[1]G. Bougas

[1]Physics Department, Missouri University of Science and Technology, Rolla,
MO, 65409, USA

2]Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38,
D-01187, Dresden, Germany

# Density effects on the interferometry of Efimov states by modulating
magnetic fields

<EMAIL_ADDRESS>S. I. Mistakidis<EMAIL_ADDRESS>P. Giannakeas
<EMAIL_ADDRESS>*  [

###### Abstract

Dynamical association of Efimov trimers in thermal gases by means of modulated
magnetic fields has proven very fruitful in determining the binding energy of
trimers. The latter was extracted from the number of remaining atoms, which
featured oscillatory fringes stemming from the superposition of trimers with
atom-dimers. Subsequent theoretical investigations utilizing the time-
dependent three-body problem revealed additional association mechanisms,
manifested as superpositions of the Efimov state with the trap states and the
latter with atom-dimers. The three atoms were initialized in a way to emulate
a thermal gas with uniform density. Here, this analysis is extended by taking
into account the effects of the density profile of a semi-classical thermal
gas. The supersposition of the Efimov trimer with the first atom-dimer remains
the same, while the frequencies of highly oscillatory fringes shift to lower
values. The latter refer to the frequencies of trimers and atom-dimers in free
space since the density profile smears out the contribution of trap states.

##  1 Introduction

Dilute ultracold gaseous matter constitutes the main platform in order to
explore unique and exotic attributes of few-body systems in a controllable
manner. For example, ultracold experiments achieved to shed light on the
existence of Efimovian triatomic molecules [1]; an exotic three-body system
which was originally perceived by V. Efimov in the context of nuclear physics
[2, 3, 4]. Specifically, it was theoretically predicted that an infinite
geometric progression of three-body bound states emerges from unbound pairwise
two-body subsystems [5, 6, 7, 8].

Recent theoretical [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and
experimental [22, 23, 24, 25, 26, 27, 28, 29, 30] advances provide an in depth
exploration of the properties and the universal idiosyncrasies of the
Efimovian physics in the realm of nuclear and atomic physics. For instance,
ultracold experiments that use radio-frequency spectroscopy or
magnetoassociation techniques have accurately measured the binding energy of
Efimov states [31, 32, 33] or interaction quench/sweeps in order to probe
their lifetime [34]. Yudkin et al. introduced a Ramsey-type protocol where a
thermal gas is exposed to a double sequence of modulated magnetic fields, i.e.
magnetic field pulses, for variable in between times [35, 36]. This Ramsey-
type interferometer allowed to probe the challenging regime below the atom-
dimer threshold without using a unitary Bose gas, accurately measuring the
trimer binding energies. However, an overall damping in the oscillations of
the number of remaining atoms was observed, where the corresponding decay time
scale exceeds the typical lifetime of Efimov trimers even for 85Rb3 [34].
Theoretical investigations demonstrated that this Ramsey-type protocol
possesses a general scope and can be used to probe Efimov states not only near
the atom-dimer threshold, but also below the break-up threshold [37].
Furthermore, Ref. [37] explicated in the case of thermal gases that the decay
time of interferometric fringes is twice as large as the lifetime of the
Efimov states. This means that exposing gases in magnetic field pulses permits
us to simultaneously assess the binding energies and lifetimes of Efimov
states.

In this work, we further investigate theoretically the coherent association of
Efimov trimers in a spherical trap, which is induced by a sequence of double
magnetic field pulses emphasizing the role of a thermal gas density component.
More specifically, in Ref. [37] in the corresponding three-body calculations
the trapping frequency was rescaled by the peak density of a thermal gas of
85Rb atoms. This means the density dependent trapping frequency ensures that
the few-body system exhibits the same diluteness as in a standard experimental
scenario. Here, we consider the same system as in Ref. [37] taking into
account a more realistic density profile, therefore going beyond previous
studies. In particular, the physically relevant quantity is the thermally
averaged probability to occupy an Efimov state which is integrated over a
profile of density dependent trapping frequencies.

