# Substrate interference and strain in the second harmonic generation from
MoSe2 monolayers

S. Puri  Department of Physics, University of Arkansas, Fayetteville, AR
72701, USA.  S. Patel  Department of Physics, University of Arkansas,
Fayetteville, AR 72701, USA.  J. L. Cabellos  Universidad Politécnica de
Tapachula. C.P. 30830. Tapachula, Chiapas, Mexico  L. E. Rosas-Hernandez
Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA.
S. Barraza-Lopez  Department of Physics, University of Arkansas, Fayetteville,
AR 72701, USA.  B. Mendoza  Centro de Investigaciones en Optica, A.C., León,
C.P. 37150. Guanajuato, Mexico  H. Nakamura<EMAIL_ADDRESS>Department of
Physics, University of Arkansas, Fayetteville, AR 72701, USA.

###### Abstract

Nonlinear optical materials of atomic thickness–such as non-centrosymmetric 2H
transition metal dichalcogenide monolayers–have a second order nonlinear
susceptibility ($\chi^{(2)}$) whose intensity can be tuned by strain. However,
whether $\chi^{(2)}$ is enhanced or reduced by tensile strain is a subject of
conflicting reports. Here, we grow high-quality MoSe2 monolayers under
controlled biaxial strain created by two different substrates, and study their
linear and non-linear optical responses with a combination of experimental and
theoretical approaches. A 15-fold overall enhancement in second harmonic
generation (SHG) intensity is observed on MoSe2 monolayers grown on SiO2 when
compared to its value when on a Si3N4 substrate. A seven-fold enhancement was
ascertained to substrate interference, and a factor of two to the enhancement
of $\chi^{(2)}$ arising from biaxial strain: substrate interference and strain
are independent handles to engineer the SHG strength of non-centrosymmetric 2D
materials.

Second harmonic generation (SHG) is a nonlinear optical phenomenon in which
two photons combine and form a single photon with twice the energy of the
original ones Boyd (2020). SHG is used to determine crystal symmetry Malard
_et al._ (2013); Kumar  _et al._ (2013); Shi  _et al._ (2017); Klimmer  _et
al._ (2021); Hsu  _et al._ (2014) and it can also probe excitonic states
Malard  _et al._ (2013); Kumar  _et al._ (2013); Shi  _et al._ (2017); Klimmer
_et al._ (2021); Hsu  _et al._ (2014).

Non-centrosymmetric monolayers (MLs) of transition metal dichalcogenides
(TMDs) have very high nonlinear susceptibilities of the order of 105 \- 108
pm2/V, making them highly efficient for SHG Kumar  _et al._ (2013); Malard
_et al._ (2013); Wen  _et al._ (2019). Furthermore, they can sustain
large–uniaxial or biaxial–tensile strain Roldán  _et al._ (2015); Naumis  _et
al._ (2023). Their crystal symmetry is lowered when under uniaxial strain,
resulting in an increased angular anisotropy in polarization-resolved SHG
measurements Mennel  _et al._ (2019, 2018); Liang  _et al._ (2017); Li  _et
al._ (2023); Khan  _et al._ (2020); Mennel  _et al._ (2020) and a decrease in
SHG intensity along the armchair direction Liang  _et al._ (2017); Mennel  _et
al._ (2020).

As it preserves the point symmetry of the crystal, biaxial strain could be
considered a more desirable tool to tune the efficiency of nonlinearity
without impacting polarization-dependent properties. Nevertheless, reports on
SHG in biaxially tensile strained TMD MLs are scarce and contradictory. For
instance, Covre et al. stated a decrease in the SHG intensity Covre  _et al._
(2022) on MoSe2 MLs, while Liu et al. documented an enhancement in SHG in a
MoS2 ML Liu  _et al._ (2023). Similarly, Shi et al. observed an increased SHG
in atomically thin WS2 produced on silicon holes, which was explained as a
result of a Fabry-Perot cavity resonance Shi  _et al._ (2022). Adding to the
conflicting nature of some reports, the tuning rate for uniaxial strain in
Reference Liang  _et al._ (2017) was used to rule out the involvement of
strain, even though the experimental geometry appears to more closely match
one in which the 2D material is under biaxial strain Shi  _et al._ (2022).

Figure 1: (a) Schematics of MoSe2 MLs without strain and under biaxial tensile
strain by a rapidly quenched growth on Si3N4 or SiO2 substrates. The
disposition of the unit cell employed in SHG calculations and cartesian axes
are depicted in the leftmost diagrams. (b) PL of MoSe2 MLs grown on Si3N4 and
SiO2. Experimental data (dots) with Lorentzian fits (solid lines) show red-
shift of the exciton peak on SiO2 from biaxial tensile strain. (c) Raman shift
in MoSe2 MLs on Si3N4 and SiO2 substrates. The inset highlights the $A_{1g}$
phonon peak for the MLs on each substrate, which is known to redshift for
biaxial strain Horzum  _et al._ (2013) but not for uniaxial strain. (d) PBE-
DFT band structures of a MoSe2 ML under no strain (solid lines) or 1% tensile
strain; the direct nature of the band gap (at the $K-$point) is apparent. (e)
Electronic band structure under a 2% and 3% tensile strain.

To resolve this puzzle, biaxially strained MoSe2 MLs were grown by physical
vapor deposition on either Si3N4 or SiO2 substrates here. Strain was induced
by a rapid thermal quenching after growth. The biaxial nature of strain as
well as its magnitude were determined by optical characterization that
included photoluminescence (PL), Raman studies [Figures 1(b) and 1(c)], and
theory. Details follow.

A high growth temperature in the PVD process efficiently induces a tensile
strain due to the thermal expansion mismatch between the ML and the substrate.
We used two substrates: 73 nm tick Si3N4 on Si and 87 nm tick SiO2 on Si. The
percent of strain induced due to the thermal expansion mismatch is given by
Ahn  _et al._ (2017):

$\mathrm{strain(\%)}=\left[{\int_{\mathrm{RT}}^{T_{\mathrm{g}}}(\alpha_{\mathrm{MoSe_{2}}}-\alpha_{\mathrm{subs}}})dT\right]\times
100,$ (1)

where $\alpha_{\mathrm{MoSe_{2}}}=7.1\times 10^{-6}$ is the thermal expansion
coefficient of bulk MoSe2 El-Mahalaey and Evans (1976) (which we assumed to be
thickness-independent here). $\alpha_{\mathrm{subs}}$ can either be
$\alpha_{\mathrm{Si_{3}N_{4}}}=3.5\times 10^{-6}$ Sinha  _et al._ (1978), or
$\alpha_{\mathrm{SiO_{2}}}=5.0\times 10^{-7}$ Rustum Roy and McKinstry (1989).
$T_{\mathrm{g}}$ is the growth temperature and $\mathrm{RT}$ stands for room
temperature. Additional details will be presented elsewhere (S. Patel,
manuscript in preparation). Our estimates yield up to 0.43%, or 0.73% tensile
strain at $T_{\mathrm{g}}=1130^{\circ}$ C when the substrate is Si3N4 or SiO2,
respectively. (See Figure 1(a) for schematics, and Methods for details.)

When contrasted with a peeled, unstrained reference MoSe2 ML sample, the PL
shows a redshift of the exciton peak for samples grown on substrates, which is
a signature of biaxial tensile strain. In turn, the Raman shift measured for
MoSe2 MLs on both Si3N4 and SiO2 substrates also shows a redshifted $A_{1g}$
phonon peak (known to be significant only under biaxial strain Horzum  _et
al._ (2013)). As depicted in Figure 1(d), MoSe2 MLs display direct bandgap at
the $K-$point when under no strain, which reduces in size under a 1% biaxial
tensile strain. As seen in Figure 1(e), further (2 and 3%) tensile strain
induces indirect bandgap (in between the $\Gamma$ and $K$ points) Ugeda  _et
al._ (2014) (see Methods).

