Light-induced states in the transient-absorption spectrum of a periodically pumped strong-field-excited system
LLight-induced states in the transient-absorption spectrumof a periodically pumped strong-field-excited system
Juliane Haug ∗ and Stefano M. Cavaletto † Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: October 23, 2018)The transient-absorption spectrum of a V -type three-level system is investigated, when this isperiodically excited by a train of equally spaced, δ -like pump pulses as, e.g., from an optical-frequency-comb laser. We show that, even though the probe pulse is not assumed to be much shorterthan the pump pulses, light-induced states appear in the absorption spectrum. The frequency- andtime-delay-dependent features of the absorption spectra are investigated as a function of several lasercontrol parameters, such as the number of pump pulses used, their pulse area, and the pulse-to-pulsephase shift. We show that the frequencies of the light-induced states and the time-delay-dependentfeatures of the spectra contain information on the action of the intense pulses exciting the system,which can thus complement the information on light-imposed amplitude and phase changes encodedin the absorption line shapes. I. INTRODUCTION
With the advent of femto- and attosecond pulses,transient-absorption spectroscopy (TAS) [1, 2] has estab-lished itself as a powerful method to study strong-fieldquantum dynamics in atoms [3–8], molecules [9–11], andsolids [12–15]. First experiments employed a traditionalpump–probe setup, where a short pump pulse is used toexcite strong-field dynamics in the system, and its timeresponse is observed by measuring the absorption spec-trum of a probe pulse at different time delays [4, 16].However, increasing attention has been received in recentyears by theoretical studies and experiments in which theprobe pulse either precedes or overlaps with the strongpump pulse [17–19]. In several attosecond transient-absorption-spectroscopy (ATAS) experiments, for exam-ple, the spectrum of an attosecond extreme-ultraviolet(XUV) pulse is observed in the presence of a subsequentstrong femtosecond infrared (IR) pulse dressing the statesof the atomic system. In the absence of the IR pulse, theXUV spectrum consists of lines centered on the transi-tion energies between the ground state and the so-calledbright states directly excited by the XUV pulse. How-ever, in the presence of a strong IR pulse coupling thesebright states to other dark levels, light-induced states(LISs) appear in the spectrum during overlap, associatedwith the dressed states of the system.The modification of the absorption line shapes and theappearance of LISs in the spectrum have emerged as keyingredients to understand and control strong-field quan-tum dynamics. As recently demonstrated, the line shapescontain the full information about the temporal responseof a strongly driven quantum system, enabling its recon-struction without scanning over time delays [20], as longas the dynamics are initiated and probed by a sufficiently ∗ On leave from: Eberhard Karls Universität Tübingen,Geschwister-Scholl-Platz, 72074 Tübingen, Germany † Email: [email protected] short pulse. However, for LISs to appear during overlap,and in order to enable real-time reconstruction of dipoleresponses directly from absorption spectra, it is necessarythat the probe pulse be much shorter than the timescaleof the observed strong-field-induced dynamics.If the duration of the probe pulse is comparable withthat of the pump pulse, the absorption spectra do notprovide the resolution necessary to access the real-timedynamics taking place during the pump pulse. This isfor example the case when optical transitions have tobe studied or controlled with optical femtosecond pumpand probe pulses of equal duration [21–24]. The Rabioscillations responsible for the appearance of LISs takeplace only within the pump pulse, and a probe pulse ofequal duration does not have the required resolution todistinguish them. Therefore, no LISs appear in the spec-trum as a signature of strong-field dynamics in such case.However, the absorption lines of the bright states stillcarry information about the total (integrated and non-time-resolved) action of the pulse on the atomic system,which is imprinted in the associated line shapes and canbe understood in terms of light-imposed amplitude andphase changes [25–29].Here, we show that, even when probe and pump pulseshave the same duration, LISs appear in the absorptionspectrum of the probe pulse if a train of pump pulses isused, employing the TAS setup shown in Fig. 1. Periodictrains of intense optical pulses, as provided by optical-frequency-comb lasers [30–32], have found numerous ap-plications in precision spectroscopy [33], the developmentof all-optical atomic clocks [34], and attosecond science[35]. Furthermore, by exploiting coherent pulse accumu-lation and quantum interference effects, they have alsobeen employed for the control of atomic coherences [36–38] and united time–frequency spectroscopy [39]. Controlof the x-ray transient-absorption spectrum with an op-tical frequency comb has also been put forward for thegeneration of an x-ray frequency comb [40, 41].We investigate the dynamics of a V -type three-levelsystem, modeling optical transitions in atomic Rb, whenexcited by ultrashort optical pump and probe pulses. We a r X i v : . [ phy s i c s . a t o m - ph ] O c t DetectorSampleprobe pump
