document: spectroscopic studies of hyperfine manifolds in alkalies , such as measurements of energy separations , have benefitted by the high precision of the experimental techniques available to interrogate atoms @xcite . 
   their hydrogen - like structure makes interpretation of experimental results straightforward in terms of electromagnetic fields generated by the valence electron and nuclear moments . 
   precise measurements in higher excited states accessible through two - step transitions@xcite have appeared in recent years . 
   this has renewed interest in improving calculations in other states where theoretical methods such as many - body perturbation theory ( mbpt ) ( see for example the recent book of w. r. johnson @xcite ) are yet to be tested against experimental results . 
   precise measurements in excited states , beyond the first one , have several experimental complications . 
   standard spectroscopic techniques rely on the high population of atoms in the ground state to guarantee a good signal to noise ratio of the fluorescence or absorption of the atomic sample . in two - step transitions 
   this is no longer the case . 
   the amount of population transferred to the intermediate level , for reasonable powers of the lasers , tends to be small , and detectors at the desired frequency might no be readily available . 
   we present in this paper two - color modulation transfer spectroscopy as a tool for studies of atomic properties of higher excited states . 
   the method consist of two lasers ( pump and probe ) counter - propagating through a thermal vapour . before being directed to the interaction region , 
   one of the lasers is modulated . 
   the first step of the transition _ 
   i.e. _ the pump , connects the ground state to a resonant intermediate state while the probe scans over the desired energy manifold . 
   we monitor the absorption of the pump laser as a function of probe laser detuning . 
   the non - linear interaction of the lasers  burns a hole " in the atomic ground state population . 
   the generated spectra presents sub - doppler peaks ( sometimes called lamb - bennett dips ) corresponding to the atomic resonances with the trademark sidebands at their side . 
   this technique overcomes the two main inconveniences of direct absorption of the probing laser _ i.e. _ low signal to noise ratio and non - availability of detectors at the desired wavelength . 
   we present two ladder systems in @xmath0rb to illustrate the main features of the technique and two different applications of the modulation . 
   we select the @xmath1 and the @xmath2 ladder transitions to illustrate their different uses . 
   the amplitude of the probe laser is modulated for the first system while the second system has its pump frequency modulated . 
   the frequency modulation of the pump laser and good signal to noise ratio allows us to lock the probe laser to the @xmath3 excited atomic resonance . in this case 
   the probe laser remains modulation free . 
   this is highly desired since the electronic modulation of the laser itself can carry unwanted effects such as sidebands at higher or lower frequencies as well as bandwidth problems . 
   the method we are presenting is , of course , not limited to these two cases and can be extended to other atomic levels . 
   the organization of the paper is as follows : section ii contains the theoretical model , section iii explains the experimental setup and results , section iv has a summary of the precise measurements using this method , and section v presents the conclusions . 
   we start with a three level model that can show some of the qualitative features of the experimental spectra . 
   we use a density matrix formalism to describe a three level atom in ladder configuration interacting with two lasers , one of which has sidebands . 
   we model our system as doppler - free ignoring zeeman sublevels to keep it tractable . 
   the experimental situation is more complex and for quantitative analysis it is necessary to take into account those same effects that we are ignoring . 
   figure [ figure energy levels theory ] shows our theoretical model . 
   we treat two cases . 
   fig [ figure energy levels theory ] ( a ) is a ladder type system with an amplitude modulated probe ( amp ) . 
   fig ( b ) presents the same system except it has a frequency modulated pump ( fmp ) . 
   the intermediate and last levels are coupled by a single laser with three frequencies : a carrier and two sidebands separated form the carrier by @xmath4 ( in mhz ) . 
   we represent the amplitude of the carrier by a rabi frequency @xmath5 and the sidebands by a modulation depth @xmath6 . the ground and intermediate states are coupled by @xmath7 . 
   the detuning of the carrier between levels @xmath8 and @xmath9 is zero in the model as it is for our experiment and we let the detuning between levels @xmath9 and @xmath10 vary as @xmath11 . 
   the total population is normalized to unity . 