We now turn to SHG measured by a setup schematically shown in Figure 2(a). The
excitation source is a Ti:Sapphire femtosecond laser (Tsunami, Spectra-
Physics) generating 200 fs pulses at a repetition rate of 80 MHz (see
Methods). An inset in Figure 2(a) shows a log-log plot of pump power versus
SHG intensity of the MoSe2 ML with a slope of 2, which is a hallmark of SHG.
For polarization-resolved SHG measurements, the sample is rotated in 5∘
increments, and a linear polarizer is used in the detection line to collect
SHG signals with a polarization parallel to that of the pump laser. The
polarization resolved SHG depicted in Figure 2(b) shows a characteristic
sixfold symmetry that arises from the $D_{3h}$ symmetry of the underlying
crystalMennel  _et al._ (2018); Malard  _et al._ (2013); Kim and Lim (2015),
even under biaxial strain.

Figure 2: (a) Sketch of the experimental setup for the collection of SHG light
in reflection geometry using a tunable femtosecond laser. The inset shown is
double logarithmic plot of pump power dependence of SHG. The slope of the plot
is 2. (b) Normalized SHG intensity as a function of sample rotation angle
$\theta$ about the $z-$axis, in which the polarization of the SHG signal is
parallel to that of the pump laser. Experimental data points are represented
by pink dots for MoSe2 MLs on SiO2 and by blue dots for MoSe2 MLs on Si3N4,
and the solid line is calculated theoretically to be proportional to
$\sin^{2}(3\theta)$.

Following Wang and Qian Wang and Qian (2017), we calculate the effective
second order susceptibility for a parallel polarization configuration. By
expanding the expression for the SHG polarization:

$P_{i,2\omega}=\epsilon_{0}\sum_{j=x,y,z}\sum_{k=x,y,z}\chi_{ijk}^{(2)}(2\omega;\omega,\omega)E_{j}E_{k},$
(2)

Where $\epsilon_{0}$ is the permittivity. There is only one independent entry
in the second-order dielectric susceptibility for point group $\bar{6}m2$
($D_{3h}$)Boyd (2020):
$\chi_{yyy}^{(2)}=-\chi_{yxx}^{(2)}=-\chi_{xxy}^{(2)}=-\chi_{xyx}^{(2)}\equiv\chi^{(2)}$,
where the $x$, $y$ and $z$ labels correspond to the laboratory axes as shown
in Figure 1(a) and we dropped the dependency on $2\omega$ and $\omega$ from
$\chi^{(2)}$. Then the second harmonic polarization takes the form:

$\displaystyle P_{x,2\omega}$
$\displaystyle=-2\epsilon_{0}\chi^{(2)}E_{x}E_{y},$ (3) $\displaystyle
P_{y,2\omega}$
$\displaystyle=\epsilon_{0}\chi^{(2)}\left(E_{y}E_{y}-E_{x}E_{x}\right),$ (4)
$\displaystyle P_{z,2\omega}$ $\displaystyle=0.$ (5)

In a polarization-resolved SHG, an analyzer is inserted to select SHG
polarized either parallel or perpendicular to that of the pump beam, while
rotating the polarization of the pump on the sample. The electric field of the
pump at an angle of $\theta$ with respect to the zigzag ($x$) direction of
MoSe2 ML can be expressed as $\mathbf{E}=(E_{0}\cos\theta,E_{0}\sin\theta,0)$.
The effective second order susceptibility in the parallel configuration
$\chi_{||}^{(2)}$ (see Ref. Wang and Qian (2017)) is obtained by applying a
projection onto a vector parallel to the polarization of the incoming beam:

$\chi_{||}^{(2)}(\theta)=(\cos{\theta},\sin{\theta},0)\cdot\frac{\mathbf{P}}{\epsilon_{0}E_{0}^{2}},$
(6)

with $\mathbf{P}$ as in Equation (3). This way:

$\chi_{||}^{(2)}(\theta)=\chi^{(2)}(-3\cos^{2}{\theta}\sin{\theta}+\sin^{3}{\theta})=-\chi^{(2)}\sin(3\theta),$
(7)

and the SHG intensity in the polar plot [solid curves in Figure 2(b)] can be
expressed as:

$I_{\mathrm{SHG}}(\theta)\propto|\chi^{(2)}|^{2}\sin^{2}(3\theta).$ (8)

The SHG is thus proportional to $|\chi^{(2)}|^{2}$, which will be later used
as a measure of SHG intensity.

At this point, we address the main contribution from this work, i.e., the
relative magnitude of the SHG signals obtained on the two biaxially-strained
samples. To this end, Figures 3(a) and 3(b) show the SHG intensity as a
function of the SHG wavelength for the tensile strained ML samples grown on
Si3N4 and SiO2 substrates, respectively. The overall increase in the SHG
intensity towards longer wavelength resembles earlier results Kikuchi  _et
al._ (2019); Miyauchi  _et al._ (2016). We also observed a shoulder on the SHG
spectrum, which is attributed to single photon resonances known as A-excitons
at pump wavelengths of 805 nm and 810 nm for MoSe2 MLs on Si3N4 and SiO2,
respectively. The different shifts in A-exciton peaks confirm the different
biaxial tensile strain on the two different substrates discussed in Figures
1(b) and 1(c).

A ratio of SHG intensity across the measured wavelength range is provided in
Figure 3(c). The main takeaway is a large, nearly 15-fold enhancement in the
SHG response of the MoSe2 ML on 87 nm thick SiO2 when compared to that of a
MoSe2 ML on 73 nm thick Si3N4. On the other hand, quantitatively comparing the
SHG intensity of strained MoSe2 MLs with the exfoliated ML poses a challenge
because (1) intrinsic material properties (such as doping) may differ between
the exfoliated versus PVD-grown MoSe2 MLs, and (2) the 285 nm thick SiO2
substrate used for the exfoliated ML happens to be close to a minimum of
interference, which makes the interference contribution highly sensitive to a
small error in the real thickness of the dielectric layer. In what follows, we
focus on PVD grown MoSe2 ML samples to determine the contribution of strain on
the observed SHG enhancement, combining experimental data and computationally
obtained SHG responses under strain.

Figure 3: (a) SHG intensity as a function of the SHG wavelength for the ML
sample on Si3N4. (b) SHG intensity as a function of the SHG wavelength for the
ML sample and SiO2. The shoulders on the SHG spectrum in (a) and (b) are
attributed to single photon resonance at A-exciton at around the pump
wavelengths of 805 nm and 810 nm for MoSe2 MLs on Si3N4 and SiO2,
respectively, which is consistent with a different biaxial tensile strain
being created by the two different substrates. (c) A 15-fold enhancement in
the SHG response from the MoSe2 ML on SiO2 when contrasted to that on Si3N4
across the measured wavelengths.

