Excited electron-bubble states in superfluid helium-4: a time-dependent density functional approach
EExcited electron-bubble states in superfluid He: a time-dependent density functionalapproach
David Mateo, Dafei Jin, Manuel Barranco, and Mart´ı Pi Departament ECM, Facultat de F´ısica, and IN UB,Universitat de Barcelona. Diagonal 647, 08028 Barcelona, Spain Department of Physics, Brown University, Providence, RI 02912, USA (Dated: November 2, 2018)We present a systematic study on the excited electron-bubble states in superfluid He using atime-dependent density functional approach. For the evolution of the 1P bubble state, two differentfunctionals accompanied with two different time-development schemes are used, namely an accuratefinite-range functional for helium with an adiabatic approximation for electron versus an efficientzero-range functional for helium with a real-time evolution for electron. We make a detailed com-parison between the quantitative results obtained from the two methods, which allows us to employwith confidence the optimal method for suitable problems. Based on this knowledge, we use thefinite-range functional to calculate the time-resolved absorption spectrum of the 1P bubble, whichin principle can be experimentally determined, and we use the zero-range functional to real-timeevolve the 2P bubble for several hundreds of picoseconds, which is theoretically interesting due tothe break down of adiabaticity for this state. Our results discard the physical realization of relaxed,metastable 2P electron-bubbles.
PACS numbers: 47.55.D-, 67.25.du, 33.20.Kf, 71.15.Mb
I. INTRODUCTION
Electron bubble (e-bubble) in liquid-helium has beenan attractive topic for numerous experimental and theo-retical studies in the past, and it has drawn again someinterest in recent years.
Density functional (DF) the-ory has proved to be a powerful tool in dealing with manyinteresting physical situations involving electron bubbles.When it is applied to optically excited e-bubbles, not onlycan it achieve quantitative agreement with experimentson the absorption spectra, it can also nicely dis-play the dynamical evolutions on the picosecond timescale, such as how the bubbles change shapes, releaseenergy, or even break into smaller bubbles.
Theselatest works likely require using different density func-tionals (finite-range or zero-range for liquid-helium) indifferent time-development schemes (adiabatic or real-time evolutions for electron). Regardless of the technicaldetails, the time-dependent density functional approachis no doubt the only workable approach at present forstudying the evolution of the excited e-bubble states inliquid-helium. The quantitative results drawn from thesesimulations can be useful to interpret the experimentalresults and predict new ones.Upon dipole excitation from the 1S ground state tothe 1P or 2P excited state, an e-bubble evolves by re-laxing its shape around the excited electron probabilitydensity. This relaxation eventually drives the e-bubbleback to the spherical 1S ground state. It has been quan-titatively shown that depending on the pressure ( P )applied to the liquid, this may happen in two differentways. At a pressure below about 1 bar, the e-bubbleundergoes damped oscillations for a period of time longenough to allow the electron to radiatively decay to thedeformed 1S state, which then evolves radiationlessly to the spherical 1S state. In contrast, at 1 bar or above, theexcited e-bubble evolves towards a configuration madeof two baby bubbles, so that the probability of findingthe electron evenly distributes between them. This two-bubble configuration is unstable against asymmetric per-turbations, and one expects this instability to cause theelectron to localize in one of the baby bubbles while theother collapses. When doing this calculation, the authors of Ref. 15naturally chose the so-called Orsay-Trento DF. Thisfunctional is finite-range and incorporates a term thatmimicks back-flow effects in order to accurately repro-duce the dispersion relation of the elementary excitationsof superfluid He. This is instrumental to properly de-scribe the energy transfer from the bubble to the liquid,that proceeds by causing all sorts of possible excitationsin the superfluid. The token one has to pay for its use isthe very high computational cost. Due to the large dif-ference between the intrinsic time scales of electron andhelium, this functional is not well adapted for fully real-time, three-dimensional evolutions. Considering this lim-itation, the