Polarization-Dependent Disappearance of a Resonance Signal -- Indication for Optical Pumping in a Storage Ring?
W. Nörtershäuser, A. Surzhykov, R. Sánchez, B. Botermann, G. Gwinner, G. Huber, S. Karpuk, T. Kühl, C. Novotny, S. Reinhardt, G. Saathoff, T. Stöhlker, A. Wolf
PPolarization-Dependent Disappearance of a Resonance Signal –Indication for Optical Pumping in a Storage Ring?
W. N¨ortersh¨auser,
1, 2, ∗ A. Surzhykov,
3, 4, 5
R. S´anchez, B. Botermann, G. Gwinner, G. Huber, S. Karpuk, T. K¨uhl, C. Novotny, S. Reinhardt, G. Saathoff, T. St¨ohlker, and A. Wolf Institut f¨ur Kernphysik, TU Darmstadt, 64289 Darmstadt, Germany Helmholtz Akademie Hessen f¨ur FAIR, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany Physikalisch–Technische Bundesanstalt, 38116 Braunschweig, Germany Institut f¨ur Mathematische Physik, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany Laboratory for Emerging Nanometrology Braunschweig, 38106 Braunschweig, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany Johannes Gutenberg-Universit¨at Mainz, Institut f¨ur Kernchemie, 55128 Mainz, Germany Dept. of Physics & Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada Johannes Gutenberg-Universit¨at Mainz, Institut f¨ur Physik, 55128 Mainz, Germany Max-Planck-Institut f¨ur Quantenoptik, 85748 Garching, Germany Max-Planck-Institut f¨ur Kernphysik, 69117 Heidelberg, Germany (Dated: January 26, 2021)We report on laser spectroscopic measurements on Li + ions in the experimental storage ring ESRat the GSI Helmholtz Centre for Heavy Ion Research. Driving the 2 s S ( F = / ) ↔ p P ( F = / ) ↔ s S ( F = / ) Λ-transition in Li + with two superimposed laser beams it was foundthat the use of circularly polarized light leads to a disappearance of the resonance structure in thefluorescence signal. This can be explained by optical pumping into a dark state of polarized ions.We present a detailed theoretical analysis of this process that supports the interpretation of opticalpumping and demonstrates that the polarization induced by the laser light must then be at leastpartially maintained during the round trip of the ions in the storage ring. Such polarized ion beamsin storage rings will provide opportunities for new experiments, especially on parity violation. I. INTRODUCTION
Since many years, the beams of spin–polarized parti-cles, ions, and atoms have attracted considerable interestboth in fundamental and applied science [1–4]. In atomic,nuclear and particle physics, for example, many experi-ments have been performed employing electrons, muons,protons and deuterons that can be obtained with highdegrees of polarization from thermal to ultrarelativisticenergies. Less progress has been achieved so far with theproduction and use of polarized ion beams . Here, most ofthe studies have been restricted to the light–ion and low–energy (keV) domain, where optical pumping is routinelyapplied [5–7]. Only few efforts have been undertakento produce high–energy beams of polarized lithium andsodium ions at tandem accelerators, e.g. in [8]. Also notmuch is presently known about beams of spin–polarizedions circulating in storage rings, even though proposalsfor their production have been discussed already at theend of the last century [9]. These beams can be employedfor investigating spin–dependent phenomena in atomicand nuclear reactions, measuring the parity violation ef-fects in few–electron systems, and even for the search of apermanent electric dipole moment [10–13]. As an exam-ple, let us consider the case of parity non-conservation:In He-like highly charged ions, levels with opposite paritycan come very close in energy for a particular Z , which ∗ [email protected] enhances parity non-conservation effects. This happensin He-like Eu , where the 2 P and the 2 S statesbecome almost degenerate [14]. If the system can be pre-pared in a polarized state, a quenching-type experimentwith interference of hyperfine- and weak-quenched tran-sitions can be performed [11]. The observable will be thedifference in countrate for opposite polarizations. Otherexamples can be found in [10].Two key issues have to be solved, however, before suchexperiments with polarized ions in storage rings will be-come feasible. These are firstly the production of po-larized ion beams and secondly the preservation of theirpolarization during the round trips in the storage ring.The second aspect is a particularly critical issue, as thepolarization of ions typically involves the Zeeman sub-states ( M F ) of hyperfine levels with large magnetic mo-menta, given by those of the bound electron. They are,hence, much more sensitive to the fields of the storagedevice than bare nuclei [15] and any excess population inan M F level is expected to be destroyed rapidly by thefluctuating fields along the stored-ion orbit.In the present work we report a joint experimentaland theoretical study that addresses both key issues. Inparticular we consider with a rate equation approachthe production and the temporal evolution of hyper-fine polarization by Zeeman optical pumping in a two-wavelength excitation scheme. We contrast the theoret-ical results with a measurement of the steady-state flu-orescence intensity from a stored ion beam subjected tothis laser excitation, which show a significant dependenceon the laser polarization. Within this framework, we con- a r X i v : . [ phy s i c s . a t o m - ph ] J a n clude that the observations may indicate a preservationof the hyperfine polarization over a considerable numberof round trips. To the best of our knowledge, no obser-vations were reported so far that may be interpreted interms of such extended hyperfine polarization in a stor-age ring.The experimental results were obtained within anexperiment to test time dilation in Special Relativitywith Li + ions stored in the Experimental Storage Ring(ESR). This experiment probed the fluorescence yieldfrom a three-state, Λ-type resonance 2 s S ( F = / ) ↔ p P ( F = / ) ↔ s S ( F = / ) in counterpropa-gating laser beams along the ion direction. While the Λ-resonance could indeed be observed and spectroscopicallyanalyzed [16], we here describe observations showing itsdepencence on the polarizations on both lasers, whichwere set as parallel linear polarizations in the previousmeasurement [16]. The model calculations are adaptedto the parameters of the ESR experiment. II. LITHIUM-II LEVEL SCHEME
The level scheme of the two-electron system Li + resem-bles that of He and is depicted in Fig. 1. Laser excitationfrom the 1 s S ground level (not shown) is not possiblesince the required light is far in the vacuum ultraviolet.But within the triplet system, optical transitions from themetastable 1 s s S to the 1 s p P J levels ( J = 0 , ,
2) areaccessible at a wavelength of λ rest = 548 . S and 2 P J to denotethese levels. The 2 S level is metastable with a lifetimeof approximately 50 s, sufficiently long for experimentsin storage rings. However, the state must be populatedalready in the ion source. To this end, a Penning ionsource (PIG) was used and operated with a CuLi alloysince it was superior to an electron cyclotron resonanceion source (ECRIS) operated with LiF with respect tothe amount of metastable ions [17]. We have used the Λ-excitation scheme connecting the 2 S ( F = / , / ) andthe 2 P ( F = / ) hyperfine levels as indicated in theright part of Fig. 1. For the sake of bookkeeping, be-low we will enumerate these levels as: 2 S ( F = / ),2 S ( F = / ) and 2 P ( F = / ). III. EXPERIMENTA. Ion beam preparation
The experimental setup was presented in detail inRefs. [17, 18]. In brief, Li + ions in the metastable 2 S level were produced in a PIG source, accelerated in theuniversal linear accelerator (UNILAC) to an energy of8 MeV/u and injected into the heavy-ion-synchrotron(SIS) for further acceleration to 58 MeV/u correspond-ing to a velocity of β = υ / c = 0 . γ = 1 .