To calculate the effect of substrate interference on the SHG response, we use
the model by Song et al. Song  _et al._ (2023). For the normally incident pump
of wavelength $\lambda$ and intensity $I_{\lambda}$ on MoSe2 ML on a
dielectric film of thickness $d$ and a Si substrate, the SHG intensity
$I_{\lambda/2}$ is given bySong  _et al._ (2023):

$\
I_{\lambda/2}=\frac{1}{2\epsilon_{0}c}|\beta_{\lambda}|^{2}\frac{2\pi\chi^{(2)}}{\lambda\epsilon_{0}}I_{\lambda}^{2},$
(9)

where $\beta_{\lambda}$ is the structure factor that encapsulates the
influences arising from the layered structure given as:

$\ \beta_{\lambda}=(1+R_{\lambda})^{2}(1+R_{\lambda/2}).$ (10)

Here $R_{\lambda}$ and $R_{\lambda/2}$ are the reflection coefficients of
whole structure of $\lambda$ and $\lambda$/2 which are given by:

$\ R_{\lambda}=r\hskip 2.84544pt+\hskip
2.84544pt\frac{R_{s,\lambda}t^{2}}{1-R_{s,\lambda}r},\hskip 8.5359pt\
R_{s,\lambda}=\frac{r_{01}+r_{12}e^{2i\omega
n_{1}d}}{1+r_{01}r_{12}e^{2i\omega n_{1}d}}.$ (11)

In addition, $r=-\eta$/(1 + $\eta$) and $t=1/(1+\eta)$ are the reflection and
transmission coefficients of a MoSe2 ML, respectively, and $\eta$ is
calculated from the effective bulk refractive index $n_{2D}$ as
$\eta=-ih\omega(n_{2D}^{2}-1)/2$ with $\omega=2\pi/\lambda$. The reflection
coefficient at the interface between two layers is given by
$r_{ij}=(n_{i}-n_{j})/(n_{i}+n_{j})$.

Figure 4: (a) Substrate interference $\beta^{2}$ versus dielectric thickness
$d$ for Si3N4 (left subplot) and SiO2 (right subplot) slabs. The gray lines
highlight the thickness of the dielectric films used in our experiments. (b)
Real and imaginary contributions to $\chi^{(2)}(\omega)$. (c) Evolution of the
SHG intensity due to biaxial strain alone, as a function of frequency.
($EF_{str}$ in the inset is the enhancement factor due to biaxial tensile
strain for $\omega=1.669$ eV.) (d) Experimental observation of SHG relative
enhancement, and elucidation of the separate effects of interference and
biaxial strain.

The enhancement factor due to interference ($\mathrm{EF_{int}}$) is thus
written as:

$\mathrm{EF_{int}}=\left|{\frac{\beta_{\lambda}(SiO_{2})}{\beta_{\lambda}(Si_{3}N_{4})}}\right|^{2},$
(12)

and displayed in Figure 4(a) for the two substrates used here. The calculation
assumes MoSe2 ML on SiO2($d$ nm)/Si or Si3N4($d$ nm)/Si substrates as a
function of $d$ under an 800 nm pump laser. The delineated gray dashed lines
serve as indicators for the enhancement resulting from interference,
corresponding to 87 nm (73 nm) thick SiO2 (Si3N4) substrates utilized in our
experiment. The results indicate that an $\mathrm{EF_{int}}$ as large as 7.2
could be attributable to constructive interference. We note that the
interference could be overestimated by about 20-30% due to the use of plane
waves in the calculation, as the experiment focuses beam through the high NA
objective lens Song  _et al._ (2023).

The real and imaginary parts of $\chi^{(2)}$ as a function of $\omega$
calculated with the TINIBA code (developed by J. L. Cabellos, T. Rangel and B.
Mendoza) are provided in Figure 4(b). In Figure 4(c), we depict the energy
dependency of $|\chi^{(2)}|^{2}$ (proportional to $I_{\mathrm{SHG}}$), which
is seen to increase monotonically with applied tensile biaxial strain at
excitation energies (1.48-1.61 eV) studied by the experiment (Figure 4(d)).
Thus, the theory reproduces the tensile-strain induced enhancement of SHG by
the experiment. Furthermore, both theory and experiment highlight the
importance of resonant features that originate from band-to-band transitions,
as seen by steeply changing SHG intensity as a function of wavelength. The
theory also suggests that even qualitatively different strain effects may be
expected in other wavelength ranges near the resonance, which may explain
widely varying strain effects in the literature.

In summary, we explain two independent contributors to the enhancement of SHG
in non-centrosymmetric crystals: biaxial tensile strain, and interference
originating at the supporting substrate. This way, we provide clarity on two
independent factors affecting the magnitude of the SHG on 2D non-
centrosymmetric crystals unequivocally. These results shed light on the
biaxial strain tuning of nonlinear optical process in atomically thin
materials, and provide invaluable guidelines for future nanoscale
nonlinear/quantum optical devices Moody  _et al._ (2022); Autere  _et al._
(2018).

##  I Methods

###  I.1 Experimental

MoSe2 MLs were grown by physical vapor deposition (PVD) at 1130∘ C. We used
MoSe2 from AlfaAser (99.9 % pure) as a precursor under argon flow. The
diameter of a quartz tube was 1 inch. We employed reverse flow of argon from
substrate towards the powder end to avoid any unwanted nucleation before the
growth temperature is reached. Once the growth temperature is reached, forward
flow is used for short period of time (typically 10s) for growth and then
switched back to the reverse flow. Rapid quenching of the growth was carried
out by sliding the quartz tube out of the hot zone of the furnace. Details on
biaxial strain induced by high temperature PVD will be presented elsewhere (S.
Patel, manuscript in preparation). As a comparison, we also measured
exfoliated MoSe2 MLs purchased from a vendor (2D semiconductors) on SiO2.

For the polarization resolved SHG setup, we used a Ti:Sapphire femtosecond
(fs) laser source (Tsunami, Spectra-Physics) generating pulses of duration 200
fs at repetition rate of 80 MHz. With the use of a prism compensator, the
pulse width was kept constant between 770 nm and 840 nm. A dichroic mirror
directs excitation laser to 40x objective lens (NA 0.6) to excite the sample
and the SHG signal is collected in the reflection geometry. A beam splitter
added on the beam path illuminates the sample with white light and another
beam splitter directs the reflected light to a CCD camera for sample imaging.
Two short pass filters were used to reject the pump beam to detect SHG signals
in a 500 mm focal spectrometer (Acton Research) equipped with a thermo-
electrically cooled CCD. For the polarization resolved SHG measurements,
sample is rotated in 5∘ step and a linear polarizer is used in the detection
line to collect SHG signals parallel to the pump polarization.

###  I.2 Theoretical

First-principles calculations were carried out using plane wave density
functional theory (DFT) as implemented in the ABINIT Gonze2002 package. The
General Gradient Approximation (GGA) with Perdew-Burke-Ernzerhof Perdew  _et
al._ (1996) (PBE) functional was considered, along with optimized norm-
conserving Vanderbilt pseudopotentials Hamann (2013). The plane-wave energy
cutoff was set to 20 Hartrees ($~{}500$ eV) and a Monkhorst-Pack Monkhorst and
Pack (1976) k-point sampling of $18\times 18\times 1$ centered in the $\Gamma$
point was used. The results indicate that the pristine MoSe2 ML has a lattice
constant of $a=3.327$ Å, a distance between selenium bonds of $d_{Se-
Se}=3.344$ Å, and a band gap of $E_{g}=1.445$ eV, which are in good agreement
with previously reported values Chang  _et al._ (2013); Horzum  _et al._
(2013). After considering a biaxial tensile strain of $1\%$, $2\%$ and $3\%$
we found that at $1\%$ the band gap remains direct as in the pristine case but
undergoes from direct to indirect as more strain is applied as predicted
before Horzum  _et al._ (2013). The second order susceptibility tensor was
calculated with the TINIBA code, developed at the Centro de Investigaciones en
Optica by B. Mendoza, J. L. Cabellos, and T. Rangel.

We used a scissor correction equal to 0.809 eV on the electronic band
structure of all calculated susceptibilities; this value was taken as the GW
gap reported in Reference Ugeda  _et al._ (2014). Our calculations are of the
single-particle type, so the calculated responses do not include excitonic
effects.

##  II Author Contributions

HN conceived the project and directed all experimental work. S Puri performed
the non-linear optical characterization and analyses. S Patel grew the MoSe2
ML samples and performed PL and Raman measurements. SBL directed the
theoretical work in which LER-H optimized the MoSe2 MLs under strain and JLC
carried out SHG calculations in a code developed by BM. LER-H crafted the
final version of all figures. All authors discussed the results. S Puri, SBL,
and HN wrote the manuscript with input from all the authors.