   [ figure energy levels theory ] ( b ) follows the same nomenclature except that the sidebands arise from frequency modulation and they appear in the pump laser @xmath7 . for the fmp systems the sidebands have the appropriate sign difference . 
   we have a set of nine linear equations for the slowly varying elements of the density matrix @xmath12 after using the rotating wave approximation with the sidebands rotating - one clockwise , one counter clockwise - at a frequency @xmath4 . 
   the equations are :    @xmath13\sigma_{nm}~+}\\    & & \frac{i}{2}\sum_{k}(\alpha_{nk}\sigma_{km}-\sigma_{nk}\alpha_{km})=\dot{\sigma}_{nm}~for~n\neq m,\nonumber\end{aligned}\ ] ]    where @xmath14 is the transition frequency , and @xmath15 is the laser frequency connecting the levels . 
   the damping rate is given by : @xmath16 and @xmath17 for the fmp system and @xmath18 for the amp system . 
   the time dependence of the rabi frequency makes the standard approach of obtaining the steady state solution of the system not feasible . 
   instead , we use a floquet basis expansion of the density matrix @xcite to solve our system of equations . 
   we replace each of the slowly rotating elements of the density matrix by :    @xmath19    where @xmath20 is the fourier amplitude of the component oscillating at @xmath21 . 
   the system is now a series of @xmath22 coupled equations for some large @xmath23 that have to be solved recursively . 
   it is necessary to set @xmath24 for some @xmath23 to cut off the infinite number of coupled equations . by solving the equations in terms of their predecessors 
   we can extract @xmath25 .    . 
   the parameters are ( in units of @xmath26 ) : @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31,width=283 ]    . 
   the parameters are ( in units of @xmath26 ) : @xmath32 , @xmath33 , @xmath34 , @xmath35 , and @xmath36,width=283 ]    for our experiment we are interested in the terms @xmath37 , @xmath38 , and @xmath39 which are proportional to the absorption of the first laser carrier and sidebands , respectively . 
   we plot the absolute value of the imaginary part as a function of @xmath11 to recover the absorption . 
   this is necessary to take into account the square - law nature of the photodiode . 
   our three level model reproduces the resonance features of the absorption observed as the second excitation goes into resonance for both amp and fmp systems ( see fig . 
   [ figure absorption am ] and fig . 
   [ figure absorption fm ] , respectively ) . 
   the demodulation of the fmp signal yields the expected error - like feature shown in fig . 
   [ figure demodulated absorption ] .    . 
   the parameters are ( in units of @xmath26 ) : @xmath32 , @xmath33 , @xmath40 , @xmath41 , and @xmath36,width=283 ]    the size of the sidebands in our model depends on the modulation index ( separation from resonance and strength ) , as well as the specific decay rates of the levels which set up the rabi frequencies @xmath42 in the amp and fmp systems . 
   figure [ figure experimental setup pump ] and fig . 
   [ figure experimental setup probe ] present block diagrams of the fmp and amp systems , respectively . 
   a coherent 899 - 01 ti : sapphire laser with a linewidth of less than 100 khz is the pump laser in both cases . a small amount of laser power from the pump laser is frequency modulated by a small bandwidth electro - optical modulator at @xmath4315 mhz and sent to a glass cell filled with rubidium at room temperature to lock the laser frequency to the @xmath44 crossover line of the @xmath45 and @xmath46 hyperfine levels for the fmp system and to the on resonance @xmath47 transition of the @xmath48 level for the amp system at 795 nm with a pound - drever - hall lock . 
   level @xmath8 in the amp system corresponds to the lower hyperfine state of the 5s@xmath49 level ( @xmath45 ) while @xmath9 is the highest hyperfine state of the 5p@xmath49 level ( @xmath50 ) of @xmath0rb . 
   the decay rate between the two levels is @xmath51 5.7  mhz @xcite . 
   we simplify the hyperfine states of the @xmath52 level to just one level with decay rate @xmath53 3.5  mhz @xcite .    for the fmp system , 
   the probe laser is an sdl diode laser with a linewidth of 5 mhz at 776 nm . 
