, reproducing key
features of interference and wavepacket recombination.

Further quantitative insight may be gained from the 1-dimensional density
$|\Psi(Q_{2})|^{2}$, obtained by omitting the $D_{1}(i\beta_{1})$
displacements from the reconstruction procedure discussed earlier. In this
case, the measurements scanned over $\beta_{2}$ are Fourier-transformed to
give $|\Psi(Q_{2})|^{2}=\int
d\beta_{2}\,e^{-i\sqrt{2}Q_{2}\beta_{2}}\chi(i\beta_{2})/\sqrt{2}\pi$. In fig.
3e, we present $|\Psi(Q_{2})|^{2}$ for seven different evolution times. A
comparison of experiment and theory shows excellent agreement in the shape and
amplitude of the measured density function. We attribute minor discrepancies
to the dephasing of the bosonic modes, miscalibrations such as uncompensated
AC Stark shifts, and technical imperfections in the protocol implementation.

## Discussion

Our approach avoids the limitations of direct experiments on molecular
systems, where only few observables—such as spectra and scattering cross
sections—can be measured. Instead, a fully controllable quantum device—such as
an ion-trap MQB simulator [38]—can, in principle, read out any observable; as
we showed here, this includes the full two-dimensional density of the ^171Yb+
ion as it moves in space and time. A further advantage comes from the ratio
($r$) of the ion’s natural timescale (ms) and the measurement speed (ns),
leading to an increase in the observable timing resolution of $r\sim 10^{6}$.
This improves the achievable resolution of chemical-dynamics measurements
relative to ultrafast observations.

A key general feature of quantum simulations is their programmability [48].
Our work is a simulation of the dynamics of the Jahn-Teller model, which is
often used to describe molecular systems. In an MQB simulator, the qudit-boson
interaction is controllable, meaning that the same device can be programmed to
simulate different molecular systems, solid-state systems, or theoretical
models that do not occur naturally. In particular, our geometric-phase
simulator could be used to simulate dynamics in molecules with conical
intersections where the interactions are not as symmetric as in
$H^{\mathrm{JT}}$, such as the general quadratic vibronic-coupling Hamiltonian
[38].

Like any analog simulation—quantum or classical—our approach is ultimately
limited by noise and uncorrected errors. In our experiment, the main sources
of decoherence and dissipation are motional dephasing and motional heating
[49, 39] (see Methods). However, in MQB simulations of molecular processes,
noise can be characterised and even amplified in order to create a realistic
model of molecular environments, such as collisions in solution. Since our
Jahn-Teller experiment shows only weak effects of decoherence over the full
period $2T$, we would need to inject additional noise to simulate conical-
intersection dynamics of real molecules in chemically realistic situations
(i.e., other than a single molecule in vacuum). In scaling up to larger
molecules, the ability to simulate dissipation would allow us to probe regimes
in nonadiabatic dynamics that are among the most difficult to simulate on
conventional computers [38].

Our methodology for probability density reconstruction enables scalability and
resource efficiency. Early techniques for motional-state tomography were
performed in the Fock basis [43, 50, 47], a process that requires many
measurements if full motional densities are sought. More recently, wavepacket-
reconstruction methods were developed based on the direct measurement of the
characteristic function, significantly reducing the number of necessary
measurements [46]. Our approach builds on the latter techniques, but has two
additional advantages. First, we extend the characteristic-function method to
multimode probability density reconstruction, while retaining both the
requirement of few measurements and the ability to use one readout qubit.
Second, using a mid-circuit measurement allows us to reuse the simulation
qubit for the reconstruction, without any ancilla qubits.

We have recently become aware of related simultaneous work on simulating a
conical intersection using a chain of trapped ions [51]. The system was
adiabatically driven to its vibronic ground state, whose reconstructed two-
dimensional density showed a node attributed to geometric phase. This work is
complementary to ours in several ways: it focused on signatures of geometric
phase in the ground state, not in the dynamics; it used Trotterised time
evolution, while we drove the two interactions simultaneously; and it used an
ancilla qubit in the reconstruction, while we used the mid-circuit-measurement
approach discussed earlier.