##  III Notes

The authors declare no competing financial interest.

###### Acknowledgements.

The nonlinear optics studies were supported by the Department of Defense under
Award No. FA9550-23-1-0500. The growth system used to prepare MLs was
supported by the MonArk NSF Quantum Foundry from the National Science
Foundation under Award No. DMR-1906383. L.E.R.-H. and S.B.-L. were funded by
the US Department of Energy (Award No. DE-SC0022120). Calculations performed
at Arkansas were carried out at the Pinnacle Supercomputer, funded by the NSF
under award OAC-2346752.

## References

  * Boyd (2020) R. W. Boyd,  _Nonlinear optics_, 4th ed. (Academic Press, 2020).
  * Malard  _et al._ (2013) L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. De Paula, Phys. Rev. B 87, 201401 (2013).
  * Kumar  _et al._ (2013) N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, Phys. Rev. B 87, 161403 (2013).
  * Shi  _et al._ (2017) J. Shi, P. Yu, F. Liu, P. He, R. Wang, L. Qin, J. Zhou, X. Li, J. Zhou, X. Sui, _et al._ , Adv. Mater. 29, 1701486 (2017).
  * Klimmer  _et al._ (2021) S. Klimmer, O. Ghaebi, Z. Gan, A. George, A. Turchanin, G. Cerullo, and G. Soavi, Nat. Photon. 15, 837 (2021).
  * Hsu  _et al._ (2014) W.-T. Hsu, Z.-A. Zhao, L.-J. Li, C.-H. Chen, M.-H. Chiu, P.-S. Chang, Y.-C. Chou, and W.-H. Chang, ACS Nano 8, 2951 (2014).
  * Wen  _et al._ (2019) X. Wen, Z. Gong, and D. Li, InfoMat 1, 317 (2019).
  * Roldán  _et al._ (2015) R. Roldán, A. Castellanos-Gomez, E. Cappelluti, and F. Guinea, J. Phys.: Condens. Matter 27, 313201 (2015).
  * Naumis  _et al._ (2023) G. G. Naumis, S. A. Herrera, S. P. Poudel, H. Nakamura, and S. Barraza-Lopez, Rep. Prog. Phys. 87, 016502 (2023).
  * Mennel  _et al._ (2019) L. Mennel, M. Paur, and T. Mueller, APL Photonics 4 (2019), 10.1063/1.5051965.
  * Mennel  _et al._ (2018) L. Mennel, M. M. Furchi, S. Wachter, M. Paur, D. K. Polyushkin, and T. Mueller, Nat. Commun. 9, 516 (2018).
  * Liang  _et al._ (2017) J. Liang, J. Zhang, Z. Li, H. Hong, J. Wang, Z. Zhang, X. Zhou, R. Qiao, J. Xu, P. Gao, _et al._ , Nano Lett. 17, 7539 (2017).
  * Li  _et al._ (2023) Z.-Y. Li, H.-Y. Cheng, S.-H. Kung, H.-C. Yao, C. R. P. Inbaraj, R. Sankar, M.-N. Ou, Y.-F. Chen, C.-C. Lee, and K.-H. Lin, Nanomaterials 13, 750 (2023).
  * Khan  _et al._ (2020) A. R. Khan, B. Liu, T. Lu, L. Zhang, A. Sharma, Y. Zhu, W. Ma, and Y. Lu, ACS Nano 14, 15806 (2020).
  * Mennel  _et al._ (2020) L. Mennel, V. Smejkal, L. Linhart, J. Burgdórfer, F. Libisch, and T. Mueller, Nano Lett. 20, 4242 (2020).
  * Covre  _et al._ (2022) F. S. Covre, P. Faria, V. O. Gordo, C. S. de Brito, Y. V. Zhumagulov, M. D. Teodoro, O. Couto, L. Misoguti, S. Pratavieira, M. B. d. Andrade, _et al._ , Nanoscale 14, 5758 (2022).
  * Liu  _et al._ (2023) B. Liu, T. Yildirim, E. Blundo, D. de Ceglia, A. R. Khan, Z. Yin, H. T. Nguyen, G. Pettinari, M. Felici, A. Polimeni, _et al._ , Appl. Phys. Rev. 10 (2023), 10.1063/5.0144641.
  * Shi  _et al._ (2022) J. Shi, X. Wu, K. Wu, S. Zhang, X. Sui, W. Du, S. Yue, Y. Liang, C. Jiang, Z. Wang, _et al._ , ACS Nano 16, 13933 (2022).
  * Horzum  _et al._ (2013) S. Horzum, H. Sahin, S. Cahangirov, P. Cudazzo, A. Rubio, T. Serin, and F. M. Peeters, Phys. Rev. B 87, 125415 (2013).
  * Ahn  _et al._ (2017) G. H. Ahn, M. Amani, H. Rasool, D.-H. Lien, J. P. Mastandrea, J. W. Ager Iii, M. Dubey, D. C. Chrzan, A. M. Minor, and A. Javey, Nat. Commun. 8, 608 (2017).
  * El-Mahalaey and Evans (1976) S. H. El-Mahalaey and B. Evans, J. Appl. Crystallography 9, 403 (1976).
  * Sinha  _et al._ (1978) A. K. Sinha, H. J. Levinstein, and T. E. Smith, J. Appl. Phys. 49, 2423 (1978).
  * Rustum Roy and McKinstry (1989) D. K. A. Rustum Roy and H. A. McKinstry, Ann. Rev. Matter. Sci. 19, 59 (1989).
  * Ugeda  _et al._ (2014) M. M. Ugeda, A. J. Bradley, S.-F. Shi, F. H. da Jornada, Y. Zhang, D. Y. Qiu, W. Ruan, S.-K. Mo, Z. Hussain, Z.-X. Shen, F. Wang, S. G. Louie, and M. F. Crommie, Nat. Mater. 13, 1091 (2014).
  * Kim and Lim (2015) D. H. Kim and D. Lim, JKPS 66, 816 (2015).
  * Wang and Qian (2017) H. Wang and X. Qian, Nano Lett. 17, 5027 (2017).
  * Kikuchi  _et al._ (2019) Y. Kikuchi, Y. Miyauchi, R. Takaoka, T. Suzuki, M. Tanaka, and S. Ohno, Phys. Rev. B 100, 075301 (2019).
  * Miyauchi  _et al._ (2016) Y. Miyauchi, R. Morishita, M. Tanaka, S. Ohno, G. Mizutani, and T. Suzuki, JJAP 55, 085801 (2016).
  * Song  _et al._ (2023) Y. Song, W. Wang, Y. Wang, Y. Shan, J. L. Cheng, and J. Sipe, Opt. Expr. 31, 19746 (2023).
  * Moody  _et al._ (2022) G. Moody, V. J. Sorger, D. J. Blumenthal, P. W. Juodawlkis, W. Loh, C. Sorace-Agaskar, A. E. Jones, K. C. Balram, J. C. Matthews, A. Laing, _et al._ , Journal of Physics: Photonics 4, 012501 (2022).
  * Autere  _et al._ (2018) A. Autere, H. Jussila, Y. Dai, Y. Wang, H. Lipsanen, and Z. Sun, Advanced Materials 30, 1705963 (2018).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Hamann (2013) D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
  * Monkhorst and Pack (1976) H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
  * Chang  _et al._ (2013) C.-H. Chang, X. Fan, S.-H. Lin, and J.-L. Kuo, Phys. Rev. B 88, 195420 (2013).