adiabatic approximation was used to updatethe electron wavefunction at every instantaneous heliumconfiguration. One of the main concerns is then to es-tablish how long the adiabatic approximation is valid for.A careful analysis led to the conclusion that it holds forthe 1P e-bubble for at least several tens of picoseconds, a period of time large enough to guarantee the reliabil-ity of the results obtained for this state. Contrarily, theadiabatic approximation breaks down very quickly forthe 2P e-bubble, implying that the existence of relaxedquasi-equilibrium 2P bubbles is questionable.The authors of Ref. 14 followed a different path. In-stead of using a finite-range DF, they employed a muchsimpler zero-range one. This allowed them to carry out a r X i v : . [ c ond - m a t . o t h e r] N ov fully real-time calculations for hundreds of picosecondswithout imposing any adiabatic assumption, at the priceof an inaccurate description of the elementary excitationsof the liquid. Although this may not qualitatively affectthe physical results, it is unclear how quantitatively reli-able are the results so obtained.In this work, we perform a more systematic study onthe relaxation of the excited e-bubbles. We first carryout a detailed comparison between the finite-range andzero-range functionals applied to the 1P bubble problem,through which we gain some insight about their strong-points and shortcomings. We then use the finite-rangefunctional to calculate the time-resolved absorption spec-trum of the 1P bubble, which can in principle be mea-sured in the experiments. Next, we use the zero-rangefunctional to real-time evolve the 2P bubble for severalhundreds of picoseconds, which is of theoretical interestas one can clearly trace how the adiabatic approximationbreaks down for this state.Our paper is organized as follows. In Sec. II, we intro-duce the theoretical framework and numerical schemes.In Sec. III, we discuss the results so obtained. Finally, asummary is presented in Sec. IV. Several movies showingthe dynamical evolution of electron bubbles can be foundin the supplementary material. II. THEORETICAL FRAMEWORK
Within the DF approach, the energy of an electron-helium system at zero temperature can be written as afunctional of the single-electron wavefunction Φ and themacroscopic helium wavefunction Ψ E [Φ , Ψ] = (cid:126) m e (cid:90) d r |∇ Φ | + (cid:126) m He (cid:90) d r |∇ Ψ | + (cid:90) d r | Φ | V e − He [ ρ ] + (cid:90) d r E He − He [ ρ ] . (1)Specifically, Ψ = √ ρ exp[ ıS ] gives the helium particledensity ρ and the superfluid velocity v = (cid:126) ∇ S/m He . V e − He [ ρ ] is the electron-helium interaction potential, and E He − He [ ρ ] is the helium-helium potential energydensity. For the sake of comparison, we choose for E He − He [ ρ ] either the finite-range Orsay-Trento (OT) den-sity functional, or the zero-range Stringari-Treiner (ST)density functional. It is known that the former one pro-vides a very accurate description of superfluid He, par-ticularly of the dispersion relation that covers all the ro-ton excitations up to the wave number q = 2 . − .In contrast, the latter one only reproduces the long-wavelength phonon excitations and is not so accurate,but it has the advantage of high computational efficiencyin dynamic evolutions. The dispersion relations obtainedfrom both functionals using the method of Ref. 21 areplotted in Fig. 1 together with the experimental results. Functional variations of the associated grand poten-tial with respect to Ψ and Φ yield the following Euler- Lagrange equation for the helium and Schr¨odinger equa-tion for the electron: − (cid:126) m He ∆Ψ + (cid:26) δ E He − He [ ρ ] δρ + | Φ | δV e − He [ ρ ] δρ (cid:27) Ψ = µ Ψ(2) − (cid:126) m e ∆Φ + V e − He [ ρ ]Φ = ε Φ , (3)where µ is the helium chemical potential and ε is theelectron eigenenergy. Throughout this paper, we treatpressure as a given external condition. The associatedchemical potential and particle density in bulk liquid areobtained from the equation of state derived from the DFbeing used.The above equations are solved numerically with 13-point finite-difference formulae. In the finite-range DFcase, we work in three-dimensional Cartesian coordinatesthat allow for an extensive use of fast Fourier transfor-mation techniques as explained in Ref. 5. Whenevernecessary, we implement a Gram-Schmidt scheme to de-termine from Eq. (3) the electron spectrum in the heliumcavity. In the zero-range DF case, we work in cylindricalcoordinates assuming azimuthal