50 s S P F
43 ns F F F FIG. 1. Level scheme and hyperfine structure of the 2 S → P transition in Li + . The F = / ↔ F = / ↔ F = / Λ transition investigated in this work is shown on the right.
Finally, the beam pulses are extracted into the experi-mental storage ring (ESR) with an efficiency of 10 %.Typical average ion currents of 20 µ A or a total numberof N ions = 1 . · singly charged particles are revolv-ing in the ESR. However, a varying fraction of up to50% of the beam was observed to be N , having avery similar mass-to-charge ratio as Li + . The residualgas pressure of less than 10 − mbar in the ESR allowsfor storage times of the ion beam of more than 100 s.The ions are continuously subject to electron cooling toreduce the momentum spread of the ion beam. This re-duces the Doppler width of the spectroscopy signal andincreases phase-space density and therefore the numberof ions that can be addressed simultaneously with a nar-rowband laser. Approximately 20 s after injection, thecooling process has reached equilibrium and laser spec-troscopy can be performed while electron cooling is con-tinued (see Sec. III C). Additionally, a weak rf signal atthe tenth harmonic of the revolution frequency of the ionsis applied to an rf-cavity at the ESR to compensate forsmall energy fluctuations of the electron cooler. This su-perimposes a bunching potential copropagating with theions, keeping them precisely at an average velocity de-fined by the rf frequency and reduces velocity diffusion.Laser spectroscopy with full Doppler broadening wasperformed in the closed 2 S ( F = / ) → P ( F = / ) hyperfine transition to locate the resonance [17, 18].For the cooled beam the Doppler-broadened FWHM wasfound to be close to 1 GHz, corresponding to a relativeion momentum spread ∆ p/p close to 1 · − . It should benoted that a strong ion-laser interaction on such a closedtransition can lead to a depletion of a velocity class due tothe recoil in repeated photon absorption-emission events.This effect on the external degrees of freedom of an atomor ion is regularly used for laser cooling [19] and hasalso been employed at storage rings [20, 21]. With thelaser powers used in our experiment, such effects have notbeen observed: the closed two-level transition exhibits aregular symmetric Gaussian line shape and for the Λ-transition the momentum transfer that can occur beforethe ion is transferred to a dark 2 S hyperfine state is verysmall. Fast cycling will only happen if both lasers talk tothe same velocity class and in this case the ion interactswith photons from opposite directions and momentumtransfer can again be neglected. Hence, we also do nottake into account velocity changes by photon recoil in ourtheoretical model in Sec. V.The 2 S lifetime in the storage ring was determinedfrom the time evolution of the fluorescence signal to bepractically identical with its 50-s radiative lifetime andthe observed resonance strength allowed us to estimatethat the metastable 2 S level was populated in less than0.1% of the Li + ions. B. Interaction Region at the ESR
The ESR is described in detail, e.g., in [22] and withrespect to laser spectroscopic experiments in [23]. Here,we will concentrate on the straight interaction region ofthe ion and the laser beam as it is shown in Fig. 2. Theion beam revolving in the ESR is superimposed with thelaser beam at the exit of the 60 ◦ -dipole magnet on theright (ESR’s south-west corner) and is separated againat the 60 ◦ -dipole magnet on the left (ESR’s north-westcorner). The laser beams are injected through viewportsthat are mounted in the straight direction at the dipolechambers. Two magnetic quadrupole doublets are lo-cated along the straight section, while in the center ofthe field-free region, a gas target is installed followed bythe (fluorescence) detection region used in this experi-ment. The detection region extends along 65 cm. Thelast mirror is about 12-m downstream the ion beam fromits first merging point with the laser close to the ESRdipole magnet. C. Laser setup
A Li + ion in the excited 2 P term decays back to 2 S with an average lifetime of τ (2 P ) = 43 ns, which atthe beam velocity corresponds to an average fluorescencedecay length of βγcτ = 4 .
