# Evaluating states in trapped ions with local correlation between internal
and motional degrees of freedom

Silpa Muralidharan<EMAIL_ADDRESS>Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, Japan  Ryutaro
Ohira  Graduate School of Engineering Science, Osaka University, 1-3
Machikaneyama, Toyonaka, Osaka, Japan  Center for Quantum Information and
Quantum Biology, Osaka University, 1-2 Machikaneyama, Toyonaka, Osaka, Japan
Shota Kume  Graduate School of Engineering Science, Osaka University, 1-3
Machikaneyama, Toyonaka, Osaka, Japan  Kenji Toyoda
<EMAIL_ADDRESS>Center for Quantum Information and Quantum
Biology, Osaka University, 1-2 Machikaneyama, Toyonaka, Osaka, Japan

###### Abstract

We propose and demonstrate a scalable scheme for the simultaneous
determination of internal and motional states in trapped ions with single-site
resolution. The scheme is applied to the study of polaritonic excitations in
the Jaynes–Cummings–Hubbard model with trapped ions, in which the internal and
motional states of the ions are strongly correlated. We observe quantum phase
transitions of polaritonic excitations in two ions by directly evaluating
their variances per ion site. Our work establishes an essential technological
method for large-scale quantum simulations of polaritonic systems.

##  I Introduction

Quantum simulations allow us to study the properties of many-body quantum
systems that are hard to investigate with classical computers [1]. A promising
platform for realizing quantum simulations is a system of trapped ions [2, 3].
Trapped ions have advantages over other systems from such perspectives as ease
of preparation and control, and we can address individual particles with
little perturbation to neighboring particles.

The Jaynes–Cummings (JC) model describes the atom–field interaction in a
combined system of a two-level atom and a quantized electric-field mode [4].
An interconnected array of two-level atoms interacting with quantized wave
modes is known as the Jaynes–Cummings–Hubbard (JCH) model, and related systems
of coupled cavity arrays have been extensively investigated [5, 6, 7, 8, 9,
10, 11].

The JCH model can be realized with arrays of cavity QED systems [12, 13] and
of circuit QED systems [14, 15, 16, 17, 18], as well as systems of trapped
ions [19, 20, 10, 21, 22, 23]. These systems can be flexibly controlled with
sets of system parameters. Therefore, they can be considered to be attractive
systems for studying quantum many-body phenomena.

In the JCH model, quasi-particles called polaritons play an essential role.
Each polariton is represented as the superposition of an internal excitation
and a photon or a motional excitation (phonon). Polaritons are well-defined
particles in the JCH model and their total number is conserved. The
implications for understanding the JCH model is apparent in the quantum phase
transition of a JCH system between the Mott-insulator (MI) and superfluid (SF)
phases, which are characterized by a drastic change in the polariton-number
variance per site [7].

In the MI phase, each polariton is fixed to a respective site, and
fluctuations of the polariton number per site vanish. In the SF phase, each
polariton is distributed over multiple sites, and the ensemble of polaritons
forms a coherent wave-like state that spreads over the entire system. In the
latter case, the variance of the polariton number per site becomes finite.
Therefore, by evaluating the polariton-number variance per site while the
system parameters are varied, the quantum phase transition from the MI phase
to the SF phase can be detected. In the case of the JCH model with trapped
ions [19], each polariton is a correlated combination of an internal
excitation and a vibrational quantum (local phonon [24, 25, 26, 27, 28, 29,
21, 30, 31, 22]). Therefore, by detecting the internal and motional states,
the states of polaritons can be precisely evaluated. It is not easy to
directly measure the motional states and they should instead be first mapped
to the internal states to be detected by, e.g., state-dependent fluorescence
detection. However, this process destroys the internal states. The internal
and motional states cannot be simultaneously determined even if we measure the
former and the latter consecutively in this order, since the measurement of
the internal states with fluorescence detection may destroy the motional
states due to photon recoil during several fluorescence cycles.

One approach to address this is a conditional measurement method [19, 32]
using levels in such ions as 40Ca+, which has a metastable state. In this
method, a random guess of the internal states (the ground state $\ket{g}$ or
the metastable state $\ket{e}$) of a particular ion is first made. If the
guess is made for the $\ket{g}$ or $\ket{e}$ state, a carrier $\pi$ pulse or
no pulse, respectively, is applied to the $\ket{g}$–$\ket{e}$ transition.
Then, laser light resonant to the transition between $\ket{g}$ and a short-
lived excited state is exposed to the ions. If the ion scatters fluorescence
at this point, the initial guess was wrong, and the measurement result is
discarded. Otherwise, the initial guess was right, and the internal state is
subsequently transferred back to the $\ket{g}$ state. An analysis of the
motional states using a blue-sideband (BSB) Rabi oscillation is then
performed. Fourier analyses on a series of these results give information on
the motional states and the internal states that correspond to the initial
guesses. This combined information retains the local correlation between the
internal and motional states in the states of interest. This method can be
applied to the case of a single ion.

In the case of an ion chain, the concept of local vibrational modes (local
phonons) becomes approximate. Each mode can be coupled with those in different
ion sites through Coulomb couplings. When a particular ion site is heated due
to photon recoils during fluorescence cycles, the heat is transferred to
different ion sites, and eventually the motional states at those sites may be
destroyed (mutual heating among ion sites).