Our analysis shows that the averaged probabilities still exhibit signatures of
coherence by varying the time between the two magnetic field pulses, i.e. the
dark time. Indeed, the corresponding Fourier spectrum of the thermally
averaged probability to occupy trimers possesses predominantly three
frequencies: (i) two high ones, corresponding to the energy difference of the
Efimov trimer with the break-up threshold, and the latter with the atom-dimer
states, and (ii) a low frequency which originates from the energy difference
between the Efimov and atom-dimer states. Comparing these results with the
predictions of Ref. [37] we conclude that the density profile of a thermal gas
only affects the high frequencies which approach those of a three-body system
without the spherical trap. Furthermore, the low frequency is independent of
the density implying that the Ramsey-type dynamical protocol can be utilized
to probe few-body state-to-state coherences without being smeared out by
density and thermal effects.

This work is structured as follows. In Sec. 2 we introduce the characteristics
of the time-dependent three-body systems with trapping frequencies associated
to the density profile of a thermal gas. In Sec. 3 we discuss the impact of
the density profile of a thermal gas on the thermally averaged probability to
occupy trimers. Finally in Sec. 4 we summarize our results and discuss further
perspectives.

##  2 Time-Dependent three-body system and correspondence to a thermal gas

Owing to the dilutness of thermal gases, one can study density effects on
three-body harmonically trapped systems by utilizing density dependent
confinement frequencies [38, 39]. As such, the density of the few-body system
matches the one of a condensate or a thermal gas. Therefore, as a paradigm in
the following we focus on the dynamics of three 85Rb atoms of mass $m$.
Importantly, we consider an ensemble of three-body systems, each trapped in a
radially symmetric trap with frequency $\omega^{(\mathbf{j})}_{r}$ where the
index $(\mathbf{j})$ labels the setup. The frequencies are chosen such that
the harmonic oscillator length
$a^{(\mathbf{j})}_{r}=\sqrt{\hbar/(m\omega^{(\mathbf{j})}_{r})}$ corresponds
to a mean interparticle distance,
$a^{(\mathbf{j})}_{r}\simeq[n^{(\mathbf{j})}]^{-1/3}$ [40, 38, 39] between
atoms in the thermal gas. The $n^{(\mathbf{j})}$ span the one-body density
profile of a non-interacting semi-classical gas trapped in a harmonic trap
with radial frequency $\Omega_{r}=2\pi\times 10\,\rm{Hz}$ and peak density at
the trap center $n_{0}=5\times 10^{12}\,\rm{cm}^{3}$, following the experiment
of Ref. [34]. In particular, the assumed densities lie within the interval
$[0.6,1]n_{0}$, and the trap frequency associated to the largest (smallest)
density is denoted by $\omega^{(\mathbf{1})}_{r}$
($\omega^{(\mathbf{c})}_{r}$). The one-body density profile reads explicitly
[41],

$n(\bm{x})=n_{0}\exp\\{-\frac{1}{2}m\Omega_{r}^{2}\bm{x}^{2}/(k_{B}\mathcal{T})\\},$
(1)

where $\mathcal{T}$ is the temperature, $k_{B}$ is the Boltzman constant, and
$\bm{x}$ are the three-dimensional coordinates of the thermal cloud. The above
expression relates the density $n^{(\mathbf{j})}$ and the spatial coordinates
of the thermal cloud, i.e. $n^{(\mathbf{j})}=n(\bm{x}^{(\mathbf{j})})$.

More concretely, for every trapping frequency
$\omega^{(\mathbf{j})}_{r}\simeq\hbar[n^{(\mathbf{j})}]^{2/3}/m$, we solve the
time-dependent three-body Schrödinger equation pertaining to the below given
Hamiltonian featuring time-dependent $s$-wave scattering length $a(t)$ [37].
Namely,

$\displaystyle\mathcal{H}^{(\mathbf{j})}(t)=\sum_{i=1}^{3}\left(\frac{-\hbar^{2}\nabla_{i}^{2}}{2m}+\frac{m[\omega_{r}^{(\mathbf{j})}]^{2}}{2}\bm{r}_{i}^{2}\right)+\sum_{i<j}\frac{4\pi\hbar^{2}a(t)}{m}\delta^{(3)}(\bm{r}_{ij})\left(\partial_{r_{ij}}r_{ij}\cdot\right),$
(2)

where $r_{ij}=\left|\bm{r}_{i}-\bm{r}_{j}\right|$, and $\bm{r}_{i}$ being the
position of the $i$-th indistinguishable particle. The $s$-wave scattering
length, $a(t)$, is modulated according to the following experimentally
realizable [35, 36, 37] double pulse Ramsey-type sequence