52 m. To perform Doppler-reduced spectroscopy in the Λ scheme described above,two laser beams were superimposed with the ion beamalong the straight section of the ESR, where the fluores-cence detection region is located as shown in Fig. 2. Asthe average fluorescence length is to short for a separa-tion between ion beam and laser beam before detectionthe lasers must traverse the detection region and back-ground from scattered laser light cannot be avoided [23].The Doppler effect at 33.8 % of the speed of light shiftsthe excitation wavelength from λ = 548 . λ p = 386 nm for the laser beam copropagating (parallel)with the ions, while excitation with the counterpropa- gating (antiparallel) laser beam has to be performed ata wavelength of λ a = 780 nm in the laboratory frame.Copropagating laser light is produced with a commer-cial tunable solid-state titanium-sapphire (Ti:Sa) ringlaser (coherent, 899-21) operated at 772 nm pumped by aNd:YVO laser (Coherent, Verdi) at 532 nm. Frequencydoubling in an enhancement cavity (Wavetrain2, Sirah)with a BBO ( β -BaB O ) crystal provided the requiredwavelength at 386 nm. The Ti:Sa laser is stabilized tothe rovibronic P(42)1-14 transition in molecular iodine I by frequency modulation saturation spectroscopy[24, 25]. The linewidth of the laser is approximately250 kHz on a 1-s integration time. It is estimated thatthis increases to about 350 kHz after frequency doubling.The counterpropagating laser beam is generated with apair of diode lasers in Littrow configuration (Toptica,DL100) of which one is locked to a hyperfine transitionat 780 nm in the D2 line Rb [26], while the second oneis tunable and referenced to the first one by a frequency-offset lock [27]. The linewidth of the counterpropagat-ing laser is about 2 MHz. More details are provided in[16, 18].Both lasers are transported to the storage ring usingpolarization-maintaining optical fibers. For the UV lightfor collinear excitation a 15-m long Thorlabs fiber (typePM-S350-HP) is used, while the IR light requires a fiberwith a length of 50 m (type PM780-HP).The fused-silica viewports at the ESR’s dipole magnetshave a surface flatness of λ /20 and can be tilted to sendthe reflections of the window surfaces out of the ESRto reduce stray light. The angles and positions of thelaser beams are controlled with accuracy of ∆ ϑ = 20 µ radand ∆ x ≈ µ m by motorized rotation and translationstages. Two scraper pairs, situated in the middle of theexperimental section, 6.5 meters apart from each otherare used to establish ion beam – laser beam overlap insidethe vacuum chamber. The scrapers can be introducedinto the beam path with a reproducibility of better than0.1 mm, allowing us to guarantee parallel beams within80 µ rad.Laser beam parameters have been determined with abeam viewer: both foci are located 17 m downstreamof the detection regions last mirror, actually outside theESR beamline, with radii (at / e intensity) of 1.10(5) mmand 2.71(10) mm for the blue and red laser beams, re-spectively. Along the 11-m long interaction path prior todetection, the beam radii vary accordingly from 3.8 mm(3.4 mm) at the dipole magnet down to 3.1 mm (2.2 mm)at the detection region for the red (blue) laser. The laserpower was adjusted to 5 mW for the red and 0.4 mWfor the blue laser. This corresponds to intensities ofroughly 3.3 to 4.9 times the saturation intensity of I =6 . / cm [28] for the red and 30-80% saturation in-tensity for the blue transition along the interaction path.In the theoretical simulation (see Sec. V), we use an aver-age size of 3.5 mm (red) and 2.8 mm (blue), respectively,and transform the intensity into the rest frame of the ion.These values are listed in Tab. I.
11 m 386 nm780 nm Dipole Quadrupoledoublet Li + beamTop viewFront view Quadrupoletriplet Detection regionlength 0.6 m FIG. 2. Top-view and front-view of the straight section of theESR where the laser and the ion beam are superimposed. Thelasers are directed through the viewports at the 60 ◦ -dipolemagnets to co- and counterpropagate with the ion beam. D. Fluorescence Detection
The system for fluorescence detection is situated in themiddle of the experimental section. It was originally builtfor laser spectroscopy on hydrogen-like Pb [29], a detaileddescription is provided in [30] and its detection proper-ties were modelled in [31]. Three photomultiplier tubes(PMTs) (two
Hamamatsu , R2256P and one
Thorn EMI,9635QB ) are used to detect the fluorescence photons. AllPMTs are equipped with cooler housings to keep the tem-perature below 5 ◦ C to reduce thermal noise. Each PMTis additionally equipped with a combination of lenses op-timized to focus photons from the beam position ontothe photocathode. The detection system can only col-lect light that is emitted under angles of at least 20 ◦ with respect to the beam direction. The fluorescence de-tection is therefore restricted to a wavelength range be-tween 410 nm and about 700 nm with a combination ofoptical filters (Schott, BG 39 & Itos KV408) to suppresslaser stray light [17]. Consequently, processes that wouldsteer the emission direction of the ions away from theseforward directions could lead to a signal reduction evenwith a constant fluorescence rate. E. Spectroscopic Procedure
Resonance spectra of the Li + ions are recorded in thefollowing way: First, the Ti:Sa laser producing the co-propagating beam (wavelength λ p ) is stabilized to theiodine line and the ion beam energy is adapted by chang-ing the electron energy at the electron cooler to res-onantly excite the ions along the 2 S ( F = / ) → P ( F = / ) transition. Then, the Λ-resonance sig-nal is observed by tuning the frequency of the counter-propagating (diode) laser beam (wavelength λ a ) acrossthe 2 S ( F = / ) → P ( F = / ) transition. While λ a is scanned, the fluorescence intensity is recorded forfour phases when the laser beams are switched on or off(“ λ a only”, “ λ p only”, “both lasers”, and “no laser”) [16] Normalized count rate f a - 3 8 4 2 2 3 1 3 0 M H z P o l a r i z a t i o n p - p s + - s + s - - s + FIG. 3. Optical resonances of Li + ions in the ESR recordedwith different laser polarizations during the “both lasers on”phase (see text). The linear-linear case was recorded withboth laser beams polarized in the horizontal (ring) plane. Forcircularly polarized light, the helicities of the UV (1 st ) andIR photons (2 nd ) are indicated. The x -axis represents thefrequency of the IR light ( λ a ) produced by the diode laser thatcounterpropagates to the ion beam and drives the 2 S ( F = / ) → P ( F = / ) transition. The UV light ( λ p ) of theTi-sapphire lasers second harmonic is copropagating and itsfrequency is fixed on the 2 S ( F = / ) → P ( F = / )resonance of the central velocity group of the revolving