symmetry around the z -axis ( r = 0) and specular symmetry about the z = 0plane. Hence, we only need to solve the equations in the r ≥ z ≥ The time evolution starts from an excited e-bubblestate, which means that the electron has been suddenlybrought from the 1S onto the 1P or 2P state of the orig-inal spherical e-bubble. From this initial configuration,the superfluid helium then evolves according to ∂ Ψ ∂t = − ı (cid:126) (cid:26) − (cid:126) m He ∆ − µ + U He − He [ ρ, v ]+ | Φ | δV e − He [ ρ ] δρ (cid:27) Ψ , (4)where the detailed form of the effective potential U He − He [ ρ, v ] can be found e.g. in Refs. 24 and 25.In the adiabatic approximation scheme, we do notevolve the electron in real-time but keep tracing the in-stantaneous eigenstates satisfying Eq. (3). In most cases,e-bubbles around excited electron states evolve towardsconfigurations that are not spherically but axially sym-metric. The originally degenerate angular momentumelectron eigenstates in the spherical bubble now split ac-cording to the projection ( m -values) on the symmetry z -axis, among which only the ± m states are still de-generate with each other. It is thus convenient to usethe notation for the orbital angular momentum of sin-gle particle states in linear molecules, i.e., σ, π, δ, φ, . . . for | m | = 0 , , , , . . . In addition to the axial symme-try, a n P bubble also keeps the original specular symme-try in the course of its evolution. Hence, one can con-struct the electron eigenbasis in such a way that the elec-tron wavefunctions satisfy Φ( r, z, θ ) = ± Φ( r, − z, θ ). Thecorrespondence between the lower lying spherically andaxially symmetric electron states is displayed in Fig. 2along with a representation of their probability densities.The superscript + ( − ) denotes specularly symmetric (an-tisymmetric) states. The adiabatic evolution is obtainedby keeping the 1P (2P) electron in the instantaneous 1 σ − (3 σ − ) eigenstate.In the real-time dynamics scheme, the electron evolvesaccording to ∂ Φ ∂t = − ı (cid:126) (cid:26) − (cid:126) m e ∆ + V e − He [ ρ ] (cid:27) Φ . (5)We employ a fourth-order Runge-Kutta method to ob-tain the first time steps, and Hamming’s method forsubsequent steps. A time step of 10 − ps is chosen forthe adiabatic evolution, and of 10 − ps for the dynamicalevolution. These very different values reflect the mass ra-tio m e /m He ∼ − . This is this reason that makes a dy-namical evolution unaffordable when E He − He [ ρ ] is finite-range, since updating the mean field is computationallyvery costly.During the bubble evolution, sound waves releasedfrom its surface eventually reach the cell boundary. Ifno action is taken, they will bounce back spoiling thecalculation. A way to handle this problem is to includesome source of damping into Eq. (4) governing the fluidevolution, see e.g. Refs. 1,27,28. We have opted by mak-ing the replacement ı −→ ı + Λ( r ) in Eq. (4). Thiscorresponds to a rotation of time axis in the complexplane by introducing a damping field Λ( r ), which takesthe form Λ( r ) = Λ (cid:20) (cid:18) s − s a (cid:19)(cid:21) , s ≡ | r | . (6)We keep the dimensionless parameter Λ (cid:39) .
6, and set a = 5 ˚A, s = 60 ˚A in the finite-range calculation, and a = 8 ˚A, s = 90 ˚A in the zero-range calculation. Theevolution is damping-free [Λ( r ) (cid:28)
1] in a sphere of ra-dius s < s − a , which is 50 ˚A in the finite-range caseand 70 ˚A in the zero-range case. From Figs. 3-6 and thesupplementary material one can see that this region islarge enough for the 1P e-bubble to expand within anundampening environment. For the 2P e-bubble evolu-tion, we use a ( r, z ) calculation box of 150 ×
150 ˚A and s = 120 ˚A, leaving ∼
100 ˚A of undampening space forthe bubble to expand.The above prescription works extremely well, as it effi-ciently dampens the excitations of the macroscopic wave- function at the cell boundaries, and does not need a largebuffer region to absorb the waves —actually we use thesame box where the starting static calculations have beencarried out. It allows us to extend the adiabatic calcu-lations of Ref. 15 from tens to hundreds of picoseconds.
III. RESULTS AND DISCUSSIONA. 1P e-bubble dynamics
1. Adiabatic versus real-time dynamical evolution
To some extent, an e-bubble in liquid-helium is nearlya textbook example of an electron confined in a spherical-square-well potential. Its static properties are fairly in-sensitive to the complexities of the chosen functionalprovided the bulk and surface properties can be wellreproduced.
