# Circuit QED with a graphene double quantum dot and a reflection-line
resonator

Guang-Wei Deng  Key Laboratory of Quantum Information, University of Science
and Technology of China, Chinese Academy of Sciences, Hefei 230026, China  Da
Wei  Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China  J.R.
Johansson  iTHES research group, RIKEN, Wako-shi, Saitama, 351-0198 Japan
Miao-Lei Zhang  Key Laboratory of Quantum Information, University of Science
and Technology of China, Chinese Academy of Sciences, Hefei 230026, China
Shu-Xiao Li  Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China  Hai-Ou
Li  Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China  Gang
Cao  Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China  Ming
Xiao  Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China  Tao Tu
Key Laboratory of Quantum Information, University of Science and Technology of
China, Chinese Academy of Sciences, Hefei 230026, China  Guang-Can Guo  Key
Laboratory of Quantum Information, University of Science and Technology of
China, Chinese Academy of Sciences, Hefei 230026, China  Hong-Wen Jiang
Department of Physics and Astronomy, University of California at Los Angeles,
California 90095, USA  Franco Nori  CEMS, RIKEN, Wako-shi, Saitama, 351-0198
Japan  Physics Department, The University of Michigan, Ann Arbor, Michigan
48109-1040, USA  Guo-Ping Guo  Corresponding author<EMAIL_ADDRESS>Key
Laboratory of Quantum Information, University of Science and Technology of
China, Chinese Academy of Sciences, Hefei 230026, China

Graphene has attracted considerable attention in recent years due to its
unique physical properties and potential applications. Graphene quantum dots
have been proposed as quantum bits, and their excited-state relaxation rates
have been studied experimentally. However, their dephasing rates remain
unexplored. In addition, it is still not clear how to implement long-range
interaction among qubits for future scalable graphene quantum computing
architectures. Here we report a circuit quantum electrodynamics (cQED)
experiment using a graphene double quantum dot (DQD) charge qubit and a
superconducting reflection-line resonator (RLR). The demonstration of this
capacitive coupling between a graphene qubit and a resonator provides a
possible approach for mediating interactions between spatially-separated
graphene qubits. Furthermore, taking advantage of sensitive microwave readout
measurements using the resonator, we measure the charge-state dephasing rates
in our hybrid graphene nanostructure, which is found to be of the order of
GHz. A spectral analysis method is also developed to simultaneously extract:
the DQD-resonator coupling strength, the tunneling rate between the DQD charge
states, and the charge-state dephasing rate. Our results show that this
graphene cQED architecture can be a compelling platform for both graphene
physics research and potential applications.

##  I Introduction

Circuit QED provides a platform for studying microwave photons and artificial
atoms in electrical circuits Xiang  _et al._ (2013); You and Nori (2011).
Fundamental physical phenomena and quantum algorithms have been demonstrated
using spatially-separated Josephson-junction qubits coupled via
superconducting microwave resonators Xiang  _et al._ (2013); You and Nori
(2011). Extending this idea, theoretical works Childress  _et al._ (2004); Guo
_et al._ (2008); Lin  _et al._ (2008); Cottet and Kontos (2010); Jin  _et al._
(2012); Bergenfeldt and Samuelsson (2013); Contreras-Pulido  _et al._ (2013);
Lambert  _et al._ (2013) on hybrid systems using superconducting transmission-
line resonators (TLRs) and semiconducting artificial atoms have been made and
there have been experiments on qubits based on carbon nanotubes Delbecq  _et
al._ (2011, 2013); Viennot  _et al._ (2013), GaAs/AlGaAs Frey  _et al._
(2012); Toida  _et al._ (2013); Basset  _et al._ (2013), and InAs nanowires
Petersson  _et al._ (2012). Recently, graphene has attracted considerable
attention for its particular properties and variety of applications Geim and
Novoselov (2007); Geim (2009); Guo  _et al._ (2009). Due to its gapless
electronic band structure and the Klein tunneling phenomena Rozhkov  _et al._
(2011); Castro Neto  _et al._ (2009), most graphene quantum dots are formed by
the shape-effect of etched nanostructures. This etching procedure introduces
new physics related to edge states and further increases the difficulties in
fabricating and manipulating the graphene nanostructures. With advanced device
designs, researchers have recently realized pulsed-gate transient spectroscopy
and a relaxation time of 100 ns has been measured Volk  _et al._ (2013) in a
graphene quantum dot device. However, the dephasing times, which may be
significantly shorter than the relaxation times, remain unexplored.
Furthermore, there are no proposals on how to couple multiple graphene qubits
of etched quantum dots.

Here we report a circuit-QED experiment with a hybrid device using a graphene
etched DQD and a superconductor reflection-line resonator Zhang  _et al._
(2013). This provides a platform for investigating the physics of graphene
nanostructures interacting with microwave photons and for exploring potential
applications. A DQD-resonator coupling strength of the order of tens of MHz is
demonstrated in this hybrid architecture, which is consistent with coupling
strengths reported in cQED experiments using GaAs Frey  _et al._ (2012); Toida
_et al._ (2013) and InAs Petersson  _et al._ (2012) quantum dots. In addition,
this DQD-resonator architecture provides access to a sensitive dispersive
microwave readout Blais  _et al._ (2004) mechanism for the graphene
nanostructures. Previously, graphene quantum dots have only been studied using
direct current (DC) transport measurements Ponomarenko  _et al._ (2008) or
quantum-point contacts for charge sensing Wang  _et al._ (2010); Güttinger
_et al._ (2008). Using a dispersive readout via the resonator, we can
simultaneously extract the tunneling rate between graphene DQD charge states,
the DQD-resonator coupling strength, and the dephasing rate, by measuring the
resonators phase response as a function of the DQD bias at multiple probe
frequencies. We find that the charge-state dephasing rates in our graphene DQD
varies between 0.5 GHz and 2 GHz for different charge states.

##  II The device

Our hybrid graphene-DQD/superconducting resonator device is shown in Fig. 1.
The coupling of cavity to randomly located graphene flasks is a technical
challenge. To meet this challenge, we have designed and fabricated a half-
wavelength superconducting reflection-line resonator consisting of two
differential microstrip lines which does not require the ground plane that is
indispensable in traditional transmission-line designs. The microwave field is
mostly confined between the two strips, which at each point along the line has
an electrical potential with opposite sign (180 degree phase shift). The RLR
is coupled to a regular transmission line via a 180 degree hybrid, which
splits the microwave signal into two opposite phases [Fig. 1(a)]. The
reflected microwave signal is measured using a network analyzer (NA). This RLR
structure is a flexible design that could accommodate the coupling of multiple
qubits (see supplementary materials).

We couple the RLR to the DQD by connecting the two strips at one end of the
RLR to the two Ti/Au plunger gates LP and RP of the graphene quantum dot, see
Fig. 1(b-c). This design of the RLR allows us to apply bias voltages through
the two strips to facilitate the needed electrostatic confinement of the
graphene DQD. The basic structure of the DQD along with an adjacent quantum
point contact channel is defined by plasma etching of a large graphene flake.
The electron numbers $(M,N)$ in the left and right dots are well defined by
the confinement potential induced by the LP and RP gates, respectively. An
electric dipole moment of $d\sim 1000$ $ea_{0}$ is formed by the change in
charge distribution as one trapped electron moves between the two potential
wells of the DQD (see the supplementary materials). Here $a_{0}$ is the Bohr
radius and $e$ is the electron charge. The DQD couples to the microwave field
generated by the superconducting resonator via this dipole moment. The sample
is mounted in a dry dilution refrigerator with a base temperature of about
$26$ mK. The resonance frequency of the RLR is $6.23896$ GHz and the quality
factor is about 1600 with all the gates of the DQD grounded.