ions. and the Λ-resonance spectrum is obtained from the dif-ference between the “both-lasers” phase and the summedsignals of the two single-laser phases. In the fluorescencesignal of the “ λ a -on” phase, a Doppler-broadened spec-trum occurs through F = / → F = / excitation fora velocity-class of the ions selected by the narrow-bandlaser. This fluorescence signal is reduced by the opticalpumping to the F = / hyperfine levels through sponta-neous emission F → F . In the “both-lasers” phase andin the signal difference representing the Λ resonance spec-trum, an additional spectral structure occurs when λ a se-lects for the F → F transition the same velocity class asthe fixed λ p laser selects for the F → F transition. Hy-perfine pumping to the F level is then avoided so thatthe average population in F and, correspondingly, thefluorescence intensity rises. Indeed, a Λ-resonance signalat a width significantly reduced compared to the Dopplerwidth could be observed and spectroscopically analyzedto yield the desired test of Special Relativity [16]. Theremaining linewidth of about 100 MHz is still more thanan order of magnitude larger than the natural linewidthof approximately 3.7 MHz. This is ascribed to velocity-changing collisions happening during many revolutions inthe ESR, shifting ions dark-pumped by one of the lasersback into resonance with the other one [16]. IV. RESULTS
In the experimental runs leading to the Relativity Test[16], also the effect of the laser polarizations on the Λ-signal was studied. In particular, the situation of parallellinear polarizations in the ESR bending plane, chosenfor the measurement, was replaced by chosing circularpolarizations for both lasers. With this goal, λ / -platesin front of the ESR windows were introduced and theirfast axis rotated to obtain circular polarization. Hereand in the following, we always refer to the helicity stateof the photon, i.e. the projection of the photon spin ontoits own momentum direction.The signals (denoted as Λ-spectra in the following)observed when both lasers are on are shown in Fig. 3for the tested laser polarizations. The black solid curverepresents the resonance with both lasers linearly polar-ized in the horizontal plane ( π – π ). This signal shows asharp structure representing the suppression of hyperfine-population pumping by a single laser, as described aboveand analyzed in Ref. [16]. A wider (Doppler-broadened)pedestal of this resonance occurs in addition. It wasexplained by the diffusion of ion velocities during theirround trips, which leads to a replenishment of ions in thelaser-accessible state from neighboring velocity classeswith state populations that have not been reduced bythe scanned (IR) laser. Similarly, hyperfine populationpumping by the Λ-type interaction with both lasers canbe suppressed over a wider range of λ a detunings whichexplains the increased linewidth of the observed Λ-signal.Such velocity changes can occur in electron-ion collisionsin the electron cooler, in ion-atom collisions with resid-ual gas atoms or can be induced by the rf cavity usedfor the weak bunching. The dash-dotted trace in redwas recorded with both laser beams having σ + polar-ization. Here, the fluorescence near the Λ-resonance be-comes much smaller, regarding both the pedestal and theresonant structure. Only a weak signal above backgroundremains. Then, the λ / -plate of the UV laser was rotatedby 90 ◦ to obtain σ − light. In this case, even the weakresonance vanishes, as displayed by the blue dotted line.We interpret the disappearence of the fluorescence nearthe Λ-resonances even under the combined effect of bothlasers λ a and λ p by optical pumping between M F Zee-man sublevels with respect to the laser and ion-beamdirection as a quantization axis, as illustrated in Fig. 4.For the σ − – σ + case (Fig. 4 (b)), ∆ M F = +1 transi-tions are indicated by the solid lines in blue and red,respectively. From all M F states of the two lower lev-els of the Λ-transition, only the M F = − / state of the2 S ( F = / ) level cannot be excited with σ − IR light,but this state is populated by partial decays from the M F = − / , − / substates of the 2 P ( F = / ) level asindicated by the dashed arrows. All other decays leading A similar signal for vertical polarization was also recorded. F = 3/2 F = 5/2 F = 5/2 -3/2 -1/2 +1/2 +3/2-5/2 +5/2 M F F = 3/2 F = 5/2 F = 5/2 -3/2 -1/2 +1/2 +3/2-5/2 +5/2 M F (a)(b) � + � + � � � + FIG. 4. Level scheme with M F substates for the explanationof the dark state in the Λ-scheme with two circularly polarizedlaser beams. The upper panel (a) displays the case of σ − – σ + polarization, while the lower panel (b) corresponds to the σ + (UV)– σ + (IR) case. back to laser-coupled M F states are not shown. Bothlasers are shifting the population towards this “dark”substate. In absence of a redistribution process betweenthe different M F states of a level, resonant laser excita-tion must eventually end with all population accumulatedin the dark substate and fluorescence will stop. Similarqualitative analysis, that takes into account transitionsbetween various hyperfine substates M F , may explain theexistence of a weak fluorescence signal, observed for the σ + – σ + case. As seen from Fig. 4 (b), in this case theUV laser excites ∆ M F = +1 transitions. Again, the M F = − / magnetic substate of the 2 S ( F = / ) levelbecomes the dark substate but this time the excitationwith the UV beam shifts the population in the oppo-site direction, ∆ M F = +1. Hence, one can expect theprocess to slow down and that it will take longer untilfluorescence vanishes.It should be noted that the case denoted as a π – π polarization of both lasers, for the M F states referring tothe lasers’ direction represents a coherent superpositionof the transitions marked in both panels in Fig. 4. Hence,repumping from the dark states is ensured and a large Λrepumping effect is observed.In both cases the population of the M F = − / darksubstate leads to an effective pumping of the hyperfine-level population to F while state F is depleted. How-ever, this optical pumping is also connected with asteady-state polarization of the stored F ions in the M F = − / state. This consequence of the observed laser-polarization dependence was not followed further in theSpecial Relativity test experiment. However, as pointedout in Sec. I, it could have exciting consequences suchthat we consider this effect more closely in the following.For the described polarization effect to yield experi-mental consequences, two scenarios can be considered:( i ) The laser beam interacts with the ions along a dis-tance of 11 m in front of the detection region, whichcorresponds to ∼ . ◦ and 80 ◦ with the highest efficiency for detection. We de-note this as the “fast pumping” scenario, in whichany polarization in the ion beam is considered tobe again destroyed during a round trip. Conser-vation of the polarization during the ion storage isirrelevant in this scheme.