In particular, a zero-range DF descrip-tion of the 1S-1P absorption energies of the e-bubble as afunction of P does not differ much from that obtained bya finite-range DF description. This means that in somesituations one may simply use a zero-range functional,which has an advantage of high computational speed.Clearly, the dynamics of an e-bubble is much moreinvolved than its statics. In our problems, the –nonspherical– squeezing and stretching of the bubble maycause its waist to shrink to a point when electron tunnel-ing plays a role, and may also dissipate a large amount ofenergy by exciting elementary modes in the surroundingliquid. So, even if the static properties of the e-bubbleare equally well described by both functionals, it is notobvious whether they yield a similar dynamical evolutionfor an excited e-bubble. This is the first issue we want toaddress.We use two different schemes to compute the relaxationof a 1P e-bubble at P = 0, 0.5, 1 and 5 bars, and comparethe results so obtained. One such scheme is the finite-range OT density functional description for helium withthe adiabatic evolution for electron. The other schemeis the zero-range ST density functional description forhelium with real-time evolution for the electron.As can be seen in Figs. 3-6, the evolution starts withthe bubble stretching along the z -axis and shrinking onits waist. After this stage, the bubble may continue os-cillating and releasing energy into the liquid, eventuallyreaching a relaxed, metastable 1P state, or may split intotwo baby bubbles due to the liquid filling-in around thebubble waist. The density waves radiated to the liquidduring this evolution take away a considerable part ofthe energy injected into the system during the absorp-tion process, i.e., 105 meV at P = 0 and 148 meV at 5bar. In Figs. 3-6 we compare the bubble evolution obtainedwithin the two frameworks for different pressures. At afirst glance, both dynamics are nearly equivalent dur-ing the first 50 ps, starting to differ from this time onalthough they are still qualitatively similar. A more de-tailed analysis, focused on three key elements of the den-sity profiles, indicates the following: a . The shape of the bubble surface, defined as thelocus where the liquid density equals half the saturationdensity value ρ , e.g., 0.0218 ˚A − at P = 0 bar. Thisshape determines the most crucial properties of an e-bubble. From this shape, we know whether the bubble issimply connected or has split. Up to t (cid:46)
50 ps, the shapeof the bubble is nearly identical in both descriptions. Atlater times, the bubble shape changes at a slower pace inthe ST than in the OT description.Figure 7 illustrates the time evolution of the bubblesurface. In particular, the top panel shows the evolu-tion of the point on the bubble surface at r = 0 with z >
0. This represents half the longitudinal extent of thee-bubble. One can see that this length oscillates in the STcalculation with a lower frequency than in the OT one.If the bubble symmetrically splits into two baby bubbles,there are two such points, as can be seen in the bottompanel for P = 1 bar. We have found that, besides themoment at which the distance between the baby bubblesincreases steadily —about 175 ps for the OT functionaland 200 ps for ST functional— there is a time intervalbetween ∼
60 and ∼
90 ps for the OT functional, and ∼
60 and ∼
110 for the ST functional, where the 1P bubbleat P = 1 bar has split but the emerging baby bubblesare “locked” by the shared electron that exerts some at-tractive force on them, forcing them back to a simplyconnected configuration. Eventually, the baby bubblesare unlocked and the distance between them grows. b . The surface thickness of the bubble, defined as thewidth of the region satisfying 0 . ρ ≤ ρ ( r ) ≤ . ρ . Thethickness of the bubble surface has been found to benearly independent of the local surface curvature at any-time during the evolution, see also Ref. 15. It is about1 ˚A larger in the ST than in the OT description, ascan be seen in Figs. 3-6 (the blurrier the bubble-heliuminterface, the larger the surface thickness). The zero tem-perature OT result, about 6 ˚A, is in agreement with theexperimental findings. c . The density oscillations traveling through the liq-uid. This is the point at which the differences betweenthe two functionals become more apparent. The densitywaves produced by the ST functional have much largerwavelengths because this approach cannot sustain shortwavelength inhomogeneities due to the huge energy costfromthe |∇ ρ ( r ) | surface energy term. The OT func-tional has not such a term and is free from this drawback.Roughly speaking, the short wavelength waves arising inthe OT approach are smeared out in a sort of big tsunami in the ST case, see for instance the panels correspondingto t = 5 and 10 ps in Fig. 6. It is worth emphasiz-ing that the wave interference pattern found in the OTdescription is not an artifact produced by waves bounc-ing back from the box boundaries, as those are alreadywashed out by the damping term. It arises from the in-terference of waves produced at different points on thee-bubble surface. To quantitatively study the nature of the waves emit-ted during the bubble evolution, we perform a Fourieranalysis of the density profile along the symmetry axis,restricting it to the region 30 ˚A ≤ z ≤