##  III Measurement of the DQD through the QPC

We first demonstrate a gate-defined graphene DQD with a QPC charge sensing
measurement. In order to study electron tunneling between the two dots, and to
form a dipole coupling to the microwave field, the tunneling barriers of the
DQD must be made large. This also makes the resistance through the DQD large,
which makes it difficult to detect DC transport through the DQD. We therefore
use a nearby QPC as a charge sensor to probe the DQD. By recording the
transconductance $dI_{\rm QPC}/dV_{\rm LP}$ as a function of the LP-RP gate
voltages using a standard lock-in amplifier technique, we can measure the
hexagon-like charge-stability diagram in a very large range of gate voltages.
The result demonstrates that a graphene double quantum dot is formed in our
device [see Fig 2(a)]. We also measured the full width at half maximum (FWHM)
of the QPC signal across the $(M+1,N)\leftrightarrow(M,N+1)$ interdot
transition line as a function of temperature, and we extracted the electron
temperature $T_{e}$, the gate lever arm $\alpha$, and interdot tunneling rate
$2t_{C}$ from the experimental data Wei  _et al._ (2013), see Fig. 2(d,e,f)
and the supplementary materials.

##  IV Measurement of the DQD through the resonator

We also probe the DQD using the RLR by applying a coherent microwave signal to
the resonator and analyzing the reflected signal. We fix the probe frequency
at 6.2385 GHz and record the amplitude $A$ and phase $\phi$ of the reflected
signal $S_{11}$, as a function of the DQD bias voltages $V_{\rm LP}$ and
$V_{\rm RP}$. Phase shifts $\Delta\phi$ and amplitude changes $\Delta A$ are
observed at the triple points and on the interdot transition lines, where the
charge states of the left and right dots are degenerate [see Fig. 2(b,c)]. On
the cotunneling lines, no phase shift or amplitude change is detected because
the charging energy (about 10 meV) of a single quantum dot is much larger than
the RLR photon energy (26 $\mu$eV). However, the RLR photon energy is close to
the interdot transition energy, and the electron transitions between the dots
can therefore be assisted by and detected through the RLR. Using the same LP-
RP gate voltage biases as in our previous QPC measurements, we can again
measure the charge-stability diagram using the phase shift and amplitude
change [see Fig. 2(a-c)]. The phase shift and amplitude change are caused by a
dispersive shift of the resonance frequency shift due to the interaction with
the off-resonant DQD. Keeping the probe frequency $\omega_{R}/2\pi$ fixed,
when $\omega_{0}/2\pi$ is shifted to lower frequencies, produces a change in
$\Delta A$ and $\Delta\phi$ [see Fig. 3(e,f)]. In order to study the dipole
coupling of this hybrid system, we record the phase and amplitude response
while we sweep the DQD gate voltages across the
$(M+1,N)\leftrightarrow(M,N+1)$ interdot transition line, corresponding to the
DQD qubit energy bias $\epsilon$ being swept from negative to positive values.
A two-level artificial atom is formed with an energy splitting of
$\Omega=\sqrt{\epsilon^{2}+4t_{C}^{2}}$, where $2t_{C}$ is the tunneling
splitting caused by the interdot coupling. The charge states hybridizes around
$\epsilon=0$ [see Fig. 3(a)]. The effective interaction strength is
characterized by the AC susceptibility ${\rm Re}(\chi)$ [see Fig. 3(b)]. We
find experimentally that the resonator and the DQD can be successfully
coupled. This is encouraging as it was not obvious previously to us that the
coupling strength between this cavity and an atomic layered material can be
sufficiently strong. We find that the phase and amplitude response sensitively
depend on the graphene DQD parameters, and these relations are analyzed
theoretically in the supplementary material. Although $2t_{C}$ generally can
be tuned in this kind of etched graphene structure using a middle gate Wei
_et al._ (2013), our setup lacks of this middle gate because the QPC is
fabricated in its place, and we therefore cannot tune $2t_{C}$ in-situ.
$2t_{C}$ is measured for a large region [see Fig. 2(a)] in our sample and is
found to be larger than $\omega_{0}$. In previous work Frey  _et al._ (2012);
Petersson  _et al._ (2012); Toida  _et al._ (2013); Basset  _et al._ (2013),
single-peak and double-peak structures in the response of the phase and
amplitude as a function of $\epsilon$ for different values of $2t_{C}$ have
been demonstrated. The observed double-peak response is due to the changing
sign of the dispersive shift when the qubit energy transition from larger to
smaller than the cavity frequency, which can occur if $2t_{C}<\omega_{0}$ when
$|\epsilon|$ is swept.

##  V Device parameters

The measured phase shift $\Delta\phi=-{\rm arg}(S_{11})$ depends on the
resonance frequency $\omega_{0}$, the internal and external resonator
dissipation rates $\kappa_{\rm i}$ and $\kappa_{\rm e}$, the DQD-resonator
coupling strength $g_{C}$, the DQD interdot tunneling rate $2t_{C}$, energy
bias $\epsilon$, energy relaxation rate $\gamma_{1}$, and dephasing rate
$\gamma_{2}$. Here $\omega_{0}$, $\kappa_{\rm i}$, and $\kappa_{\rm e}$ can be
obtained by fitting the phase response as a function of probe frequency [see
Fig. 4(a)] (see the supplementary materials), $g_{C}$ can be calculated using
a capacitance model, $t_{C}$ can be extracted from measurements at varying
temperature Wei  _et al._ (2013), and $\epsilon$ can be calibrated from the
gate voltage lever arm measurements (6%) that is also obtained from the
measurements when varying the temperature. Previous work on graphene quantum
dots has reported $\gamma_{1}$ to be about 100 MHz Volk  _et al._ (2013). This
leaves $\gamma_{2}$ as the only remaining unknown parameter. As a example,
near the DQD bias region $V_{\rm LP}=325$ mV and $V_{\rm RP}=268$ mV, where
$g_{C}=15$ MHz and $2t_{C}=8$ GHz, by fitting the phase shift as a function of
$\epsilon$, we obtain the $\gamma_{2}$ for these charge states to be about
$1.7\pm 0.1$ GHz. Actually, $g_{C}$ and $2t_{C}$ can also be extracted from
this fitting. Previous work Basset  _et al._ (2013) has proven that using the
resonator is more precise to measure the tunneling rates when they approach
the resonator eigenfrequency $\omega_{0}$. $2t_{C}$ in our device is larger
than $\omega_{0}$ so that double peak in phase response is not observed.
Moreover, $g_{C}$ is found to be different for various DQD bias regions
Viennot  _et al._ (2013). Depending on the setting of the two plunger gates,
we get $g_{C}$ ranging from 6.5 MHz to 20 MHz. It is particularly worth noting
that the experimentally discovered $g_{C}$ of the hybrid structures for
graphene qubits is comparable to that for superconducting qubits and
semiconductor qubits. Here the probe frequency is fixed at $\omega_{0}$. Later
we will discuss a method to extract $\gamma_{2}$ more precisely with varying
probe frequency.