In conclusion, our experiment represents the direct observation of wavepacket
interference caused by geometric phase in dynamics around a conical
intersection. Our approach to quantum simulation using an MQB trapped-ion
system makes chemical dynamics that are otherwise unmeasurable directly
accessible in the laboratory. This is a key demonstration of the utility of
small-scale quantum computational devices to offer practical insights into
chemical dynamics and resolve intractable problems in chemical physics.

###### Acknowledgements.

We thank Jacob Whitlow and Kenneth Brown for valuable discussions. We were
supported by the U.S. Office of Naval Research Global (N62909-20-1-2047), by
the U.S. Army Research Office Laboratory for Physical Sciences
(W911NF-21-1-0003), by the U.S. Intelligence Advanced Research Projects
Activity (W911NF-16-1-0070), by Lockheed Martin, by the Australian
Government’s Defence Science and Technology Group, by the Sydney Quantum
Academy (VCO, ADR, MJM, and TRT), by a University of Sydney-University of
California San Diego Partnership Collaboration Award (JBPS, JYZ, and IK), by
H. and A. Harley, and by computational resources from the Australian
Government’s National Computational Infrastructure (Gadi) through the National
Computational Merit Allocation Scheme.

## Author Contributions Statement

RJM, IK, CH and TRT conceived the original idea; VCO, RJM, JBP, JY, and IK
developed the theoretical methods; CHV, TN, ADR, MJM, and TRT developed and
carried out the experiments; CHV, VCO, TRT, and IK wrote the manuscript with
feedback from all authors; all authors discussed the results and interpreted
the data.

## Competing Interests Statement

The authors declare no competing interests.

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## Methods

### Experimental setup

The ^171Yb^+ ion is confined in a Paul trap with radial mode oscillation
frequencies of $2\pi\times$1.34\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ and
$2\pi\times$1.47\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$, corresponding to
bosonic mode B1 and B2. The qubit is encoded in the two magnetically
insensitive hyperfine levels of the ^2S_1/2 ground state, where we assign the
labels $\ket{\downarrow}=\ket{F=0,m_{F}=0}$ and
$\ket{\uparrow}=\ket{F=1,m_{F}=0}$.

We use two laser beams derived from a $355\text{\,}\mathrm{n}\mathrm{m}$
pulsed laser to coherently control the qubit and bosonic modes via stimulated
Raman transitions within the ^2S_1/2 ground state. The two Raman beams are
orthogonal to one another, and configured so that they can be coupled to both
radial vibrational modes. Each Raman beam passes through an acousto-optical
modulator (AOM), which allows the phase, frequency and amplitude of the beam
to be adjusted by altering the RF signal driving the AOM. One of the RF
signals is generated by an arbitrary waveform generator (Keysight M8190A),
allowing multiple phase-coherent tones to be imprinted on one of the laser
beams. We ensure phase coherence between all pulses in the experimental
sequence by tracking the phase (relative to the beginning of the pulse
sequence) and applying appropriate corrections.

By tuning the frequency difference of the Raman beams, one can drive carrier,
red- and blue-sideband transitions. Qubit rotations are obtained by driving
carrier transitions, while an SDF
$H^{\mathrm{SDF}}_{j,\phi_{s}}(0,\Omega,\phi_{m})$ arises from combining the
red- and blue-sideband transitions. Applying this interaction for a duration
$\tau$ with the qubit in an eigenstate of $\sigma_{\phi_{s}}$ displaces
bosonic mode $j$ by $D_{j}(\alpha)=\exp(\alpha
a^{\dagger}_{j}-\alpha^{*}a_{j})$, where $\alpha=-i\Omega\tau
e^{i\phi_{m}}/2$. The amplitude and phase-space direction of the displacement
are adjusted by varying $\tau$ and $\phi_{m}$, respectively.