70 ˚A, away fromthe bubble location to avoid uncontroled effects arisingfrom the bubble itself. The Fourier transform of the den-sity fluctuation is shown in Fig. 8 at t = 8 . P = 0 bar. While both functionals generate low- q den-sity waves in the phonon region (see Fig.1) the ST ap-proach does not display any structure, whereas in the OTapproach one can identify two distinct peaks. The higherone is located at q ∼ . − near the maxon region, andthe lower one is located at q ∼ . − close to the rotonminimum.With these results on the evolution of the 1P e-bubblein mind, we can state with some confidence that the STdescription is accurate enough for describing the fate ofthe e-bubble, yielding the appropriate final topology ata given pressure, and a more than qualitative picture ofits evolution. The shape of the cavity, which is respon-sible for most electron properties, is essentially the samein both ST and OT descriptions. The different way ofenergy release associated with their each kind of elemen-tary excitation may yield somewhat diverse evolutions atlonger times, but it has little relevance for the problemsat hand. One should keep in mind however, that if theactual subject of the study are the elementary excitationsof the bulk liquid, the use of the OT functional is thenunavoidable.We also want to stress that computing the 1P e-bubbledynamics in real time for the ST functional has allowedus to explicitly check the adiabatic approximation in theelectron evolution during the time interval relevant forthe e-bubble “fission”. We have computed the overlapbetween the time-evolving electron wave function and theinstantaneous eigenstate 1 σ − , and have found it to beequal to unity at all times, meaning that the adiabaticapproximation holds. As we will discuss later on, this isnot the case for the 2P e-bubble.
2. Time-resolved absorption spectrum
Within the OT functional plus adiabatic approxima-tion scheme, we have studied the excitation of 1P bub-bles by photoabsorption either to the m = 0 component(2 σ + ), or to the m = ± π − ), arising fromthe splitting of the originally spherical 1D state, see Fig.2. In principle, this can be measured in a pump-probeexperiment by which the e-bubble is excited by two con-secutive laser pulses. The delay set between these pulsescorresponds to the time interval between the excitationand the measurement, which is the same as the time de-fined in our calculations. The intensity of the absorptionlines is characterized here by their oscillator strength cal-culated in the dipole approximation f ab = 2 m e (cid:126) ( E a − E b ) (cid:12)(cid:12) (cid:104) a | r | b (cid:105) (cid:12)(cid:12) . (7)We recall that this oscillator strength fulfills the sum rule (cid:80) a f ab = 1, but is generally not positive-definite. If theinitial state is not the ground state, a partial sum maybe greater than unity.Starting from the 1P electron state 1 σ − ( m = 0), thetwo possible photoexcitation transitions are 1 σ − → σ + and 1 σ − → π − , see Fig. 2. The specularly asymmetricstates have a nodal point on the z = 0 symmetry plane,implying that they are rather insensitive to the presenceof helium in that plane. Therefore, the transition energyfor excitations between two asymmetric states should notdepend much on whether the bubble has split or not.Contrarily, the specularly symmetric states do not havesuch a nodal point, and so are more sensitive to splitting.The lowest-lying transition connecting specularly asym-metric with specularly symmetric states may thus probethe topology of the bubble, since the absorption energyfor this transition should increase by a sizeable amountwhen the bubble splits. The level structure at the rightpart of Fig. 2 may help understanding these issues.The time-resolved absorption energies and oscillatorstrengths of the evolving 1P bubble for P = 0 . P = 0 . P = 1 bar at t (cid:39)
170 ps. The bubble splittingyields a clear signature in the energies and the oscillatorstrengths: the evolution of the transition energies is sim-ilar for P = 0 . P = 1 bar before the splitting, butwhen the bubble splits at P = 1 bar, the 1 σ − → σ + energy rapidly increases by ∼
70 meV, becoming compa-rable to the 1 σ − → π − energy. This is a consequence ofthe change in the bubble topology, which makes the finalsymmetric and antisymmetric states nearly degenerate.A conspicuous pattern also appears in the evolutionof the oscillator strength. The strength for the specu-larly asymmetric transition 1 σ − → π − remains nearlyconstant at f ∼ .