In this article, we demonstrate the measurement of local motional states
conditioned on the internal states by shelving part of the ions to a long-
lived internal state. In this method, fluorescence cycles are not used to
determine the internal states, and the probability amplitudes that are of no
interest are simply hidden in the auxiliary long-lived internal state.
Therefore, this method does not suffer from mutual heating among ion sites as
explained above, and is applicable to the measurement of local motional states
in a multi-ion crystal. In this work, this method is applied to determine the
polariton number and its variance in the JCH model while the system parameters
are changed. The method is scalable with respect to the number of ion sites,
i.e., the same time sequence can be used in principle for an arbitrary number
of ions in the chain.

##  II JCH model

The JC model [4] was initially proposed by E. Jaynes and F. Cummings in 1963.
It is a quantum optics model that describes the interaction of a two-level
atom with a single quantized mode of an optical cavity electromagnetic field.
The model is widely used in such fields as cavity QED and circuit QED. The
Hamiltonian for the JC model is given as follows:

$H_{\mathrm{JC}}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\hbar\omega_{0}\hat{\sigma}_{z}+\hbar
g(\hat{\sigma}_{+}\hat{a}+\hat{\sigma}_{-}\hat{a}^{\dagger}),$ (1)

where $\omega$ is the oscillation frequency of the electromagnetic field,
$\hat{a}^{\dagger}$, $\hat{a}$ are the creation and annihilation operators for
the quantized mode of the electromagnetic field, $\omega_{0}$ is the resonance
frequency of the two-level atom, $\hat{\sigma}_{z}$ is the Pauli operator in
the $z$ direction, $g$ is the JC coupling coefficient and $\hat{\sigma}_{+}$,
$\hat{\sigma}_{-}$ are the atomic raising and lowering operators. This model
introduces superposition states corresponding to dressed atoms or polaritons
and a semi-infinite series of eigenenergies called the JC ladder [4]. It leads
to such phenomena as a vacuum-field Rabi oscillation or collapses and revivals
of atomic oscillations. The analogy between a cavity QED system and a trapped-
ion system leads to implementing the JC model in trapped-ion experiments for
quantum information processing.

When trapped ions in an ion chain with relatively tight confinement along the
radial directions [24] are used and optical fields resonant to the red-
sideband (RSB) transitions are applied, the two-level systems in the ions
interacting with radial local phonons can be considered as an interconnected
array of JC systems, which can be described with the JCH model. The
Hamiltonian for the JCH model in a system of two trapped ions is given as
follows [19]:

$\displaystyle H_{\mathrm{JCH}}$ $\displaystyle=$
$\displaystyle\hbar\frac{\kappa}{2}(\hat{a}^{\dagger}_{1}\hat{a}_{2}+\hat{a}_{1}\hat{a}^{\dagger}_{2})+\hbar\Delta\sum_{j=1,2}\ket{e_{j}}\bra{e_{j}}$
(2) $\displaystyle+\hbar
g\sum_{j=1,2}(\hat{a}^{\dagger}_{j}\hat{\sigma}^{-}_{j}+\hat{a}_{j}\hat{\sigma}^{+}_{j}),$

where $\kappa$ is the hopping rate of the radial local phonons,
$\hat{a}^{\dagger}_{j}$ and $\hat{a}_{j}$ are the creation and annihilation
operators for phonons at the $j$th ion, $\Delta$ is the amount of detuning of
the optical field from the RSB transition, $\ket{e_{j}}$ is the internal
excited state in the $j$th ion, and
${\sigma}^{+}_{j}\equiv\ket{e_{j}}\bra{g_{j}}$,
${\sigma}^{-}_{j}\equiv\ket{g_{j}}\bra{e_{j}}$ are the raising and lowering
operators for the internal states in the $j$th ion, where $\ket{g_{j}}$ is the
internal ground state in the $j$th ion. In this JCH system, quantum phase
transitions between the MI and SF phases are expected to occur [5, 7]. A
quantum phase transition is similar to a classical phase transition, while the
cause of the transition is directly related to quantum fluctuations instead of
such parameters as pressure or temperature.

##  III MI-to-SF quantum phase transitions in the JCH model

In the JCH model, a quantum phase transition from the MI phase to the SF phase
in a trapped-ion chain is realized by the adiabatic transfer process [6, 5].
Here, we consider that the ions are prepared in
$\ket{\psi_{\mathrm{MI}}}=\ket{e_{1}}\ket{e_{2}}\ket{0}_{1}\ket{0}_{2}$. This
state corresponds to a ground state, the MI state, in which the excitations
are localized. The parameters $\Delta$ and $g$ can be changed by varying the
frequency and amplitude of the laser, respectively. We linearly sweep $\Delta$
from a negative value to a positive one and change the amplitude (and hence
$g$) following a Gaussian shape. In this case, the Hamiltonian changes time-
dependently and an adiabatic transfer is induced [33]. At the final stage of
this transfer, a ground state called the SF phase is created which has the
form
$\ket{\psi_{\mathrm{SF}}}=\ket{g_{1}}\ket{g_{2}}\otimes[(1/\sqrt{2})\ket{1}_{1}\ket{1}_{2}-(1/2)\ket{2}_{1}\ket{0}_{2}-(1/2)\ket{0}_{1}\ket{2}_{2}]$
[34]. In the intermediate region of the adiabatic transfer, polaritonic MI and
SF phases exist.

