# Floquet engineering a bosonic Josephson junction

Si-Cong Ji  Vienna Center for Quantum Science and Technology (VCQ),
Atominstitut, TU Wien, Vienna, Austria  Thomas Schweigler  Vienna Center for
Quantum Science and Technology (VCQ), Atominstitut, TU Wien, Vienna, Austria
JILA, University of Colorado, Boulder, Colorado, USA  Mohammadamin Tajik
Federica Cataldini  Vienna Center for Quantum Science and Technology (VCQ),
Atominstitut, TU Wien, Vienna, Austria  João Sabino  Vienna Center for Quantum
Science and Technology (VCQ), Atominstitut, TU Wien, Vienna, Austria
Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal
Instituto de Telecomunicações, Physics of Information and Quantum Technologies
Group, Lisbon, Portugal  Frederik S. Møller  Sebastian Erne  Jörg Schmiedmayer
<EMAIL_ADDRESS>Vienna Center for Quantum Science and Technology
(VCQ), Atominstitut, TU Wien, Vienna, Austria

###### Abstract

We study Floquet engineering of the tunnel coupling between a pair of one-
dimensional bosonic quasi-condensates in a tilted double-well potential. By
modulating the energy difference between the two wells, we re-establish tunnel
coupling and precisely control its amplitude and phase. This allows us to
initiate coherence between two initially uncorrelated Bose gases and prepare
different initial states in the emerging sine-Gordon Hamiltonian. We fully
characterize the Floquet system and study the dependence of both equilibrium
properties and relaxation on the modulation.

Introduction.—Periodic driving, i.e. Floquet engineering, offers a wide range
of pathways to design effective Hamiltonians which are otherwise difficult to
achieve or even unrealizable in static laboratory systems [1, 2, 3]. For
ultracold atoms, Floquet engineering has been used in optical lattices to
coherently control tunneling between sites [4, 5, 6, 7, 8], applied to
generate large artificial gauge fields [9, 10, 11] and to simulate the quantum
Hall effect [12, 13, 14, 15]. Proposals for continuous systems range from
implementing the Pokrovsky-Talapov model [16, 17, 18], used to describe the
Commensurate-Incommensurate phase transition [19], to analogue simulators for
pre-heating [20] and false-vacuum decay [21]. Experimental studies on
continuous interacting many-body Floquet systems and their equilibration
process are, however, limited. In this letter we present an experimental study
of Floquet engineering tunneling and phase locking in a pair of continuous
one-dimensional superfluids in a tilted double well (DW).

Experimental setup and Floquet engineering— Our experiment (Fig. 1) starts
with a pair of tunnel-coupled one-dimensional (1D) Bose gases of 87Rb atoms
trapped in a DW potential created on an atom chip [22, 23] by radio-frequency
(RF) dressing [24, 25]. For both wells, the trapping frequencies are
$\omega_{\bot}=2\pi\times 1.4$ kHz transversely and
$\omega_{\mathrm{z}}=2\pi\times 10$ Hz longitudinally. We prepare the system
(total atom number $N\approx 10^{4}$, peak density $\rho_{0}\approx 50$
µ$\mathrm{m}^{-1}$) through evaporative cooling in a balanced DW potential.
The initial temperature of the samples is $T_{\text{i}}\approx
37(5)\,\mathrm{nK}$ as measured by the two-point density correlation function
after $11.2\,\mathrm{ms}$ Time-of-Flight (ToF) [26]. In the longitudinal
direction, the spatially resolved relative phase between the superfluids is
extracted from matter-wave interference after $15.6\,\mathrm{ms}$ ToF. For the
further analysis we use the central $50\,$µ$\mathrm{m}$ (variation of density
less than 15%). Expectation values are calculated by averaging over 30-60
experimental realizations.

Figure 1:  (a) Two atomic clouds in a tilted double-well potential, where
$\Delta E$ is the energy difference between the two minima. For $\Delta E=0$
the barrier results in tunnel coupling $J$, completely suppressed for $\Delta
E\gg\hbar\omega_{J}$ (see text). Floquet engineering is realized by
periodically modulating the energy difference $\Delta E$ with an amplitude
$V_{\mathrm{drive}}$. (b) Typical matter-wave interference pattern after
$15.6\,\mathrm{ms}$ Time-of-Flight with atom density depicted in color and (c)
the extracted relative phase $\phi_{r}(z)$ between the two superfluids.