##  VI Measurements at multiple probe frequencies

From the theoretical analysis, we find that across the interdot transition
line where $2t_{C}>\omega_{0}$, a double-peak response can also be observed
with a suitable choice of probe frequency (see the supplementary materials).
The narrower structures of the double-peak response, compared to the single-
peak response, are more sensitive to a variety of device parameters, and
therefore more suitable for parameter extraction. We therefore developed a
method where mutiple probe frequencies are applied to the RLR (Fig. 4(c)),
which spans across the region where the double-peaked phase-shift response is
observed, as a function of $\epsilon$ and for $2t_{C}>\omega_{0}$ (Fig 3). We
would like to point out here this multiple probe frequency technique is
particularly useful for our graphene DQD, and other systems, where the qubit
parameters cannot be varied in a broad range. We extracted the phase error of
our measurement setup from the measurement data. Based on this error, a
simulation shows the extraction of $\gamma_{2}$ at the double-peak region is
more precise than single-peak region for $2t_{C}>\omega_{0}$ (see the
supplementary materials). We therefor fit the DQD parameters at double-peak
region as $2t_{C}$ in our device is larger than $\omega_{0}$. For example,
near the DQD bias region $V_{\rm LP}=302$ mV and $V_{\rm RP}=244$ mV, we
obtain $g_{C}=16.4\pm 0.4$ MHz, $2t_{C}=10.3\pm 0.1$ GHz and
$\gamma_{2}=1.6\pm 0.1$ GHz. In another DQD bias regime, $V_{\rm LP}=283$ mV
and $V_{\rm RP}=212$ mV, we obtain $g_{C}=6.7\pm 0.4$ MHz, $2t_{C}=7.3\pm 0.1$
GHz, and $\gamma_{2}=0.65\pm 0.1$ GHz. Errors of the fitted results are small,
as data was the measured and averaged until smooth curves were obtained. The
variances we use here are obtained from the least-square fit, and are subject
to the assumption that the model used correctly describes the measured data.
The main error of this fitting comes from the converting from gate voltage to
$\epsilon$ Wei  _et al._ (2013); DiCarlo  _et al._ (2004), this may cause an
error of about 20 percent for $\epsilon$.

##  VII Discussion

In Ref. Basset  _et al._ (2013), the $\gamma_{2}$ in a GaAs DQD system was
found to depend strongly on $2t_{C}$. Here we have measured different $2t_{C}$
in different charge-state regions and found that both $g_{C}$ and $\gamma_{2}$
depend on the bias conditions, using a mutiple-probe-frequency method. In the
supplementary materials, we have analyzed the double peak region for the phase
response as a function of $\epsilon$, and we found that it is a sensitive
region for extracting parameters. In contrast to previous work Frey  _et al._
(2012); Basset  _et al._ (2013), where the resonantor response as a function
of $\epsilon$ has been used to extract parameters by varying $2t_{C}$, in our
method we only tune the probe frequency. However, theoretically we find that
tuning any of the free parameters, for example $2t_{C}$, $g_{C}$, $\gamma_{2}$
or $\omega$, can result in a double-peaked phase response. In our experiment
we tune the probe frequency $\omega$ because it is easy to control and can be
tuned much more accurately than other parameters. Also, our method of
measuring the phase shift at multiple probe frequencies could have an
advantage since it does not induce variations in the DQD parameters due to
changes in the DQD bias conditions (while, for example, tuning $2t_{C}$ might
Frey  _et al._ (2012); Basset  _et al._ (2013)). In our hybrid DQD-resonator
device, we demonstrate that $g_{C}$ varies from 6.5 MHz to 20 MHz, and
$\gamma_{2}$ from 0.5 GHz to 2 GHz, as the DQD bias conditions are changed.
Since $g_{C}\ll\gamma_{2}$ in our device, we do not reach the strong coupling
regime and we therefore do not observe vacuum Rabi splitting Wallraff  _et
al._ (2004). It is therefore important to analyze dephasing time of the
graphene DQD. The dephasing time however cannot be easily obtained by normal
means because paddles and edge states Wang  _et al._ (2010); Molitor  _et al._
(2010); Evaldsson  _et al._ (2008); Gallagher  _et al._ (2010) in graphene can
mask-off its determination in charge transport based measurements. The
resonant cavity, on the other hand, is primarily sensitive to the electrical
dipole of the DQD and is affected substantially less by the electrostatic
disorders. Indeed, the previously unknown $\gamma_{2}$ has been extracted for
the first time in our experiment. Reducing $\gamma_{2}$ and reaching the
strong coupling regime remains an important goal for future work.

In conclusion, we have designed and fabricated a superconducting reflection-
line resonator, and for the first time coupled a resonator to a graphene
double quantum dot. This provides a platform for studying the physics of
light-matter interaction with graphene devices in the microwave regime. In the
future, long-distance and scalable quantum information processing with
graphene qubits may be possible using this circuit quantum electrodynamics
architecture. We demonstrate a graphene-qubit/resonator coupling rate of
around tens of MHz in this hybrid device. This is consistent with results
obtained in previous experiments using semiconducting quantum dots and
transmission-line resonators Frey  _et al._ (2012); Toida  _et al._ (2013). By
fitting the phase shift as a function of the graphene-qubit energy splitting,
we have accurately extracted device parameters and dephasing rates of the
hybrid nanostructure using multiple probe-frequency measurements. For the
first time, we have measured the dephasing rate in a graphene double quantum
dot, which was observed to range from 0.5 GHz to 2 GHz depending on the
graphene-qubit bias conditions.

## Acknowledgements

This work was supported by the National Fundamental Research Programme (Grant
No. 2011CBA00200), and National Natural Science Foundation (Grant Nos.
11222438, 10934006, 11274294, 11074243, 11174267 and 91121014).

##  VIII contributions

G.W.D., D.W., S.X.L., M.L.Z. and H.W.J. fabricated the samples and performed
the measurements. J.R.J., F.N., H.O.L., G.C., T.T. and G.C.G. provided
theoretical support and analysed the data. G.P.G. supervised the project. All
authors contributed to the writing of this paper.