65, whereas the strength for thespecularly symmetric transition 1 σ − → σ + oscillateswhen the bubble is simply connected but falls down to f ∼ .
32 when the bubble splits. This is again a conse-quence of the near degeneracy of symmetric and asym-metric states in the split-bubble regime. The oscillatorstrength for the antisymmetric transition is a factor oftwo larger than that of the symmetric transition in thesplit-bubble regime because the final state 1 π − is twofolddegenerate. We thus conclude that time-resolved absorption ener-gies are of practical interest because they bring rich in-formation on the bubble shape and can be determinedin experiments. This may shed light on the longstand-ing question about whether 1P e-bubbles under pressuredo really “fission” into two baby bubbles as our calcu-lations indicate, and how the electron wavefunction col-lapses into one of them, without violating the quantummeasurement axiom.
B. 2P e-bubble dynamics: the breakdown ofadiabaticity
Our previous analysis of the dynamics of the 1P e-bubble has shown that one does not need to use the ac-curate OT functional to describe this process. The muchsimpler ST approach already yields a fair description.This is particularly useful when we move to the study ofthe 2P e-bubble dynamics. An attempt to simulate thisevolution has been made within the OT approach and theadiabatic approximation. This could only be performedfor a few picoseconds, as it was shown that the adiabaticapproximation fails at t (cid:39) . σ − (arising from the spherical2P level) and 2 σ − (arising from the spherical 1F level)energy levels. By using the efficient real-time ST scheme,we now relax the adiabatic approximation, following theevolution of the 2P e-bubble for several hundreds of pi-coseconds. We keep refering to this bubble as a “2P e-bubble”, but should have in mind that once the adiabaticapproximation breaks down, the electron is no longer inthe original eigenstate. Generally, it is in a superpositionof states that have the same quantum numbers as theinitial state, meaning that it can be in any superpositionof σ − states.The 2P bubble evolution is shown in Fig. 11. For thefirst 100 ps, the shape evolution of the 2P bubble is sim-ilar to that of the 1P bubble, as it expands along thesymmetry z -axis while its waist shrinks in the perpen-dicular plane. From this point on, the bubble oscillatesback and forth in a kind of four-lobe shapes quite differ-ent from those seen in the 1P bubble. We attribute theseconspicuous shape variations to the breaking down of theadiabatic approximation as the electron moves from aeigenstate to a nontrivial superposition of those compat-ible with the symmetries of the system. After evolvingfor ∼
325 ps, the 2P bubble splits into two baby bubbles.We present in Fig. 12(a) the evolution of the instan-taneous eigenenergies of the first σ − states. As can beseen in panel (b), the 2 σ − and 3 σ − states nearly meet at t (cid:39) . we have found that thissituation corresponds to an avoided crossing. Panel (c)shows the overlap of the evolving electron wavefunctionwith the relevant instantaneous eigenstates. The electronis initially in a 3 σ − state (the overlap is unity), but atthe point of avoided crossing the adiabaticity is lost: theelectron state is a superposition of the 2 σ − ( ∼ σ − ( ∼ ∼
155 and ∼
180 ps when the 2P bubble at P = 0 bar has split butthe emerging baby bubbles do not go away. When thishappens, the nσ − and nσ + states should be degenerate.This is illustrated in panel (a) of Fig. 12 for the n = 1states. Notice from Fig. 11 that in the 50 ps (cid:46) t (cid:46)
100 psinterval the bubble is simply connected and the apparentdegeneracy displayed in Fig. 12(c) is due to the energyscale. The same thing happens around t ∼
230 ps.
IV. SUMMARY
We have thoroughly studied the dynamical evolution of1P and 2P excited electron bubbles in superfluid He atzero temperature. To this end, we have resorted to zero-and finite-range density functionals, establishing how re-liable the former is by comparing its results with thoseobtained with the latter.Although the results obtained for the 1P bubble evolu-tion from these two functionals show some quantitativedifferences, especially for long-time evolutions, they arequalitatively equivalent. In particular, both lead to theconclusion that 1P bubbles “fission” at pressures above 1bar. The ST functional result is of particular relevance,as it has been obtained by a real-time evolution, with-out assuming the adiabaticity of the process. This con-firms the previous results obtained using the finite-rangeOT functional and the adiabatic approximation for muchshorter periods of time than in the present work. Some indirect experimental evidence indicates achange in the de-excitation behavior of the 1P e-bubbleas pressure increases.