### Experimental protocol

Preparation. The bosonic modes are cooled in two stages. First, they are
Doppler cooled using a $369.5\text{\,}\mathrm{n}\mathrm{m}$ laser red-detuned
from the ^2S_1/2 $\rightarrow$ ^2P_1/2 transition. Second, resolved sideband
cooling is used to reach their motional ground states, achieving temperatures
of $\bar{n}=0.04$ measured via sideband thermometry [1]. The qubit is prepared
in its ground state $\ket{\downarrow}$ via optical pumping, using another
$369.5\text{\,}\mathrm{n}\mathrm{m}$ laser resonant with the ^2S_1/2
$\ket{F=1}$ $\rightarrow$ ^2P_1/2 $\ket{F=1}$ transition.

Initialisation. To initialise B1, we apply an SDF interaction
$H^{\mathrm{SDF}}_{1,y}(0,\Omega_{1},\pi/2)$ for a duration $\tau$, which
gives a displacement $D_{1}(\alpha)$ where $\alpha=\Omega_{1}\tau/2$. Setting
$\tau=\sqrt{2}\kappa/\omega\Omega_{1}$ so that $\alpha=\kappa/\sqrt{2}\omega$
displaces the mode from $Q_{1}=0$ to $Q_{1}=-\kappa/\omega$ because
$D_{1}(\alpha)Q_{1}D_{1}(\alpha)^{\dagger}=Q_{1}-\sqrt{2}\operatorname{Re}\alpha=Q_{1}-\kappa/\omega$.
The qubit is first mapped into the SDF interaction basis ($\ket{+}_{y}$) with
an $R_{x}(\pi/2)$ rotation, and is returned to $\ket{\downarrow}$ after the
displacement with an $R_{x}(-\pi/2)$ rotation. The Rabi frequency of the SDF
interaction was frequently recalibrated and on average we measured
$\Omega_{1}=2\pi\times$2.23\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

Time evolution. Two SDF interactions on B1 and B2 are applied during the time
evolution. Their measured Rabi frequencies were, on average,
$\sqrt{2}\kappa=2\pi\times$1.42\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and
are calibrated within 2% of each other. The duration $T$ of the geometric-
phase dynamics is scaled according to the calibrated Rabi frequency.

