\section*{Methods}

\subsection{Details of the experiment}

The granular medium was made of a number $N \in [300,2600]$ of spheres
of diameter $d=4$ mm made of non-magnetic steel (mass of each sphere:
$0.267$ g), glass (mass $0.0854$ g), or
delrin\textsuperscript{\textregistered} (mass $0.0462$ g). They were
housed in a plexiglas\textsuperscript{\textregistered} cylinder with a
conical-shaped floor (diameter $90$ mm, minimum height $28,5$ mm,
maximum height $47.5$ mm) in which a plexiglas vane (height $15$ mm,
width $6$ mm, length $35$ mm) was suspended in order to be in contact
with the granular medium and not with the container~\cite{gnoli14}.  The container
was vertically vibrated by an electrodynamic shaker (LDS V450) fed
with an acceleration signal $a(t)$. In most of the experiments $a(t)$
is a white noise with a band-pass filter between $200$ Hz and $400$
Hz, while in the lowest $p_{00}$ case ($p_{00}=78$ Pa) we used a
sinusoidal signal at frequency $53$ Hz. This choice is motivated by
two empirical observations: 1) a lower number of particles (as in the
case of low $p_{00}$) requires a larger energy input to be
homogeneously fluidized and to reach the blade, and this can be
obtained by supplying energy through a sinusoidal signal
at low frequency; 2) in dense cases a sinusoidal signal induces
spurious resonances, while in diluted cases such resonances are never
observed. We have checked that performing the same experiments with
noise signal for $a(t)$ (pushing the shaker to its working limits)
gives flow curves with the same shape.  An accelerometer placed on the
container side measured $a(t)$, allowing us to define
$\Gamma=\sqrt{2\langle a^2(t)\rangle}/g$.
The vane, mounted through its rotation axis to a rotary encoder, was
also connected to a dc motor (typical working voltage $12$ V) as the
source of the driving torque. The motor was directly fed by a dc
voltage supply in the range $0$ to $7$ V. No limit was set for the
maximum current absorbed by the motor that, averaged on the duration
of the experiment, was never higher than $450$ mA. A data acquisition
system collected data for the angular position/velocity of the vane,
the effective motor voltage, the current circulating in the motor and
the root-mean-square vertical acceleration of the container. A
procedure of calibration allowed us to translate average values of
current into average values of applied torque. The same procedure
helped determining the moment of inertia of the rotating block,
$3.2\times 10^2 \textrm{g mm}^2$ (the blade with its axis and the
gears linking it to the motor). The typical experiment, at a given
$\Gamma$ and applied motor voltage, was $3600$ s long, with the
granular always ``reset'' at the beginning of each run for $30$ s at
high shaking intensity ($\Gamma=42$) and motor off. This procedure -
together with periodic replacement of used spheres - guaranteed
reproducible results at a distance of several weeks. Packing fractions
was non-homogeneous (it was larger in regions far from the borders of
the container): its value at rest was estimated to be in the range
$55\% - 70\%$, while it decreased when vibration was switched on. In
the analysis we have identified the shear rate $\dot\gamma$ with the
average of the angular velocity $\omega(t)$ of the rotating blade,
i.e. $\dot\gamma=\langle \omega(t) \rangle$, while the shear stress
$\sigma$ is proportional to the average of the applied torque $T(t)$
through the shear stress constant $\kappa$, i.e. $\sigma=\kappa
\langle T(t) \rangle$ with $\kappa=2 \pi R^2 H$ (with $R$ and $H$ the
blade half-length and height, respectively)~\cite{dzuy83}.

\subsection{Details of data analysis}

Velocity power density spectra (VPDS) are defined as $S(f)=(1/(2
t_{TOT}) |\int_0^{t_{TOT}} \omega(t) e^{i(2 \pi f) t} dt|^2$ with
$t_{TOT}$ the time-length of an experiment ($=3600$ seconds). Some
examples of $S(f)$ curves are shown in the Fig.~1 of Supplementary
Information (SI). In~\cite{scalliet15} VPDS in a similar vibro-fluidized
experimental setup, without applied torque ($\sigma=0$), has been
investigated. In the dilute or gas-like limit, e.g. low number of
spheres at high shaking, the VPDS takes a simple Lorentzian shape
$S(f)=D/[1+(2 \pi f /f_{visc})^2]$, with $D$ the asymptotic (long
time) diffusivity and $f_{visc}$ the effective viscosity due to
granular gas-vane collisions. When the number of particles (density)
is increased and/or the intensity of shaking ($\Gamma$) is reduced,
the system approaches a slow liquid regime and the VPDS develops a
wide bump (or smooth peak) with a maximum near $f \sim 20$ Hz, which
is associated to oscillations of the velocity autocorrelation induced
by liquid cages. At much smaller frequencies the VPDS reaches a
plateau whose height, $\lim_{f \to 0} S(f)$, corresponds to diffusivity
$D$: indeed the blade is not trapped in a cage forever, eventually it
manages to explore a much larger phase space and reaches normal
diffusion. From the low frequencies plateau of VPDS we have extracted
values of $D$ for Figure~\ref{fig:fluct}. We have defined the
cage-exit frequency $f_{cage}$ as the x-position, in the VPDS plot, of the
minimum separating the cage bump from the low-frequency diffusive
plateau (see filled circles in Fig.~1 of SI).

\vspace{.5cm}

{\bf Acknowledgements}

We thank Giorgio Pontuale for useful discussions and suggestions. We
also thank Rachele De Felice and Francisco Vega Reyes for reading 
the manuscript.

\vspace{.5cm}

{\bf Corresponding author}

Correspondence to: Andrea Puglisi, andrea.puglisi@roma1.infn.it