# A fast tunable 3D-transmon architecture for superconducting qubit-based
hybrid devices

Sourav Majumder  Department of Physics, Indian Institute of Science,
Bangalore-560012 (India)  Tanmoy Bera  Department of Physics, Indian Institute
of Science, Bangalore-560012 (India)  Ramya Suresh  Department of Physics,
Indian Institute of Science, Bangalore-560012 (India)  Vibhor Singh
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science,
Bangalore-560012 (India)

###### Abstract

Superconducting qubits utilize the strong non-linearity of the Josephson
junctions. Control over the Josephson nonlinearity, either by a current bias
or by the magnetic flux, can be a valuable resource that brings tunability in
the hybrid system consisting of superconducting qubits. To enable such a
control, here we incorporate a fast-flux line for a frequency tunable transmon
qubit in 3D cavity architecture. We investigate the flux-dependent dynamic
range, relaxation from unconfined states, and the bandwidth of the flux-line.
Using time-domain measurements, we probe transmon’s relaxation from higher
energy levels after populating the cavity with $\approx 2.1\times 10^{4}$
photons. For the device used in the experiment, we find a resurgence time
corresponding to the recovery of coherence to be 4.8 $\mu$s. We use a fast-
flux line to tune the qubit frequency and demonstrate the swap of a single
excitation between cavity and qubit mode. By measuring the deviation in the
transferred population from the theoretical prediction, we estimate the
bandwidth of the flux line to be $\approx$ 100 MHz, limited by the parasitic
effect in the design. These results suggest that the approach taken here to
implement a fast-flux line in a 3D cavity could be helpful for the hybrid
devices based on the superconducting qubit.

Superconducting qubit, hybrid devices

Josephson circuits are the ideal candidates to realize a wide range of quantum
technologies. Low dissipation and the ability to implement tailored
Hamiltonians in a quantum circuit have led to a wide range of matured
platforms, such as quantum-noise limited amplifiers [1, 2, 3], circuit-QED
systems [4, 5], and hybrid devices [6]. While a wide variety of interactions
can be implemented by designing the static nonlinearity using Josephson
junctions [7, 8], a class of interaction Hamiltonians requires the application
of resonant or off-resonant pumps [9, 10, 11]. One such application of c-QED
platform is towards the hybrid devices, where it can be used as an auxiliary
mode. Hybrid systems based on the mechanical oscillators [12, 13, 14, 15],
electron spins [16, 17], surface acoustic waves [18, 19, 20], and magnons [21]
have been investigated. Recent developments on the hybrid devices, based on
electrostatic coupling with the nanomechanical oscillator [22], and with
acoustic resonator [23, 24], operating in the number resolved limit have
further raised the interest towards the c-QED based hybrid devices [6].

In hybrid devices, often the requirement of high pump power for the
enhancement of the parametric coupling, renders the integration of the c-QED
system incompatible. In a high power regime, for example, the transmon qubit
decouples from the cavity mode and gets excited to the unconfined-states [25].
The critical power necessary to operate the transmon within the few energy-
level subspace can be increased by an inductive shunt but not without
compromising the underlying non-linearity [26]. Another useful feature would
be the ability to adjust qubit-cavity coupling from dispersive to resonant
limits by rapidly tuning the qubit frequency with magnetic flux. While such
fast-flux bias lines are straightforward to design in planer devices,
integrating them into the 3D-cavity is challenging [27, 28].

With these challenges in mind, investigating the performance of transmon qubit
in 3D architecture with a fast-flux line could still have practical importance
[29, 30, 31, 32, 33, 34]. For example, consider a low frequency mechanical
oscillator coupled to a microwave cavity. In such a device, the re-
thermalization time, defined as the time taken to reach the mean phonon
occupation of one after initialization to the quantum ground state, can be
large due to the high quality factor of the mechanical oscillator. Therefore
in a hybrid device, if the relaxation time of the qubit from unconfined states
remains smaller than the re-thermalization time, both the systems, qubit and
the mechanical oscillator can be initialized to their quantum ground state
without a significant loss to the state fidelity. Thus, such initialization in
the quantum limit can be used for controlled interaction between the two
modes.

Here we incorporate a fast-flux line for a frequency tunable transmon qubit in
3D cavity architecture. We investigate three aspects of such design which is
required for the hybrid devices. The dynamic range of the system for various
flux bias is probed first. We measure the timescale associated with the
recovery of coherence in the system after a strong pump as the transmon
relaxes from highly excited states and pump photon leaves the cavity. The
initialization to the ground state is probed by performing vacuum-Rabi
measurements while varying the delay between the pump and the control signals.
Finally, the fast-flux is used to demonstrate the single excitation swap
between the cavity and transmon mode.

Figure 1: (a) A schematic of the device showing frequency tunable transmon
qubit coupled to a 3D cavity. The SQUID loop is positioned in a small recess
machined inside the cavity wall to incorporate a port for flux tuning. (b) An
optical image of the sample showing the fabricated transmon qubit and the flux
line. The thin vertical electrode provides the necessary qubit capacitance and
couples to the fundamental mode of the 3D cavity. (c) Scanning electron
microscope image of the SQUID loop. (d) Transmission through the cavity
$|S_{21}|$ as the magnetic flux threaded by the SQUID loop is varied. (e)
Qubit spectroscopy at the high drive power when qubit is flux biased at
$\Phi=0$. Various qubit transitions are labeled accordingly. We determine the
qubit anharmonicity to be $\approx$ 225 MHz.

Unlike the conventional 3D transmon the position of the SQUID loop is shifted
away from the center of the cavity into a recess created in the cavity
wall[35, 36]. As shown in Fig. 1(a), the SQUID is shifted to a recess designed
inside the cavity wall. An antenna pointing towards the cavity center provides
the necessary capacitance to the qubit mode and the coupling to the cavity
mode. The transmon design was simulated using the black-box quantization
technique [37]. Positioning the SQUID loop in a recess allows us to
incorporate a local flux line near the SQUID loop to tune the qubit frequency
rapidly. Such integration is also compatible with high coherence cavities
designed from superconductors [27, 28, 38].

The flux control line is designed to avoid any perturbation to the cavity mode
while minimizing the relaxation of qubit mode to the flux-drive port. Fig.
1(b) shows an optical microscope image of the fabricated device using the
shadow evaporation technique on an intrinsic silicon substrate. Fig. 1(c)
shows the scanning electron microscope image of the SQUID loop. The patterned
device is placed inside an oxygen-free high thermal conductivity (OFHC) copper
cavity and cooled down to 20 mK. Our measurement setup consists of various
absorptive/reflective filters. Detailed schematic of the measurement setup is
shown in the supplementary information (SI). We use a 1 GHz low-pass
reflective filter on the flux-line and place it close to the flux drive port.
The entire cavity assembly is placed inside the magnetic and infrared
radiation shields.

Figure 2: (a) Shift in the dressed cavity frequency as the input power to the
cavity is increased for variable magnetic flux. The minimum power -129 dBm
corresponds to the mean cavity occupation of 0.17 photons. The solid lines are
numerical fits to extract the cavity nonlinearity. (b) The loaded (total)
quality factor of the dressed mode for different input power to the cavity at
multiple flux bias points.

We begin by performing spectroscopy measurements on the device. Fig. 1(d)
shows the cavity spectroscopy measurement as the magnetic flux through the
SQUID loop is changed by varying the current through the flux-line. An avoided
crossing, signifying the strong coupling, between the qubit mode and the
cavity mode is clearly visible. From the qubit spectroscopy, shown in Fig.
1(e), we determine the maximum qubit frequency (ground to first excited state
transition) to be $\omega_{q}^{0}/2\pi\sim 7.203$ GHz and corresponding
dressed cavity frequency for the ground state as $\omega_{c}/2\pi\sim 5.996$
GHz. We measure the coupling between two modes to be $g/2\pi\sim 87$ MHz,
which is close to the designed value [37]. While the maximum qubit frequency
depends on the total critical current of the two junctions, the minimum qubit
frequency depends on the asymmetry of the two junctions. From the two-tone
spectroscopy measurements, we could tracked the qubit frequency down to 4 GHz
while tuning it with the flux. Various device parameters are summarized in
Table 1.