In an analogy to second-order phase transitions in statistical physics,
quantum phase transitions in the current system can be understood by the
emergence and disappearance of certain ordered phases. The MI states are
considered to have an order in the excitation (polariton) number per site, in
close analogy to the Bose–Hubbard systems [35, 36]. The SF states in this
system are believed to have coherence among multiple sites. In the case of two
ions, the phonon and polaritonic SF states can be represented as
superpositions of product states over multiple sites [34]. Although it is not
straightforward to define an order parameter that embodies the coherence among
multiple sites, the order in the polariton number per site can be readily
defined.

Angelakis et al. proposed using the polariton-number variance per site to
determine the different phases of the system [7]; it tends to be zero for the
MI phase, which has perfect number order, while it takes finite values for the
SF phase. Accordingly, we use the variance for the total excitation number
(polariton number) per site to determine quantum phase transitions in the JCH
model. Other excitation-number variances (atomic and phonon) can be used for
purposes such as classifying the ground states of the system.

The variance for the total excitation number per site (total variance) is
given by
$\Delta\hat{N}_{j}^{2}\equiv\langle\hat{N}_{j}^{2}\rangle-\langle\hat{N}_{j}\rangle^{2}$,
where $\hat{N}_{j}=\ket{e_{j}}\bra{e_{j}}+\hat{a}_{j}^{\dagger}\hat{a}_{j}$.
The variance for the atomic excitation number per site (atomic variance) is
given by
$\Delta\hat{N}_{a,j}^{2}\equiv\langle\hat{N}_{a,j}^{2}\rangle-\langle\hat{N}_{a,j}\rangle^{2}$,
where $\hat{N}_{a,j}=\ket{e_{j}}\bra{e_{j}}$. This helps to identify
polaritons. Additionally, the variance for the phonon number per site (phonon
variance) is defined as
$\Delta\hat{N}_{p,j}^{2}\equiv\langle\hat{N}_{p,j}^{2}\rangle-\langle\hat{N}_{p,j}\rangle^{2}$,
where $\hat{N}_{p,j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$. This follows an
increasing trend similar to that for the total variance in MI-to-SF quantum
phase transitions, while its detailed behavior in the intermediate region is
different from that for the total variance. The values for the total and
atomic variances and the corresponding classes of the ground states are
summarized in Table 1.

|  $\Delta N_{a,k}=0$ |  $\Delta N_{a,k}\neq 0$
---|---|---
$\Delta N_{t,k}=0$ |  Atomic MI |  Polaritonic MI
$\Delta N_{t,k}\neq 0$ |  Phonon SF |  Polaritonic SF
Table 1: Variances and classes of the ground states.

##  IV Conditional measurement of internal and motional states

The levels used for the conditional-measurement scheme and the probability
amplitudes associated with the levels are depicted in Fig. 1(a). In this
scheme, an auxiliary long-lived internal state (a sublevel in the Zeeman
manifold of the excited state; denoted as $\ket{a}$) is used. The levels in
the internal two-level system $\\{\ket{g},\ket{e}\\}$ are connected by a
carrier transition (denoted as “Carrier” in Fig. 1), while the internal ground
state $\ket{g}$ and the auxiliary state $\ket{a}$ are connected by another
carrier transition (denoted as “Shelving” in Fig. 1). We assume that the
motional states can be treated as a set of local vibrational modes. This is an
approximate picture, which is valid as long as the sideband Rabi frequencies
used for the analysis of the motional states during the conditional
measurements are much larger than the inter-site coupling rates of the
motional states.

Figure 1: (a) Level scheme and probability amplitudes associated with the
levels before a conditional measurement. (b) Levels and probability amplitudes
after conditioning the system to $\ket{g}$ [just after applying the first
three pulses in Fig. 2(b)]. (c) Levels and probability amplitudes after
conditioning the system to $\ket{e}$ [just after applying the first two pulses
in Fig. 2(c)].

The time sequences for the conditional-measurement scheme are given in Fig. 2.
In the overall time sequence given in Fig. 2(a), an initial state is first
prepared. This involves cooling the system to the vibrational ground state and
optical pumping to a particular Zeeman sublevel in the internal ground-state
manifold. Then a quantum simulation (equivalent to the adiabatic transfer
mentioned above) and a conditional measurement are performed.

The conditional measurement is performed in two different ways: one is
conditioned on $\ket{g}$ [Fig. 2(b)] and the other on $\ket{e}$ [Fig. 2(c)].
We explain these two cases using a formulation, assuming a chain of $N$ ions,
where each ion has three internal states and an associated local vibrational
mode. The Coulomb couplings between different ion sites are ignored
temporarily for simplicity. The state of the $\it{i}$th ion is expressed as
follows:

$\ket{\psi_{i}}=\sum_{x=g,e}\ket{\psi_{x}}_{i},$ (3)

where

$\left\\{\begin{aligned}
\ket{\psi_{g}}_{i}&=&\sum_{n=0}^{m}C_{g,n}\ket{g,n}_{i},\\\
\ket{\psi_{e}}_{i}&=&\sum_{n=0}^{m}C_{e,n}\ket{e,n}_{i}.\end{aligned}\right.$
(4)

Here, $\ket{g,n}_{i}\equiv\ket{g}_{i}\ket{n}_{i}$ and
$\ket{e,n}_{i}\equiv\ket{e}_{i}\ket{n}_{i}$, where $n$ is the phonon number.
It is assumed that the maximum number of phonons in each vibrational mode is
$m$. The coefficients $C_{g,n}$ and $C_{e,n}$ are probability amplitudes
satisfying $\sum_{n=0}^{m}(|C_{g,n}|^{2}+|C_{e,n}|^{2})=1$. [See Fig. 1(a) for
the initial arrangement of the probability amplitudes.]