The RF dressing allows precise control over the DW potential, determining the
barrier height, i.e. the tunnel coupling $J$, and the potential energy
difference $\Delta E$ between the two wells (Fig. 1(a)), thereby realizing an
extended Josephson junction (JJ). For $\Delta E=0$ such a system is a quantum
simulator for the sine-Gordon (SG) quantum field theory [27, 28, 29]. The
Josephson frequency $\omega_{J}=\sqrt{4J\mu/\hbar}$ is determined by the on-
site interaction energy $\mu$ and the tunneling strength $J$ in the balanced
(untilted) DW. Tunneling between the two superfluids can be completely
suppressed by tilting the DW with an energy difference $\Delta
E\gg\hbar\omega_{J}$. Floquet engineering enables us to revive the tunnel
coupling within the tilted DW through a periodic modulation $\Delta
E(t)=\Delta E_{0}+2V_{\mathrm{drive}}\operatorname{sin}(\omega
t+\varphi_{\mathrm{drive}})$.

If the modulation frequency $\omega$ is near resonant with $\Delta E_{0}$,
Floquet assisted tunneling will re-couple the two superfluids, even if the
amplitude $V_{\mathrm{drive}}\ll\Delta E_{0}$. After time-averaging over this
fast modulation, the system can effectively be treated as a static balanced
DW. In the rotating Floquet frame, the time averaged Hamiltonian in the two-
mode approximation is given by [30, 29]

$\displaystyle\langle\widetilde{H}(t)\rangle=-\hbar\widetilde{J}\left(\begin{array}[]{cc}0&\mathrm{e}^{-i(\varphi_{\mathrm{drive}}+\pi/2)}\\\
\mathrm{e}^{+i(\varphi_{\mathrm{drive}}+\pi/2)}&0\end{array}\right)~{},$ (1)

where
$\widetilde{J}=J\times\mathcal{B}_{1}(\frac{2V_{\mathrm{drive}}}{\hbar\omega})$
is the revived effective tunneling strength and $\mathcal{B}_{1}$ is the first
order Bessel function. The ground state of the Hamiltonian (1) takes the form

$\displaystyle\lvert\widetilde{\Psi}_{\mathrm{ground}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\operatorname{exp}[-i(\frac{\varphi_{\mathrm{drive}}}{2}+\frac{\pi}{4})]\\\
\operatorname{exp}[+i(\frac{\varphi_{\mathrm{drive}}}{2}+\frac{\pi}{4})]\end{array}\right)\;~{},$
(2)

The amplitude and phase of the tunneling term in the Floquet Hamiltonian Eq.
(1) can be controlled through the amplitude $V_{\mathrm{drive}}$ and phase
$\varphi_{\mathrm{drive}}$ of the modulation. For the ground state Eq. (2),
the latter results in a non-zero relative phase
$\widetilde{\phi}_{\mathrm{r}}=\varphi_{\mathrm{drive}}+\pi/2$ between the two
superfluids when $\varphi_{\mathrm{drive}}\neq-\pi/2$. Transforming the
relative phase $\widetilde{\phi}_{\mathrm{r}}$ in the rotating Floquet frame
back to the laboratory frame, the measured relative phase between the two
wells is given by

$\phi_{\mathrm{r}}=\widetilde{\phi}_{\mathrm{r}}+\Delta
E_{0}t/\hbar-\frac{2V_{\mathrm{drive}}}{\hbar\omega}\operatorname{cos}(\omega
t+\varphi_{\mathrm{drive}})~{}.$ (3)

Hence, when $\varphi_{\mathrm{drive}}=-\pi/2$, the initial state with
$\phi_{r}=0$ in lab frame is also the ground state with
$\widetilde{\phi}_{r}=0$ in the Floquet frame. On the other hand, for an
arbitrary modulation phase $\varphi_{\mathrm{drive}}$, the state $\phi_{r}=0$
is, in the Floquet frame, mapped to a JJ with a non-vanishing starting phase
$\Delta\widetilde{\phi}_{r,0}=\frac{2V_{\mathrm{drive}}}{\hbar\omega}\operatorname{cos}(\varphi_{\mathrm{drive}})-\varphi_{\mathrm{drive}}-\pi/2$.
For later convenience, we introduce
$\Delta\widetilde{\phi}_{r}=\widetilde{\phi}_{r}-\varphi_{\mathrm{drive}}-\pi/2$,
which shifts the ground state of the JJ back to zero phase. The ability to
tune the coupling strength and initial relative phase of such an extended
bosonic JJ establishes the building block for more elaborate Floquet
engineering.