( ii ) In another scenario the buildup of polarization ac-cording to scheme ( i ) would not be sufficient toexplain the observations. Then, at least a part ofthe polarization must be conserved after the roundtrip such that the polarization can slowly build upover several round trips.In the following theoretical section, we will investigatewhich of these two scenarios is more suitable for explain- ing our observations. Therefore, we analyze the laser-ioninteraction with a rate-equation approach, taking all rel-evant experimental boundary conditions into account. V. THEORYA. System of rate equations for Li + ions In order to better understand the experimental resultsfrom the previous section, we have performed a theoret-ical analysis of the population dynamics of Li + ions inthe presence of two superimposed laser fields. Generally,such an analysis would require solution of the Liouville–von Neumann equation that describes the time evolutionof the ionic density matrix [32]. For a discussion as itis intended here, just aiming at a decision which pump-ing scheme introduced above is supported by theory, wecan restrict ourselves to a less computationally demand-ing approach based on a system of rate equations. Therate–equation approach has been widely employed in thepast for the investigation of the optical pumping of ionsin storage rings [10, 33]. It predicts how the populationsof magnetic substates of an ion, as described by the di-agonal elements of the density matrix, evolve over thetime due to transitions from/to another substates. Thefocus on the substate populations implies neglecting thecoherence between the ionic (sub–) states, which is char-acterized by the non–diagonal density–matrix elements.This assumption is well justified for the analysis of thepresent ESR experiment in which fast spontaneous decayof the excited state 2 P as well as ionic collisions andinteractions with the external fields in the ring lead tothe loss of the coherence.In the present work we solve the system of coupleddifferential equations:d N F ,M F d t = − N F ,M F (cid:88) M F Γ exc F M F ,F M F + (cid:88) M F N F ,M F Γ dec F M F ,F M F , (1a)d N F ,M F d t = − N F ,M F (cid:88) M F Γ exc F M F ,F M F + (cid:88) M F N F ,M F Γ dec F M F ,F M F , (1b)d N F ,M F d t = − N F ,M F (cid:88) M F Γ dec F M F ,F M F + (cid:88) M F Γ dec F M F ,F M F + (cid:88) M F N F ,M F Γ exc F M F ,F M F + (cid:88) M F N F ,M F Γ exc F M F ,F M F , (1c)that describes the populations N F,M F ( t ) of the hy-perfine substates | F , M F (cid:105) ≡ (cid:12)(cid:12) S : F = / , M F (cid:11) , | F , M F (cid:105) ≡ (cid:12)(cid:12) S : F = / , M F (cid:11) and | F , M F (cid:105) ≡ (cid:12)(cid:12) P : F = / , M F (cid:11) of Li + ions. Here F is the totalangular momentum of the ion and M F is its projectiononto the beam propagation direction, chosen as the quan-tization axis.In Eqs. (1), moreover, Γ exc F i M Fi ,F M F is the rate forthe photoexcitation | F i , M F i (cid:105) + (cid:126) ω → | F , M F (cid:105) , whileΓ dec F M F ,F i M Fi = Γ dec , sp F M F ,F i M Fi + Γ dec , ind F M F ,F i M Fi with i =1 , hyperfine substates can be easily traced back to the rateof the spontaneous 2 P → S fine–structure decaythat we denote as Γ P , S . For example, with the helpof the angular momentum algebra one can findΓ dec , sp F M F ,F i M Fi = 5(2 F i + 1) (cid:104) F i M F i LM | F M F (cid:105) × (cid:26) F F i L I (cid:27) Γ P , S , (2)where I = 3 / L = 1, for the photon–ion interaction isassumed. By employing this expression and the theoryof Einstein coefficients, we obtain next the rate for thelaser–induced decay asΓ dec , ind F M F ,F i M Fi = 5(2 F i + 1) (cid:26) F F i L I (cid:27) × (cid:32) (cid:88) p = ± c p p (cid:104) F i M F i L p | F M F (cid:105) (cid:33) × (cid:18) I π c (cid:126) ω ∆ ω (cid:19) Γ P , S . (3)Here, I , ω and ∆ ω are the intensity, angular frequencyand (frequency) width of the incident laser radiation,whose values are displayed in Table I. Moreover, the co-efficients c p characterize the photon polarization vector (cid:15) , written in the helicity ( p = ±
1) representation as (cid:15) = c (cid:15) + c − (cid:15) − , | c | + | c − | = 1 . (4)By choosing these parameters as c = c − = 1 / √ c ± = 1 one can “construct” π –linearly– and σ ± –circularly polarized light, respectively. Finally, the pho-toexcitation rate Γ exc F i M Fi ,F M F is directly related toΓ dec , ind F M F ,F i M Fi by the principle of the detailed balance.In what follows we will apply the developed theoreticalapproach to investigate the population dynamics of Li + hyperfine levels for two scenarios. Thus, in Sec. V B thenaive single–run scenario will be considered in which theions enter the laser–overlap region at t = 0 and move init infinitely long. This academic case will allow us to de-termine the characteristic time–scales of the populationdynamics and to discuss how they depends on the polar-ization of incident laser light. Later in the Sec. V C, wewill “simulate” the ESR experiment in which the Li + ionscirculate in the ring, entering and leaving periodically thelaser beams. To describe this realistic multi–cycle sce-nario we will apply the rate equations (1) in which theinduced rates Γ dec , ind F M F ,F i M Fi and Γ dec , ind F M F ,F i M Fi are givenby Eq. (3) for the laser–interaction region and set to zero when the ions leave this region. Intensity of blue laser I blue = 1 . Intensity of red laser I red = 52 . Laser angular frequency ω = 3.4 × HzLaser width of blue laser ∆ ω blue = 1.5 × HzLaser width of red laser ∆ ω red = 1.8 × HzRate of the 2 P → S decay Γ P , S = 2 . × HzTABLE I. Parameters of the incident laser radiation and ofthe Li + ion used in the evaluation of the transition rates. Allparameters are given in the rest frame of an ion, moving withthe relativistic velocity β = 0.338. B. Single–run analysis