$\displaystyle a(t)=a_{bg}+a_{m}\cos{(\Omega
t)}\Big{[}\chi(t)+\chi(t-t_{d}-2t_{0}-\tau)\Big{]},$ (3)
$\displaystyle\rm{with~{}envelope}~{}~{}\chi(t)=\begin{cases}\sin^{2}\left(\frac{\pi
t}{2t_{0}}\right),&0\leq t<t_{0}\\\ 1,&t_{0}\leq t<t_{0}+\tau\\\
\sin^{2}\left(\frac{\pi(t-\tau)}{2t_{0}}\right),&t_{0}+\tau\leq t\leq
2t_{0}+\tau\\\ 0,&\rm{otherwise}\end{cases}.$ (4)

It becomes evident that the instantaneous scattering length $a(t)$ oscillates
around the background scattering length $a_{bg}$ with driving amplitude
$a_{m}\sim 0.2\,a_{bg}$, and frequency $\Omega$. The two pulses are
characterized by a time-dependent envelope $\chi(t)$, where $t_{0}$ and $\tau$
signify the ramp on/off times and length of the pulse envelope, respectively.
The pulse characteristics, which will be explicated in what follows, are
designed such that Efimov trimers and energetically low atom dimers are
coherently populated [37, 35, 36]. During the dark time $t_{d}$, referring to
the time-interval in between the two pulses, the three-body system evolves
freely.

Figure 1: Rescaled energy spectrum of three atoms in a trap with radial
frequency $\omega^{(1)}_{r}$, corresponding to the peak density $n_{0}$. The
eigenstates fall into three categories, trimers (T), atom-dimers (AD) and
three atom states (A).

To solve Eq. (2), we transform the above-described three-body Hamiltonian from
the lab to the center-of-mass frame,
$\mathcal{H}^{(\mathbf{j})}(t)=\mathcal{H}^{(\mathbf{j})}_{\rm{cm}}+\mathcal{H}^{(\mathbf{j})}_{\rm{rel}}(t)$.
Here, only the relative Hamiltonian $\mathcal{H}^{(\mathbf{j})}_{\rm{rel}}(t)$
depends on the time-dependent scattering length. Subsequently, a change from
cartesian to hyperspherical coordinates, see also Refs. [5, 6, 7] for
technical details, is performed. In this context, the relative degrees of
freedom of the system consist of the hyperradius $R$ that controls the overall
system size, and the five hyperangles $\bm{\varpi}$, which determine the
relative distance between the three atoms. When cast in this coordinate frame,
the relative Hamiltonian reads

$\displaystyle\mathcal{H}^{(\mathbf{j})}_{\rm{rel}}(t)=-\frac{\hbar^{2}}{2\mu}\frac{1}{R^{5/2}}\frac{\partial^{2}}{\partial
R^{2}}\left(R^{5/2}\cdot\right)+\frac{15\hbar^{2}}{8\mu
R^{2}}+\frac{\hbar^{2}\bm{\Lambda}^{2}}{2\mu R^{2}}$
$\displaystyle+\frac{1}{2}\mu[\omega^{(\mathbf{j})}_{r}]^{2}R^{2}+a_{bg}V(R;\bm{\varpi})+V(R;\bm{\varpi})[a(t)-a_{bg}]$
$\displaystyle=\mathcal{H}^{(j)}_{bg}(R;\bm{\varpi})+a_{m}V(R;\bm{\varpi})\cos{(\Omega
t)}\Big{[}\chi(t)+\chi(t-t_{d}-2t_{0}-\tau)\Big{]},$ (5)

where $\mu=m/\sqrt{3}$ is the three-body reduced mass and $\bm{\Lambda}^{2}$
is the grand angular momentum operator describing the total angular momentum
of three atoms [42]. Also, the $V(R;\bm{\varpi})$ term is the regularized
contact interaction potential expressed in hyperspherical coordinates. Note
that we have set apart the time-dependence in the last term of Eq. (5).