As a preparation step for performing an analysis of the motional state in the
conditional-measurement scheme, we transfer the population in either $\ket{g}$
or $\ket{e}$ to a long-lived auxiliary state $\ket{a}$, depending on which
state we want to analyze.

To analyze the probabilities associated with $\ket{g}$, we use the time
sequence given in Fig. 2(b). We first apply a $\pi$ pulse resonant to the
transition $\ket{g}\leftrightarrow\ket{e}$. This is followed by a $\pi$ pulse
resonant to the $\ket{g}\leftrightarrow\ket{a}$ transition and a second $\pi$
pulse resonant to the $\ket{g}\leftrightarrow\ket{e}$ transition. These three
pulses effectively shelve populations originally in $\ket{e}$ to $\ket{a}$,
and keep those originally in $\ket{g}$ in the same state. The state after the
application of these pulses is represented as follows:

$\ket{\psi_{i}}=\sum_{x=g,a}\ket{\psi_{x}}_{i},$ (5)

where

$\left\\{\begin{aligned}
\ket{\psi_{g}}_{i}=\sum_{n=0}^{m}C_{g,n}\ket{g,n}_{i},\\\
\ket{\psi_{a}}_{i}=\sum_{n=0}^{m}C_{e,n}\ket{a,n}_{i}.\end{aligned}\right.$
(6)

[See Fig. 1(b) for the arrangement of the probability amplitudes in this
case.]

By applying a BSB pulse to this state, whose variable length $\tau$ is swept
from 0 to a certain time, Rabi oscillations between
$\ket{g,n}\leftrightarrow\ket{e,n+1}$ can be observed. By performing Fourier
analysis on the results, it is possible to deduce the probabilities
$|C_{g,n}|^{2}$ originally associated with $\ket{g}$.

To measure the probabilities associated with $\ket{e}$, we use a slightly
different time sequence, shown in Fig. 2(c). In this case, the first $\pi$
pulse used in the previous case is omitted, while the following sequence is
used as-is. Before applying the BSB pulse, the state is similar to Eq. (6),
where $C_{g,n}$ and $C_{e,n}$ are replaced with each other. [See Fig. 1(c) for
the arrangement of the probability amplitudes in this case.] By applying a BSB
pulse with a variable length $\tau$ and performing Fourier analysis on the
result, the probabilities $|C_{e,n}|^{2}$ originally associated with $\ket{e}$
can be obtained.

Collecting the results of these measurements, it is possible to evaluate the
internal and motional states. For example the probability for states with
$\it{k}$ polaritons is given as follows:

$P^{k}_{\mathrm{pol}}=\begin{cases}|C_{g,k}|^{2},&(k=0)\\\
|C_{g,k}|^{2}+|C_{e,k-1}|^{2}.&(k\geq 1)\end{cases}$ (7)

Figure 2: Time sequences for conditional measurements. (a) Total time
sequence. First, the initial state is prepared. Then a quantum simulation (an
adiabatic transfer) and a conditional measurement are performed. (b) Time
sequence for a measurement conditioned on $\ket{g}$. $\tau$ is the variable
length of the BSB pulse, which is swept to obtain the BSB oscillation signals.
(c) Time sequence for a measurement conditioned on $\ket{e}$. $\tau$ is a
variable length similar to the one above.

We should mention the effective fraction of measurements in this scheme. In
contrast to the conditional-measurement schemes used previously [19, 32],
which involve a dedicated measurement of the internal state in advance of the
measurement of the motional state in each sequence, our scheme uses shelving
of the probability amplitudes that are of no interest to an auxiliary internal
state. This helps avoid unwanted effects of heating from adjacent ion sites
during fluorescence cycles, and omits post-selections. However, the effective
amount of information acquired is reduced in this scheme due to the fact that
part of the probability amplitudes are hidden in the auxiliary level. In this
sense this scheme is not superior to those methods that use post-selections
[19, 32]. The overall loss of information on average is estimated to be 50%.
For example, if we perform 100 measurement trials conditioned on $\ket{g}$ and
the same number of trials conditioned on $\ket{e}$, the average net amount of
effective trials would be 100 in total, and the respective ratio of the
effective trials over the total number may differ between $\ket{g}$ and
$\ket{e}$, depending on the internal-state population. The effective fraction
of 50% does not depend on the number of ions involved. Therefore, it can be
asserted that this scheme is scalable with respect to the number of ions in
this sense.

This loss of information manifests in the observed results as a reduction in
the contrasts in the BSB Rabi oscillation signals and associated offsets
corresponding to the populations hidden to the long-lived auxiliary state.
(The populations in the auxiliary state cannot be distinguished from those in
the excited state in our case.) The contrast in the results conditioned on
$\ket{g}$ or $\ket{e}$ should be proportional to the population in that state
before the conditional-measurement sequence. Therefore, a fitting process
should be adapted to such signals with different contrasts and offsets. We
will discuss this in Sec. VII.1.