Reconstruction measurement. After the simulated time evolution, the system is
in the entangled state
$\ket{\Psi}=a_{\downarrow}\ket{\downarrow}\ket{\psi_{\downarrow}}_{B_{1}}\ket{\xi_{\downarrow}}_{B_{2}}+a_{\uparrow}\ket{\uparrow}\ket{\psi_{\uparrow}}_{B_{1}}\ket{\xi_{\uparrow}}_{B_{2}}$.
In preparation of the reconstruction, a mid-circuit measurement projects the
qubit state to either $\ket{\downarrow}$ or $\ket{\uparrow}$ through state-
dependent fluorescence induced by a $369.5\text{\,}\mathrm{n}\mathrm{m}$ laser
beam resonant with the ^2S_1/2 $\ket{F=1}$ $\rightarrow$ ^2P_1/2 $\ket{F=0}$
transition. The qubit states are inferred by thresholding the number of
photons collected on an avalanche photodiode (measured state preparation and
measurement fidelity of 99.5%), and the outcomes of the measurement determine
the probabilities $p_{\downarrow}=|a_{\downarrow}|^{2}$ and
$p_{\uparrow}=|a_{\uparrow}|^{2}$. A measurement outcome of $\ket{\uparrow}$
induces significant decoherence of the bosonic modes due to photon recoils.
Therefore, the reconstruction only proceeds if the measurement outcome is
$\ket{\downarrow}$, for which no photon is emitted. Doing so projects the
bosonic modes to
$\ket{\psi_{\downarrow}}_{B_{1}}\ket{\xi_{\downarrow}}_{B_{2}}$. To retrieve
$\ket{\psi_{\uparrow}}_{B_{1}}\ket{\xi_{\uparrow}}_{B_{2}}$ instead, we insert
an $R_{x}(\pi)$ pulse that flips the qubit before the measurement. After the
mid-circuit measurement, the characteristic functions
$\chi_{\downarrow}(i\beta_{1},i\beta_{2})$ and
$\chi_{\uparrow}(i\beta_{1},i\beta_{2})$ corresponding to each qubit state are
measured as described in the main text. The full characteristic function is
then the sum of both contributions,
$\chi(i\beta_{1},i\beta_{2})=p_{\downarrow}\chi_{\downarrow}(i\beta_{1},i\beta_{2})+p_{\uparrow}\chi_{\uparrow}(i\beta_{1},i\beta_{2})$.
The values $\beta_{j}=\Omega_{2}\tau_{j}$ are scanned by varying the SDF-pulse
duration $\tau_{j}$. The Rabi frequency was recalibrated between experiments
and, on average,
$\Omega_{2}=2\pi\times$2.31\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
resulting in combined pulse durations of up to
$553\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. We measured
$\chi(i\beta_{2})$ for $\beta_{2}\in[0,5]$ and $\chi(i\beta_{1},i\beta_{2})$
for $\beta_{1},\beta_{2}\in[0,4]$ by varying the SDF pulse durations. Since
the characteristic function is Hermitian,
$\chi(i\beta_{1},i\beta_{2})^{*}=\chi(-i\beta_{1},-i\beta_{2})$, we used
symmetry to find $\chi(i\beta_{2})$ for $\beta_{2}<0$ and
$\chi(i\beta_{1},\beta_{2})$ for $\beta_{1}<0$ or $\beta_{2}<0$. We did not
measure the vanishing imaginary part of $\chi(i\beta_{2})$ nor
$\chi_{\downarrow}$ at $t=0$. The measured characteristic functions are shown
in Extended Data Fig. 1.

Data acquisition. The characteristic functions were measured in a way to
average out the effects of drift. In each run of the experiment, we randomised
the order of the displacement-pulse durations in which $\chi$ was
reconstructed. For each run, the quantum circuit to obtain $\chi_{\downarrow}$
and $\chi_{\uparrow}$ was repeated until the mid-circuit measurement succeeded
500 times, resulting in 500 measurement repetitions of the reconstruction
routine and 1000 measurements to obtain a value of the full $\chi$.
Furthermore, the order of the displacements was randomised. Overall, each of
the 1- and 2-dimensional experiments was repeated, respectively, four and two
times and the results of the runs averaged for a total of 2000 and 1000
measurements for each duration. The bosonic mode frequencies were calibrated
every $6\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$, while the full system
parameters were recalibrated after the second experimental run. The
1-dimensional and the four 2-dimensional ($t=\\{0,0.9T,T,2T\\}$) experiments
were done on five separate days with total durations of 15.6, 2.8, 8.8, 8.7
and 10.6 hours, respectively.

Noise sources Decoherence of the bosonic modes, made up of motional heating
and dephasing, was the dominant noise mechanism in our system. Motional
heating was caused by electric field noise at the radial mode frequency, while
motional dephasing arose from fluctuations in the harmonic trapping potential
strength [2]. We measured the heating rate and the motional dephasing time of
B1 (representative for both modes) to be
$\dot{\bar{n}}=$0.2\text{\,}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\leavevmode\nobreak\
\mathrm{s}^{-1}$$ and $T_{2}^{*}=$35\text{\,}\mathrm{m}\mathrm{s}$$ [39]. The
motional coherence was limited by noise in the radio frequency (RF) trapping
voltage, which was actively stabilised to mitigate fluctuations in the radial
mode frequencies. Slow drifts of the trapping voltage were compensated for
with frequent motional mode recalibrations.