Figure 1. Experimental setup for the detection of the optical-density transient-absorption spectrum of a transmitted probepulse (light blue) in the presence of an additional pump field(red) consisting of a train of pulses in a noncollinear geometry. model the pump field as a periodic train of N identi-cally spaced pulses, showing that, for sufficiently largevalues of N , LISs appear in the absorption spectrum ofthe probe pulse. The spectra are investigated as a func-tion of the delay between the probe and pump fields, inthe case of a probe pulse preceding, in between, or fol-lowing the pump pulses. We show that the strong-fieldaction of the pump pulses is encoded in the central fre-quencies of the LISs appearing in the spectrum, and intheir time-delay-dependent periodic properties. This en-ables the extraction of information about the intensity-dependent action of the pump pulses on the system di-rectly from the frequency of the LISs. It can thus be usedto complement the information obtained in the case of asingle pump pulse, where no LISs appear and the actionof the intense pump pulse is exclusively encoded in theline shapes of the bright states.The paper is organized as follows. Section II intro-duces the theoretical model used to describe the V -typethree-level system and its interaction with the pump andprobe fields (II A), and the transient-absorption spec-trum (II B). In Sec. II C, the dynamics of the system andthe associated spectra are calculated for a train of N equally distant δ -like pump pulses, for a probe–pump(II C 1), pump–probe (II C 2), and pump–probe–pumpsetup (II C 3), depending on the position of the probepulse with respect to the train of pump pulses. The re-sulting spectra are presented and discussed in Sec. IIIfor different values of the laser control parameters. Inparticular, we investigate the appearance of LISs for anincreasing number of pump pulses (III A), focusing on N → ∞ (III B), for which we highlight the frequency- andtime-delay-dependent features exhibited by the spectra,in Secs. III B 1 and III B 2, respectively. Additional math-ematical details are included in the Appendixes. Atomicunits are used throughout unless otherwise stated. II. THEORETICAL MODELA. Three-level model and equations of motion
The TAS geometry is displayed in Fig. 1. It features aprobe pulse, whose absorption spectrum is detected upontransmission through the atomic sample, and an addi-tional pump field, consisting of a train of pulses, whichmodifies the dipole response of the atomic system. Thepulses considered in the following have the form [42] E ( t, t c , φ ) = E ( t, t c , φ ) ˆ e z = E ( t − t c ) cos[ ω c ( t − t c ) + φ ] ˆ e z , (1)where E ( t, t c , φ ) is the amplitude of the field and ˆ e z isthe direction of linear polarization. Here, we have in-troduced the central time of the pulse t c , its carrier fre-quency ω c , envelope function E ( t ) , and carrier-envelopephase (CEP) φ .The time-dependent pump field [30–32] E pu ( t ) = E pu ( t ) ˆ e z = N − (cid:88) n =0 E , pu ( t − nT p ) cos[ ω c ( t − nT p ) + φ , pu + n ∆ φ ] ˆ e z , (2)consists of a train of N equally spaced pulses, centeredat times t n = nT p , n ∈ { , , . . . , N − } , separated by arepetition period T p , and with envelope function E , pu ( t ) as shown in Fig. 2(a). In the following, we will refer toa pulse centered on t n as the n th pulse—for instance,the th pulse will always denote the first-arriving pumppulse centered on t . The CEP of the n th pulse is givenby φ , pu + n ∆ φ , where the CEP φ , pu of the initial thpulse and the constant pulse-to-pulse phase shift ∆ φ areboth φ , ∆ φ ∈ [0 , π ] .We define the Fourier transform of a generic time-dependent function g ( t ) as ˜ g ( ω ) = (cid:90) ∞−∞ g ( t ) e − i ωt d t. (3)For a single pulse, N = 1 , the Fourier transform ˜ E pu ( ω ) of the pump field E pu ( t ) is obtained by the Fourier trans-form ˜ E , pu ( ω − ω c ) of the envelope function shifted bythe carrier frequency ω c . However, for an infinite trainof pulses, N → ∞ , ˜ E pu ( ω ) consists of a set of equallyspaced lines centered on the frequencies ω m = ω o + mω r , m ∈ Z (4)with the repetition frequency and offset frequency ω r = 2 πT p , ω o = ∆ φT p , (5)respectively [30–32]. The strength of the lines is modu-lated by ˜ E , pu ( ω − ω c ) . This is shown in Fig. 2(b) andfurther discussed in Appendix A. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● (a)(b) Figure 2. (a) Time-dependent pump field (light blue) withenvelope function (dark blue). (b) Spectrum of a pump-pulsetrain for N → ∞ . In addition to a train of pump pulses, a weak probepulse is used, E pr ( t ) = E , pr ( t − τ ) cos[ ω c ( t − τ ) + φ , pr ] ˆ e z , (6)whose absorption spectrum is measured upon interactionwith the atomic sample, as shown in Fig. 1. The probepulse is assumed to be linearly polarized, with envelope E , pr ( t ) , CEP φ , pr , and is centered on τ . This representsthe time delay between E pr ( t ) and the initial pulse in thetrain of pulses E pu ( t ) . A negative time delay τ < mod-els a probe–pump experimental setup in which the probepulse precedes the train of pump pulses. In contrast, pos-itive time delays can either model a pump–probe–pumpsetup, in which the probe pulse is preceded and followedby pump pulses; or a pump–probe setup, where the probepulse excites the system after the total (and finite) num-ber N of pump pulses.The pulses excite the V -type three level system shownin Fig. 3, with electric-dipole-( E -)allowed transitions | (cid:105) → | k (cid:105) , k ∈ { , } . This is here used to model s S / → p P / and s S / → p P / transi-tions in Rb atoms between the ground state and thefine-structure-split excited states [43, 44], with transi-tion energies ω k = ω k − ω and dipole-moment matrixelements D k = D k ˆ e z . For the Rb atomic implemen-tation, ω = 1 .
56 eV and ω = 1 .
59 eV , whereas D k are well approximated by their nonrelativistic values [45], D = D √ , i.e., D = 1 .
75 a . u . and D = 2 .