We have explored here the pos-sibilities offered by the photoabsorption spectrum of the1P e-bubble to disclose whether such a bubble de-excitesby “fission” or by a more conventional radiative decay,and have obtained the signatures that would help distin-guish between both decay channels. Although far from trivial, a pump-probe experiment may detect a changein the absorption spectrum of the 1P bubble associatedwith the appearance of the two baby bubble de-excitationchannel.Finally, we have studied the evolution of the 2P e-bubble in real-time within the ST functional approach.We have dynamically found that the adiabatic approxi-mation does not hold at any positive pressure confirm-ing the results obtained within the OT plus adiabaticapproximation approach. Negative pressures, as thoseattained in cavitation experiments, have not been stud-ied. The physical realization of a relaxed, metastable 2Pconfiguration is discarded.
Acknowledgments
The authors wish to thank Humphrey Maris for help-ful discussions. This work was performed under GrantsNo. FIS2008-00421/FIS from DGI, Spain (FEDER),and 2009SGR1289 from Generalitat de Catalunya. D.Mateo has been supported by the Spanish MEC-MICINN through the FPU fellowship program, GrantNo. AP2008-04343. D. Jin has been supported bythe National Science Foundation of the United Statesthrough Grant No. DMR-0605355. J. Eloranta and V. A. Apkarian, J. of Chem. Phys. ,10139 (2002). A. Ghosh and H. J. Maris, Phys. Rev. Lett. , 265301(2005). V. Grau, M. Barranco, R. Mayol, and M. Pi, Phys. Rev.B , 064502 (2006). M. Rosenblit and J. Jortner, J. Chem. Phys. , 194505(2006); ibid. , 194506 (2006). M. Pi, R. Mayol, A. Hernando, M. Barranco, and F. An-cilotto, J. Chem. Phys. , 244502 (2007). L. Lehtovaara and J. Eloranta, J. Low. Temp. Phys. ,43 (2007). H. J. Maris, J. Phys. Soc. Jpn. , 1 (2008). W. Guo, D. Jin, G. M. Seidel, and H. J. Maris, Phys. Rev.B , 054515 (2009). F. Ancilotto, M. Barranco, and M. Pi, Phys. Rev. B ,174504 (2009). D. Mateo, A. Hernando, M. Barranco, and M. Pi, J. LowTemp. Phys. , 397 (2010). D. Jin and H. J. Maris, J. Low Temp. Phys. , 317(2010). C. C. Grimes and G. Adams, Phys. Rev. B , 6366 (1990). C. C. Grimes and G. Adams, Phys. Rev. B , 2305 (1992). D. Jin, W. Guo, W. Wei, and H. J. Maris, J. Low Temp.Phys. , 307 (2010). D. Mateo, M. Pi, and M. Barranco, Phys. Rev B , 74510(2010). H. J. Maris, A. Ghosh, D. Konstantinov, and M. Hirsch,J. Low Temp. Phys. , 227 (2004). F. Dalfovo, A. Lastri, L. Pricaupenko, S. Stringari, and J. Treiner, Phys. Rev. B , 1193 (1995). E. Cheng, M. W. Cole, and M. H. Cohen, Phys. Rev. B , 1136 (1994); Erratum ibid. , 16 134 (1994). S. Stringari and J. Treiner, Phys. Rev. B , 16 (1987); J.Chem. Phys. , 5021 (1987). D. Mateo, J. Navarro, and M. Barranco, Phys. Rev. B ,134529 (2010). R. J. Donnelly and J. A. Donnelly, R.N. Hills, J. LowTemp. Phys. , 471 (1981) M. Frigo and S. G. Johnson, Proc. IEEE , 216 (2005). L. Giacomazzi, F. Toigo, and F. Ancilotto, Phys. Rev. B , 104501 (2003). L. Lehtovaara, T. Kiljunen, and J. Eloranta, J. Comp.Phys. , 78 (2004). A. Ralston and H. S. Wilf,
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Atoms and Molecules (Academic Press,New York, 1978). Fig. 10 displays the oscillator strenght per energy level andnot per state . π states are twofold degenerate due to the twopossible l -values, l = ±
1, while σ states are non-degenerate as l = 0. For this reason, when these states become nearlydegenerate in the split-bubble regime, one transition has f ∼ .