Floquet assisted tunneling.— We first consider the simplest case, preparing
the initial state as the ground state in the Floquet frame, i.e. choosing the
modulation phase $\varphi_{\mathrm{drive}}=-\pi/2$: We start in a strongly
phase locked initial state with $\phi_{\mathrm{r}}\approx 0$, prepared by
cooling into a strongly coupled, balanced DW. We then completely suppress the
tunneling between the two wells rapidly by introducing an energy difference
$\Delta E_{0}=h\times 411\,\mathrm{Hz}\gg\hbar\omega_{J}$ and, at the same
time, begin the periodic modulation $\Delta E(t)$. We choose the driving
frequency $\omega=\Delta E_{0}/\hbar$ resonant with the energy detuning and a
driving amplitude $V_{\mathrm{drive}}=h\times 85\,\mathrm{Hz}$, leading to an
expected revived tunneling strength of $\widetilde{J}\approx 0.2J$.

Figure 2:  (left) Floquet driven DW: $\Delta E=h\times 411\>$ Hz with Floquet
drive $\omega=2\pi\times 411$ Hz, starting phase
$\varphi_{\mathrm{drive}}=-\pi/2$ and modulation amplitude
$V_{\mathrm{drive}}=h\times 85\>$ Hz: (a) Evolution of the relative phase
$\phi_{\mathrm{r}}$, calculated via the ensemble and spatially averaged
circular mean of $\phi_{\mathrm{r}}(z)$ and shifted by $2\pi\mathbb{N}$; (b)
difference $\Delta\widetilde{\phi}_{\mathrm{r}}$ between the measured relative
phase $\phi_{\mathrm{r}}$ and the estimate from Floquet theory; (c) Coherence
factor. (right) (d-f) Comparison to a static tilted DW with $\Delta E=h\times
258$ Hz. Blue dots: experimental data. Red line: prediction from Floquet
theory. The insets in (a) and (d) show the experimental distribution of
$\Delta\widetilde{\phi}_{\mathrm{r}}$(z) and the light green region in (b),
(d) and (e) display its standard deviation. The error bars give the standard
error of the mean.

In Figure 2 we present such a Floquet assisted tunneling experiment (left
column) and compare it to the static tilted DW (right column). Figure 2(a)
shows the time evolution for the spatially averaged relative phases (blue
dots), in good agreement with the theoretical predictions (red line) given by
Eq. (3) with $\widetilde{\phi}_{\mathrm{r}}=0$. The insets show the full
distribution functions of $\widetilde{\phi}_{\mathrm{r}}$ for the initial and
final states. The observed broadening of the distribution reflects the
relaxation in the SG model following a quench to a lower coupling
$\widetilde{J}$. The small, random deviations $\Delta\widetilde{\phi}_{r}$
(Fig. 2(b)) represent the fluctuations of the field within the Floquet
engineered SG Hamiltonian.

The revived tunneling strength $\widetilde{J}$ can be quantified via the
coherence factor
$\mathcal{C}=\langle\operatorname{cos}(\widetilde{\phi}_{\mathrm{r}})\rangle$,
depicted in Fig. 2(c). The initial state with $\phi_{\mathrm{r}}\approx 0$
shows almost perfect phase coherence $\mathcal{C}>0.9$, whereas for
$\mathcal{C}\approx 0$ the relative phase is completely random. At early
times, $t\lesssim 5\,\mathrm{ms}$, we observe a fast decrease of coherence due
to quasi-particle dephasing caused by the quench to a smaller effective
tunneling coupling $\widetilde{J}$. For $t\gtrsim 30\,\mathrm{ms}$ the system
reaches a quasi-steady state with an average coherence factor
$\mathcal{C}\approx 0.38$ (Fig. 2(c)). This demonstrates that the system
retains finite coherence due to a non-vanishing Floquet assisted tunneling
coupling, even after relaxation.