47 a . u . [43, 44]. The left side of Fig. 3 introduces the effectivedetuning δ r = ω − (cid:22) ω ω r (cid:23) ω r , (7) S p e c t r u m o f t h e p u m p p u l s e t r a i n ~~ ~~ A t o m i c l e v e l s t r u c t u r e ~~ ~~~~ Figure 3. V -type three-level scheme (blue), with transitionenergies ω and ω , used to model Rb atoms interacting withbroadband laser pulses. The red lines on the right display thespectrum of the pump field in the case of a train of δ pulsesequally separated by the repetition period T p = 2 π/ω r . Theblack lines on the left introduce the ω r -dependent effectivedetuning δ r . where (cid:98) x (cid:99) denotes the floor function. Notice that (cid:98) ω /ω r (cid:99) ω r is the greatest frequency consisting of a mul-tiple of ω r which, at the same time, is smaller than orequal to ω . For comparison, the central frequencies ofthe lines in the spectrum of the pump-pulse train areshown on the right side of Fig. 3.The time evolution of the state of the system | ψ ( t ) (cid:105) = (cid:80) i =1 c i ( t ) | i (cid:105) is determined by the Schrödinger equation i d | ψ ( t ) (cid:105) d t = ˆ H ( t ) | ψ ( t ) (cid:105) , (8)with the total Hamiltonian ˆ H ( t ) = ˆ H + ˆ H totint ( t ) con-sisting of the unperturbed atomic Hamiltonian ˆ H = (cid:80) i =1 ω i | i (cid:105)(cid:104) i | and the total E light-matter interactionHamiltonian ˆ H totint ( t ) in the rotating-wave approximation[46–48]. For a single pulse as described by Eq. (1), theinteraction Hamiltonian reads ˆ H int ( t, t c , φ ) = − (cid:88) k =2 Ω k ( t − t c ) e i ω c ( t − t c ) e i φ | (cid:105)(cid:104) k | +H . c ., (9)where we have introduced the time-dependent Rabi fre-quencies Ω k ( t ) = D k E ( t ) . (10)The total interaction Hamiltonian ˆ H totint ( t ) = ˆ H pr ( t ) + N − (cid:88) n =0 ˆ H pu ,n ( t ) , (11)including the action of the probe and pump pulses, ˆ H pr ( t ) = − (cid:88) k =2 Ω pr ,k ( t − τ ) e i ω c ( t − τ ) e i φ , pr | (cid:105)(cid:104) k | + H . c ., (12) ˆ H pu ,n ( t ) = − (cid:88) k =2 Ω pu ,k ( t − nT p ) e i ω c ( t − nT p ) e i φ , pu e i n ∆ φ | (cid:105)(cid:104) k | + H . c ., (13)respectively, can then be defined in terms of the probeand pump Rabi frequencies Ω pr / pu ,k ( t ) = D k E , pr / pu ( t ) .In the following, we will analytically study the case ofultrashort pulses approximated by Dirac δ peaks, Ω k ( t ) = ϑ k δ ( t ) , (14)with pulse areas ϑ k = (cid:90) Ω k ( t ) d t. (15)For the three-level system of interest, we introduce theeffective pulse area ϑ = (cid:113) ϑ + ϑ (16)and the angle α = arctan( ϑ /ϑ ) , (17) such that ϑ = ϑ sin( α ) , ϑ = ϑ cos( α ) . (18)The time evolution of | ψ ( t ) (cid:105) = ˆ U ( t, t ) | ψ ( t ) (cid:105) froman initial state at time t , t > t , can be expressedas the action of an evolution operator ˆ U ( t, t ) = (cid:80) i,j =1 U ij ( t, t ) | i (cid:105)(cid:104) j | , of elements U ij ( t, t ) , which is asolution of i d ˆ U ( t, t )d t = ˆ H ( t ) ˆ U ( t, t ) , ˆ U ( t , t ) = ˆ I, (19)where ˆ I is the identity matrix. In the case of interest,consisting of pump and probe δ pulses, the evolution ofthe system can be split into intervals of free evolution,characterized by the operator ˆ V ( t ) = e − i ˆ H t = diag(1 , e − i ω t , e − i ω t ) , (20)separated by the instantaneous action of a δ pulse [23].The pump- and probe-pulse interaction operators de-scribing this instantaneous action can be obtained afterrewriting the interaction Hamiltonian in Eq. (9) in thecase of a single δ pulse as ˆ H int ( t, t c , φ ) = ˆ F † ( φ ) ˆ B ( ϑ, α ) ˆ F ( φ ) δ ( t − t c ) , (21)in terms of the unitary matrix ˆ F ( φ ) = diag(1 , e i φ , e i φ ) (22)accounting for the phase of the pulse, and the operator ˆ B ( ϑ, α ) = − ϑ α ) cos( α )sin( α ) 0 0cos( α ) 0 0 (23)including the dependence upon the pulse strength. Anexplicit solution of Eq. (19) in this single-pulse case allowsone to introduce the probe- and pump-pulse interactionoperators ˆ U pr ( ϑ pr , α, φ , pr ) = ˆ F † ( φ , pr ) ˆ A ( ϑ pr , α ) ˆ F ( φ , pr ) , (24) ˆ U pu ,n ( ϑ pu , α, φ , pu , ∆ φ ) = ˆ F † ( φ , pu ) (cid:2) ˆ F † (∆ φ ) (cid:3) n ˆ A ( ϑ pu , α ) (cid:2) ˆ F (∆ φ ) (cid:3) n ˆ F ( φ , pu ) , (25)respectively, both modeling the instantaneous action of the associated δ pulse and defined in terms of [49] ˆ A ( ϑ, α ) = e − i ˆ B ( ϑ,α ) = cos( ϑ/