32 and the other f ∼ .
65, a factor of two larger. E. Teller, J. Phys. Chem. ,1 (1937) q (˚A − ) ¯ h ω ( K ) FIG. 1: (Color online) Dispersion relation of the elementaryexcitation in bulk liquid He at T = 0. Solid line: the OTfinite-range functional results. Dashed line: the ST zero-rangefunctional results. Dots: the experimental data from Ref. 22. σ + σ − π + σ + π − δ + { { FIG. 2: (Color online) Splitting of electronic levels along witha representation of their probability densities once the spher-ical symmetry is broken. The states in the spherical configu-ration (left) are labeled in the standard n L way. In the axi-ally symmetric case (right) the label is nl sz , where s = +( − )stands for symmetric (antisymmetric) states under specularreflection. -40 -20 0 20 40 -40-2002040 x (˚A) z ( ˚A ) -40-2002040-40-2002040-40 -20 0 20 40 t = FIG. 3: (Color online) Evolution of the 1P e-bubble at P = 0bar. The left-hand side of each panel shows the results for theOT finite-range density functional plus adiabatic approxima-tion for the electron. The right-hand side part of each panelshows the results for the ST zero-range density functional plusreal-time evolution for the electron. -40 -20 0 20 40 -40-2002040 x (˚A) z ( ˚A ) -40-2002040-40-2002040-40 -20 0 20 40 t = FIG. 4: (Color online) Same as Fig. 3 at P = 0 . -40 -20 0 20 40 -40-2002040 x (˚A) z ( ˚A ) -40-2002040-40-2002040-40 -20 0 20 40 t = FIG. 5: (Color online) Same as Fig. 3 at P = 1 bar. -40 -20 0 20 40 -40-2002040 x (˚A) z ( ˚A ) -40-2002040-40-2002040-40 -20 0 20 40 t = FIG. 6: (Color online) Same as Fig. 3 at P = 5 bar. z ( ˚A ) P = 0 bar t (ps) z ( ˚A ) FIG. 7: (Color online) Evolution of the extent of the bubblealong the z axis at P = 0 and 1 bar. The solid line is the OTfinite-range result, and the dashed line is the ST zero-rangeresult. q (˚A − ) I ( a r b . un i t s ) P = 0 bar t = 8 . FIG. 8: (Color online) Fourier transform of the density fluc-tuation along the z -axis within the region 30 ˚A ≤ z ≤
70 ˚Afor the expansion process of the 1P e-bubble at P = 0 barand t = 8 . E n e r g y ( m e V ) time (ps) P = 1 bar5010015020025050100150200 P = 0 . σ − → σ + σ − → π − FIG. 9: (Color online) Time-resolved absorption energies at P = 0 . t = 170 ps indicates the time at which thebubble splits in the P = 1 bar case. O s c ill a t o r S tr e n g t h P = 1 bar P = 0 . σ − → σ + σ − → π − FIG. 10: (Color online) Time-resolved absorption oscillatorstrengths at P = 0 . t = 170 ps indicates the time at whichthe bubble splits at P = 1 bar -80 -40 0 40 80 x (˚A) z ( ˚A ) -80-4004080
25 ps 50 ps150 ps300 ps 200 ps350 ps -80-4004080-80-4004080-80 -40 0 40 80 t = -80 -40 0 40 80 FIG. 11: (Color online) Evolution of the 2P e-bubble at P = 0bar using real-time dynamics and the ST zero-range func-tional. time (ps) σ − σ − σ − σ − E n e r g y ( m e V ) E n e r g y ( m e V ) time (ps) 00.20.40.60.81 O v e r l a p σ + (b) (c)(a) FIG. 12: (Color online) (a): Lower-lying instantaneous σ − eigenstates together with the 1 σ + eigenstate of the 2P e-bubble at P = 0 bar as a function of time. The thin verticalline at t = 325 ps indicates the time at which the bubblesplits. (b) Enlarged view of the region where the 3 σ − and2 σ − states repel each other. (c) Overlap of the time-evolvingelectron state onto the 3 σ − (solid line) and 2 σ − eigenstates(dashed line), |(cid:104) Φ( r , t ) | nσ − (cid:105)|2