### Characteristic functions

Extended Data Fig. 1:  Characteristic functions of the wavepacket measured for
various evolution times. a-d Joint two-dimensional characteristic function
$\chi(i\beta_{1},i\beta_{2})=p_{\downarrow}\chi_{\downarrow}(i\beta_{1},i\beta_{2})+p_{\uparrow}\chi_{\uparrow}(i\beta_{1},i\beta_{2})$
measured at times $t=\\{0,0.9T,T,2T\\}$ using the full pulse sequence of fig.
2. The real (left) and imaginary (right) parts are measured with and without
an $R_{x}(\pi/2)$ pulse in the reconstruction. The top row shows theoretical
predictions and the bottom experimental results. $\chi(i\beta_{1},i\beta_{2})$
were measured in the range $\beta_{1},\beta_{2}\in[0,4]$ with $11\times 11$
equidistant samples (dashed quadrant). Values in the remaining three quadrants
are obtained from the symmetry of $\chi(i\beta_{1},i\beta_{2})$. e One-
dimensional characteristic functions $\chi_{\downarrow}(i\beta_{2})$ and
$\chi_{\uparrow}(i\beta_{2})$ obtained by omitting displacements on B1 in the
reconstruction. $\beta_{2}$ was uniformly sampled in the range $[0,5]$ with 26
points. Each two- and one-dimensional characteristic function was averaged
over 1000 and 2000 measurements, respectively. Error bars in e represent one
standard deviation based on quantum projection noise.

Figure 1 shows the measured characteristic functions used to reconstruct the
wavepacket probability densities. The two-dimensional characteristic functions
(fig. 1a–d) require four measurements at each $t$ to obtain the real and
imaginary parts of $p_{\downarrow}\chi_{\downarrow}(i\beta_{1},i\beta_{2})$
and $p_{\uparrow}\chi_{\uparrow}(i\beta_{1},i\beta_{2})$. The one-dimensional
characteristic functions (fig. 1e) require two measurements to determine
$p_{\downarrow}\chi_{\downarrow}(i\beta_{2})$ and
$p_{\uparrow}\chi_{\uparrow}(i\beta_{2})$, as the vanishing imaginary part is
not measured. In both one- and two-dimensional reconstructions, only positive
values of $\beta_{1}$ and $\beta_{2}$ are sampled; the characteristic function
in other ranges is obtained by symmetry.

We performed post-processing to remove artifacts associated with Fourier
transformations between a characteristic function and its probability density.
A non-zero DC offset appearing as background noise in the characteristic
functions propagates into the probability densities at the origin [46]. Since
background noise with a non-zero mean is a technical imperfection and is
independent of the geometric-phase evolution, we correct for it in post-
processing. We estimate the mean of the background noise by averaging
$\chi(i\beta_{1},i\beta_{2})$ with $\sqrt{\beta_{1}^{2}+\beta_{1}^{2}}\geq
3.6$ for the two-dimensional and $\chi(i\beta_{2})$ with $\beta_{2}\geq 3.6$
for the one-dimensional case, and offset the data by the negative of this
average. This baseline correction was on average 0.02, with the largest
correction of 0.03.

### Phase coherence in the pulse sequence

This appendix describes the experimental procedure to track the qubit and
motional phases, ensuring phase coherence between sequential spin-motional
interactions.

The laser-induced excitations interacting with an ion with a qubit frequency
$\omega_{0}$ and a motional mode with frequency $\omega_{m}$ are the carrier
(c), red-sideband (rsb), and blue-sideband transitions (bsb). Their
interaction Hamiltonians, after dropping high-frequency terms, are