2) i sin( α ) sin( ϑ/
2) i cos( α ) sin( ϑ/ α ) sin( ϑ/
2) sin ( α ) cos( ϑ/
2) + cos ( α ) sin( α ) cos( α ) [cos( ϑ/ − α ) sin( ϑ/
2) sin( α ) cos( α ) [cos( ϑ/ −
1] sin ( α ) + cos ( α ) cos( ϑ/ . (26)As described in the following, for the calculation ofthe absorption spectrum it is convenient to introduce the associated density matrix ˆ ρ ( t ) = (cid:80) i,j =1 ρ ij | i (cid:105)(cid:104) j | ,of elements ρ ij and given by ˆ ρ ( t ) = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | = ˆ U ( t, t ) ρ ( t ) ˆ U † ( t, t ) in the case of a pure state. Bydefining the nine-dimensional column vector (cid:126)R =( ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ ) T , i.e., the row-ordered vectorization of the density matrix, with ele-ments R i ( t ) , i ∈ { , . . . , } , its time evolution (cid:126)R ( t ) =ˆ U ( t, t ) (cid:126)R ( t ) can be written in terms of the × matrix ˆ U ( t, t ) = ˆ U ( t, t ) ⊗ ˆ U ∗ ( t, t ) , (27)where ˆ U ∗ is the complex conjugate of ˆ U and where ⊗ denotes the Kronecker product [50] ˆ U = U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ U ˆ U ∗ . Due to the mixed-product property, ( ˆ U ˆ U ) ⊗ ( ˆ U ∗ ˆ U ∗ ) =( ˆ U ⊗ ˆ U ∗ )( ˆ U ⊗ ˆ U ∗ ) , whenever the evolution operator ˆ U = ˆ U ˆ U is equal to the product of two terms ˆ U and ˆ U ,then ˆ U = ˆ U ⊗ ˆ U ∗ = ˆ U ˆ U is also equal to the product ofthe associated matrices ˆ U = ˆ U ⊗ ˆ U ∗ and ˆ U = ˆ U ⊗ ˆ U ∗ . B. Transient-absorption spectrum
Experimental optical-density absorption spectra canbe simulated via calculation of the single-particle dipoleresponse of the system [2] S (¯ ω, τ ) ∝ − ω Im (cid:20) (cid:88) k =2 D ∗ k (cid:90) ∞−∞ ρ k ( t, τ ) e − i¯ ω ( t − τ ) d t (cid:21) , (28)with the Fourier transform centered on the arrival time ofthe measured probe pulse. The above expression is validfor low densities and small medium lengths, where the ef-fect of the propagation of the pulses through the mediumcan be neglected. The transient-absorption spectrumprovides access to the dipole response of the system viathe coherences ρ k ( t, τ ) , i.e., off-diagonal terms of thedensity matrix.In order to effectively account for broadening effects inthe experiment, which determine the finite linewidth ofthe absorption lines, the Fourier transform in Eq. (28)will be evaluated at the complex frequency ¯ ω = ω − i γ/ .Here, ω is the real frequency of the photons detectedby the spectrometer, while γ accounts for the experi-mental linewidth. Evaluating Eq. (28) at this complexfrequency is equivalent to calculating the Fourier trans-form of ρ k ( t, τ ) e − γ ( t − τ ) / , i.e., of an effectively decayingdipole. This is also equivalent to convolving S ( ω, τ ) witha Lorentzian function of width γ/ . It is also importantto stress that the poles of S (¯ ω, τ ) lie on the real axis, aswe will show in Sec. II C and Appendix E. If we evalu-ated S (¯ ω, τ ) for ¯ ω = ω ∈ R , the spectrum would divergeat the frequencies corresponding to these poles. By eval-uating the spectra at the complex frequency ¯ ω = ω − i γ/ ,however, these divergences reduce to peaks of width γ/ . The poles of S (¯ ω, τ ) are then associated with the centralfrequencies of the peaks appearing in the spectrum.In the following, we will set γ (cid:28) Γ k , i.e., much smallerthan the spontaneous decay rates Γ k of the excited statesto the ground state. As a result, during the time scales ofinterest as defined by the exponential function e − γ ( t − τ ) / ,spontaneous decay can be safely neglected in the equa-tions of motion, thus justifying the pure-state approachused to derive the equations of motion of ˆ ρ ( t, τ ) . At thesame time, we will set γτ (cid:28) , γT p (cid:28) , such that thedipole response of the system can be controlled by thesequence of pump pulses within its decay.Alternatively, one could have effectively includedbroadening effects via an atomic Hamiltonian ˆ H withcomplex eigenenergies ω k − i γ k / , i.e., by including theeffective decay of the coherences ρ k ( t, τ ) directly in theequations of motion. However, for the parameters cho-sen, and in particular when γτ (cid:28) , we tested that thereis no appreciable difference between results obtained withthese two alternative approaches. Using Eq. (28) with acomplex frequency ¯ ω will allow us to significantly