   the lasers overlap inside an independent rubidium glass cell at room temperature wrapped in @xmath54-metal in lin - perp - lin polarization configuration . 
   their @xmath55 power diameter of the laser beams is 1 mm . 
   we scan the probe laser over the @xmath56 level hyperfine manifold and observe the absorption of the pump laser as a function of the probe laser detuning using a fast photodetector . 
   we send the signal to a bias - t and record the dc and demodulated ac components with a wavesurfer digital oscilloscope with an 8-bit resolution from lecroy . 
   we keep the power of the pump laser and the modulation depth fixed to a value of 100 @xmath54w and @xmath57 , respectively . 
   we change the power of the probe beam and observe its influence on the spectra . 
   it is possible to observe the resonant features of the @xmath56 hyperfine manifold with little as 100 @xmath54w of probe power . 
   higher probe power increases the signal size and the width of the features . playing with the polarization and powers we also observe eit features @xcite . 
   we restrict ourselves to a space parameter where these very narrow features are absent . 
   figure [ figure absorption with sidebands ] ( fmp ) and [ figure absorption ] ( fmp ) show typical experimental traces of the absorption of the 780 nm laser . 
   the spectrum has been offset to zero transmission for convenience . 
   the first of these , fig . 
   [ figure absorption with sidebands ] , has the dc component of the absorption with the sidebands appearing on both sides of the main resonances . 
   no doppler background is observed for any of the experimental conditions explored , showing that this is a doppler free spectrum . 
   [ figure absorption ] ( a ) shows the lower hyperfine states of the @xmath56 level manifold with no sidebands for clarity . fig .  [ figure absorption](b ) has the demodulated ac component of the absorption . 
   the dashed lines identify the error - like features with their corresponding hyperfine levels . 
   we use this spectrum to stabilize the frequency of the probe laser . 
   level in @xmath0rb . 
   it presents the main resonances as well as the indicated sidebands ( sb ) . 
   the power of the probe and pump beam are 4.3 mw and 100 @xmath54w , respectively . , width=283 ]     resonances in @xmath0rb of ( a ) absorption without sidebands and ( b ) demodulated absorption of 780 nm laser as a function of detuning of the 776 nm laser . 
   the power of the probe and pump beam are 4.3 mw and 100 @xmath54w , respectively.,width=283 ]    we monitor the laser frequency of the probe beam using a coherent confocal fabry - perot cavity with a free spectral range of 1.5 ghz to test the performance of the laser lock . 
   [ figure error signal ] shows the fringe - side transmission of the probe laser through the cavity . 
   we monitor the behavior of the laser before and after it has been locked . 
   the reduction of the frequency excursions is quite evident as the laser is locked to the atomic resonance . under normal experimental conditions 
   we have observed locking times of 30 minutes , and a significant reduction of the rms noise of more than a factor of seven . 
   excited atomic transition using fmp.,width=283 ]    a thick glass plate splits into two the main beam at 795 nm in the amp system before entering an independent rubidium vapor glass cell inside a three layered magnetic shield . 
   a grating narrowed diode laser at 1.324 @xmath54 m ( from here on referred to as 1.3 @xmath54 m laser ) with a linewidth better than 500 khz excites the second transition . 
   we scan the frequency of the 1.3 @xmath54 m laser over the hyperfine manifold of the @xmath52 level . a fiber - coupled semiconductor amplifier increases the power of the 1.3 @xmath54 m laser before it goes to a large bandwidth ( @xmath4310 ghz ) electro - optic modulator ( eom ) that generates the sidebands . 
   the power of each 795 nm beam is approximately 10  @xmath54w with a diameter of 1 mm . 
   we operate in the low intensity regime to avoid power broadening , differential ac stark shifts and line splitting effects such as the autler - townes splitting . 
   both beams are circularly polarized by a @xmath58 waveplate . 
   the counter propagating 1.3 @xmath54 m laser beam with a power of 4 mw and approximately equal diameter overlaps one of the 795  nm beams . after the glass cell an independent photodiode detects each 795  nm beam . 
   the outputs of the detectors go to a differential amplifier to reduce common noise . 
   a digital oscilloscope records the output signal for different values of modulation . 