## References

  * Xiang  _et al._ (2013) Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013).
  * You and Nori (2011) J. Q. You and F. Nori, Nature 474, 589 (2011).
  * Childress  _et al._ (2004) L. Childress, A. Sørensen, and M. Lukin, Phys. Rev. A 69, 042302 (2004).
  * Guo  _et al._ (2008) G.-P. Guo, H. Zhang, Y. Hu, T. Tu, and G.-C. Guo, Phys. Rev. A 78, 020302 (2008).
  * Lin  _et al._ (2008) Z.-R. Lin, G.-P. Guo, T. Tu, F.-Y. Zhu, and G.-C. Guo, Phys. Rev. Lett. 101, 230501 (2008).
  * Cottet and Kontos (2010) A. Cottet and T. Kontos, Phys. Rev. Lett. 105, 160502 (2010).
  * Jin  _et al._ (2012) P.-Q. Jin, M. Marthaler, A. Shnirman, and G. Schön, Phys. Rev. Lett. 108, 190506 (2012).
  * Bergenfeldt and Samuelsson (2013) C. Bergenfeldt and P. Samuelsson, Phys. Rev. B 87, 195427 (2013).
  * Contreras-Pulido  _et al._ (2013) L. D. Contreras-Pulido, C. Emary, T. Brandes, and R. Aguado, New J. Phys. 15, 095008 (2013).
  * Lambert  _et al._ (2013) N. Lambert, C. Flindt, and F. Nori, EPL 103, 17005 (2013).
  * Delbecq  _et al._ (2011) M. R. Delbecq, V. Schmitt, F. D. Parmentier, N. Roch, J. J. Viennot, G. Fève, B. Huard, C. Mora, A. Cottet, and T. Kontos, Phys. Rev. Lett. 107, 256804 (2011).
  * Delbecq  _et al._ (2013) M. Delbecq, L. Bruhat, J. Viennot, S. Datta, A. Cottet, and T. Kontos, Nat. Commun. 4, 1400 (2013).
  * Viennot  _et al._ (2013) J. J. Viennot, M. R. Delbecq, M. C. Dartiailh, A. Cottet, and T. Kontos, arXiv:1310.4363 (2013).
  * Frey  _et al._ (2012) T. Frey, P. J. Leek, M. Beck, A. Blais, T. Ihn, K. Ensslin, and A. Wallraff, Phys. Rev. Lett. 108, 046807 (2012).
  * Toida  _et al._ (2013) H. Toida, T. Nakajima, and S. Komiyama, Phys. Rev. Lett. 110, 066802 (2013).
  * Basset  _et al._ (2013) J. Basset, D. Jarausch, A. Stockklauser, T. Frey, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Phys. Rev. B 88, 125312 (2013).
  * Petersson  _et al._ (2012) K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M. Taylor, A. A. Houck, and J. R. Petta, Nature 490, 380 (2012).
  * Geim and Novoselov (2007) A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007).
  * Geim (2009) A. K. Geim, Science 324, 1530 (2009).
  * Guo  _et al._ (2009) G.-P. Guo, Z.-R. Lin, T. Tu, G. Cao, X.-P. Li, and G.-C. Guo, New J. Phys. 11, 123005 (2009).
  * Rozhkov  _et al._ (2011) A. V. Rozhkov, G. Giavaras, Y. P. Bliokh, V. Freilikher, and F. Nori, Phys. Rep. 503, 77 (2011).
  * Castro Neto  _et al._ (2009) A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
  * Volk  _et al._ (2013) C. Volk, C. Neumann, S. Kazarski, S. Fringes, S. Engels, F. Haupt, A. Muller, and C. Stampfer, Nat. Commun. 4, 1753 (2013).
  * Zhang  _et al._ (2013) M.-L. Zhang, G.-W. Deng, S.-X. Li, H.-O. Li, G. Cao, T. Tu, M. Xiao, G.-C. Guo, H.-W. Jiang, I. Siddiqi, and G.-P. Guo, arXiv:1309.0178 (2013).
  * Blais  _et al._ (2004) A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).
  * Ponomarenko  _et al._ (2008) L. A. Ponomarenko, F. Schedin, M. I. Katsnelson, R. Yang, E. W. Hill, K. S. Novoselov, and A. K. Geim, Science 320, 356 (2008).
  * Wang  _et al._ (2010) L.-J. Wang, G. Cao, T. Tu, H.-O. Li, C. Zhou, X.-J. Hao, Z. Su, G.-C. Guo, H.-W. Jiang, and G.-P. Guo, Appl. Phys. Lett. 97, 262113 (2010).
  * Güttinger  _et al._ (2008) J. Güttinger, C. Stampfer, S. Hellmüller, F. Molitor, T. Ihn, and K. Ensslin, Appl. Phys. Lett. 93, 212102 (2008).
  * Wei  _et al._ (2013) D. Wei, H.-O. Li, G. Cao, G. Luo, Z.-X. Zheng, T. Tu, M. Xiao, G.-C. Guo, H.-W. Jiang, and G.-P. Guo, arXiv:1307.5663 (2013).
  * DiCarlo  _et al._ (2004) L. DiCarlo, H. Lynch, A. Johnson, L. Childress, K. Crockett, C. Marcus, M. Hanson, and A. Gossard, Phys. Rev. Lett. 92, 226801 (2004).
  * Wallraff  _et al._ (2004) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004).
  * Molitor  _et al._ (2010) F. Molitor, C. Stampfer, J. Güttinger, A. Jacobsen, T. Ihn, and K. Ensslin, Semicond. Sci. Technol. 25, 034002 (2010).
  * Evaldsson  _et al._ (2008) M. Evaldsson, I. V. Zozoulenko, H. Y. Xu, and T. Heinzel, Phys. Rev. B 78, 161407 (2008).
  * Gallagher  _et al._ (2010) P. Gallagher, K. Todd, and D. Goldhaber-Gordon, Phys. Rev. B 81, 115409 (2010).

Figure 1:  Hybrid graphene DQD/superconducting RLR device. (a) Circuit
schematic and micrograph of the hybrid device. The half-wavelength reflection-
line resonator is connected to a graphene DQD at the end of its two
striplines. A microwave signal is applied to the other end of the resonator,
and the reflected signal is detected using a network analyzer. The DC voltage
used to control the electron numbers in the DQD is applied via the two DC pads
directly connected to the resonator striplines. (b) Micrograph of the DQD gate
structure. (c) Scanning-electron micrograph of a typical sample of our device.

Figure 2:  Measurements of the graphene DQD charge-stability diagram. (a) The
charge-stability diagram measured using a quantum-point contact. (b-c) The
charge-stability diagram measured by the amplitude (b) and phase (c) response
of the reflection-line resonator. The three charge-stability diagrams show a
close correspondence. (d) A charge-stability diagram of a weak tunnel coupling
region, used to measure the full width at half maximum (FWHM) of the QPC
signal. (e) FWHM measured at the base temperature. We fit the data to a
Lorentzian. (f) FWHM as a function of the lattice temperature, measured by
varying the temperature of the mixing chamber. The high temperature region
shows a linear dependence and $2t_{C}$, $\alpha$, and $T_{e}$ can be extracted
by fitting the FWHM as a function of the lattice temperature.

Figure 3:  Measurements of the DQD-resonator coupling. [note: inaccurate
figure title] (a) The DQD energy levels. (b) AC susceptibility, ${\rm
Re}(\chi)$, as functions of the DQD detuning $\epsilon$. (c) The RLR resonance
frequency $f_{0}=\omega_{0}/2\pi$, compared to the DQD qubit transition
frequency, $\Omega$, for different interdot tunneling rates $2t_{C}$. (d) The
phase response of the RLR as a function of gate voltages $V_{\rm LP}$ and
$V_{\rm RP}$ near the $(M+1,N)\leftrightarrow(M,N+1)$ interdot transition
line, measured at a fixed probe frequency $\omega_{R}/2\pi=6.2385$ GHz. (e-f)
The spectrum of the phase (e) and amplitude (f) response for $\epsilon=0$
(blue) and for very large $\epsilon$ (red). (g) The phase response as a
function of DQD detuning $\epsilon$, in the signel-peak region (upper panel)
and the double-peak region (lower panel).  Figure 4:  Phase response. (a)
Best-fit of the phase vs frequency curve. Quality factor and resonance center
can be obtained. (b) $\gamma_{2}$ sensitivity to the fitting. Blue dot line is
the measured data, red line shows the best-fit curve, green and yellow lines
are the results with changing $\gamma_{2}$ at a small value
$\Delta\gamma_{2}$. (c) Experimental data of the phase shift $\Delta\phi$, as
a function of the DQD detuning $\epsilon$, collected for the same interdot
transition line as shown in Fig. 3(d). Each measurement is taken at a
different probe frequency $f_{R}$, which has a detuning of the cavity $\Delta
f=f_{R}-f_{0}$. The theoretical model used in the fitting is described in the
supplementary materials. The free fitting parameters were $2t_{C}$, $g_{C}$
and $\gamma_{2}$, and other DQD and resonator parameters were assumed to be
known from other measurements and calibrations. The extracted parameters are
given in the text.