For comparison we show in Fig. 2(d)-(f) the evolution for a static tilted DW
with $\Delta E_{0}=h\times 258\,\mathrm{Hz}$ and no periodic modulation (i.e.
$V_{\mathrm{drive}}=0$). The evolution of the relative phase within the first
$10\,\mathrm{ms}$ is in good agreement with Eq. (3). The constant energy
difference $\Delta E_{0}$ only leads to a monotonic accumulation of a relative
phase. For $t\gtrsim 20\,\mathrm{ms}$ the total suppression of the initial
tunneling leads to a fully random phase distribution (Fig. 2(d), inset) and
consequently a vanishing coherence factor $\mathcal{C}\approx 0$. The detuning
$\Delta E_{0}$ was chosen to be the minimal energy difference between the two
wells in the periodically modulated case. Consequently $\mathcal{C}\to 0$ in
Fig. 2(f) verifies the complete suppression of static coupling in the Floquet
system at any time and $\mathcal{C}\gg 0$ in Fig. 2(c) can only come from
tunneling revived through Floquet engineering.

In the next experiment, we demonstrate that Floquet engineering can be used to
recouple two independent superfluids, leading to the build up of coherence
between the two wells. As before, we prepare the system in a balanced DW, but
now with a high barrier and an initial coupling $J\approx 0$. Hence, the
initial state after cooling is completely uncorrelated, i.e. the relative
phase distribution is fully random and $\mathcal{C}\approx 0$. We subsequently
lower the barrier height between the two wells while at the same time
suppressing the coupling $J$ by introducing an energy difference $\Delta
E_{0}=h\times 380\,\mathrm{Hz}$. The reshaping of the DW potential is done
within $5\,\mathrm{ms}$, short compared to the tunneling time ($\sim
2\pi/\omega_{J}$) but sufficiently long to avoid radial excitations ($\sim
2\pi/\omega_{\bot}$). Thereafter, we proceed with our usual Floquet modulation
with frequency $\omega=\Delta E_{0}/\hbar$ and amplitude
$V_{\mathrm{drive}}=h\times 190\,\mathrm{Hz}$ which recouples the DW.

Figure 3:  The evolution of (a) relative phase $\phi_{r}$ and (b) coherence
factor $\mathcal{C}$ of two initially uncorrelated Bose gases after switching
on Floquet assisted tunneling. The driving is resonant with the energy
difference of the DW: $\omega=2\pi\times 380\,\mathrm{Hz}$ and its amplitude
is: $V_{\mathrm{drive}}=h\times 190\,\mathrm{Hz}$.

Figure 3 shows the time evolution of the relative phase $\phi_{r}$ and the
coherence factor $\mathcal{C}$ for such a Floquet recoupling experiment. The
evolution of the relative phase $\phi_{\mathrm{r}}$ shows good agreement with
the theoretical predictions. The initial distribution of
$\widetilde{\phi}_{\mathrm{r}}$ is uniform but rapidly narrows around zero
(see insets), illustrating the buildup of coherence between the two wells.
This is also clearly visible in the increase of the coherence factor
$\mathcal{C}$ depicted in Fig. 3(b). Within the first $5\,\mathrm{ms}$,
corresponding to only 2 periods of the modulation, $\mathcal{C}$ increases to
its plateau value $\mathcal{C}\approx 0.5$. This demonstrates fast phase
locking by tunneling restored through Floquet engineering.

Tunneling strength and heating.— Having established Floquet engineered
tunneling, we proceed to a detailed experimental analysis of the tunneling
strength $\widetilde{J}$ and the long time evolution of the system. Figure
4(a-f) shows the time evolution of the coherence factor $\mathcal{C}$ for six
different driving amplitudes $V_{\mathrm{drive}}/h$ ranging from 0 to
$155\,\mathrm{Hz}$. The experimental sequence and all other parameters are the
same as for Fig. 2(a-c). Qualitatively, the increase of the Floquet engineered
coupling $\widetilde{J}$ is readily apparent from the increase of
$\mathcal{C}$ with the amplitude $V_{\mathrm{drive}}$.

The time evolution shown in Fig. 4(b-f) can clearly be divided into two
stages: Within the first $5\,\mathrm{ms}$, the quench of the tunneling
coupling from ${J}$ to $\widetilde{J}<J$ leads to quasi-particle dephasing and
results in a rapid decline of $\mathcal{C}$. This is consistent with previous
observations for the relaxation following a quench in static DW potentials
[31, 32, 33]. In the second stage, $t\gtrsim 5\,\mathrm{ms}$, we observe a
further slow decline of coherence due to heating.

Figure 4:  (a-f). The evolution of the coherence factor $\mathcal{C}$ under
Floquet driving with different amplitudes. The red lines are linear fits
highlighting the two different time regimes discussed in the text. (g) The
heating rates for the driven systems in (b-f). (h) Dependence of the Floquet
induced Josephson tunneling on the driving amplitude. Blue dots: experiment;
light green region: estimation from Floquet theory.