simplifythe presentation of the analytical calculations in Sec. II Cand Appendix E.We finally notice that in the following we will calculateand show spectra S (¯ ω, τ ) assuming the noncollinear ge-ometry depicted in Fig. 1. In transient-absorption spec-troscopy experiments, this geometry is employed to mea-sure the spectrum of the probe pulse independent of thepump pulse, and thus separate the contributions frompulses with the same laser frequency. In this geometry,however, fast oscillations of the absorption spectrum as afunction of time delay are effectively averaged out in anexperiment [21, 24]. We will account for this by identi-fying and selectively removing fast time-delay-dependentoscillating terms in the resulting single-particle absorp-tion spectra S (¯ ω, τ ) : S (¯ ω, τ ) = (cid:104) S (¯ ω, τ ) (cid:105) τ , (29)where (cid:104)· · · (cid:105) τ denotes averaging over τ . C. Dynamics of the system and associatedspectrum
In the following, we will obtain analytical expressionsfor the time evolution of the state (cid:126)R ( t ) , when it is excitedby a probe pulse centered on τ and by a sequence of N pump pulses centered on t n = nT p , n ∈ { , . . . , N − } ,up to the limit of N → ∞ . The initial state of the systemis (cid:126)R = (1 , , T ⊗ (1 , , T = (1 , , , , , , , , T , (30)i.e., the system is initially in its ground state. Theseanalytical expressions will then be used to calculate theassociated transient-absorption spectrum via Eq. (28),after eliminating fast oscillations in τ . For this purpose,we introduce the × operators ˆ A ( ϑ, α ) = ˆ A ( ϑ, α ) ⊗ ˆ A ∗ ( ϑ, α ) (31)and ˆ F ( φ ) = ˆ F ( φ ) ⊗ ˆ F ∗ ( φ )= diag(1 , e − i φ , e − i φ , e i φ , , , e i φ , , . (32)The instantaneous interaction with the pump and probe δ pulses can then be modeled by the interaction operators ˆ U pr = ˆ U pr ⊗ ˆ U ∗ pr = ˆ F † , pr ˆ A pr ˆ F , pr (33)and ˆ U pu ,n = ˆ U pu ,n ⊗ ˆ U ∗ pu ,n = ˆ F † , pu ( ˆ F † ∆ ) n ˆ A pu ( ˆ F ∆ ) n ˆ F , pu , (34)where we have simplified the notation by introducing ˆ F , pr / pu = ˆ F ( φ , pr / pu ) , (35) ˆ F ∆ = ˆ F (∆ φ ) , (36) ˆ A pr / pu = ˆ A ( ϑ pr / pu , α ) . (37)The free evolution of the system between two consecutivepulses is modeled by the × free-evolution operator ˆ V ( t ) = ˆ V ( t ) ⊗ ˆ V ∗ ( t )= diag(1 , e i ω t , e i ω t , e − i ω t , , e i ω t , e − i ω t , e − i ω t , (38) in order to describe the free-evolution in the period T p between two pump pulses we define ˆ V p = ˆ V ( T p ) . (39)Depending on the position of the probe pulse, three ex-perimental setups can be distinguished. When τ < , theprobe pulse completely precedes the sequence of pumppulses, while it fully follows the train of pump pulseswhen τ > ( N − T p . The general structure of the ab-sorption spectrum for these two experimental setups waspreviously investigated for a single pump pulse [21, 23],also in the presence of an intense probe pulse [24]. Inthe following, we will show how the formulas presentedtherein can be modified in order to account for a sequenceof pump pulses, and how this is imprinted in the shape ofthe absorption spectra for increasing values of N . For thecase of a train of pump pulses, a new pump–probe–pumpsetup also exists for < τ < ( N − T p , i.e., wheneverthe probe pulse lies in between two pump pulses. Wewill show that the structure of the spectrum in this gen-eral case shares several elements with the pump–probeand probe–pump setups mentioned previously. For allthe above cases, we will show that the pulse-to-pulsephase shift ∆ φ provides an important additional degreeof freedom to shape the absorption spectrum and gainunderstanding of the evolution of the system in the pres-ence of a periodic external excitation from absorption lineshapes.