At low probe power, the cavity frequency shifts to a dressed frequency due to
its interaction with the qubit. Beyond a critical power, the cavity jumps to
its bare frequency defining the dynamic range of the system. In this limit,
the phase difference across the junctions evolves continuously. This has been
attributed to the excitation of the qubit to the unconfined states lying
outside the cosine potential well [25]. We use scqubits package to compute the
energy eigen-states using the measured device parameters and find that there
are approximately 10 confined states within the cosine potential well [39].
The higher transmon levels exhibit larger charge dispersion [40]. At large
probe powers, the higher transmon levels become important, and the coupled
system must be treated by including the Kerr-nonlinearity terms. In the
dispersive limit, the Hamiltonian of the system can be written as
$\hat{H}_{sys}/\hbar=\omega_{c}\hat{a}^{\dagger}\hat{a}+\omega_{q}\hat{b}^{\dagger}\hat{b}-\frac{1}{2}\alpha_{c}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}-\frac{1}{2}\alpha_{q}\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b}+\chi\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}$,
where $\omega_{c}$ ($\omega_{q}$) is the cavity (transmon) frequency,
$\alpha_{c}$ ($\alpha_{q}$) is the cavity (transmon) Kerr-nonlinearity and
$\chi$ is the dispersive shift. Due to the qubit-induced Kerr-nonlinearity,
even in the dispersive regime the dynamic range of the cavity gets limited
significantly. Fig. 2(a) shows the experimental results of the change in the
dressed cavity frequency with probe power at the device. As the probe power is
increased, the dressed cavity frequency changes as
$\omega_{c}(\bar{n})=\omega_{c}(0)-2\alpha_{c}\bar{n}$ [11], where $\bar{n}$
is average number of photons in the cavity. From an independent calibration of
$\bar{n}$ using ac-Stark shift, we deduce $\alpha_{c}$ for different qubit
detunings. Additional dataset on ac-Stark shift is included in SI. For zero
magnetic flux ($\Phi~{}=~{}0$), when qubit detuning
$\Delta=\omega_{q}-\omega_{r}\approx 2\pi\times$1.2 GHz, we estimated the the
cavity non-linearity to be $\alpha_{c}/2\pi=-3.2$ kHz. As the qubit mode is
tuned closer to the cavity $\Delta=-2\pi\times$ 600 MHz, $\alpha_{c}/2\pi$
increases to $-27.8$ kHz, which is also indicated by the reduced dynamic range
shown in the Fig. 2(a). As expected, the maximum dynamic range is achieved
when the qubit is detuned furthest to the cavity frequency at
$\Phi/\Phi_{0}=0.5$. At this flux-operating point, we also observe a reduction
in the cavity frequency resulting from the asymmetry in the critical currents
of the SQUID junctions. Similar behavior is observed in the corresponding
quality factor of the dressed mode, as shown in the Fig. 2(b).

Figure 3: (a) A schematic of the pulse sequence used in the measurement. The
cavity is driven with a strong pump pulse, which creates a large population of
photons in the cavity and excites the transmon to unconfined states.
Subsequently, Rabi measurement protocol with varying delays $\tau_{d}$ is
followed. For control, we use a pulse that has a rise and fall parts defined
by a gaussian function and central part of the pules is rectangular. The rise
(and fall) part of the pulse is defined by a Gaussian function of length 35 ns
and a $\sigma$ of 9 ns, indicated by the arrow. (b) Cavity transmission as
probe power is increased while biasing the qubit at zero flux quantum. The
white dotted line indicates the power used for the high power pulsed
measurements. (c) Rabi measurement for different delays between the high power
pump and qubit control pulse. A plot of Rabi oscillation without any high
power pump is included as a reference. The pump power in the steady-state
corresponds to a photon number $n_{d}$ of 2.1$\times 10^{4}$. (d) Amplitude of
Rabi oscillation measured for different delays between the pump and the
control pulse. Oscillation amplitude when the pump pulse is in off condition
is denoted by the black dotted line. Statistical uncertainties from the fits
are smaller than the marker size.

After the basic characterization of the device, we investigate the high power
response in the time domain. The high pump power excites the qubit to
unconfined states. It could be accompanied by the creation of quasi-particles,
which could take a long time to relax. We perform time-domain measurements to
probe the resurgence of coherence after subjecting the system to a strong
pump.

Using the flux-bias, the qubit frequency is tuned to the maximum frequency
$\Delta=\omega_{q}-\omega_{r}\approx 2\pi\times$1.2 GHz. A control pulse at
the qubit frequency is applied. It is followed by a measurement pulse at the
dressed cavity frequency, corresponding to a steady-state cavity occupation of
$\approx$ 5 photons. The transmitted signal from the cavity is amplified at 4
K and at room temperature. The amplified signal is then down-converted to an
intermediate frequency. Both quadratures of the IF signal are recorded as a
function of time with a lock-in amplifier. To improve the readout signal
contrast, we average fifty thousand time-traces of in-phase and quadrature
streams of the readout signal.

Such an ensemble average of time traces can then be used to determine the
qubit state. We follow an approach similar to Ref. [41] and define a
normalized integrated signal $V_{H}$ as
$V_{H}=\frac{1}{2}\frac{\sum_{i}\left(V_{g}(t_{i})-V_{m}(t_{i})\right)\Delta
t}{\sum_{i}\left(V_{g}(t_{i})-V_{s}(t_{i})\right)\Delta t}$, where $\Delta t$
is the resolution of time-axis. $V_{g}$ ($V_{s}$) represents the averaged
signal traces corresponding to the qubit in the ground state (in an equal
mixture of ground and the first excited state). $V_{m}$ represents signal for
the unknown qubit state that is being measured. Here, we effectively use the
saturation control pulse for normalization. It is important to emphasize here
that $V_{H}$ slightly deviates from the first excited state probability due to
loss of population during the measurement process.

Figure 4: (a) A schematic of the pulse sequence used in the measurement. The
interaction time ($\tau_{int}$) between cavity and qubit is varied for
different values of $\tau_{d}$. (b) Normalized integrated signal showing the
swap of excitation between the qubit and the cavity mode as the interaction
length is varied. The black dotted line indicates the time to swap one
excitation. (c) The colorplot of the normalized integrated signal as the
detuning between the cavity mode and qubit is varied. The results were
obtained from a different device with similar parameters. (d) The colorplot of
the simulated qubit population in the excited state while varying the
interaction length and relative detuning $\Delta=\omega_{q}-\omega_{c}$.

To probe the relaxation of qubit from the higher energy levels to the ground
state, a strong pump tone is applied at the bare cavity frequency, followed by
the pulsed control and readout scheme as described before. A schematic of the
pulse sequence is shown in Fig. 3(a). In the presence of a strong pump, the
transmon gets excited to the unconfined states resulting in the maximum
transmission through the cavity at its bare frequency. Fig. 3(b) shows the
measurement of transmission through the cavity as power of the probe signal is
increased. The dressed mode shows the characteristic frequency shift due to
the presence of the qubit. Beyond a critical power, the maximum transmission
jumps to the bare cavity frequency. For the pump pulse, corresponding mean
occupation of $n_{d}\sim$2.1$\times$104 photons in the cavity indicated by the
dotted line in Fig. 3(b). The calibration of the pump photons is performed by
using the ac Stark measurements made at low probe powers. We have calibrated
the total attenuation in the line and calculated the total number of photon
with respect to bare cavity frequency. This strong pump pulse excites the
qubit to higher unconfined states. By varying the length of the qubit control
pulse, we perform the vacuum Rabi-oscillation measurement for different delay
time ($\tau_{d}$) between the pump and qubit control.