We self-consistently extract the Floquet tunneling coupling $\widetilde{J}$
and the time dependent temperature (Fig. 4(g)) by comparing the measured
correlation functions after dephasing to SG model predictions in thermal
equilibrium [34, 28, 35] under the assumption that the Floquet tunneling
strength $\widetilde{J}$ stays constant during the driving. Remarkably the
derived heating rate $\Gamma\sim 0.6\,\mathrm{nK/}$µ$\mathrm{s}$ is rather
independent of the driving amplitude $V_{\mathrm{drive}}$ (see S1).

Equivalently, using the initial temperature and correlation functions we
determine the initial tunneling coupling $J=23\pm 5\,s^{-1}$ in the initial
balanced DW. In Fig. 4(h) we compare the measured $\widetilde{J}$ to the
calculated Floquet tunneling strengths Eq. (1). We find good agreement between
experimental results and Floquet theory.

The common mode temperature of the system before and after $30\,\mathrm{ms}$
of Floquet driving was measured by density correlation functions after
$11.2\,\mathrm{ms}$ ToF [26]. We found no significant heating in the common
mode (see S1 and [29] for details).

Josephson oscillations and relaxation.— Finally we consider different driving
phases $\varphi_{\mathrm{drive}}$ of the modulation, which imprints an initial
relative phase difference $\Delta\widetilde{\phi}_{\mathrm{r},0}$ directly in
the Floquet frame. This realizes an extended JJ and permits precise control
over the initial conditions. In the experiments we implement the same
experimental Floquet sequence as before, starting from a strongly tunneling
coupled state with $\phi_{\mathrm{r}}\approx 0$ and $\mathcal{C}\approx 0.9$,
detuning $\Delta E_{0}=h\times 436\,\mathrm{Hz}$, and driving amplitude
$V_{\mathrm{drive}}=h\times 93\,\mathrm{Hz}$, but now considering different
driving phases $\varphi_{\mathrm{drive}}=0,\,\pm\pi/6,\,\pm\pi/3,\,\pm\pi/2$.
Using Eq. (3), we get the initial phase differences
$\Delta\widetilde{\phi}_{\mathrm{r,0}}\in[-\pi,0]$.

Figure 5(a-f) shows the time evolution of the relative phase for different
$\varphi_{\mathrm{drive}}$. In all cases we find strongly damped Josephson
oscillations, in accordance with previous experiments in static DW potentials
[36].

Figure 5:  (a-f) Josephson oscillations starting from different initial
relative phases $\Delta\widetilde{\phi}_{r,0}$ in the rotating Floquet frame.
The red line is fitted with the assumption that the Josephson frequencies
$\omega_{J}$ and the decay rates $\tau$ are independent of the initial phase
$\Delta\widetilde{\phi}_{r,0}$. (g) Coherence factor and (h) temperature of
the system after $20\,\mathrm{ms}$ of driving for different initial relative
phase $\Delta\widetilde{\phi}_{r,0}$.

We quantify the evolution by fitting a damped oscillation in order to extract
the Josephson frequency $\omega_{J}$ and the characteristic damping time
$\tau$. From the previous section, we expect a Floquet tunneling strength
$\widetilde{J}\approx 4.8\,s^{-1}$ and $\omega_{J}\approx 2\pi\times
46\,\mathrm{Hz}$, independent of $\varphi_{\mathrm{drive}}$. We therefore fit
the experimental data with a single Josephson frequency and damping time,
leading to $\omega_{J}^{\mathrm{fit}}=2\pi\times 40(3)\,\mathrm{Hz}$ and
$\tau^{\mathrm{fit}}=8.8(1.2)\,\mathrm{ms}$ respectively. Note that
$\omega_{J}^{\mathrm{fit}}$ agrees with the theoretical expectations within
the statistical error, thus being in good agreement with the exact Floquet
predictions for $\widetilde{J}$. The damping time $\tau^{\mathrm{fit}}$ is
compatible with damping times observed in static DW potentials [36, 37].