   [ figure whole scan ] shows an absorption spectrum of the 795  nm laser as a function of the detuning of the 1.3 @xmath54 m laser that shows the signature sidebands of the technique . 
   [ figure linear regression ] shows a plot of the distance between the sidebands as a function of the modulation of the 1.3 @xmath54 m laser . 
   the sidebands that appear on the absorption spectra provide _ in situ _ 
   calibration for the energy spacing of the hyperfine splittings . 
   this effectively translates a measurement of energy spacings from the optical region to a much easier measurement in the radio frequency range . 
   we observe a rich atomic behavior such as eit and reversal of the peaks that depends on the power of the lasers , relative polarization and magnetic field intensity ( see for example fig [ figure bad absorption ] ) . 
   this points towards a stringent control on experimental parameters for precision studies of energy separations .    , 
   f=1 and f=2 hyperfine states of @xmath0rb with sidebands . 
   the big sideband belongs to the f=1 peak . 
   the small feature on the side of the f=2 peak corresponds to the second sideband of the f=1 peak . 
   the glass cell is in a magnetic field of 0.37 g.,width=283 ]    m laser for @xmath0rb . 
   the arrow point to the value of the modulation that corresponds to the overlap of the sidebands and half the hyperfine splitting of the @xmath52 level hyperfine splitting.,width=283 ] 
   table [ table theory and experiment ] shows the values of the magnetic dipole constants using relativistic mbpt @xcite with single double ( sd ) and single double partial triple ( sdpt ) wave functions and values extracted from measurements of the hyperfine splitting in other electronic states currently in the literature for @xmath59=1/2 @xcite . 
   we have not been able to find in the literature values for higher levels with adequate precision to include them in the figure . 
   the agreement of the theory with the experiment , for @xmath59=1/2 levels , is well within the 1% level . 
   the sdpt relativistic wave functions do indeed improve the accuracy of the calculations of the single double wave functions .    .single 
   double ( sd ) and partial triple ( sdpt ) excitation calculated from _ 
   ab intio _ mbpt in ref . 
   @xcite and experiment magnetic dipole constants for the first @xmath59=1/2 levels in @xmath60rb . 
   ( adapted from ref . 
   @xcite ) . [ cols="<,^,^,^",options="header " , ]     the accuracy of the 6@xmath61 measurement is high enough to extract a hyperfine anomaly @xcite in an excited state , which shows that the effect is independent of the n state of the level , as originally predicted by bohr and weiskopf @xcite . 
   we have presented two - color modulation transfer spectroscopy as reliable and simple method for studies of atomic properties in excited states . 
   the characteristic sidebands appearing the spectra have the two - fold utility of working as an _ in situ _ 
   ruler for measurements of energy separations or to lock the frequency of a laser to an excited transition . 
   the good quality of the data presented is due to monitoring of the absorption of the pump beam instead of direct absorption of the probe beam . 
   the absorption of the pump beam ( or lack thereof ) , is always guaranteed since a vast amount of atoms are always in the ground state and even small changes _ 
   i.e. _ excitation to the last step of the transition , will be noticeable even for small powers of the pump beam . 
   in addition , the spectra does not show a doppler background due to the lack of an equilibrium thermal population in the intermediate state . 
   it is the hope that the method will stimulate studies of atomic properties of excited states and further push the experimental precision and theoretical work in excited atomic states . 
   work supported by nsf . a.p.g . would like to thank e. gomez for discussions on the subject of this article and p. barberis for help on the theory of three level atoms .
summary: we present two - color modulation transfer spectroscopy as a tool for precision studies of atomic properties of excited states . 
   the bi - colored technique addresses a narrow set of velocity groups of a thermal atomic vapour using a two - step transition to 
   burn a hole " in the velocity distribution . 
   the resulting spectrum presents sub - doppler linewidths , good signal to noise ratio and the trademark sidebands that work as an _ in situ _ 
   ruler for the energy spacing between atomic resonances . 
   the spectra obtained can be used for different applications such as measurements of energy splittings or stabilization of laser frequencies to excited atomic transitions . 
   = 10000