#  From the Boltzmann equation with non-local correlations to a standard non-
linear Fokker-Planck equation

Airton Deppman  Instituto de Física - Universidade de São Paulo, São Paulo
05508-090, Brazil<EMAIL_ADDRESS>Alireza K. Golmankhaneh  Department of
Physics, Urmia Branch, Islamic Azad University, Urmia, Iran
<EMAIL_ADDRESS>Eugenio Megías  Departamento de Física Atómica,
Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain
<EMAIL_ADDRESS>Roman Pasechnik  Department of Physics, Lund University, 221 00
Lund, Sweden<EMAIL_ADDRESS>
###### Abstract

In this work, we study the formal connections between the non-linear Fokker-
Planck Equation associated with the non-additive entropy and the Boltzmann
Equation with the non-additive correlation functional. The collisional term
following the $q$-algebra is adopted. In the derivation of the non-additive
Fokker-Planck Equation, two constraints are imposed on the final result: i)
that the entropic index $q$ is a characteristic parameter of the non-additive
systems with a value that does not change with time, and ii) that for
$q\rightarrow 1$ a smooth transition for the standard Fokker-Planck Equation
is obtained.

The Fokker-Planck Equation (FPE) has been extensively used in many branches of
Physics, describing the time evolution of the probability density for dynamic
systems. The Non-linear FPE (NLFPE) appears in several phenomena, e.g.,
anomalous diffusion. Systems governed by the entropy $S_{q}$ [1, 2] have been
associated with a particular NLFPE [3], where non-local effects give rise to a
particular form of NLFPE, which has applications in the dynamics of self-
gravitating systems [4, 5], overdamped systems [6] and super- and sub-
diffusive transport regimes [7]. That equation was used to study granular
medium [8], and the predictions presented in that work were experimentally
proven to be correct [9]. In this work, FPE associated with the non-additive
systems will be called non-additive FPE or simply qFPE.

In recent years, both FPE and NLFPE have found increasing interest in High
Energy Physics (HEP) because it is a tool to investigate the dynamics of heavy
quarks in the Quark Gluon Plasma (QGP) [10, 11]. The HEP data have evidenced
the heavy-tailed momentum distributions that are described by the
q-exponential function, and therefore the HEP is associated with the
nonadditive entropy, $S_{q}$. The emergency of the nonextensive statistics in
non-Abelian Quantum Field Theory (QFT) has been associated with the
thermofractal structures [12, 13] resulting from the scaling properties of the
field theory that are manifested in the renormalization group equation [14].
As a consequence of the fractal structure, the QCD running-coupling is
obtained by an analytical formula. The parameter $q$, that in the nonextensive
statistics represents the entropic index, can be calculated in terms of the
QCD fundamental parameters, which are the number of colours, $N_{c}$, and the
number of flavours, $N_{f}$. Hereby, the qFPE might be a useful tool for
describing the dynamics of quarks in the QGP.

In the present work, we study formal connections between the qFPE and the
standard Boltzmann Equation endowed with a correlation term proposed in Refs.
[15, 16]. The stationary solutions for the generalized equation are
$q$-exponentials and are described by the $S_{q}$ statistics. The origins of
the new correlation term are associated with the finite size of the system
[17] and, mathematically, with the $q$-algebra [18].

The dynamics of complex systems following the non-additive entropy must
necessarily be different from those following the Boltzmann statistics, and
the main difference dwells in the collisional term of the dynamical equation.
As a consequence, the Fokker-Planck Equation must be modified, but one can
expect that both the dynamical equation and the Fokker-Planck Equation recover
the correspondent equation for the Boltzmann statistics in the limit of
$q\rightarrow 1$.

Starting from the Boltzmann Equation, if $f(\mathbf{x},\mathbf{p},t)$ is the
probability density for a system over the phase-space, the Boltzmann Equation
is given by

$\frac{\partial f(\mathbf{p},t)}{\partial
t}=-\nabla_{\mathbf{p}}f(\mathbf{p},t)\cdot\mathbf{F}+C[f]\,,$ (1)

where $C[f]$ is the integral collision term, and

$f(\mathbf{p},t)=\int d^{3}xf(\mathbf{x},\mathbf{p},t)\,.$ (2)

Here, it is assumed that the density $f(\mathbf{x},\mathbf{p},t)$ vanishes for
$|\mathbf{x}|\rightarrow\infty$. In the following, the symbol $t$ in the
argument of the density will be omitted for convenience. In the absence of
external force, $\mathbf{F}=0$, and the equation reduces to

$\frac{\partial f(\mathbf{p})}{\partial t}=C[f]\,.$ (3)

For a quantum system, the collision term is expressed in terms of the
transition matrix, $\cal M$, as

$\begin{split}&C[f(\mathbf{p})]=\frac{1}{2\omega_{p}}\int\frac{d^{3}p^{\prime}}{2\omega_{p^{\prime}}}\int\frac{d^{3}q}{2\omega_{q}}\int\frac{d^{3}q^{\prime}}{2\omega_{q^{\prime}}}|{\cal
M}|^{2}\times\\\
&\;\delta^{(3)}\left((\mathbf{p}^{\prime}-\mathbf{p})-(\mathbf{q}-\mathbf{q}^{\prime})\right)\left(h[f(\mathbf{p}),f(\mathbf{q})]-h[f(\mathbf{p}^{\prime}),f(\mathbf{q}^{\prime})]\right)\,,\end{split}$
(4)

where, usually, one has

$h[f(\mathbf{p}),f(\mathbf{q})]=f(\mathbf{p})f(\mathbf{q})\,.$ (5)

At the stationary state, the density will be described by an exponential
function. Initially, distribution functions of the form $f\equiv
f(\bm{\beta}\cdot\mathbf{p})$, with
$\bm{\beta}=\beta_{x}\mathbf{i}+\beta_{y}\mathbf{j}+\beta_{z}\mathbf{k}$ will
be considered.

The correlation term in Eq. (5) plays a central role in the derivation.
Defining the transition rate, $w(\mathbf{p},\mathbf{k})$ as

$w(\mathbf{p},\mathbf{k})=\frac{1}{2\omega_{p}}\frac{1}{2\omega_{p^{\prime}}}\frac{1}{2\omega_{q^{\prime}}}\int\frac{d^{3}q}{2\omega_{\mathbf{q}}}|{\cal
M}|^{2}f(\mathbf{q})\,$ (6)

with $\mathbf{k}=\mathbf{p}^{\prime}-\mathbf{p}$, it results 111Notational
remark: $(wf)(\mathbf{a})=w(\mathbf{a})f(\mathbf{a})$. Wherever possible, the
second argument, $\mathbf{k}$, will be omitted in $w$ and $(wf)$.

$C[f(\mathbf{p})]=\int
d^{3}k\,[(wf)(\mathbf{p}+\mathbf{k})-(wf)(\mathbf{p})]\,.$ (7)

Expanding the product $(wf)(\mathbf{p})$ up to the second order, to satisfy
the Pawula theorem [19], the usual Fokker-Planck Equation is obtained.