We consistently observe relaxation to a stationary state after $t\gtrsim
20\,\mathrm{ms}$. This indicates transfer of the initial potential energy,
driving the coherent Josephson oscillation, into fluctuations of the relative
phase [38]. In Fig. 5(g,h) we show the time-averaged coherence factor
$\mathcal{C}$ and final temperature of the system for $t>20\,\mathrm{ms}$,
respectively. The decrease (increase) of coherence (temperature) with
$\Delta\widetilde{\phi}_{\mathrm{r},0}$ reflects the increased potential
energy
$\sim\widetilde{J}\operatorname{cos}(\Delta\widetilde{\phi}_{\mathrm{r,0}})$
of the initial state. Notably, we again find the common mode temperatures to
remain approximately constant throughout the evolution.

Conclusion.— Floquet engineering allows to revive the tunnel coupling in a
tilted double well, creating an extended bosonic Josephson junction. We find
excellent quantitative agreement between the experimental results and
theoretical predictions. The periodic modulation technique developed in this
work will greatly expand the freedom of manipulating such double well systems
and open up new ways to study many intriguing fundamental phenomena with broad
relevance ranging from condensed matter physics to cosmology.

###### Acknowledgements.

We thank I. Mazets, B. Rauer and M. Serbyn for helpful discussions. This work
is supported by the DFG/FWF Collaborative Research Centre ‘SFB 1225
(ISOQUANT)’, and by the Wiener Wissenschafts- und TechnologieFonds (WWTF),
Project No.MA16-066 (SEQUEX). F.C., F.S.M., and J. Sabino acknowledge support
by the Austrian Science Fund (FWF) in the framework of the Doctoral School on
Complex Quantum Systems (CoQuS). T.S. acknowledges support from the Max Kade
Foundation through a postdoctoral fellowship. S.-C.J. and S.E. acknowledges
support through an ESQ (Erwin Schrödinger Center for Quantum Science and
Technology) fellowship funded through the European Union’s Horizon 2020
research and innovation program under Marie Skłodowska-Curie Grant Agreement
No 801110. This project reflects only the author’s view, the EU Agency is not
responsible for any use that may be made of the information it contains. ESQ
has received funding from the Austrian Federal Ministry of Education, Science
and Research (BMBWF). J. Sabino acknowledges support by the Fundação para a
Ciência e a Technologia (PD/BD/128641/2017).

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SUPPLEMENTAL MATERIAL

$V_{\mathrm{drive}}/h$ (Hz) |  $J_{\mathrm{exp}}$ ($s^{-1}$) |  $J_{\mathrm{Flo}}$ ($s^{-1}$) |  $T_{\mathrm{r}}^{f}$ (nK) |  $\Gamma$ (nK/ms) |  $T_{\mathrm{c}}^{f}$ (nK)
---|---|---|---|---|---
31 |  3.2(0.7) |  1.7(0.4) |  42(3) |  0.50 |  35(3)
62 |  3.7(0.9) |  3.4(0.7) |  32(2) |  0.50 |  34(5)
93 |  4.8(1.1) |  5.0(1.1) |  37(3) |  0.60 |  26(4)
124 |  6.7(1.7) |  6.6(1.4) |  32(2) |  0.69 |  28(5)
155 |  7.9(1.9) |  8.1(1.7) |  35(3) |  0.64 |  38(5)
Table S1:  Results for different modulation amplitudes $V_{\mathrm{drive}}$
corresponding to Fig. 4. Depicted values are the experimentally measured
($J_{\mathrm{exp}}$) and theoretically expected ($J_{\mathrm{Flo}}$, see Eq.
(S10)) tunneling coupling, the heating rate ($\Gamma$), and the final
temperature for both the relative ($T_{\mathrm{r}}^{f}$) and common
($T_{\mathrm{c}}^{f}$) degrees of freedom.

##  Appendix A Floquet Hamiltonian engineering of DW system

The derivation of the Floquet Hamiltonian for our DW system is based on the
single-particle picture

$\displaystyle\widehat{H}(t)$ $\displaystyle=\widehat{H}_{0}+\widehat{H}_{t}$
(S1) $\displaystyle=\left(\begin{array}[]{cc}\Delta E_{0}&-\hbar J\\\ -\hbar
J&0\end{array}\right)+V_{\mathrm{drive}}\sin(\omega
t+\varphi_{\mathrm{drive}})\cdot\sigma_{\mathrm{z}}~{},$ (S4)

where $J$ is the tunneling strength in the balanced double well, $\Delta
E_{0}$ is the energy difference between the two trap bottoms, $\omega$ is the
driving frequency, $\varphi_{\mathrm{drive}}$ is the starting phase of the
Floquet modulation, and $\sigma_{\mathrm{z}}$ is the Pauli matrix.