Fig. 3(c) shows the measurements of normalized integrated signal $V_{H}$ for
different delays. The horizontal axis corresponds to the duration of the qubit
control pulse. For comparison, a measurement made in the absence of the pump
pulse is included as well. For $\tau_{d}=$ 2.64 $\mu$s, we see small
oscillations in the measurement indicating the coherent pouplation transfer
between the ground and the first excited state. For such short delay time
$\tau_{d}$, there are two effects that reduce the contrast of oscillations.
First, the qubit population in the ground or first excited state could be low
due to its excitation to higher levels. Second, for short times, the pump
photon occupancy in the cavity can be substantial leading to dephasing. We use
slowly varying control pulse and therefore rule out any leakage of qubit to
the higher levels by the control pulse. For $\tau_{d}=$ 8.8 $\mu$s, the
oscillations closely resemble the result obtained with pump maintained in off-
condition. We systematically measure the amplitude of Rabi oscillations for
different delay time between pump and the control pulse. Fig 3(d) shows the
plot of oscillation amplitude for different delay times, showing the clear
resurgence of the coherence in the device. Additional dataset is included in
the SI. By fitting it to $\mathcal{F}(1-e^{\frac{t-t_{0}}{\tau}})$, we extract
the characteristic timescale, $t_{0}+\tau\approx$ 4.8 $\mu$s for the
relaxation from unconfined states. We point out here that such a timescale
involves contributions from the relaxation of qubit from the higher excited
states and from the dephasing due to occupancy of the cavity by pump-photons.

Device parameter | Symbol | Value
---|---|---
Bare cavity frequency | $\omega_{c}^{0}/2\pi$ | 6.002 GHz
Maximum qubit frequency | $\omega_{q}^{0}/2\pi$ | 7.203 GHz
Kerr-nonlinearity | $\alpha_{q}/2\pi$ |  $-225$ MHz
Maximum Joesphson energy | $E_{J}^{0}/h$ | 30.65 GHz
Cavity linewidth at zero flux | $\kappa/2\pi$ | 1.38 MHz
Qubit relaxation time at zero flux | $T_{1}$ | 2.11 $\mu$s
Qubit cavity coupling | $g/2\pi$ | 87 MHz
Table 1: Summary of device parameters studied in the main text

After characterizing the response of the device under high power, next, we
perform the characterization of the flux bias port. We utilize the high
bandwidth of the flux drive line and create a single-photon state in the
cavity. The pulse protocol for such scheme is shown in Fig. 4(a). It consists
of initializing the qubit to the first excited state by applying a
$\pi$-pulse. The qubit frequency is then rapidly tuned to bring it in
resonance with the cavity. The modes are maintained in resonance for a
variable time $\tau_{int}$ and then the qubit mode is brought back to the
original frequency followed by a measurement pulse. During the interaction
period, the qubit and the cavity modes exchange the single excitation
coherently. To understand the applicability of this scheme in a strongly
driven system, we follow this protocol after a strong pump (at the bare cavity
frequency) pulse with varied delay time $\tau_{d}$.

The current in the flux loop, controlling the qubit detuning, is applied using
an arbitrary waveform generator. The shape of the flux-pulse is rectangular
with rising and falling segments set as the half-gaussian with standard
deviation of 1.1 ns. The width and the amplitude of the flux pulse are varied
to control the interaction time and the qubit frequency, respectively. Fig.
4(b) shows $V_{H}$ as the interaction length is varied. For delay time of 8.8
$\mu$s ($>$ $t_{0}+\tau$), we observe that the qubit regains the coherence and
oscillation due to the swapping of a single excitation can be clearly seen.
Due to the strong coupling between the qubit and the cavity, it takes
approximately 3 ns to transfer the single-photon from the qubit mode to the
cavity mode indicated by the black dotted line.

Fig. 4(c) shows the colorplot of $V_{H}$ as the qubit detuning and interaction
duration is varied. The oscillation frequency of the single excitation swap
changes as the relative detuning between the qubit and cavity mode frequency
is varied. At zero detuning, the oscillation frequency is minimum at 2$g$ and
increases to $\sqrt{4g^{2}+\Delta^{2}}$ with detuning [13]. The deviation from
the ideal chevron pattern suggests that the flux-pulse disperses as it travels
down the sample. The initial change in the flux pulse is not able to tune the
qubit in resonance with the cavity for short interaction time. We observe a
small distortion in the chevron pattern for time-scales shorter than 10 ns.
From this deviation, we concluded the bandwidth of the flux-line to be
approximately 100 MHz. The bandwidth of flux line is limited by the parasitic
capacitance and self-inductance of the current loop patterned near the SQUID
loop. While we try to maintain a 50 $\Omega$ environment till the connector on
the cavity, the impedance of the flux line on the silicon chip deviates from
50 $\Omega$ and this limits the bandwidth. Such distortions in flux-pulse, in
principle, could be improved by using pre-compensated flux-pulses. To better
understand the experimental results, we numerically simulate the system by
solving the Lindblad master equation with the flux pulse sequence used in the
experiment [42]. The simulated outcome of the excited state population is
plotted in Fig. 4(d) with variable detuning in the vertical axis. The
difference between the simulation and experimental plots can be understood
from the distorted flux-pulse at the sample, as discussed above.

To summarize, we demonstrated a design of a fast-tunable transmon qubit in a
3D waveguide cavity architecture. We characterized its relaxation from
unconfined states to the ground state after a high power drive pulse. We
measure a resurgence time of 4.8 $\mu$s. We characterize the fast-flux line
and find a bandwidth of $\approx$ 100 MHz. These performance benchmarking
results provide the design guidelines for hybrid systems intended to integrate
additional degrees of freedom with the circuit-QED platform [31, 32].

## Acknowledgment

This material is based upon work supported by the Air Force Office of
Scientific Research under award number FA2386-20-1-4003. V.S. acknowledge the
support received under the Young Scientist Research Award by the Department of
Atomic Energy and support received under the Core Research Grant by the
Department of Science and Technology (India). The authors acknowledge device
fabrication facilities at CeNSE, IISc Bangalore, and central facilities at the
Department of Physics funded by DST.