After the evaluation of the transition rates Γ dec andΓ exc , one can integrate the system of coupled equations(1) to find the populations of the hyperfine substates N F,M F ( t ) as a function of time. Of course, to performthese calculations one has to define first the polarizationof both laser beams and the initial population of Li + ions.In Fig. 5, for example, we display the results obtained forpopulations that are initially statistically distributed inthe “ground–state” hyperfine levels: (cid:12)(cid:12) S : F = / , M F (cid:11) : N F ,M F (0) = 1 / , (5a) (cid:12)(cid:12) S : F = / , M F (cid:11) : N F ,M F (0) = 1 / , (5b) (cid:12)(cid:12) P : F = / , M F (cid:11) : N F ,M F (0) = 0 , (5c)for all M F i = − F i , ..., F i . These initial conditions corre-spond to the scenario where unpolarized Li + ions in themetastable 2 S state enter the (laser–ion) interactionregion. Moreover, the calculations have been carried outfor the horizontally linearly polarized (upper panel of thefigure) and circularly polarized σ − – σ + and σ + – σ + (mid-dle and lower panels) counter–propagating lasers beams.We have chosen these polarization states in order to re-produce the experimental setup as described in Sec. III.The numerical evaluation of the system (1) for theinitial conditions (5) and for two polarized laser beamsallows us to explore the population of all sixteen hy-perfine substates, involved in the 2 s S ( F = / ) ↔ p P ( F = / ) ↔ s S ( F = / ) Λ–transition of in-terest. For the easier visualization of the results, we donot present in Fig. 5 all sixteen N F,M F ( t )’s but depictinstead the total populations N F i ( t ) = (cid:80) M Fi N F i ,M Fi ( t )of the (cid:12)(cid:12) S : F = / , M F (cid:11) (green dash–dotted lineline), (cid:12)(cid:12) S : F = / , M F (cid:11) (blue dashed line line) and (cid:12)(cid:12) P : F = / , M F (cid:11) (black solid line) hyperfine levelmanifolds. As seen from the figure, the time evolutionof the total populations is very sensitive to the polar-ization state of the lasers. If, for example, both lasersare linearly polarized in the horizontal (storage–ring)plane, all three N F i ( t )’s converge to large non–zero val-ues at t ≈ × − s, see upper panel of Fig. 5. In -10 -8 -6 -4 Time (s)00.51 P opu l a ti on o f hyp e rf i n e l e v e l s π - πσ + - σ + σ - - σ + FIG. 5. The relative population of the 2 S : F = / (greendash–dotted line), 2 S : F = / (blue dashed line) and2 P : F = / (black solid line) hyperfine levels of Li + ionsas a function of time. The calculations have been performedfor two counter–propagating laser beams that are both eitherlinearly polarized in the horizontal plane (upper panel) orcircularly polarized. For the latter case we consider two beamswith the opposite σ − – σ + (middle panel) and the same σ + – σ + (lower panel) polarizations. Finally, the red dotted verticalline marks the time t det , that Li + ions, moving with velocity v = 0 . c , need to reach the center of the detection regionfrom the point where they enter the laser–overlap region. this case, approximately 23 % of Li + ions from the beamare excited to the 2 P ( F = / ) level. The sponta-neous decay of this level leads to the emission of the2 P ( F = / ) → S ( F , = / , / ) photons. Bymaking use of density matrix theory [34], we found thatthese photons will be emitted most likely under forward angles with respect to the beam direction and, hence, canbe efficiently recorded at the detection region. Since ionswith velocity v = 0 . c reach the detection region inabout t det ≈ − s after they enter the laser interactionregion, i.e. when the population N F is almost saturated,our calculations are consistent with the experimental ob-servation of the resonance structure in the spontaneousemission spectrum in Fig. 3.Qualitatively different behaviour of the populations N F i ( t ) is observed if the 2 S ( F = / ) ↔ P ( F = / ) ↔ S ( F = / ) Λ–transition is induced by twocircularly polarized lasers instead of linearly polarizedones. In order to better illustrate this difference, wesolved the system of the rate equations (1) for two sce-narios in which counter–propagating photons have (i) theopposite σ − – σ + and (ii) the same σ + – σ + circular polar-ization states. The results of the calculations are pre-sented in the middle and lower panels of Fig. 5. Asseen from these panels, the populations of the hyper-fine levels exhibit saturation at later times and at verydifferent values if comparing to the linear polarizationcase. For example, almost all Li + ions, interacting withoppositely polarized ( σ − – σ + ) photons, can be found intheir (cid:12)(cid:12) S : F = / (cid:11) level only after 5 × − s. Aneven longer time, t ∼ − s, is needed to transfer theentire population to the same (cid:12)(cid:12) S : F = / (cid:11) level ifboth lasers are right–handed circularly polarized. Thevanishing population in the 2 P level can be easily un-derstood from the fact that absorption of circularly polar-ized σ + – σ + and σ − – σ + photons leads to the formation ofthe dark substate (cid:12)(cid:12) S : F = / , M F = − / (cid:11) , see alsoFig. 6. This substate can not be further excited to the (cid:12)(cid:12) P : F = / (cid:11) level because of missing M F substatesto fulfill the selection rule ∆ M F = − σ + red lightas indicated in Fig. 4. As a result, there will be no spon-taneous decay 2 P ( F = / ) → S ( F , = / , / )and, hence, no signal at the photomultipliers. Fig. 4 alsoprovides intuitive understanding of the fact that the dis-appearance of the fluorescence signal is observed laterfor the σ + – σ + polarization. In this case photons fromthe counter–propagating lasers induce ∆ M F = +1 and∆ M F = − σ − – σ + case.The complete transfer of the population to the darksubstate (cid:12)(cid:12) S : F = / , M F = − / (cid:11) happens muchlater than the time t ≈ − s when the ions reach the de-tection region, as can be seen from the middle and lowerpanels of Fig. 5. Moreover, our theoretical analysis againshowed that during all the (saturation) time the spon-taneous decay 2 P ( F = / ) → S ( F , = / , / )leads to the photon emission predominantly in the for-ward beam direction, i.e. under the angles at which thedetection region is most sensitive. Based on these obser-vations one would expect that a clear fluorescence signalwill be detected for the interaction of Li + ions with cir-cularly polarized lasers. This contradicts, however, the t = 10 -8 s t = 10 -7 s t = 10 -6 s t = 10 -5 s t = 10 -4 s P opu l a ti on o f hyp e rf i n e s ub s t a t e s -5/2 -3/2 -1/2 1/2 3/2 5/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 Projection of total angular momentum M F -5/2 -3/2 -1/2 1/2 3/2 5/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 π - π σ - - σ + σ + - σ + FIG. 6. The relative population N F M F of the hyperfine substates (cid:12)(cid:12) S : F = / , M F (cid:11) at times t = 10 − s, t = 10 − s, t = 10 − s, t = 10 − s and t = 10 − s after initially unpolarized Li + ions enter the laser–overlap region. Calculations havebeen performed for two counter–propagating laser beams that are both either linearly polarized in the horizontal plane (upperpanel) or circularly σ − – σ + (middle panel) and σ + – σ + (lower panel) polarized. For the latter two polarization cases one canobserve the formation of the dark substate (cid:12)(cid:12) S : F = / , M F = − / (cid:11) . This formation, however, takes much longer time forthe σ + – σ + polarization case where