\section{Methods Summary}

The Vulcan Petawatt Laser Facility at Rutherford Appleton Laboratory provides a beam using optical parametric 
chirped pulse amplification (OPCPA) technology with a central wavelength of 1054~nm and a pulse 
full-width-half-maximum (FWHM) duration of 0.7~ps. The OPCPA approach enables an amplified spontaneous 
emission (ASE) to peak-intensity contrast ratio exceeding $1:10^9$
several nanoseconds before the peak of the laser pulse.  
The laser pulse  was focused with an $f/3$ off-axis parabola 
providing up to 160~J on the target. The maximum laser intensity of $3\times10^{20}$ W/cm$^2$
was achieved with a laser focus containing approximately 30\% of the energy in an 8~$\mu$m
(FWHM) diameter spot. The p-polarized laser beam was incident on target 
at $40^{\circ}$ from the target surface normal

The spectra were measured using a FSSR (focusing spectrometer with spatial resolution) equipped with spherically bent 
(radius of curvature $R=150$~mm) mica crystals. The FSSR spectrometers
were aligned to observe radiation in the wavelength range from $7.0$ to $8.4$ \AA~(energy from $1.47$ to $1.77$~keV) 
containing the K-shell spectra from multi-charged and neutral Al ions. The spectra were acquired from the
front, laser irradiated surface, at 45$^{\circ}$ to the target normal as schematically shown in Fig.~1(c). 
Background fogging and crystal fluorescence due to intense 
fast electrons were reduced using a pair of 0.5~T permanent magnets that formed a slit 10~mm wide in front of the crystal. 
Spectra were recorded on Kodak Industrex AA400 photographic X-ray film. 


Preliminary time-dependent
calculations using a simplified atomic model were performed to determine the bulk plasma conditions. We then performed detailed steady-state
calculations at various plasma conditions to compute the emission spectrum as a function of electron temperature and density and radiation
temperature, where we assumed that the radiation field seen by the plasma was a Planckian distribution. Our preliminary time-dependent
calculations show that, at the high electron densities
under consideration here the system reaches steady-state very quickly.
Our calculations include
all relevant atomic processes, that is, photoionization, collisional ionization, autoionization, collisional and radiative excitation and de-excitation,
and all recombination processes. The atomic structure and collisional cross section calculations were made using the Los Alamos suite of 
atomic physics codes$^{30}$. The resulting rate
equations are solved to produce a set of ion populations that depends sensitively on the electron temperature and density, the radiation
field, and the fraction of energetic electrons included in our model. Opacity effects are included via escape factors. The resulting
level populations are used to produce emission spectra, where the effects of detailed lines are included via a mixed-UTA (unresolved-transition-array)
approach. To correctly model the exotic plasma created experimentally, it is necessary to include a very large number of configurations,
representing Al ions in which up to six electrons have been removed from the $K$ and $L$ shells and placed in excited states. Such multiply excited
states were found to be important in accurately reproducing the observed emission spectra. In particular, we find that the radiative decay
from the $L$ to $K$ shell in any one specific configuration results in a relatively weak line, 
but that the very large number of configurations with different combinations
of spectator electrons in excited states, in which this radiative transition occurs, allows the observed transition to become prominent.  


\section{Acknowledgments}

We thank the Vulcan technical and target preparation teams at the Central Laser Facility for their support during the experiments. 
The research leading to these results has received funding from the Science and Technology Facilities Council, 
and the Engineering and Physical Science Research Council (Grant No. EP/E048668/1) of the United Kingdom.
The Los Alamos National Laboratory is operated by Los Alamos
National Security, LLC for the NNSA of the U.S. DOE under Contract No.
DE-AC5206NA25396.
The work of the JIHT RAS team was supported by RF President Grant \# MK-4725.2012.8 and RAS Presidium Program for Basic Research \#2.

\section{Author Contributions}

N.C.W., E.W., N.B., R.G.E., I.U and S.A.P. conceived this project. Interpretation of the spectroscopic data was carried out 
by J.C., J.A, A.Y.F., and I.Y.S. The experimental data was analysed by A.Y.F., S.A.P., and R.J.D. 
The Vulcan experiment was carried out by E.W., O.C., R.J.D., A.K.R., C.R.D.B., S.A.P, K.S.S, T.K., I.U., N.B, K.L.L., R.J.G., and N.C.W. 
Additional theoretical support was provided by A.G.Z. and additional experimental support was provided by P.M. and D.J.H. 
All authors contributed to the work presented here and to the final paper.



 