To subsequently tackle the time-dependent relative Hamiltonian, we employ the
basis of the field-free Hamiltonian $\mathcal{H}^{(\mathbf{j})}_{bg}$. The
latter is diagonalized within the adiabatic hyperspherical approach [43, 7, 5,
6, 8], yielding the field-free eigenstates, $\\{\ket{n^{(\mathbf{j})}}\\}$
with corresponding energy $E^{(\mathbf{j})}_{n}$. The main idea behind this
approach is to write the eigenstates in a basis of hyperangular functions
$\Phi^{(\mathbf{j})}_{\nu}(R;\bm{\varpi})$, treating $R$ as an adiabatic
parameter,

$\braket{R,\bm{\varpi}}{n^{(\mathbf{j})}}=R^{-5/2}\sum_{\nu}F^{(n,\mathbf{j})}_{\nu}(R)\Phi^{(\mathbf{j})}_{\nu}(R;\bm{\varpi}),$
(6)

where $F^{(n,\mathbf{j})}_{\nu}(R)$ are the so-called hyperradial channels.
The $\Phi^{(\mathbf{j})}_{\nu}(R;\bm{\varpi})$ basis functions are obtained by
solving the field-free relative Hamiltonian at fixed hyperradius $R$, i.e.
omitting the derivatives,

$\mathcal{H}^{(\mathbf{j})}_{bg}(R=const;\bm{\varpi})\Phi^{(\mathbf{j})}_{\nu}(R=const;\bm{\varpi})=U^{(\mathbf{j})}_{\nu}(R=const)\Phi^{(\mathbf{j})}_{\nu}(R=const;\bm{\varpi}),$
(7)

resulting in a set of potential curves, $U^{(\mathbf{j})}_{\nu}(R)$.
Subsequently, plugging in the expansion (6) into the field-free Hamiltonian
and integrating over the hyperangles $\bm{\varpi}$, a set of coupled
differential equations [5] is obtained for the hyperradial channels
$F^{(n,\mathbf{j})}_{\nu}(R)$,

$\displaystyle-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dR^{2}}F^{(n,\mathbf{j})}_{\nu}(R)+U^{(\mathbf{j})}_{\nu}(R)F^{(n,\mathbf{j})}_{\nu}(R)$
$\displaystyle-\frac{\hbar^{2}}{2\mu}\sum_{\nu^{\prime}}\Bigg{[}\braket{\Phi^{(\mathbf{j})}_{\nu}(R;\bm{\varpi})}{\frac{\partial^{2}}{\partial
R^{2}}\Phi^{(\mathbf{j})}_{\nu^{\prime}}(R;\bm{\varpi})}_{\bm{\varpi}}F^{(n,\mathbf{j})}_{\nu^{\prime}}(R)$
$\displaystyle+2\braket{\Phi^{(\mathbf{j})}_{\nu}(R;\bm{\varpi})}{\frac{\partial}{\partial
R}\Phi^{(\mathbf{j})}_{\nu^{\prime}}(R;\bm{\varpi})}_{\bm{\varpi}}\frac{d}{dR}F^{(n,\mathbf{j})}_{\nu^{\prime}}(R)\Bigg{]}=E^{(\mathbf{j})}_{n}F^{(n,\mathbf{j})}_{\nu}(R),$
(8)

where the braket notation, $\braket{}_{\bm{\varpi}}$, refers to the
integration only over the hyperangles.

The resulting field-free eigenstates are categorized into three classes as can
be readily seen in the three-body spectrum depicted in Fig. 1 as a function of
the $s$-wave scattering length [44]. Namely, Efimov trimers (T) (orange solid
line in Fig. 1), atom-dimers (AD) (blue solid lines) and three-atom states (A)
(black dashed lines). For positive $a_{bg}$, the excited Efimov trimer
approaches the atom-dimer states and eventually dissociates [19, 14]. This
interaction regime was employed in [35, 36] in order to create coherent
superposition of the first excited Efimov trimer with the first atom-dimer.
Apart from the driving frequency
$\Omega=\left|E^{(1)}_{AD}-E^{(1)}_{A}\right|/h$, the pulse envelope is chosen
such that both the first excited Efimov state and the first atom-dimer are not
energetically resolved by the pulse [35]. This condition refers to the full-
width-at-half-maximum of the Fourier transform of $\chi(t)$ being comparable
to $\left|E^{(2)}_{T}-E^{(1)}_{AD}\right|/h$. Note that the driving frequency
and the pulse characteristics depend on the considered trapping frequency
$\omega_{r}^{(\mathbf{j})}$. However, these parameters do not strongly vary
since the Efimov trimer and first atom-dimer states are energetically deep,
and hence they are not affected by the trap. In contrast, the three-atom and
highly-excited atom-dimer states are the only ones that are strongly
influenced by the trap frequency.