blue and red lasers transfer the substate populations in opposite directions, see lower panelof Fig. 4. experimental observations, which show no signal for the σ − – σ + case and a very weak resonance for the σ + – σ + po-larization state in Fig. 3. In order to explain this discrep-ancy we recall that our calculations have been performedfor the initial conditions (5), i.e. for initially unpolarized Li + ions in their “ground” levels (cid:12)(cid:12) S : F , = / , / (cid:11) .Such a choice of the initial populations N F,M F ( t = 0) isbased on the assumption that any non–statistical pop-ulation of hyperfine levels will be destroyed after ionspass the bending magnets of the ESR storage ring. Theobserved discrepancy between experimental findings andrate–equation predictions leads us to question this as-sumption. Namely, the weak or even vanishing fluores-cence signal, recorded at the detection region from Li + ions interacting with circularly polarized photons, can beunderstood if the ions are already longitudinally polar-ized when they enter the laser field. Unless this polariza-tion was produced by the bending magnets, it is causedby the optical pumping during the previous passages ofthe ion–laser interaction region. C. Multi–cycle analysis
In order to support our interpretation of continuous optical pumping of Li + ions during their circulationsin the ESR, we have simulated the multi–cycle state-population dynamics. Within one cycle of this simula-tion, the system of rate equations is used to compute thepopulations N F,M F ( t ) in the interval 0 ≤ t ≤ t ring with t ring ≈ − s being the duration of a single round trip inthe ring. Moreover, to account for the fact that ions in-teract with the laser beams only part of their trip we haveintroduced in Eq. (1) the time–dependent laser–induced(decay and excitation) rates:Γ exc ( t ) = (cid:40) Γ exc , ≤ t ≤ t laser , t laser < t ≤ t ring , (6a)Γ dec , ind ( t ) = (cid:40) Γ dec , ind , ≤ t ≤ t laser , t laser < t ≤ t ring , (6b)0 P opu l a ti on o f t h e s t a t e P F = / σ + - σ + σ - - σ + π - π FIG. 7. The relative population of the excited level 2 P : F = / of Li + ions as a function of the number of circula-tions in the storage ring. N P : F = / ( t ) is calculated for themoment when the ions pass the detection region and for thevarious polarization states of the applied lasers. where t laser ≈ × − s specifies the time during whichions move in the presence of the laser fields. Similarto the t det above, both times t ring and t laser have beenestimated for the ESR geometry and for the ion velocity v = 0 . c .The evaluation of the system of rate equations in thetime interval 0 ≤ t ≤ t ring and with the modified in-duced rates (6) allows us to simulate the state popula-tion dynamics during a single round trip in the ESR. Inorder to understand how N F,M F ( t )’s will change for therealistic experimental scenario of multiple circulations ofions in the ring, we (i) employ the populations obtainedat the end of a previous cycle, i.e. at t = t ring , as ini-tial conditions for the next cycle, (ii) solve the system(1) with these conditions, and (iii) repeat the procedurecycle–by–cycle. We have applied this multi–step analy-sis to investigate how the population of the excited state (cid:12)(cid:12) P : F = / (cid:11) , if observed at the moment when theions pass the detection region, evolves with the numberof cycles. The results of our calculations are displayedin Fig. 7 for the cases where the Li + ions are pumpedby linearly polarized (black solid line) as well as by cir-cularly polarized σ − – σ + (blue dotted line) and σ + – σ + (red dash–dotted line) lasers. We have assumed, more-over, that in the beginning of the very first cycle of thesimulation the ions are unpolarized and, hence, their sub-level population is described by Eq. (5).Figure 7 depicts the cycle–by–cycle evolution of the to-tal population N F at the time when the ions pass thefluorescence detection region during their multiple turns.It is obvious that the behavior is quite distinct for the dif-ferent polarizations of the laser beams. If, for example,both lasers are linearly polarized within the ESR plane,the population of the excited level (cid:12)(cid:12) P : F = / (cid:11) re- mains virtually constant, N F ≈ .
23. This is expectedfrom the “single–run” results which are presented in theupper panel of Fig. 5 and indicate that the level popula-tions are saturated to large non–zero values in 2 × − s,i.e. approximately at the moment when the ions reachthe detection region. Moreover, since the intensity ofthe 2 P ( F = / ) → S ( F , = / , / ) spontaneousemission is proportional to N F , our calculations sup-port the experimental observation of the pronounced res-onance in the fluorescence signal shown in Fig. 3.In contrast to the linear–polarization case, the popu-lation of the excited level (cid:12)(cid:12) P : F = / (cid:11) generally de-creases with the number of cycles if the ions are pumpedby circularly polarized laser beams. One can understandthis based on the “single–run” analysis from Sec. V B aswell as from our assumption about the preservation of thepolarization of ions during their round trips. Namely, asseen from Fig. 5, the time t laser ≈ × − s of a sin-gle passage of the ions through the laser interaction re-gion is shorter comparing to the time of depopulation ofthe (cid:12)(cid:12) P : F = / (cid:11) level, but is sufficient to produce apartial polarization of the Li + ions. If this polarizationis preserved—at least to some extent—as the ions con-tinue to move in the ring, the population of the hyperfinesubstates in the beginning of the next ion–laser interac-tion period will deviate from Eq. (5). During this nextround the lasers will even further polarize the ions andthe procedure will continue until the formation of thedark substate (cid:12)(cid:12) S : F = / , M F = − / (cid:11) and, hence,full depopulation of the (cid:12)(cid:12) P : F = / (cid:11) level.From our calculations and the discussion above we con-clude that the population saturation time in the “single–run” scenario is directly related to the number of cy-cles needed to polarize the ions in the “multi–cycle”case. For example, the pumping of Li + ions by twolaser beams with opposite circular polarization, σ − – σ + , results in a relatively fast saturation and, conse-quently, to the formation of the respective dark substate (cid:12)(cid:12) S : F = / , M F = − / (cid:11) just after 20 cycles in thering. In contrast, very long saturation time, t ∼ − s,and more than 300 round trips are required to polarizeLi + interacting with σ + – σ + beams. These qualitativepredictions, based on the assumption of multi–step opti-cal pumping of Li + ions, allow us to better understandthe experimental