##  V Experimental Procedure

Two 40Ca+ ions are trapped in a linear Paul trap. An rf voltage of 23 MHz is
applied for radial confinement. An rf quadrupole electric field with a
frequency of 23 MHz is applied for radial confinement, and a dc potential is
applied for axial confinement. The secular frequencies are
$(\omega_{x},\omega_{y},\omega_{z})/2\pi=(2.6,2.4,0.15)$ MHz. The distance
between the two ions is 20 $\mu$m and the hopping rate of local phonons [24,
37] is 3.8 kHz. The ions are first cooled by Doppler cooling along all
directions, and then their vibrational motion along the two radial directions
($x$ and $y$) is reduced to the ground state by sideband cooling [38, 39].
Doppler cooling is realized with 397-nm ($S_{1/2}$–$P_{1/2}$) and 866-nm
($D_{3/2}$–$P_{3/2}$) laser beams and sideband cooling is realized with 729-nm
($S_{1/2}$–$D_{5/2}$) and 854-nm ($D_{5/2}$–$P_{3/2}$) laser beams. Sideband
cooling is performed with a repetition of a pulse section 20 times, in each of
which 729- and 854-nm pulses are simultaneously applied for 400 $\mu$s. The
ions are intermittently optically pumped to $S_{1/2}$ by a 397-nm beam with
$\sigma^{-}$ polarization before, between and after the pulse sections for
sideband cooling. The final vibrational quantum numbers along the radial
directions after sideband cooling are
$\\{\bar{n}_{x},\bar{n}_{y}\\}\sim\\{0.2,0.1\\}$.

The $y$ radial direction in the ion crystal is used for the motional degree of
interest in this work. There are two collective modes in this direction, the
center-of-mass (COM) and rocking modes, which are separated from each other in
frequency by a spacing similar to the hopping rate. The heating rate for the
$y$ direction is estimated to be $\sim$5 quanta/s [30].

Figure 3:  Level scheme for a trapped 40Ca+ with internal ground
($\ket{g,n}$), excited ($\ket{e,n}$) and auxiliary ($\ket{a,n}$) states, where
$n$ represents the motional quantum number in a local vibrational mode.

Excitation beams at 729 nm for $S_{1/2}$–$D_{5/2}$ are used to induce JC
coupling, as well as to prepare and analyze the internal and motional states.
The level scheme of 40Ca+ relevant for the manipulation of the internal and
motional states is shown in Fig. 3. (In this figure the motional state is
assumed to be that of a local vibrational mode in view of applying this to the
measurement scheme.)

The excitation beams at 729 nm are individually applied to each of the two
ions. The relative intensities of the beams are adjusted to optimize equal
illumination of the two ions, by balancing the maximum Rabi frequencies
between the two ions.

The fluctuations in the radial secular frequencies are reduced by introducing
a feedback-control system for the rf amplitude [40].

The internal state of the two ions is determined by illuminating them with
lasers at 397 and 866 nm and by detecting fluorescence photons with a
photomultiplier tube or an electron multiplying charge-coupled-device camera.

In the quantum simulation of the JCH model, first the system is prepared in
the atomic MI state
($\ket{\psi_{\mathrm{MI}}}=\ket{e_{1}}\ket{e_{2}}\ket{0}_{1}\ket{0}_{2}$) by
applying a carrier $\pi$ pulse at 729 nm after sideband cooling. Then an
adiabatic transfer is performed, with a duration of typically 960 $\mu$s. The
duration is determined to satisfy the adiabaticity in the internal states,
while the adiabaticity in the motional state is not fully satisfied (this will
be discussed in Sec. VII.3). The internal state during this adiabatic transfer
can be detected by truncating it and by illuminating the ions with the lasers
at 397 and 866 nm. A conditional measurement, as explained above, involving
the excitation of the BSB transition, can also be performed by truncating the
adiabatic transfer.

##  VI Measurement of internal-state populations

The results of internal state measurements from 0 $\mu$s to 960 $\mu$s are
shown in Fig. 4. The measured population (blue circles with error bars) and a
numerical simulation for the population (red dashed curve) are shown in Fig.
4(a). The time dependence of the parameters used in the numerical simulation
in Fig. 4(a) is plotted in Fig. 4(b). The conditions for the numerical
simulation are summarized as follows. The hopping rate $\kappa/2\pi$ is
3.8kHz. $2g/2\pi$ is swept in a Gaussian form with a maximum value of 10.0
kHz. $\Delta/2\pi$ is varied from $-$50 kHz to 50 kHz during a time of 960
$\mu s$. The Lindblad master equation [41] is used for the numerical
simulation, in which transverse relaxation for the BSB transition, for example
due to fluctuations of the sideband resonance frequencies, is taken into
account by using $\gamma_{\mathrm{T}}=2\pi\times 0.19$ kHz as the rate. The
initial populations in the internal states are set to be 0.95 in $\ket{e}$ and
0.05 in $\ket{g}$ 111The cause of the reduction by 5 % in the population of
$\ket{e}$ can be explained as being due to the incomplete preparation of the
atomic MI state by the carrier $\pi$ pulse. We use the individual illumination
of each ion with a dedicated beam at 729 nm. Each beam is passed through a
dedicated polarization-maintaining optical fiber to assure path stability.
However, this fiber may produce a drift in the optical phases. Fluctuations in
interference between the beams caused by this drift of optical phases, as well
as those in the pointing of the beams, lead to relatively large intensity
fluctuations at the positions of the ions and affect the fidelity of rotations
even for the carrier transition..