Figure 2: (a) Comparison of the ratio of thermally averaged probabilities
taking into account a realistic density profile for the thermal gas (blue
line), or just the peak density (yellow line) at $\mathcal{T}=270\,\rm{nK}$.
High-amplitude oscillations appear in region I, which are subsequently damped
in region II due to thermal effects. (b) Frequency spectrum stemming from
$\langle\mathbb{P}_{T}(t_{d})\rangle$. The low-frequency peak (black dashed
line) remains unchanged regardless of whether we consider the realistic
density profile or solely the peak density. The other two high frequencies are
slightly shifted from the corresponding values with just the peak density
(yellow dashed lines), towards frequencies corresponding to the first atom-
dimer and first excited Efimov state in the absence of any trap (black dashed-
dotted lines).

For every trap frequency, $\omega_{r}^{(\mathbf{j})}$, the three-body system
is initialized in an ensemble of three-atom states, populated according to the
Maxwell-Boltzmann distribution at temperature $\mathcal{T}$. In this way,
thermal effects are taken into account [45, 46]. Subsequently, the time-
dependent relative wavefunction assigned to an initial three atom-state
$\alpha$ is expanded in the field-free basis,
$\ket{\Psi^{(\alpha,\mathbf{j})}_{\rm{rel}}(t)}=\sum_{n}c^{(\alpha,\mathbf{j})}_{n}(t)\ket{n^{(\mathbf{j})}}$,
where $c^{(\alpha,\mathbf{j})}_{n}(t)$ is the probability amplitude of the
$n$-th stationary state.

Plugging the above expression into the time-dependent Schrödinger equation
(TDSE) results in a matrix differential equation for the probability
amplitudes [37],

$i\hbar\frac{d\bm{c}^{(\alpha,\mathbf{j})}(t)}{dt}=\bm{\mathcal{H}}^{(\mathbf{j})}_{rel}(t)\cdot\bm{c}^{(\alpha,\mathbf{j})}(t),$
(9)

where $\bm{\mathcal{H}}^{(\mathbf{j})}_{rel}(t)$ is the relative Hamiltonian
matrix expressed in the field-free basis. A second-order split-operator method
is utilized to solve the above equation [47].

Since the aim is to monitor the dynamical population of Efimov trimers in a
thermal gas, we introduce the ratio of the thermally averaged probability to
occupy trimers after the second pulse (nominator in Eq. (10a)) with respect to
the end of the first pulse (denominator in Eq. (10a)). Specifically, this
observable takes the form

$\displaystyle\mathbb{P}^{(\mathbf{j})}_{T}(t_{d})=\frac{\sum_{\alpha\in
A}\sum_{k\in\,T}e^{-\frac{E^{(\mathbf{j})}_{\alpha}}{k_{B}\mathcal{T}}}\left|c^{(\alpha,\mathbf{j})}_{k}(2\tilde{\tau}+t_{d})\right|^{2}}{\sum_{\alpha\in
A}\sum_{k\in\,T}e^{-\frac{E_{\alpha}^{(\mathbf{j})}}{k_{B}\mathcal{T}}}\left|c^{(\alpha,\mathbf{j})}_{k}(\tilde{\tau})\right|^{2}},$
(10a) $\displaystyle
c^{(\alpha,\mathbf{j})}_{k}(2\tilde{\tau}+t_{d})=\sum_{n}U^{(\mathbf{j})}_{kn}(2\tilde{\tau}+t_{d},\tilde{\tau}+t_{d})e^{-iE^{(\mathbf{j})}_{n}t_{d}/\hbar}U^{(\mathbf{j})}_{n\alpha}(\tilde{\tau},0).$
(10b)

In the above expressions, $\tilde{\tau}=2t_{0}+\tau$ is the pulse duration and
$U^{(\mathbf{j})}_{ij}(\cdot,\cdot)$ represents the three-body evolution
operator (in terms of the field-free basis) during a single pulse.