For non-additive systems, the correlation function is given by

$h[f(\mathbf{p}),f(\mathbf{q})]_{q}=f(\mathbf{p})\otimes_{q}f(\mathbf{q})\,,$
(8)

where the symbol $\otimes_{q}$ represents the $q$-product [18]. In Refs. [15,
16] the authors show that the $q$-exponential distribution can be obtained at
the stationary state if, instead of Eq. (5), the collision term is expressed
in terms of

$h[f(\mathbf{p}),f(\mathbf{q})]_{q}=\left[f(\mathbf{p})^{1-q}+f(\mathbf{q})^{1-q}-1\right]^{\frac{1}{1-q}}\,.$
(9)

The term on the right-hand side of the equation above is the algebraic
representation of the $q$-product, and it results in the regular product of
functions when $q\rightarrow 1$.

The main aspects related to the non-additive functional in Eq. (9) were
discussed in Ref. [20], and in Ref. [17] it was shown that the origin of the
correlation might be related with topological characteristics of the phase-
space. Because in the non-additive case the correlation term changes, it is
necessary to derive the extended Fokker-Planck Equation for this case. This
will be done in this work

It will be considered that $\mathbf{k}$ can be made sufficiently small. One
can observe, from Eq. (8), that $f(\mathbf{k})\rightarrow 1$ for
$\mathbf{k}\rightarrow 0$. Using the first-order approximation of the right-
hand side of Eq. (9) it results

$f(\mathbf{p})\otimes_{q}f(\mathbf{k})=f(\mathbf{p})+f(\mathbf{p})^{q}{\bm{\beta}}\cdot{\mathbf{k}}\,,$
(10)

where $\bm{\beta}\equiv\frac{\partial
f}{\partial\mathbf{k}}\big{|}_{\mathbf{k}=0}$ is the linear coefficient of the
first order approximation of the function $f(\mathbf{k})$. According to the
rules of the $q$-algebra, if $\mathbf{k}$ is an infinitesimal vector, the
$q$-product is222The $q$-calculus was introduced in Ref. [18], where the
$q$-product is defined in accordance with the r.h.s. of Eq. (9) in the present
paper. In Eq. (11), we use the dual derivative operator, defined by Eq. (20)
in Ref. [18].

$f(\mathbf{p})\otimes_{q}f(\mathbf{k})=f(\mathbf{p})\left[1+\frac{\bm{\beta}\cdot\mathbf{k}}{1+(1-q)\bm{\beta}\cdot\mathbf{p}}\right]\,.$
(11)

Comparing the last two equations, it results in

$f(\mathbf{p})^{q}=\frac{f(\mathbf{p})}{1+(1-q)\bm{\beta}\cdot\mathbf{p}}\,.$
(12)

The group of transformations of thermofractals is isomorphic with the
$q$-algebra [21], so its evolution is also described by Eq. (12). Moreover,
due to the fractal structure with scaling invariance, it also follows that the
product $f(\mathbf{p})\otimes_{q}f(\mathbf{k})=f(\mathbf{p}+\mathbf{k})$.
Here, we assume that this equation is valid in general, for all non-additive
systems. From Eq. (10) and using the result above, it results that

$f(\mathbf{p}+\mathbf{k})-f(\mathbf{p})=\frac{\partial
f}{\partial\mathbf{p}}\cdot\mathbf{k}=f(\mathbf{p})^{q}\bm{\beta}\cdot\mathbf{k}\,,$
(13)

therefore

$\frac{\partial
f(\mathbf{p})}{\partial\mathbf{p}}=f(\mathbf{p})^{q}\bm{\beta}\,.$ (14)

The equation above, if understood in a proper way, is the central point of the
formal connection between the Boltzmann Equation and the non-additive Fokker-
Planck Equation. Notice that, already in the Boltzmann equation, there is the
introduction of the $q$-product in the place of the regular product. From an
epistemological point of view, this is the necessary gap for going from the
Boltzmann statistics to the non-additive statistics. However, the regular
product plays a central role in the derivation of the FPE, so it is necessary
to replace the $q$-product by the regular product. In doing so, the
epistemological gap must be placed at another point, and Eq. (14) will be used
for that. Here, Eq. (14) is to be understood as valid for any function $f$,
and so it imposes constraints on the derivative operator that will be
incorporated in the qFPE to be derived.

The correlation function in Eq. (8) can be written in the form of a regular
product if one constrains the distribution $f(\mathbf{p})$ to satisfy the
constraint on the derivative given by Eq. (14), since if the latter equation
is valid, the first one necessarily holds. Then, the collision term of the
Boltzmann Equation assumes the same form for the non-additive systems as for
the systems following the Boltzmann statistics, and the transition rate,
$w(\mathbf{p},\mathbf{k})$, can be written in the same way as in Eq. (6). For
clarity, the symbol $\bar{\partial}/\partial\mathbf{p}$ will be used to
identify the special derivative operator which satisfies the non-additivity
condition in Eq. (14). This symbol will be eliminated in favour of the
standard derivative in the process to obtain the qFPE, and its formal
structure will be described by the relations (23), below.

Expanding the product $wf$ up to the second order, the terms depending on
$\mathbf{p}+\mathbf{k}$ result in

$\begin{split}(wf)(\mathbf{p}+\mathbf{k})-(wf)(\mathbf{p})=&\sum_{i}k_{i}\frac{\bar{\partial}(wf)(\mathbf{p})}{\partial
p_{i}}+\\\ &\frac{1}{2}\sum_{i,j}k_{i}k_{j}\frac{\bar{\partial}^{2}}{\partial
p_{i}\partial p_{j}}(wf)(\mathbf{p})\,.\end{split}$ (15)

Because only the transition rate depends on $\mathbf{k}$, it is possible to
integrate over $d^{3}k$, obtaining

$\frac{\partial f}{\partial t}-\frac{\bar{\partial}}{\partial
p_{i}}\left[A_{i}f+\frac{\bar{\partial}(B_{ij}f)}{\partial p_{j}}\right]=0\,,$
(16)

where

$A_{i}=\int d^{3}k\,w(\mathbf{p},\mathbf{k})k_{i}\,,$ (17)

and

$B_{ij}=\frac{1}{2}\int d^{3}k\,w(\mathbf{p},\mathbf{k})k_{i}k_{j}\,.$ (18)

The summation rule over repeated indexes is adopted in Eq. (16).

The result obtained in the last equation is, formally, the usual term for the
standard Fokker-Planck Equation, unless by the use of the special derivatives.
The objective of the present is to obtain the form of the Fokker-Planck
Equation for non-additive systems in terms of the standard derivatives. This
task is essentially different from the one carried out in Ref. [22]. Here, the
non-additive Fokker-Planck Equation will be derived from the properties of the
correlation functional. The following constraints are imposed on the qFPE:

  1. 1.

that the fundamental aspects of the non-additive systems can be completely
incorporated into the mathematical description of the dynamics by imposing the
relations of the derivatives of the distribution presented in Eq. (14);

  2. 2.

that the parameter $q$ is a well-defined number during the diffusive process,
that is, $q$ is constant and unique for a given system;

  3. 3.

that the generalized FPE smoothly recover the standard form of the Boltzmann
statistics when $q\rightarrow 1$.

It will be shown that the non-linear Fokker-Planck Equation proposed in Ref.
[3] is obtained from the Boltzmann Equation by solely assuming the non-
additive correlation function and the $q$-algebra.