The greater than zero final population can be explained mainly by the effect
of transverse relaxation. In our estimation, the effect of diabaticity is not
significant for this particular case.

The transfer of population from $\ket{e}$ to $\ket{g}$ observed in Fig. 4(a)
can be considered to be evidence that a transition from the atomic MI state to
the phonon SF state has occurred [20].

Figure 4:  (a) Experimentally observed adiabatic-transfer curve (blue circles
and error bars) and numerically simulated results (red dashed curve). Each
experimental point is the average of 100 experiments, and the error is
calculated by assuming a binomial distribution. (b) Time dependence of the
parameters for the optical pulse assumed in the numerical simulation in (a):
$g$ (JC-coupling constant) divided by $\pi$ (red solid curve) and $\Delta$
(detuning) divided by $2\pi$ (blue dashed curve) for the numerical simulation.

##  VII Conditional measurements and evaluation of variances

###  VII.1 Evaluation of populations from fitting

In performing the conditional measurements to evaluate the variances, we
applied the two different time sequences given in Fig. 2(b) and (c) in
separate experimental runs, and the BSB Rabi oscillations in both cases were
measured. By analyzing the Rabi oscillation results in both cases via fitting
with model functions, and by combining the acquired information, the state
populations were inferred and the variances were then evaluated.

The atomic variance can be estimated if the internal state population is
known. For the particular conditional measurements in this work, we used the
first points of the BSB Rabi oscillations, for which no BSB pulse of a finite
length is applied, to infer the population in the internal states. These are
equivalent to the internal-state populations just before the application of
the BSB pulses in Fig. 2(b) and (c). Those populations are denoted as $P_{g}$
and $P_{e}$ for the ground and excited states, respectively. Here, we assume
that the system is symmetric with respect to the permutation of the two ions,
and hence that the populations for the two ions are identical to each other.

For the estimation of the phonon and total variances, especially for the
latter, the information obtained from conditional measurements is fully
exploited. The obtained BSB Rabi oscillation signals are analyzed by fitting
the obtained results with model functions derived from those used in standard
analyses on sideband Rabi oscillations [43]:

$\displaystyle\Pi_{e}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}a\left(1+\sum_{n=0}^{m}\tilde{P}_{n}\cos^{2}(\Omega_{n,n+1}t/2)e^{-\gamma_{\mathrm{R}}t}\right)$
$\displaystyle+(1-a).$

Here, $\Pi_{e}(t)$ is the time-dependent excited-state population during the
BSB excitation, $m$ is the maximum Fock-state quantum number assumed,
$\tilde{P}_{n}$ is the population in each motional Fock state before the BSB
excitation, $\Omega_{n,n+1}\equiv\eta\Omega_{0}\sqrt{n+1}$ is the BSB Rabi
frequency ($\eta$ is the Lamb Dicke factor and $\Omega_{0}$ is the resonant
Rabi frequency for an ion without motion; the terms of higher orders in $\eta$
are ignored here), and $\gamma_{\mathrm{R}}$ is the common relaxation rate
reflecting decoherence and dephasing processes in sideband Rabi oscillations
222 In principle, the previously quoted transverse-relaxation rate
$\gamma_{\mathrm{T}}$ in the Lindblad master equation and this
$\gamma_{\mathrm{R}}$ should have a certain relation. If the latter is assumed
to be determined purely from the former, $\gamma_{\mathrm{R}}\sim
0.25\times\gamma_{\mathrm{T}}$ is expected, which is confirmed by numerical
simulations. In that case, $\gamma_{\mathrm{R}}\sim 0.05$ kHz is expected if
$\gamma_{\mathrm{T}}\sim 2\pi\times 0.19$ kHz is satisfied. On the other hand,
the actual value of $\gamma_{\mathrm{R}}$ determined from the chi-square
minimization, as explained in Sec. VII.1, is $2\pi\times 1.1$ kHz, which is
more than an order of magnitude larger than the above-mentioned value. This
discrepancy may be explained by the concrete nature of the fluctuations. If an
adiabatic-transfer process like that in Fig. 4(a) is used and the typical
timescale for the fluctuations is much longer than the pulse duration, the
adiabatic transfer process is not affected significantly by the fluctuations
thanks to their robustness against parameter variations. This may be why the
assumed relaxation rate that matches the experimental results is much smaller
than that for the numerical simulation results shown in Fig. 4(a). . $a$
($0\leq{a}\leq 1$) is a factor that takes into account the reduction of
contrasts and the associated offsets as discussed above. Here, $a$ is
determined from the estimated internal-state populations as explained in the
last paragraph.