To simplify this Hamiltonian, we perform a unitary transformation to the
Floquet frame

$\widetilde{H}(t)=\widehat{R}(t)\widehat{H}(t)\widehat{R}^{\dagger}(t)-i\hbar\widehat{R}(t)\frac{d\widehat{R}^{\dagger}(t)}{dt}~{},$
(S5)

with the unitary operator

$\widehat{R}(t)=\left(\begin{array}[]{cc}e^{i\frac{\Delta
E_{0}}{\hbar}t-i\mathcal{V}(t)}&0\\\
0&e^{+i\mathcal{V}(t)}\end{array}\right)~{}.$ (S6)

Here we defined

$\mathcal{V}(t)=\frac{V_{\mathrm{drive}}}{\hbar\omega}\cos(\omega
t+\varphi_{\mathrm{drive}})$ (S7)

to shorten the notation. This eliminates the diagonal term in Eq. (S1) leading
to:

$\widetilde{H}(t)=-\hbar
J\left(\begin{array}[]{cc}0&\mathrm{e}^{+i\frac{\Delta
E_{0}}{\hbar}t-i2\mathcal{V}(t)}\\\ \mathrm{e}^{-i\frac{\Delta
E_{0}}{\hbar}t+i2\mathcal{V}(t)}&0\end{array}\right)~{}.$ (S8)

Based on the Floquet theory, we can achieve the effective Hamiltonian by
calculating the time-average of Eq. (S8) in one modulation period. When the
driving frequency is resonant with the energy difference, i.e.
$\hbar\omega=\Delta E_{0}$, the time-averaged Floquet Hamiltonian is given by

$\langle\widetilde{H}(t)\rangle=-\hbar\widetilde{J}\left(\begin{array}[]{cc}0&e^{-i(\varphi_{\mathrm{drive}}+\pi/2)}\\\
e^{+i(\varphi_{\mathrm{drive}}+\pi/2)}&0\end{array}\right)~{},$ (S9)

where

$\widetilde{J}=J\cdot\mathcal{B}_{1}(\frac{2V_{\mathrm{drive}}}{\hbar\omega})$
(S10)

is the effective tunneling strength determined by the first order Bessel
function
$\mathcal{B}_{1}(x)=\frac{1}{2\pi}\int^{\pi}_{-\pi}\mathrm{d}\tau\,e^{i(x\sin\tau-\tau)}$.
From Eq. (S9), the eigenstates and corresponding eigenvalue can be easily
solved:

$\displaystyle\lvert\widetilde{\Psi}_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\operatorname{exp}[-i(\frac{\varphi_{\mathrm{drive}}}{2}+\frac{\pi}{4})]\\\
\pm\operatorname{exp}[+i(\frac{\varphi_{\mathrm{drive}}}{2}+\frac{\pi}{4})]\end{array}\right)\;$
(S13) $\displaystyle\varepsilon_{\pm}$ $\displaystyle=$
$\displaystyle\mp\hbar\widetilde{J}$ (S14)

As discussed in the main text, the relative phase of the eigenstate in the
Floquet frame depends on the driving phase $\varphi_{\mathrm{drive}}$, which
can be controlled experimentally through the RF dressing of the DW potential.
The time evolution for an arbitrary state in the Floquet frame can readily be
calculated from Eq. (S9). In order to compare with the results measured in the
experiment, we calculate the relative phase evolution of
$\lvert\widetilde{\Psi}_{+}\rangle$ (red line in Fig. 2(a) and in Fig. 3(a))
and afterwards transform $\widetilde{\phi}_{r}$ from the Floquet frame back to
the laboratory frame via Eq. (3).

##  Appendix B Sine-Gordon model for tunnel-coupled superfluids

For a static, balanced DW potential the sine-Gordon model was proposed [27]
and in thermal equilibrium experimentally verified [28] to be the low energy
effective description of a pair of tunneling coupled quasicondensates. A
complete derivation of the Floquet engineered interacting many-body system
would go far beyond the current paper. For the sake of completeness we give
here a brief derivation for our static DW system, assumed to describe the
time-independent Floquet Hamiltonian.