To incorporate the deformed algebra in terms of the standard derivative
operators, note that from Eq. (14) it results that

$\frac{\partial f}{\partial p_{i}}=-\beta_{i}f^{q}=-\beta_{i}f^{q-1}f\,,$ (19)

and hereby

$\frac{\partial}{\partial p_{j}}\left(f^{1-q}\frac{\partial f}{\partial
p_{i}}\right)=-\beta_{i}\frac{\partial f}{\partial p_{j}}\,.$ (20)

Considering that

$f^{1-q}\frac{\partial f}{\partial p_{i}}=\frac{1}{(2-q)}\frac{\partial
f^{2-q}}{\partial p_{i}}\,,$ (21)

we have

$\beta_{i}\frac{\partial f}{\partial
p_{j}}=-\frac{1}{(2-q)}\frac{\partial^{2}f^{2-q}}{\partial p_{i}\partial
p_{j}}\,.$ (22)

All constraints on the qFPE can be satisfied by noting that, near the
stationary state, the solutions of the qFPE are expected to be of the
$q$-exponential form. Then, all terms in the qFPE must return a function that
has the same exponent, and from Eqs. (19) to (22) it results that one can make
the substitutions

$\begin{cases}{\bar{\partial}f}/\partial p_{i}\rightarrow\partial f/\partial
p_{i}\\\ \bar{\partial}^{2}f/(\partial p_{i}\partial
p_{j})\rightarrow\partial^{2}f^{2-q}/(\partial p_{i}\partial
p_{j})\,.\end{cases}$ (23)

With those substitutions, the qFPE will be

$\frac{\partial f}{\partial t}-\frac{\partial}{\partial
p_{i}}\left[A_{i}f+\frac{\partial(B_{ij}f^{2-q})}{\partial p_{j}}\right]=0\,.$
(24)

Observe that the equation above results in the standard FPE when $q=1$, and
all terms will contribute with the same value for $q$, because of the result
obtained in Eq. (22).

If $\mathbf{p}\sim 0$ then $A_{i}\rightarrow\gamma p_{i}$ and
$B_{ij}\rightarrow D\delta_{ij}$, and the qFPE reduces to

$\frac{\partial f}{\partial
t}-\gamma\frac{\partial}{\partial\mathbf{p}}\cdot(\mathbf{p}f)-D\frac{\partial^{2}f^{2-q}}{\partial\mathbf{p}^{2}}=0\,.$
(25)

It is possible to show that the qFPE above has a stationary solution of the
form (for 1-dimensional cases)

$f(p)=A\left[1-(1-q)\beta(p-p_{M})\right]^{\frac{1}{1-q}}\,,$ (26)

where $p_{M}$ is the centre of the distribution at the stationary regime. The
qFPE can be generalized by using $\bm{\beta}=\sum_{i}\beta_{i}\mathbf{e}_{i}$,
where $\mathbf{e}_{i}$ are unitary vectors. The general solution for the
equation will be similar to the one above.

For more complex cases, a stationary solution can be obtained only if the
external force, $\mathbf{F}$, in the Boltzmann Equation is not null. As an
example, if $\mathbf{F}=\mathbf{a}+b\mathbf{p}$, a general solution for the
qFPE can be obtained in the form

$f(\mathbf{p})=A\left[1-(1-q)\bm{\beta}\cdot(\mathbf{p}-\mathbf{p_{M}}(t))^{2}\right]^{\frac{1}{1-q}}\,,$
(27)

that represents a $q$-Gaussian function centered at $\mathbf{p_{M}}(t)$ [8].
This solution presents an asymptotically stationary state for
$t\rightarrow\infty$.

To complete the derivation of the qFPE, it will be shown that Eq. (24) is the
only equation that satisfies the constraints imposed initially. One could
consider a general NLFPE of the form

$\frac{\partial f^{a}}{\partial t}=\gamma\frac{\partial
f^{b}}{\partial\mathbf{p}}+D\frac{\partial^{2}f^{c}}{\partial p^{2}}\,,$ (28)

for $a,b,c>0$. Considering the second constraint we imposed for the derivation
of the qFPE and the properties of the derivative operations described by Eqs.
(19)-(22), the constants $a$, $b$ and $c$ must satisfy the relation

$\frac{a}{1-q}=\frac{b}{1-q}=\frac{c}{1-q}-1\,.$ (29)

These equalities imply that $a=b$ and $c=a+(1-q)$. For $a=1$, it results

$\frac{\partial f}{\partial t}=\gamma\frac{\partial f}{\partial
p}+D\frac{\partial^{2}f^{2-q}}{\partial p^{2}}\,.$ (30)

The equation above is exactly that proposed in Ref. [3]. However, it is always
possible to define $q^{\prime}$ such that

$\frac{1}{1-q^{\prime}}=\frac{a}{1-q}\,,$ (31)

and from here one gets that

$\frac{c}{1-q}=\frac{2-q^{\prime}}{1-q^{\prime}}\,,$ (32)

resulting in the same Fokker-Planck generalization as that in Eq. (30), but in
terms of $q^{\prime}$ instead of $q$, and $f$ satisfying the $q$-algebra with
entropic index $q^{\prime}$. It is important to remind that the scale $\beta$
changes also to $\beta^{\prime}=\beta a$.

For the case $a,b,c<0$, one also obtains a similar form of the generalised
Fokker-Planck Equation but associated with a different form of the
$q$-exponential equation, according to the transformations discussed in Ref.
[21]. In this case, non-integrable solutions might be found. This result shows
that Eq. (24) is the only possible form for the Fokker-Planck Equation that
satisfies the constraint defined in the present work.

Concluding, the present work reports the formal connections between the
Boltzmann Equation generalized to include systems following the non-additive
entropy $S_{q}$, and the non-linear Fokker-Planck Equation proposed in Ref.
[3]. The approach used here starts by considering the Boltzmann Equation with
a collisional term that incorporates the $q$-algebra in the correlation
functional. By considering the properties of the $q$-product and the fact that
for a non-additive system the value of the entropic index $q$ represents a
well-defined quantity that remains unchanged during the dynamical evolution of
that system, the generalized Fokker-Planck Equation is derived. It is shown
that the only possible equation satisfying those constraints is the
generalized Fokker-Planck Equation given by Eq. (24).

This work evidences some aspects of the qFPE, proposed in Ref. [3], by
clarifying its relations with the Boltzmann Equation. The equation obtained
has already been tested experimentally, since the work in Ref. [9] confirms
the predictions done in Ref. [8] by using Eq. (24). The result obtained here
is of interest in any application of the dynamical evolution for systems that
are non-extensive, such as rock blasting [23], and fundamental aspects of
dynamical systems [24, 25, 26],

With the increasing interest in the propagation of quarks in the QGP, the qFPE
can be an important tool to model the quark interaction with the medium [11,
27, 28], and multiparticle production [29, 30, 31]. The latter application
will be considered in future works. In this particular domain, it is
fundamental to reckon with the effects of finite quark masses, which leads to
a set of coupled equations to tackle the different particle species. This type
of phenomenon has been recently addressed, in the context of the non-linear
Fokker-Planck Equation, by the work in Ref. [32].

Both the nonlinear Fokker-Planck equation and the Boltzmann equation with non-
local correlations are subjects of intense research eﬀorts, with applications
to various branches of physics, including Astrophysics and High Energy
Physics. The study of these two evolution equations and their respective
applications has remained, up to now, research areas largely independent of
each other. It is, therefore, a remarkable achievement to establish a link
between these active lines of inquiry. Although the present work has been
motivated by and aimed at applications in high energy physics, the results
obtained have a general and fundamental character that makes them relevant
also for other areas of physics, particularly in connection with the study of
complex systems.

Acknowledgements

The authors thank Dr Constantino Tsallis and Dr Ernesto Borges for
enlightening discussions on the subject of this work. The authors would like
to thank the anonymous reviewer of this paper. The last sentence in the
conclusions is taken from his comments on the present work.