## References

  * [1] Castellanos-Beltran, M. A. and Lehnert, K. W.  Applied Physics Letters 91(8), 083509 August (2007).
  * [2] Yamamoto, T., Inomata, K., Watanabe, M., Matsuba, K., Miyazaki, T., Oliver, W. D., Nakamura, Y., and Tsai, J. S.  Applied Physics Letters 93(4), 042510 July (2008).
  * [3] Macklin, C., O’Brien, K., Hover, D., Schwartz, M. E., Bolkhovsky, V., Zhang, X., Oliver, W. D., and Siddiqi, I.  Science 350(6258), 307–310 October (2015).
  * [4] Devoret, M. H. and Martinis, J. M.  Quantum Information Processing 3(1-5), 163–203 October (2004).
  * [5] Blais, A., Gambetta, J., Wallraff, A., Schuster, D. I., Girvin, S. M., Devoret, M. H., and Schoelkopf, R. J.  Physical Review A 75(3), 032329 March (2007).
  * [6] Clerk, A. A., Lehnert, K. W., Bertet, P., Petta, J. R., and Nakamura, Y.  Nature Physics 16(3), 257–267 (2020).
  * [7] Buluta, I., Ashhab, S., and Nori, F.  Reports on Progress in Physics 74(10), 104401 September (2011).
  * [8] Xiang, Z.-L., Ashhab, S., You, J. Q., and Nori, F.  Reviews of Modern Physics 85(2), 623–653 April (2013).
  * [9] Flurin, E., Roch, N., Mallet, F., Devoret, M. H., and Huard, B.  Physical Review Letters 109(18) October (2012).
  * [10] Abdo, B., Sliwa, K., Schackert, F., Bergeal, N., Hatridge, M., Frunzio, L., Stone, A. D., and Devoret, M.  Physical Review Letters 110(17), 173902 April (2013).
  * [11] Leghtas, Z., Touzard, S., Pop, I. M., Kou, A., Vlastakis, B., Petrenko, A., Sliwa, K. M., Narla, A., Shankar, S., Hatridge, M. J., Reagor, M., Frunzio, L., Schoelkopf, R. J., Mirrahimi, M., and Devoret, M. H.  Science 347(6224), 853–857 February (2015).
  * [12] LaHaye, M. D., Suh, J., Echternach, P. M., Schwab, K. C., and Roukes, M. L.  Nature 459(7249), 960–964 June (2009).
  * [13] O’Connell, A. D., Hofheinz, M., Ansmann, M., Bialczak, R. C., Lenander, M., Lucero, E., Neeley, M., Sank, D., Wang, H., Weides, M., Wenner, J., Martinis, J. M., and Cleland, A. N.  Nature 464(7289), 697–703 April (2010).
  * [14] Lecocq, F., Teufel, J. D., Aumentado, J., and Simmonds, R. W.  Nature Physics 11(8), 635–639 August (2015).
  * [15] Pirkkalainen, J.-M., Cho, S. U., Li, J., Paraoanu, G. S., Hakonen, P. J., and Sillanpää, M. A.  Nature 494(7436), 211–215 February (2013).
  * [16] Zhu, X., Saito, S., Kemp, A., Kakuyanagi, K., Karimoto, S.-i., Nakano, H., Munro, W. J., Tokura, Y., Everitt, M. S., Nemoto, K., Kasu, M., Mizuochi, N., and Semba, K.  Nature 478(7368) (2011).
  * [17] Kubo, Y., Grezes, C., Dewes, A., Umeda, T., Isoya, J., Sumiya, H., Morishita, N., Abe, H., Onoda, S., Ohshima, T., Jacques, V., Dréau, A., Roch, J.-F., Diniz, I., Auffeves, A., Vion, D., Esteve, D., and Bertet, P.  Physical Review Letters 107(22), 220501 (2011).
  * [18] Gustafsson, M. V., Aref, T., Kockum, A. F., Ekström, M. K., Johansson, G., and Delsing, P.  Science 346(6206), 207–211 (2014).
  * [19] Manenti, R., Kockum, A. F., Patterson, A., Behrle, T., Rahamim, J., Tancredi, G., Nori, F., and Leek, P. J.  Nature Communications 8(1), 975 (2017).
  * [20] Bolgar, A. N., Zotova, J. I., Kirichenko, D. D., Besedin, I. S., Semenov, A. V., Shaikhaidarov, R. S., and Astafiev, O. V.  Physical Review Letters 120(22), 223603 (2018).
  * [21] Tabuchi, Y., Ishino, S., Noguchi, A., Ishikawa, T., Yamazaki, R., Usami, K., and Nakamura, Y.  Science 349(6246), 405–408 July (2015).
  * [22] Viennot, J., Ma, X., and Lehnert, K.  Physical Review Letters 121(18), 183601 (2018).
  * [23] Chu, Y., Kharel, P., Yoon, T., Frunzio, L., Rakich, P. T., and Schoelkopf, R. J.  Nature 563(7733), 666 (2018).
  * [24] Arrangoiz-Arriola, P., Wollack, E. A., Wang, Z., Pechal, M., Jiang, W., McKenna, T. P., Witmer, J. D., Van Laer, R., and Safavi-Naeini, A. H.  Nature 571(7766), 537–540 (2019).
  * [25] Lescanne, R., Verney, L., Ficheux, Q., Devoret, M. H., Huard, B., Mirrahimi, M., and Leghtas, Z.  Physical Review Applied 11(1), 014030 (2019).
  * [26] Verney, L., Lescanne, R., Devoret, M. H., Leghtas, Z., and Mirrahimi, M.  Physical Review Applied 11(2), 024003 February (2019).
  * [27] Gargiulo, O., Oleschko, S., Prat-Camps, J., Zanner, M., and Kirchmair, G.  Applied Physics Letters 118(1), 012601 January (2021).
  * [28] Reshitnyk, Y., Jerger, M., and Fedorov, A.  EPJ Quantum Technology 3(1), 1–6 December (2016).
  * [29] Yuan, M., Singh, V., Blanter, Y. M., and Steele, G. A.  Nature Communications 6, 8491 October (2015).
  * [30] Noguchi, A., Yamazaki, R., Ataka, M., Fujita, H., Tabuchi, Y., Ishikawa, T., Koji Usami, and Nakamura, Y.  New Journal of Physics 18(10), 103036 (2016).
  * [31] Gunupudi, B., Das, S. R., Navarathna, R., Sahu, S. K., Majumder, S., and Singh, V.  Physical Review Applied 11(2), 024067 February (2019).
  * [32] Peterson, G., Kotler, S., Lecocq, F., Cicak, K., Jin, X., Simmonds, R., Aumentado, J., and Teufel, J.  Physical Review Letters 123(24), 247701 December (2019).
  * [33] Ofek, N., Petrenko, A., Heeres, R., Reinhold, P., Leghtas, Z., Vlastakis, B., Liu, Y., Frunzio, L., Girvin, S. M., Jiang, L., Mirrahimi, M., Devoret, M. H., and Schoelkopf, R. J.  Nature 536(7617), 441–445 August (2016).
  * [34] Heeres, R. W., Reinhold, P., Ofek, N., Frunzio, L., Jiang, L., Devoret, M. H., and Schoelkopf, R. J.  Nature Communications 8(1), 94 July (2017).
  * [35] Paik, H., Schuster, D. I., Bishop, L. S., Kirchmair, G., Catelani, G., Sears, A. P., Johnson, B. R., Reagor, M. J., Frunzio, L., Glazman, L. I., Girvin, S. M., Devoret, M. H., and Schoelkopf, R. J.  Physical Review Letters 107(24), 240501 December (2011).
  * [36] Juliusson, K., Bernon, S., Zhou, X., Schmitt, V., le Sueur, H., Bertet, P., Vion, D., Mirrahimi, M., Rouchon, P., and Esteve, D.  Physical Review A 94(6), 063861 December (2016).  Publisher: American Physical Society.
  * [37] Nigg, S. E., Paik, H., Vlastakis, B., Kirchmair, G., Shankar, S., Frunzio, L., Devoret, M. H., Schoelkopf, R. J., and Girvin, S. M.  Physical Review Letters 108(24), 240502 June (2012).
  * [38] Navau, C., Prat-Camps, J., Romero-Isart, O., Cirac, J., and Sanchez, A.  Physical Review Letters 112(25), 253901 June (2014).
  * [39] Groszkowski, P. and Koch, J.  Quantum 5, 583 November (2021).
  * [40] Koch, J., Yu, T. M., Gambetta, J., Houck, A. A., Schuster, D. I., Majer, J., Blais, A., Devoret, M. H., Girvin, S. M., and Schoelkopf, R. J.  Physical Review A 76(4), 042319 October (2007).
  * [41] Bianchetti, R., Filipp, S., Baur, M., Fink, J. M., Göppl, M., Leek, P. J., Steffen, L., Blais, A., and Wallraff, A.  Physical Review A 80(4), 043840 October (2009).
  * [42] Johansson, J. R., Nation, P. D., and Nori, F.  Computer Physics Communications 183(8), 1760–1772 August (2012).