$\displaystyle H_{\textrm{c}}$
$\displaystyle=\frac{\Omega_{c}}{2}\sigma^{+}e^{i\phi_{c}}e^{i(\omega_{c}-\omega_{0})t}+\textrm{h.c.},$
(7) $\displaystyle H_{\textrm{rsb}}$
$\displaystyle=\frac{\eta\Omega_{r}}{2}\sigma^{+}ae^{i\phi_{r}}e^{i(\omega_{r}-\omega_{0}+\omega_{m})t)}+\textrm{h.c.},$
(8) $\displaystyle H_{\textrm{bsb}}$
$\displaystyle=\frac{\eta\Omega_{b}}{2}\sigma^{+}a^{\dagger}e^{i\phi_{b}}e^{i(\omega_{b}-\omega_{0}-\omega_{m})t)}+\textrm{h.c.},$
(9)

where $\eta$ is the Lamb-Dicke parameter and $\Omega_{c,b,r}$ are the
respective Rabi frequencies. $\omega_{c,b,r}$ and $\phi_{c,b,r}$ correspond,
respectively, to the frequency differences and the phase differences of the
two orthogonal Raman beams. Simultaneously driving the red- and blue-sidebands
with $\Omega=\eta\Omega_{r}=\eta\Omega_{b}$ gives

$\displaystyle H_{\textrm{SDF}}$ $\displaystyle=H_{\textrm{rsb}}+$
$\displaystyle H_{\textrm{bsb}}$ $\displaystyle=\frac{\Omega}{2}\sigma^{+}[$
$\displaystyle ae^{i\phi_{r}}e^{i(\omega_{r}-\omega_{0}+\omega_{m})t}+{}$
$\displaystyle
a^{\dagger}e^{i\phi_{b}}e^{i(\omega_{b}-\omega_{0}-\omega_{m})t}+\textrm{h.c.}].$
(10)

We consider $\omega_{r}$ and $\omega_{b}$ to be set near-resonant with the
red- and blue-sideband transitions,

$\displaystyle\omega_{r}$
$\displaystyle=\omega_{0}-\omega_{m}-\delta\omega_{m}+\delta\omega_{0},$ (11)
$\displaystyle\omega_{b}$
$\displaystyle=\omega_{0}+\omega_{m}+\delta\omega_{m}+\delta\omega_{0},$ (12)

where $\delta\omega_{0}$ is an asymmetrical (center-line) detuning from the
qubit frequency, and $\delta\omega_{m}$ is a symmetrical detuning from the
motional mode frequency. With the spin phase $\phi_{s}=(\phi_{r}+\phi_{b})/2$
and the motional phase $\phi_{m}=(\phi_{b}-\phi_{r})/2$, eq. 10 can be
rewritten as

$\displaystyle
H_{\textrm{SDF}}=\frac{\Omega}{2}\sigma^{+}e^{i(\delta\omega_{0}t+\phi_{s})}[$
$\displaystyle ae^{-i(\delta\omega_{m}t+\phi_{m})}+$ $\displaystyle
a^{\dagger}e^{i(\delta\omega_{m}t+\phi_{m})}]+\textrm{h.c.}$ (13)

This Hamiltonian corresponds to eq. 3 in the main text by setting
$\delta\omega_{0}=0$ and $\delta\omega_{m}=\delta$. The motional phase
$\phi_{m}$ can be adjusted to selectively displace a mode along $Q$ or $P$.
Equation 13 shows that non-zero or miscalibrated $\delta\omega_{0}$ and
$\delta\omega_{m}$ introduce a time-dependent phase offset to $\phi_{s}$ and
$\phi_{m}$ which, if uncorrected, will lead to incorrect interactions.

The qubit frequency detuning is, from eq. 11 and eq. 12,
$\delta\omega_{0}=(\omega_{b}+\omega_{r})/2-\omega_{0}$. To avoid phase lags
associated with $\phi_{s}$, we enforce
$(\omega_{b}+\omega_{r})/2=\omega_{c}=\bar{\omega}_{0}$ for all pulses
throughout the entire circuit, namely single-qubit rotations and SDF
interactions on B1 and B2. Here, $\bar{\omega}_{0}$ indicates the qubit
frequency measured via a Ramsey sequence in a separate calibration experiment.