E.M. would like to thank the Instituto de Física and the Instituto de
Astronomia e Geofísica of the Universidade de São Paulo, Brazil, for their
hospitality during the final stages of this work. A.D. is supported by the
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil),
grant 304244/ 2018-0, by Project INCT- FNA Proc. No. 464 898/2014-5, and by
FAPESP, Brazil grant 2016/17612-7. The work of E.M. is supported by the
project PID2020-114767GB-I00 funded by MCIN/AEI/10.13039/501100011033, by the
FEDER/ Junta de Andalucía-Consejería de Economía y Conocimiento 2014-2020
Operational Program under grant A-FQM178-UGR18, and by Junta de Andalucía
under grant FQM-225. The research of E.M. is also supported by the Ramón y
Cajal Program of the Spanish MCIN under grant RYC-2016-20678. R.P. is
supported in part by the Swedish Research Council grants, contract numbers
621-2013-4287 and 2016-05996, as well as by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 668679).

## References

  * [1] C. Tsallis, Possible Generalization of the Boltzmann-Gibbs Statistics, Journal of Statistical PhysicsJ 52 (1-2) (1988) 479–487.
  * [2] C. Tsallis, Introduction to the Nonextenssive Statistical Mechanics, Springer, New York, 2009.
  * [3] A. R. Plastino, A. Plastino, Non-extensive statistical mechanics and generalized Fokker-Planck equation, Physica A: Statistical Mechanics and its Applications 222 (1) (1995) 347–354.  doi:https://doi.org/10.1016/0378-4371(95)00211-1.
  * [4] M. Shiino, Stability analysis of mean-field-type nonlinear Fokker-Planck equations associated with a generalized entropy and its application to the self-gravitating system, Phys. Rev. E 67 (2003) 056118.  doi:10.1103/PhysRevE.67.056118.
  * [5] P.-H. Chavanis, Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dimensional turbulence, Phys. Rev. E 68 (2003) 036108.
  * [6] A. R. Plastino, R. S. Wedemann, C. Tsallis, Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction, Symmetry 13 (9) (2021).  doi:10.3390/sym13091621.
  * [7] G. A. Casas, F. D. Nobre, Nonlinear Fokker-Planck equations in super-diffusive and sub-diffusive regimes, Journal of Mathematical Physics 60 (5) (2019) 053301\.  doi:10.1063/1.5001189.
  * [8] C. Tsallis, D. J. Bukman, Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54 (1996) R2197–R2200.  doi:10.1103/PhysRevE.54.R2197.
  * [9] G. Combe, V. Richefeu, M. Stasiak, A. P. F. Atman, Experimental validation of a nonextensive scaling law in confined granular media, Phys. Rev. Lett. 115 (2015) 238301.  doi:10.1103/PhysRevLett.115.238301.
  * [10] B. Svetitsky, Diffusion of charmed quarks in the quark-gluon plasma, Phys. Rev. D 37 (1988) 2484–2491.  doi:10.1103/PhysRevD.37.2484.
  * [11] D. B. Walton, J. Rafelski, Equilibrium distribution of heavy quarks in Fokker-Planck dynamics, Phys. Rev. Lett. 84 (2000) 31–34.  arXiv:hep-ph/9907273, doi:10.1103/PhysRevLett.84.31.
  * [12] A. Deppman, Thermodynamics with fractal structure, Tsallis statistics, and hadrons, Physical Review D 93 (5) (2016).
  * [13] A. Deppman, T. Frederico, E. Megias, D. P. Menezes, Fractal Structure and Non-Extensive Statistics, Entropy 20 (9) (2018).
  * [14] A. Deppman, E. Megias, D. P. Menezes, Fractals, nonextensive statistics, and QCD, Physical Review D 101 (3) (2020).
  * [15] J. A. S. Lima, R. Silva, A. R. Plastino, Nonextensive thermostatistics and the H theorem, Phys. Rev. Lett. 86 (2001) 2938–2941.  arXiv:cond-mat/0101030, doi:10.1103/PhysRevLett.86.2938.
  * [16] A. Lavagno, Relativistic nonextensive thermodynamics, Phys. Lett. A 301 (2002) 13–18.  doi:10.1016/S0375-9601(02)00964-7.
  * [17] E. Megias, J. A. S. Lima, A. Deppman, Transport Equation for Small Systems and Nonadditive Entropy, Mathematics 10 (10) (2022) 1625.  arXiv:2205.04947, doi:10.3390/math10101625.
  * [18] E. P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A: Statistical Mechanics and its Applications 340 (1) (2004) 95–101.
  * [19] R. F. Pawula, Approximation of the Linear Boltzmann Equation by the Fokker-Planck Equation, Phys. Rev. 162 (1967) 186–188.  doi:10.1103/PhysRev.162.186.
  * [20] C. Tsallis, Enthusiasm and skepticism - two pillars of science: A nonextensive statistics case, Physics 4 (2) (2022) 609–632.  doi:10.3390/physics4020041.
  * [21] A. Deppman, Thermofractals, non-additive entropy, and q-calculus, Physics 3 (2) (2021) 290–301.
  * [22] F. D. Nobre, E. M. Curado, G. Rowlands, A procedure for obtaining general nonlinear Fokker–Planck equations, Physica A: Statistical Mechanics and its Applications 334 (1) (2004) 109–118.
  * [23] M. Cardu, D. Coragliotto, P. Oreste, Analysis of predictor equations for determining the blast-induced vibration in rock blasting, International Journal of Mining Science and Technology 29 (6) (2019) 905–915.
  * [24] B. G. da Costa, I. S. Gomez, E. P. Borges, Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass, Physical Review E 102 (6) (2020).
  * [25] R. Onofrio, B. Sundaram, Relationship between nonlinearities and thermalization in classical open systems: The role of the interaction range, Physical Review E 105 (5) (2022).
  * [26] P.-H. Chavanis, The generalized stochastic smoluchowski equation, Entropy 21 (10) (2019).
  * [27] P. K. Srivastava, B. K. Patra, Drag and diffusion of heavy quarks in a hot and anisotropic qcd medium, European Physical Journal A 53 (6) (JUN 6 2017).  doi:10.1140/epja/i2017-12299-0.
  * [28] A. S. Parvan, O. V. Teryaev, J. Cleymans, Systematic comparison of tsallis statistics for charged pions produced in pp collisions, European Physical Journal A 53 (5) (MAY 22 2017).  doi:10.1140/epja/i2017-12301-y.
  * [29] G. Wilk, Z. Wlodarczyk, Some intriguing aspects of multiparticle production processes, International Journal of Modern Physics A 33 (10) (APR 10 2018).  doi:10.1142/S0217751X18300089.
  * [30] C.-Y. Wong, G. Wilk, L. J. L. Cirto, C. Tsallis, From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy pp and p(p)over-bar collisions, Physical Review D 91 (11) (2015).
  * [31] H. Berrehrah, P. B. Gossiaux, J. Aichelin, W. Cassing, E. Bratkovskaya, Dynamical collisional energy loss and transport properties of on- and off-shell heavy quarks in vacuum and in the quark gluon plasma, Physical Review C 90 (6) (2014).
  * [32] A. R. Plastino, R. S. Wedemann, F. D. Nobre, H-theorems for systems of coupled nonlinear fokker-planck equations, EPL 139 (1) (JUL 2022).  doi:10.1209/0295-5075/ac7c30.