In the actual fitting process, the maximum Fock-state quantum number $m$ is
set to 2 333 The factors that may make this assumption invalid are the
residual phonon number after sideband cooling (estimated to be $\sim 0.2$ in
the present case), heating during adiabatic transfer ($<\sim 5\times 10^{-3}$
quanta), and off-resonance excitation of the carrier transition by the red-
sideband optical pulse for the adiabatic transfer ($<0.03$). Here, we ignore
their effects. . The base Rabi frequency $\Omega_{0,1}$ and
$(\tilde{P}_{0},\tilde{P}_{1},\tilde{P}_{2})$
($\tilde{P}_{0}+\tilde{P}_{1}+\tilde{P}_{2}=1$) for each result, as well as
the overall relaxation rate $\gamma_{\mathrm{R}}$, are determined iteratively.
First, we use empirically known fixed values of
$(\tilde{P}_{0},\tilde{P}_{1},\tilde{P}_{2})$ for each result, and perform a
fitting of all the results with the least-squares method to determine
$\Omega_{0,1}$ for each result and $\gamma_{\mathrm{R}}$ (the latter is
determined by finding the value that gives the minimum value for the sum of
the chi-squares in fitting of all the results). We then treat these values as
fixed values and repeat the fitting process, thereby determining the refined
values for $(\tilde{P}_{0},\tilde{P}_{1},\tilde{P}_{2})$. This whole process
is repeated multiple times to obtain the final values for
$(\tilde{P}_{0},\tilde{P}_{1},\tilde{P}_{2})$. By multiplying $a$ by these
values, the actual populations in the combined internal and motional state
basis states are determined. For example, in the case for conditioning on
$\ket{g}$ or $\ket{e}$, we obtain $(P_{g,0},P_{g,1},P_{g,2})$ or
$(P_{e,0},P_{e,1},P_{e,2})$, respectively, which is equal to
$(a\tilde{P}_{0},a\tilde{P}_{1},a\tilde{P}_{2})$. Here,
$P_{x,n}\equiv|C_{x,n}|^{2}$ ($x=g,e$) is the population in each combined
internal and motional basis state $\ket{x,n}$. We also evaluate the error for
each of the populations, which is calculated as that propagating from the
parameter errors in the fitting process.

###  VII.2 Evaluation of variances

The variances defined in Sec. III, including the total, atomic and phonon
variances, are obtained from the combined information acquired in both the
cases conditioned on $\ket{g}$ and $\ket{e}$. In this subsection we present
the method to calculate the variances using populations based on the method in
the previous subsection. As mentioned above, we assume that the system is
symmetric with respect to the permutation of the two ions, and omit the
subscript $j=1,2$ representing the index for the ion number in the following.
The atomic variance is obtained as follows:

$\displaystyle\Delta\hat{N}_{a}^{2}$ $\displaystyle=$
$\displaystyle\mathrm{tr}(\rho\hat{N}_{a}^{2})-\mathrm{tr}(\rho\hat{N}_{a})^{2}$
$\displaystyle=$ $\displaystyle P_{e}-P_{e}^{2},$

where $\rho$ is the density operator for the total system and
$P_{e}=\bra{e}\rho\ket{e}$ (this is equal to $\sum_{n=0}^{2}P_{e,n}$ when the
phonon number $n$ is limited to $0\leq{}n\leq 2$). The phonon variance is
obtained as follows:

$\Delta\hat{N}_{p}^{2}=\sum_{n=0}^{2}n^{2}P_{n}-\left(\sum_{n=0}^{2}nP_{n}\right)^{2},$

where $P_{n}\equiv{}P_{g,n}+P_{e,n}$ ($n=0,1,2$). The total variance can be
expressed as follows:

$\displaystyle\Delta\hat{N}^{2}$ $\displaystyle=$
$\displaystyle\langle(\hat{N}_{a}+\hat{N}_{p})^{2}\rangle-\langle\hat{N}_{a}+\hat{N}_{p}\rangle^{2}$
$\displaystyle=$
$\displaystyle(\langle\hat{N}_{a}^{2}\rangle-\langle\hat{N}_{a}\rangle^{2})+(\langle\hat{N}_{p}^{2}\rangle-\langle\hat{N}_{p}\rangle^{2})$
$\displaystyle+2(\langle\hat{N}_{a}\hat{N}_{p}\rangle-\langle\hat{N}_{a}\rangle\langle\hat{N}_{p}\rangle)$
$\displaystyle=$
$\displaystyle\Delta\hat{N}_{a}^{2}+\Delta\hat{N}_{p}^{2}+2\mathrm{Cov}(\hat{N}_{a},\hat{N}_{p}),$

where

$\displaystyle\mathrm{Cov}(\hat{N}_{a},\hat{N}_{p})$ $\displaystyle\equiv$
$\displaystyle\langle\hat{N}_{a}\hat{N}_{p}\rangle-\langle\hat{N}_{a}\rangle\langle\hat{N}_{p}\rangle$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{2}nP_{e,n}-P_{e}\sum_{n=0}^{2}nP_{n}$

is the covariance between the atomic excitation number and the phonon number.
We also evaluate the error for each of the variances, which is calculated as
that propagating from those of the evaluated values for the populations
444Here, for simplicity, we assume that the errors are symmetric for the
positive and negative directions, and nonlinear dependences of the deviations
around the estimated values are ignored. This could lead to the appearance of
unphysical values for the bounds of the confidence intervals: the negative
values in Fig. 6(a) and (b) at 96 and 192 $\mu$s are considered to be due to
this..