1. Probe–pump setup ( τ < For negative time delays, when the probe pulse pre-cedes the train of pump pulses, the evolution of the sys-tem (cid:126)R ( t ) from the initial state (cid:126)R [Eq. (30)] in the pres-ence of a finite number N of pump pulses reads: (cid:126)R ( t ) = (cid:126)R , t < τ, ˆ V ( t − τ ) ˆ U pr (cid:126)R , τ < t < , ˆ V ( t − lT p ) ˆ F † , pu ( ˆ F † ∆ ) l ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) l ˆ F , pu ˆ V ( − τ ) ˆ U pr (cid:126)R , lT p < t < ( l + 1) T p , ˆ V ( t − ( N − T p ) ˆ F † , pu ( ˆ F † ∆ ) N − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − ˆ F , pu ˆ V ( − τ ) ˆ U pr (cid:126)R , t > ( N − T p , (40)where the third line describes the dynamics of the systemin the interval [ t l , t l +1 ] , l ∈ { , . . . , N − } , in betweenthe l th and the ( l +1) th pulse. In Appendix B, we presentthe evolution of a system between a general a th and ageneral b th pulse, with ≤ a ≤ b ≤ N − . The third(fourth) line in Eq. (40) are thus obtained from Eq. (B1)with a = 0 and b = l ( b = N − ). The last two lines areaffected by the number N of pump pulses. The third lineis only present for N > , since it describes the dynamicsof the system in between two pump pulses. The fourthline is only present for a finite number of pulses, sinceit describes the free evolution of the system following interaction with the last pulse centered on t N − .The two density-matrix elements ρ k ( t ) = R k ( t ) = (cid:126)v k (cid:126)R ( t ) , k ∈ { , } of interest for the calculation of theabsorption spectrum are then obtained by multiplyingthe row-vector (cid:126)v k =(1 , , ⊗ (0 , δ k , δ k )=(0 , δ k , δ k , , , , , , (41)with (cid:126)R ( t ) , as shown in Eq. (C1). The integral in Eq. (28)can then be performed in each one of the intervals iden-tified in Eq. (40), leading to S N (¯ ω, τ ) ∝ − ω Im (cid:26) (cid:88) k =2 D ∗ k i(¯ ω − ω k ) (cid:126)v k (cid:20)(cid:0) − e i(¯ ω − ω k ) τ (cid:1) + (cid:0) − e − i(¯ ω − ω k ) T p (cid:1) ˆ A pu (cid:104) ˆ I − (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − (cid:105) (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − e i¯ ωτ ˆ G ( τ )+ ˆ A pu (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − e i¯ ωτ ˆ G ( τ ) (cid:21) ˆ U pr (cid:126)R (cid:27) , (42)where we have introduced ˆ I = ˆ I ⊗ ˆ I and used the fact that N − (cid:88) l =0 (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) l = (cid:104) ˆ I − (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − (cid:105) (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − , (43)where ¯ ω = ω − i γ/ . The subscript N in S N (¯ ω, τ ) and S N (¯ ω, τ ) indicates their dependence upon the number ofpulses. In S N (¯ ω, τ ) we have also averaged over fast os-cillations as a function of τ , i.e., removed fast time-delayoscillating terms e ± i ω k τ appearing in S N (¯ ω, τ ) for thefrequencies of interest ¯ ω ≈ ω k − i γ/ . This is accountedfor by the operator e i¯ ωτ ˆ G ( τ ) . = (cid:104) e i¯ ωτ ˆ V ( − τ ) (cid:105) τ = e i¯ ωτ diag(0 , e − i ω τ , e − i ω τ , , , , , , . (44)We notice that the resulting spectrum is indepen-dent of the initial-pump-pulse CEP φ , pu due to e i φ , pu ˆ F , pu ˆ G ( τ ) = ˆ G ( τ ) . By introducing the operator ˆ D N (¯ ω ) = ˆ A pu (cid:110) ˆ I − (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − e − i(¯ ω − ω k ) T p (cid:104) ˆ I − (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − (cid:105)(cid:111) × (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − , (45)the spectrum can be written as S N (¯ ω, τ ) ∝ − ω Im (cid:26) (cid:88) k =2 D ∗ k i(¯ ω − ω k ) (cid:126)v k × (cid:20)(cid:0) − e i(¯ ω − ω k ) τ (cid:1) + ˆ D N (¯ ω ) e i¯ ωτ ˆ G ( τ ) (cid:21) ˆ U pr (cid:126)R (cid:27) , (46)with the second term in the sum highlighting how the sequence of N pump pulses acts on the system and shapesthe resulting absorption spectrum. For a single pumppulse, ˆ D N (¯ ω ) reduces to ˆ D (¯ ω ) = ˆ A pu , whereas for aninfinite train of pump pulses it reads ˆ D ∞ (¯ ω ) = (cid:0) − e − i(¯ ω − ω k ) T p (cid:1) ˆ A pu × (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − . (47)For large numbers of pump pulses, and particularly in thelimit N → ∞ , the frequency-dependent operator ˆ D N (¯ ω ) causes the appearance of LISs in the spectrum. Theseadditional peaks are due to the presence of the inverseoperator (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − and are there-fore centered on frequencies which are determined by theeigenvalues of ˆ F ∆ ˆ V p ˆ A pu . The appearance of these ad-ditional lines is the main signature of the pump-pulse-induced periodic excitation of the system in the probe–pump setup: in this case, the initial dipole generated bythe probe pulse is subsequently modified by the periodicsequence of pump pulses, and these strong-field periodicdynamics are imprinted into the spectrum via the ap-pearance of LISs.