findings from Fig. 3. Indeed, since forthe σ − – σ + case the ionic population is fully locked in thedark substate (cid:12)(cid:12) S : F = / , M F = − / (cid:11) just after acouple of dozend cycles, no signal from the spontaneousdecay 2 P ( F = / ) → S ( F , = / , / ) can be ob-served at the detection region. On the other hand, aweak resonance signal recorded for the σ + – σ + case canbe seen as a consequence of a sluggish decrease of thepopulation of the excited state (cid:12)(cid:12) P : F = / (cid:11) duringthe circulations of the ions in the ring.Even though the σ + – σ + scheme takes longer to trans-fer the population into the dark substate, a time spanof 0.1 ms, being a fraction of less than 1% of the dwelltime cannot create the remaining signal strength in the1 P opu l a ti on o f t h e s t a t e P : F = / δ = 0 % σ + - σ + σ - - σ + σ + - σ + (+ mix π ) σ - - σ + (+ mix π ) P L = 0 %P L = 5 %P L = 10 %P L = 20 % δ = 1 % δ = 5 % δ = 10 % FIG. 8. The same as Fig. 7, but for the cases where thepopulation of hyperfine substates is partially distorted by themagnets in the storage ring and collisions (upper panel), andthe σ polarization of the red laser has an admixture of the π linearly polarization component (lower panel). In orderto estimate the effect of the depolarization in the magnetsand collisions, a particular fraction δ of the population ofeach substate | F M F (cid:105) was evenly redistributed among ground–level substates after each round trip. The imperfection of thecircular polarization is quantified by the Stokes parameter P L ,or degree of linear polarization, of the π component. spectrum. We shall assume, therefore, that there is acontinuous transfer from the dark substate back into themultitude of bright states. In order to understand pos-sible reasons for this “back transfer” we remind that thetheoretical calculations presented in Fig. 7 are based ontwo very naive assumptions. Namely, we have presumedthat (a) the population of hyperfine substates, as pro-duced during the interaction with lasers and by subse-quent spontaneous decay, is neither affected by the mag-nets of the storage ring nor by collisions and (b) bothlasers are ideally ( σ –) polarized. Both conditions are notfulfilled in the real experiment. Firstly, the polarizationof the ions might be modified by collisions with coolerelectrons and rest–gas atoms as well as by the magneticfields of the dipoles and quadrupoles along the circum-ference of the storage ring. Secondly, the light polariza-tion – even if it is perfectly circular after the λ / plate –might obtain a linearly polarized component, e.g., from stress-induced birefringence in the vacuum windows. Adetailed analysis of the ion (de–) polarization dynamicsdue to collisions and interactions with magnetic fields isa rather complicated task which depends, moreover, onthe setup of a particular experiment. This analysis is outof scope of the present theoretical study and will be pre-sented in a forthcoming publication. Here we just use avery crude model in which the depolarization of ions inmagnets and due to collisions is estimated by redistribut-ing a particular fraction δ of the population of each M F state across ground–level substates after each revolution.We found that the system is quite forgiving in the σ − − σ + case, where even a depolarization of δ = 10% per revolu-tion still allows a strong depopulation of the excited level2 P : F = / and, hence, a fluorescence signal reduc-tion of about 60 % after just a dozen cycles, as can be seenin the upper panel of Fig. 8. In contrast, in the σ + − σ + case a depolarization of only 1% leads to a roughly 50 %signal recovery compared to the calculations neglectingany population redistribution, i.e. for δ = 0 %. As seenfrom the lower panel of Fig. 8, imperfect laser polariza-tion may also lead to the “back transfer” from the darkstate. Again, it is the σ − − σ + case that is more robust:Even with the significant admixture of the π componentin the red laser beam, quantified by the Stokes parameter P L = 20 %, the fluorescence signal recovers to only 13%of its original strength. For σ + − σ + , in contrast, already10% π -light is sufficient for recovering 50% of the signalstrength. The latter result shows that the small remain-ing signal in Fig. 3 might not necessarily be caused bydepolarization effects of the ion beam, but could as wellbe caused by non-ideal light polarization. More rigorousinvestigation, experimentally as well as theoretically arerequired and will be carried out in the future. VI. CONCLUSION
We have shown in our theoretical analysis that opticalpumping in the 2 s S ( F = / ) ↔ p P ( F = / ) ↔ s S ( F = / ) Λ–transition of Li + ions at the ESRcan only lead to the observed disappearance of the reso-nance signal if a considerable fraction of the induced po-larization survives the revolution along the storage ring.However, with the limited experimental information ob-tained so far, we cannot conclude that this process hasreally happened but further investigation of this systemis of high interest. Due to the restricted amount of beam-time at the ESR and the excessive number of beamtimeapplications, we have started to carry out investigationsat CRYRING [35] using Mg + ions. These can be injectedinto CRYRING from a local ion source without the neces-sity of operating and blocking the entire GSI acceleratorchain. A Λ-scheme in Mg + ions will be addressed withcw lasers but we will also test broad-band optical pump-ing with pulsed lasers. Those studies serve also for theestablishment of laser spectroscopy at CRYRING in gen-eral [36]. It should be noted that a non-observation of2optical pumping at this machine has only limited signifi-cance for disproving the explanations of the ESR resultsdue to the completely different lattice and ion dynamicsin the two storage rings. Therefore, further experimentsat the ESR are also planned. There, a convincing signa-ture could be obtained, e.g. , if the resonance disappearsagain under the conditions of the last experiment but canbe reestablished with another laser beam that interactswith the ions on the opposite side of the storage ring andis employed to depopulate the dark–state. One interest-ing application of optical pumping at the ESR would bethe hydrogen-like ion Pr [37]. Here, laser excitationacross the hyperfine splitting in the electronic ground-state, will affect the nuclear lifetime [38]. The directionof neutrino emission can be determined in the Schottkyspectrum after the electron capture process [39]. By op- tical pumping in the hyperfine transition the emission di-rection can be determined either in forward or backwarddirection and be directly observed. ACKNOWLEDGEMENT
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