Next, the thermally averaged quantity in Eq. 10a is averaged also over the
one-body density profile of the thermal gas [48] according to

$\braket{\mathbb{P}_{T}(t_{d})}=\dfrac{\sum_{\mathbf{j}}\Delta\bm{x}^{(\mathbf{j})}n(\bm{x}^{(\mathbf{j})})\mathbb{P}^{(\mathbf{j})}_{T}(t_{d})}{\sum_{\mathbf{j}}\Delta\bm{x}^{(\mathbf{j})}n(\bm{x}^{(\mathbf{j})})}.$
(11)

It is convenient to perform two variable substitutions
$\left|\bm{x}^{(\mathbf{j})}\right|\to
n^{(\mathbf{j})}\to\omega_{r}^{(\mathbf{j})}$ and express Eq. 11 with respect
to the trap frequency yielding the following relation

$\braket{\mathbb{P}_{T}(t_{d})}=\dfrac{\sum_{\mathbf{j}}\Delta\omega_{r}^{(\mathbf{j})}\sqrt{\omega_{r}^{(\mathbf{j})}}\sqrt{\ln(\omega_{r}^{(\mathbf{1})}/\omega_{r}^{(\mathbf{j})})}\mathbb{P}^{(\mathbf{j})}_{T}(t_{d})}{\sum_{\mathbf{j}}\Delta\omega_{r}^{(\mathbf{j})}\sqrt{\omega_{r}^{(\mathbf{j})}}\sqrt{\ln(\omega_{r}^{(\mathbf{1})}/\omega_{r}^{(\mathbf{j})})}}.$
(12)

For the above formula the explicit one-body density profile [Eq. (1)] has been
employed, as well as the connection between the trap frequency and the density
introduced before Eq. (2). Importantly, Eq. 12 incorporates the thermal and
density effects on the population of Efimovian trimer states induced by the
Ramsey-type dynamics induced through Eq. (2).

Figure 3: Position of the highest frequency peak originating from the
thermally averaged probability $\braket{\mathbb{P}_{T}(t_{d})}$ within the
three-level model with respect to the frequency related to the smallest
density, $\omega^{(\mathbf{c})}_{r}$. Note that $\omega^{(\mathbf{1})}_{r}$ is
associated to the peak density. The dash-dotted (dashed) horizontal line
corresponds to the energy difference between $E^{(2)}_{T}$ and $E^{(1)}_{A}$
($E^{(2)}_{T}$ and the energy of the three-atom dissociation threshold in free
space).

##  3 Effect of the thermal gas density distribution

To highlight the impact of the density profile of a thermal gas, the thermally
averaged probability of trimers $\braket{\mathbb{P}_{T}(t_{d})}$ (see Eq. 12)
obtained within the full three-body model is compared with the thermally
averaged probability ratio pertaining only to $\omega^{(\mathbf{1})}_{r}$,
$\mathbb{P}^{(\mathbf{1})}_{T}(t_{d})$, (see Eq. 10a) at $\mathcal{T}=270\,K$
[blue versus yellow lines in Fig. 2(a)]. Clear oscillatory fringes manifest in
both probability ratios, and the peak-to-peak amplitude is slightly smaller in
the case where the density profile of a thermal gas is taken into account, due
to the density average [Eq. (12)]. In line with the results presented in Ref.
[37], two distinct regimes appear during the free evolution time $t_{d}$. The
first one [region $I$ in Fig. 2(a)], refers to high-amplitude high-frequency
oscillations. At later dark times the amplitude drops due to thermal effects,
revealing low-frequency low-amplitude oscillations [region $II$ in Fig. 2(a)].