2. Pump–probe setup [ τ > ( N − T p ] For a finite number N of pump pulses, a pump–probesetup is possible, in which the probe pulse encounters theatomic system at τ > ( N − T p , following the completepump-pulse sequence. In this case, the evolution of thesystem is given by (cid:126)R ( t ) = (cid:126)R , t < , ˆ V ( t − lT p ) ˆ F † , pu ( ˆ F † ∆ ) l ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) l ˆ F , pu (cid:126)R , lT p < t < ( l + 1) T p , ˆ V ( t − ( N − T p ) ˆ F † , pu ( ˆ F † ∆ ) N − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − ˆ F , pu (cid:126)R , ( N − T p < t < τ, ˆ V ( t − τ ) ˆ U pr ˆ V ( τ − ( N − T p ) ˆ F † , pu ( ˆ F † ∆ ) N − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − ˆ F , pu (cid:126)R , t > τ, (48)where the second line, describing the dynamics of the sys-tem in the interval [ t l , t l +1 ] , l ∈ { , . . . , N − } , is present only if N > . We first observe that ˆ F , pu (cid:126)R = (cid:126)R forthe initial state in Eq. (30). By further multiplying therow-vector (cid:126)v k with (cid:126)R ( t ) , as shown in Eq. (C2), the in-tegral in Eq. (28) can then be performed in each one ofthe intervals identified in Eq. (48). Integrals in [ t l , t l +1 ] and in [ t N − , τ ] feature fast time-delay-dependent oscil-lations for ¯ ω ≈ ω k − i γ/ , due to the fast oscillatingfactor e i¯ ωτ , and therefore do not contribute to S N (¯ ω, τ ) .The spectrum thus results from the dynamics of the sys-tem only for t > τ , with the periodic sequence of pumppulses determining the state of the system encounteredby the probe pulse at τ − . This is a typical feature of thespectra in a pump–probe setup for a noncollinear geom-etry, which was already recognized for the single-pump-pulse case [23, 24]. In contrast to the probe–pump case,where the periodic excitation of the system following theprobe pulse causes the appearance of LISs, here the trainof pulses preceding the probe pulse only determines thestate in which the system is prepared. The spectrum thusreads S N (¯ ω, τ ) ∝ − ω Im (cid:20) (cid:88) k =2 D ∗ k i(¯ ω − ω k ) (cid:126)v k ˆ U pr × ˆ W ( τ − ( N − T p ) ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − (cid:126)R (cid:21) , (49)where we have removed the fast time-delay-dependentoscillations by introducing ˆ W ( τ ) . = (cid:104) ˆ V ( τ ) (cid:105) τ = diag(1 , , , , , e i ω t , , e − i ω t , (50)and where we have taken advantage of ˆ W ( τ − ( N − T p ) ˆ F † , pu ( ˆ F † ∆ ) N − = ˆ W ( τ − ( N − T p ) , i.e., the resulting spectrum is also in this case indepen-dent of the initial-pump-pulse CEP φ , pu .The pump–probe setup described above is present onlyif the pump field consists of a finite number of pulses. Inthis case, the operator ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − in Eq. (49)contains all the information on the action of the trainof pump pulses which is encoded in the absorption spec-trum. This operator clearly reduces to the single-pump-pulse operator ˆ A pu for N = 1 . We stress again that theabove formulas can be used only if γτ (cid:28) , i.e., for timedelays that allow one to neglect the amplitude change ofthe dipole response due to the decay rate γ .
3. Pump–probe–pump setup [0 < τ < ( N − T p ] The final setup we are going to consider, present onlyfor
N > , consists of a probe pulse exciting the system inbetween two pump pulses in E pu ( t ) . This pump–probe–pump setup shares features with both cases discussedabove: as in the pump–probe case, also here the actionof the pump pulses preceding the probe pulse is encodedin the state of the system encountered by the probe pulse;in analogy with the probe–pump term, the pump pulsesfollowing the probe pulse actively modify the dipole re-sponse of the system and shape the absorption spectruminto additional LISs.In order to highlight these properties, we first considerthe dynamics in a pump–probe–pump system, which canbe divided into different intervals as follows: (cid:126)R ( t ) = (cid:126)R , t < , ˆ V ( t − pT p ) ˆ F † , pu ( ˆ F † ∆ ) p ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) p ˆ F , pu (cid:126)R , pT p < t < ( p + 1) T p , ˆ V ( t − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − ˆ F , pu (cid:126)R , ( M τ − T p < t < τ, ˆ V ( t − τ ) ˆ U pr ˆ V ( τ − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − ˆ F , pu (cid:126)R , τ < t < M τ T p , ˆ V ( t − qT p ) ˆ F † , pu ( ˆ F † ∆ ) q ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) q − M τ ( ˆ F ∆ ) M τ ˆ F , pu × ˆ V ( M τ T p − τ ) ˆ U pr ˆ V ( τ − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − ˆ F , pu (cid:126)R , (cid:41) qT p < t < ( q + 1) T p , ˆ V ( t − ( N − T p ) ˆ F † , pu ( ˆ F † ∆ ) N − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − M τ − ( ˆ F ∆ ) M τ ˆ F , pu × ˆ V ( M τ T p − τ ) ˆ U pr ˆ V ( τ − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − ˆ F , pu (cid:126)R , (cid:41) t > ( N − T p . (51)In Eq. (51), we have introduced M τ − (cid:98) τ /T p (cid:99) (52)with the floor function (cid:98) x (cid:99) . The first four lines in Eq. (51) are identical to the pump–probe case analyzedpreviously. Here, however, the second line is presentonly if N > and τ > T p , since it describes thedynamics of the system in the interval [ t p , t p +1 ] , with p ∈ { , . . . , M τ − } . The fifth line accounts for the dy-namics of the system in the interval [ t q , t q +1 ] , where nowthe index q ∈ { M τ , . . . , N − } is associated with oneof the pump pulses