For the temperatures and atom numbers considered, both condensates, located in
the left and right minimum of the DW potential, fulfill the 1D condition
$\mu,k_{\mathrm{B}}T\ll\hbar\omega_{\perp}$. Since dynamics along the radial
directions are frozen out, we can proceed with the common dimensional
reduction by integrating over the tightly confined radial directions.
Tunneling through the DW barrier couples the two quantum wires, leading to the
effective one-dimensional Hamiltonian

$\displaystyle H=$ $\displaystyle\sum_{j=1}^{2}\int
dz\Bigg{[}\frac{\hbar^{2}}{2m}\frac{\partial\psi_{j}^{\dagger}}{\partial
z}\frac{\partial\psi_{j}}{\partial
z}+\frac{g_{\mathrm{1D}}}{2}\partial\psi_{j}^{\dagger}\partial\psi_{j}^{\dagger}\partial\psi_{j}\partial\psi_{j}$
$\displaystyle+$ $\displaystyle U(z)\psi_{j}^{\dagger}\psi_{j}\Bigg{]}-\hbar
J\int
dz\Bigg{[}\psi_{1}^{\dagger}\psi_{2}+\psi_{2}^{\dagger}\psi_{1}\Bigg{]}~{}.$
(S15)

Here $m$ is the atomic mass, $g_{\mathrm{1D}}=2\hbar
a_{\mathrm{s}}\omega_{\perp}$ is the 1D effective interaction strength,
$a_{\mathrm{s}}$ is the s-wave scattering length, $U$ is the longitudinal
potential, and $2\hbar J$ is the single particle tunneling-coupling energy.
The field operators fulfill the bosonic commutation relative
$[\psi_{j}(z),\psi_{j^{\prime}}^{\dagger}(z^{\prime})]=\delta_{jj^{\prime}}\delta(z-z^{\prime})$.

Figure S1:  Schematic of the potential energy for the sine-Gordon Hamiltonian
(S17). The red spheres show the initial states for the different Floquet
driving phases in Fig. 5(a)-(f). Note, that $\phi_{\mathrm{r}}$ here
corresponds to the shifted relative phase
$\Delta\widetilde{\phi}_{\mathrm{r}}$, defined in the main text.

Expressing the wave function in terms of density and phase fluctuations

$\psi_{j}(z)=\mathrm{exp}[i\theta_{j}(z)]\sqrt{n_{\mathrm{1D}}+\delta\rho_{j}(z)}~{},$
(S16)

with canonical commutators
$[\delta\rho_{j}(z),\theta_{j^{\prime}}(z^{\prime})]=i\delta_{jj^{\prime}}\delta(z-z^{\prime})$,
the low-energy effective theory can be derived by expanding the Hamiltonian
(B) to second order in the small density perturbations $\delta\rho_{j}$ and
phase gradients $\partial_{z}\theta_{j}$. This separates the Hamiltonian (B)
in a weakly coupled sum $H=H_{s}+H_{r}+V_{c,r}$ for the common (symmetric,
’s’) and relative (anti-symmetric, ’r’) degrees of freedom (DoF), defined as

$\displaystyle\delta\rho_{s}(z)=\delta\rho_{1}(z)+\delta\rho_{2}(z)~{},~{}\phi_{s}(z)=\frac{1}{2}[\theta_{1}(z)+\theta_{2}(z)]~{},$
$\displaystyle\delta\rho_{r}(z)=\frac{1}{2}[\delta\rho_{1}(z)-\delta\rho_{2}(z)]~{},~{}\phi_{r}(z)=\theta_{1}(z)-\theta_{2}(z)~{}.$

Experiments in static DW potentials showed that in thermal equilibrium the
coupling $V_{c,r}$ is negligible for a wide range of parameters [28], such
that the common and relative DoF are described by the Luttinger-Liquid and
sine-Gordon Hamiltonian, respectively. The latter is given by

$\displaystyle
H_{\mathrm{r}}=\\!\int\\!dz\Big{[}g\delta\rho_{\mathrm{r}}^{2}+\frac{\hbar^{2}n_{\mathrm{1D}}}{4m}\left(\partial_{z}\phi_{\mathrm{r}}\right)^{2}-2\hbar
Jn_{\mathrm{1D}}\operatorname{cos}(\phi_{\mathrm{r}})\Big{]}\,,$ (S17)

where, for simplicity, we consider the long wavelength limit (i.e. neglecting
derivatives of the density fluctuations). The first two terms represent the
Luttinger-Liquid Hamiltonian describing massless phononic excitations. A
schematic of the SG potential is depicted in Fig. S1, including the initial
states considered in Fig. 5(a)-(f). Their initial potential energy is
transferred to fluctuations of the SG field, leading to the relaxation of the
coherent Josephson oscillation in this extended bosonic JJ.