Likewise, eq. 11 and eq. 12 give the motional detuning as
$\delta\omega_{m}=(\omega_{b}-\omega_{r})/2-\omega_{m}$. To avoid unwanted
phase lags associated with $\phi_{m}$, we enforce
$(\omega_{b}-\omega_{r})/2=\bar{\omega}_{m}$ for all SDF pulses throughout the
circuit, where $\bar{\omega}_{m}$ is the experimentally measured motional
frequency. There is an unavoidable phase lag due to the detuning $\delta$
required in the SDF interactions during the time evolution. To correct this,
we add a motional phase offset of $\tau_{1}\delta$ to the SDF interaction
during the initial displacement, where $\tau_{1}$ is the duration of the
initialisation. Furthermore, a motional phase offset of $(t+\tau_{1})\delta$
is added to the reconstruction SDF pulses, where $t$ is the duration of the
time evolution.

### Calibration of motional frequencies

Extended Data Fig. 2: Frequency drifts of radial motional modes. a Time series
of motional frequencies $\omega_{1,2}$ corresponding to B1,2, measured using
the calibration routine detailed in the text and plotted as the frequency
offset from $\omega_{1}$ measured at $t=0$. b Allan deviation of $\omega_{1}$,
and of the difference between the two frequencies ($\omega_{1}-\omega_{2}$).

We used a calibration scheduling routine to recalibrate parameters during each
experiment and ensure high-fidelity implementations of the pulse sequence.
Moreover, we optimised the scheduler to maximise the experimental duty cycle
by analysing the temporal noise behaviours.

The data quality of the reconstructed densities depends on correctly setting
the laser frequencies for the motional sideband interactions that enact SDF
interactions. The motional frequencies $\omega_{1}$ and $\omega_{2}$
associated with B1 and B2 are calibrated as previously reported [39]. To do
so, both SDF interactions are applied, but we set the fields associated with
$\omega_{2}$ to be sufficiently off resonant while calibrating $\omega_{1}$.
We prepare the state $\ket{\downarrow}\ket{0}_{\textrm{B}_{1}}$ and apply two
sequential SDF pulses with a relative phase shift $\phi_{m}=\pi$. In the
absence of frequency errors, the mode returns to its original state and a
qubit measurement yields zero population in $\ket{\uparrow}$. However, in the
presence of errors, the motion remains entangled with the qubit, giving a non-
zero measured probability. The SDF fields’ frequencies are then scanned, and a
fit to the measurements yields the correct mode frequency. We repeat this
procedure to calibrate $\omega_{2}$ by setting the SDF field associated with
$\omega_{1}$ to be off resonant.

Figure 2 shows the drifts in the radial mode frequencies over time, which
varied in a range of $2\pi\times$2.2\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$
over 2 days. From numerical simulations, we determined that an error tolerance
of about $10\%$ is required for the detuning ($\delta\simeq
2\pi\times$667\text{\,}\mathrm{H}\mathrm{z}$$) in the time evolution to obtain
adequate results. Given that typical experiments lasted tens of hours,
frequent recalibrations of the motional mode frequencies were necessary. To
this end, we implemented a scheduling algorithm to interleave calibrations and
experiments [3]. The scheduling rate was determined by choosing a time
interval for which the Allan deviation was sufficiently small. From fig. 2b,
we chose an interval of 6 minutes, corresponding to an Allan deviation of
$2\pi\times$26\text{\,}\mathrm{H}\mathrm{z}$$ and satisfying the required
tolerance. We also found highly correlated noise between the radial modes (see
Extended Data Fig. 2), suggesting a common noise source (e.g., trap RF
amplitude fluctuations). Therefore, to increase the experiment duty cycle,
frequency offsets measured on B1 were also used to correct for B2.

## Data Availability

A repository containing data plotted in Fig. 3 and in Extended Data Fig. 1 is
available at https://doi.org/10.5281/zenodo.7955887 ([4])

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