###  VII.3 Experimental results

The experimental Rabi oscillation results measured after truncating the
adiabatic transfer at certain selected points in time (48, 432, 480, 528 and
912 $\mu$s), as well as the results of fitting performed based on the above-
mentioned procedure, are shown in Fig. 5. Values for the variances and
internal-state population estimated by processing the experimental Rabi
oscillation results, as well as those obtained in a numerical simulation, are
shown in Fig. 6. The atomic variance [blue circles with error bars and dashed
curve in Fig. 6(a)] shows a peak in the center. In this region, the
excitations of the system take the form of polaritons, each of which is the
superposition of an internal excitation and a phonon. In this case, the
internal-excitation number fluctuates, resulting in the appearance of a peak
in the atomic variance. The phonon number variance [black circles with error
bars and dashed curve in Fig. 6(b)] increases around the center of the sweep.
This indicates the emergence of the phonon SF state. The total variance [red
circles with error bars and dashed curve in Fig. 6(c)] follows a similar
trend.

In the numerically simulated results in Fig. 6(b–c), at the region after 500
$\mu$s, oscillatory behavior with a period of $\sim$150 $\mu$s is seen in both
the phonon and total variances. This feature may have arisen from the
diabaticity in the motional states. Since the total sweep time of 960 $\mu$s
is not sufficiently long compared with the hopping time constant (defined as
the inverse of the hopping rate, $\sim$260 $\mu$s), the system does not
completely adiabatically follow the ground states and additional phonon
excitations may be generated. This effect may be relaxed by choosing a longer
sweep time. We suppose that the relatively large fluctuations in the
experimental phonon and total variances [Fig. 6(b–c)] observed in the region
after 500 $\mu$s have an origin related to this effect.

The internal-state population [cyan crosses with error bars and dash-dot curve
in Fig. 6(a)] shows a relatively smooth and solid transition. This indicates
that adiabaticity of the internal states is satisfied. This experiment
confirms that the method for conditional measurements described in this work,
which is capable of measuring the internal and motional states simultaneously
without mutual interference, is sufficient for investigating the quantum phase
transitions in the current system.

Figure 5: BSB Rabi oscillations (blue points) obtained by truncating the
quantum simulation (adiabatic transfer) and by illuminating using a square
pulse with a variable duration. Here, only certain values for the duration of
the quantum simulation are picked up. Each experimental point is the average
of 100 experiments, and the error is calculated by assuming a binomial
distribution. The results for the fitting of the Rabi-oscillation signals with
model functions are also shown (red curves). The durations for the quantum
simulation (adiabatic transfer) are (a–b) 48, (c–d) 432, (e–f) 480, (g–h) 528
and (i–j) 912 $\mu$s from the first column to the fifth. The results in the
upper row [(a), (c), (e), (g) and (i)] are taken from measurements conditioned
on $\ket{g}$, and those in the lower row [(b), (d), (f), (h) and (j)] on
$\ket{e}$. Figure 6:  (a) Atomic variance, excited-state population $P_{e}$,
(b) phonon variance and (c) total variance, which are obtained from an
analysis involving the fitting of experimental results (points with error
bars) and numerical simulation (dashed/dash-dot curves). The irregularly high
experimental values ($\sim 0.3$) at 0 $\mu$s in (b) and (c) are considered to
be artifacts due to technical imperfections.

##  VIII Discussion

The study of the ground-state properties in a JCH system, which can be dealt
with by the conditional-measurement method in this work, may provide a basis
for dynamical studies of the system as well as many-body quantum dynamics. The
study of correlated particles such as polaritons may lead us toward more
profound quantum mechanical properties or phenomena.

The conditional-measurement method described in this work can be applied to
larger numbers of ions in a straightforward manner. By increasing the number
of ions, a steep transition from one quantum phase to another is expected
[47]. In a previous study using a Bose-particle system, a sudden change of
quantum phases against parameter changes was observed [36]. The JCH system is
expected to show a similar steep variation of quantum phases against parameter
changes if an increased number of ions is used.

The fitting model functions used in this article assume that sideband Rabi
oscillations are bound within the state manifold for each ion, and mutual
interactions between ions are not considered. Although this assumption
simplifies the analysis, it can lead to discrepancy between the experimental
and fitted results. This discrepancy could be overcome if the full JCH model
is used to obtain the fitting models in numerical simulations. However, this
way of analysis is not considered to be scalable, since the numerical
simulation of the full JCH model becomes increasingly difficult for larger
numbers of ion sites and is effectively intractable 555 In our estimation,
numerical simulation using the full JCH model with more than 17 ion sites and
the commensurate filling of one excitation per ion site requires memory that
can hold a number of basis states as large as that in a 40-spin system and
cannot be easily solved with existing classical computers.. In addition,
fitting necessitates setting suitable initial values, which may allow
arbitrariness to enter into the analysis process.

Another option that can overcome the shortcomings of using fitting for the
analysis would be the use of discrete Fourier transforms. By not relying on
fitting or numerical simulations of physical models to deduce phonon
distributions, this method of analysis may be applied to larger numbers of ion
sites with less difficulties and arbitrariness.

##  IX Conclusions

The system of polaritons in the JCH model that undergoes quantum phase
transitions is studied with a trapped-ion system. An MI-to-SF transition
occurs via polaritonic intermediate states. The polariton states are measured
by conditional measurements in which the internal state and motional state can
be measured simultaneously.

## Acknowledgments

This article was supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP)
Grant Number JPMXS0118067477.

###

S.M. and R.O. contributed equally to this article.

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