To assess and characterize the participating frequencies, the spectrum of the
$\braket{\mathbb{P}_{T}(t_{d})}$ signal is presented in Fig. 2(b). To
understand the role of the density profile on the dynamical trimer
occupations, the three dominant frequencies pertaining to
$\mathbb{P}^{(\mathbf{1})}_{T}(t_{d})$ in region $II$ are also provided with
vertical dashed yellow lines in Fig. 2(b). The lower one at $6.5\,\rm{kHz}$
corresponds to the difference between the first excited Efimov state and the
first atom-dimer [37]. A superposition of two such states was identified in
Refs. [35, 36], where the Ramsey-type interferometer has been employed for 7Li
at repulsive interactions. The resulting oscillatory fringes then allowed the
precise measurement of the binding energy of the Efimov state. At the high
frequency end of the spectrum, two pronounced peaks appear at $63.8\,\rm{kHz}$
and $70.4\,\rm{kHz}$. These refer to $\left|E^{(1)}_{AD}-E^{(1)}_{A}\right|/h$
and $\left|E^{(2)}_{T}-E^{(1)}_{A}\right|/h$ respectively [vertical dashed
yellow lines in Fig. 2(b)]. When analyzing the frequency spectrum in region
$I$ though, a single pronounced peak appears, referring to
$\left|E^{(2)}_{T}-E^{(1)}_{A}\right|/h$. It is therefore possible to measure
the binding energy of Efimov trimers without requiring an atom-dimer as a
reference state. This behavior grants the applicability of the interferometry
protocol to attractive interactions as well, where atom-dimers are absent.

Turning to the frequency spectrum stemming from
$\braket{\mathbb{P}_{T}(t_{d})}$ [blue solid line in Fig. 2(b)], a low
frequency peak appears also in the same position as for
$\mathbb{P}^{(\mathbf{1})}_{T}(t_{d})$, i.e.
$\left|E^{(1)}_{AD}-E^{(2)}_{T}\right|/h$. Essentially, the density profile of
a semi-classical thermal gas does not induce any significant shift in the
frequency related to the superposition of the Efimov trimer with the atom-
dimer. This outcome could contribute to understanding the impact of an
interacting thermal gas on the interference fringes [49]. The two high
frequencies, however, are slightly shifted to lower values with respect to the
spectra associated to the peak density [yellow vertical dashed lines in Fig.
2(b)]. In fact, they are shifted to values close to
$\left|E^{(2)}_{T}\right|/h$ and $\left|E^{(1)}_{AD}\right|/h$ [vertical black
dash-dotted lines in Fig. 2(b)].

Such frequencies correspond to the energy difference of the Efimov trimer and
atom-dimer with the energy of the three-atom dissociation threshold in free
space [7]. Since the two former states are deeply bound, their energy has a
weak dependence on the trap, assuming almost the same value in free space. On
the other hand, the energy of the first three-atom state highly depends on the
trap frequency, and in the non interacting case it explicitly reads
$3\hbar\omega^{(\mathbf{j})}_{r}$. Therefore, similarly to a thermal average
[37], the density average in Eq. (12) smears out all contributions stemming
from three-atom states. Eventually only oscillatory fringes survive, with
frequencies associated to the energies of the trimer and first atom-dimer.
Since the two latter states are deeply bound, the above mentioned frequency
shifts are small. In order to observe such shifts, the $F(\omega)$ spectrum is
shown in Fig. 2(b) for both regions $I$ and $II$. In region $I$, a single
dominant peak arises due to the superposition of the first excited trimer and
the first three-atom state pertaining to $\omega^{(1)}_{r}$, i.e. associated
to the peak density $n_{0}$. At such small dark times $t_{d}$, mostly high
frequencies are contributing to the density average in Eq. (12), and thus the
impact of lower trap frequencies is not appreciable in this region.

Subsequently, the dependence of the above identified shifts on the frequency
$\omega^{(\mathbf{c})}_{r}$ corresponding to the lowest density of the thermal
gas is investigated. This is carried out within the three-level model, which
was shown to be an adequate qualitative description of the dynamical
association processes in Ref. [37]. Moreover, it can qualitatively capture the
characteristics of the frequency spectrum presented in Fig. 2(b). The main
premise of the model is to take into account only three states out of the
entire spectrum [Fig. 1]; the first excited Efimov trimer, the first atom-
dimer and the first three-atom state. The three possible superpositions of
pairs of these states give rise to the three dominant frequencies in the
spectra [vertical dashed yellow lines in Fig. 2(b)]. Here, we extend the
three-level model to three particles trapped with frequencies associated to
densities lower than $n_{0}$. Essentially, for all distinct
$\omega^{(\mathbf{j})}_{r}$, only the three-atom states are significantly
different in energy out of the three levels. The resulting
$\mathbb{P}^{(\mathbf{j})}_{T}(t_{d})$ are subsequently averaged according to
Eq. (12).

The position of the high frequency peak $\omega/(2\pi)$ close to
$70\,\rm{kHz}$ stemming from the frequency spectrum of the extended three-
level model is then investigated with respect to
$\omega^{(\mathbf{c})}_{r}/\omega^{(\mathbf{1})}_{r}$ [blue solid line in Fig.
3]. The horizontal dash-dotted and dashed lines are associated to the energy
gaps of the trimer with the first three-atom state
($\left|E^{(2)}_{T}-E^{(1)}_{A}\right|/h$) and the trimer with the three-atom
threshold ($\left|E^{(2)}_{T}\right|/h$) respectively. These values act as a
reference for identifying changes in the peak position. When
$\omega^{(\mathbf{c})}_{r}/\omega^{(\mathbf{1})}_{r}\simeq 1$, a frequency
similar to $\left|E^{(2)}_{T}-E^{(1)}_{A}\right|/h$ is identified [dash-dotted
line in Fig. 3]. As $\omega^{(\mathbf{c})}_{r}$ further diminishes,
$\omega/(2\pi)$ decreases as well in a step-wise manner. This behavior is due
to the finite resolution of the Fourier transform, which cannot capture small
gradual frequency changes for the final dark times considered here. For
$\omega^{(\mathbf{c})}_{r}/\omega^{(\mathbf{1})}_{r}\lesssim 0.7$, the
frequency reaches a plateau below $\left|E^{(2)}_{T}-E^{(1)}_{A}\right|/h$,
but above the frequency associated to just the energy of the Efimov trimer
[dashed line in Fig. 3]. This suggests that as the lowest density dependent
frequency $\omega^{(\mathbf{c})}_{r}$ further decreases past this threshold,
the high-frequency peak shifts towards the value $\left|E^{(2)}_{T}\right|/h$.

##  4 Conclusions

We have addressed the impact of the density profile of a thermal gas on the
dynamical association of Efimov trimers by means of a Ramsey-type protocol
[35, 36]. To this end, the time-dependent three-body problem is solved for an
ensemble of density dependent trapping frequencies. The latter correspond to a
range of density values, spanning the density profile of a thermal gas. In
consistence with the results observed exclusively for the peak density [37],
three dominant frequencies appear in the frequency spectrum of the normalized
probability to occupy Efimov trimers. The lowest one corresponds to the
superposition of the first excited Efimov trimer with the first atom-dimer,
being the same as the one stemming only from the peak density. Essentially,
the density profile of a thermal gas does not affect the frequency associated
to this superposition, which was instrumental in measuring the binding energy
of Efimov trimers [35, 36]. On the other hand, the two high frequencies are
slightly shifted from the values pertaining only to the peak density, i.e. the
energy difference between the trimer and the first three-atom state, and
between the latter and the first atom-dimer. It is shown that the high
frequencies are slightly shifted to lower values, towards the energies of the
trimer and the first atom-dimer in free space. Such a behavior is attributed
to the smearing effect that the density average has on the highly-oscillatory
fringes associated with three-atom states.

A further question that arises from our study is whether the density profile
of a thermal gas will affect the decay times of the oscillatory fringes due to
the lifetime of Efimov states. Concretely, whether the decay times will be
twice as long as the lifetime of trimers, a result established only in the
case where the peak density is considered [37]. Moreover, an important future
aspect of our work is to address the impact of the medium on Efimov states.
Such a study requires not only the consideration of the density profile, but
also the role of interactions and correlations of the host gas.

Acknowledgements

We acknowledge useful discussions with J. P. D’Incao.

## Declarations

  * •

Conflict of interest/Competing interests The authors declare no conflict of
interest.

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