following the probe pulse, providedthat N > and τ < ( N − T p . The sixth line describesthe free evolution of the system after interaction with thewhole train of pump pulses, present only if N is finite.The fifth (sixth) line has been obtained from Eq. (B1)with a = M τ and b = q ( b = N − ). The dipole responseof the system is provided in Eq. (C3).For the same reasons described for the pump–probesetup, the integral of Eq. (28) in t < τ does not contributeto the absorption spectrum after averaging over fast time- delay-dependent oscillations, while the contribution for t ∈ [ τ, t M τ ] can be obtained by following the same stepsleading to Eq. (49). To account for the terms in thespectrum resulting from the integrals in [ t q , t q +1 ] and for t > t N − , we first introduce the operator e − i¯ ω ( T p − τ (cid:48) ) ˆ Z ( τ (cid:48) ) . = (cid:104) e − i¯ ω ( T p − τ (cid:48) ) ˆ V ( T p − τ (cid:48) ) ˆ U pr ˆ V ( τ (cid:48) ) (cid:105) τ (cid:48) , (53)where τ (cid:48) = τ − ( M τ − T p = τ − (cid:98) τ /T p (cid:99) T p , τ (cid:48) ∈ [0 , T p ] .Due to averaging over fast time-delay-dependent oscilla-tions, several matrix elements of ˆ Z ( τ (cid:48) ) vanish, as shownin Appendix D explicitly. Using Eq. (43), the spectrumfinally reads S N (¯ ω, τ ) ∝ − ω Im (cid:26) (cid:88) k =2 D ∗ k i(¯ ω − ω k ) (cid:126)v k (cid:20)(cid:0) − e − i(¯ ω − ω k )( T p − τ (cid:48) ) (cid:1) ˆ U pr ˆ W ( τ (cid:48) )+ (cid:0) − e − i(¯ ω − ω k ) T p (cid:1) ˆ A pu (cid:104) ˆ I − (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − M τ − (cid:105) (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − e − i¯ ω ( T p − τ (cid:48) ) ˆ Z ( τ (cid:48) ) ˆ F ∆ + ˆ A pu (cid:0) e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) N − M τ − e − i¯ ω ( T p − τ (cid:48) ) ˆ Z ( τ (cid:48) ) ˆ F ∆ (cid:21) ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R (cid:27) , (54)where we have used the fact that e i φ , pu e i M τ ∆ φ ( ˆ F ∆ ) M τ ˆ F , pu ˆ Z ( τ (cid:48) ) ˆ F † , pu ( ˆ F † ∆ ) M τ − = ˆ Z ( τ (cid:48) ) ( ˆ F † ∆ ) − = ˆ Z ( τ (cid:48) ) ˆ F ∆ , such that also in this case the CEP φ , pu does not influ-ence the absorption spectrum in a noncollinear geometry.In analogy to the pump–probe case, the operator ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − describes the state of the sys-tem prepared by the initial sequence of M τ pumppulses preceding the probe pulse. The term in thefirst line of Eq. (54) has then the same structure asthe pump–probe spectrum of Eq. (49), with the factor (cid:0) − e − i(¯ ω − ω k )( T p − τ (cid:48) ) (cid:1) due to the finite duration of theinterval [ τ, t M τ ] . The second and third lines in Eq. (54)clearly show a structure similar to the probe–pump spec-trum of Eq. (42), which becomes even more apparent byusing the operator ˆ D N (¯ ω ) defined in Eq. (45) to writethe pump–probe–pump spectrum as S N (¯ ω, τ ) ∝ − ω Im (cid:26) (cid:88) k =2 D ∗ k i(¯ ω − ω k ) (cid:126)v k × (cid:2)(cid:0) − e − i(¯ ω − ω k )( T p − τ (cid:48) ) (cid:1) ˆ U pr ˆ W ( τ (cid:48) )+ ˆ D N − M τ (¯ ω ) e − i¯ ω ( T p − τ (cid:48) ) ˆ Z ( τ (cid:48) ) ˆ F ∆ (cid:3) × ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R (cid:27) . (55)Similarly to the probe–pump case, also here the periodicexcitation of the system by N − M τ pulses following theprobe pulse shapes the absorption spectrum, causing theappearance of LISs. We stress again that the above for-mulas can be used only if γτ (cid:28) and γT p (cid:28) , i.e., for time delays and repetition periods that allow one toneglect the amplitude change of the dipole response dueto the decay rate γ . III. RESULTS AND DISCUSSION
The formulas obtained in the previous section will beused in the following to characterize the main features ofthe transient-absorption spectra in the presence of a pe-riodic pump excitation for different setups, i.e., differentvalues of the time delay τ . We assume a repetition fre-quency of the train of pulses ω r = ω / , correspondingto a period T p . = 2 π/ω r = 4 π/ω = 280 fs , and pulseareas ϑ ∈ [0 , π ] . We also notice that, for a δ pulse,the spectrum ˜ E , pu ( ω ) is a constant function, so that thespectrum of a train of δ pulses is given by a set of equallyspaced, equally intense lines as shown in Fig. 3. The mod-ulation of the spectrum around ω c displayed in Fig. 2(b)is absent in our case, which explains why the formulas ob-tained in Sec. II are independent of the carrier frequency.The results, however, do depend explicitly on δ r and ω o .If Gaussian pulses with a duration of
30 fs were consid-ered instead of δ pulses, then a pulse area of π wouldcorrespond to a peak intensity of ≈ × W / cm .The spectra are studied in the interval τ ∈ [ − , , assuming an experimental width of γ =0 . − , such that both γτ (cid:28) and γT p (cid:28) hold.For the atomic implementation in Rb, where D k arewell approximated by their nonrelativistic values [45], D = D √ , it follows that α = arctan( √ / . Weassume a weak probe pulse with vanishing CEP, i.e.,0 F r e q u e n c y ( e V ) S ( ω , τ = π / ω ) ( a ) S ( ω , τ ) N = ( b ) ( c ) N = ( d ) ( e ) N = ( f ) - ( arb. u. ) ( g ) - -
500 0 500 1000 1500 2000Time delay ( fs ) N = ∞ ( h ) - - - A b s o r p t i o n ( a r b . u . ) Figure 4. Transient-absorption spectra for different numbers N of pump pulses, for fixed pulse area ϑ = 3 π/ and pulse-to-pulsephase shift ∆ φ = 0 . The number of pump pulses is [(a),(b)] N = 1 , [(c),(d)] N = 4 , [(e),(f)] N = 10 , and [(g),(h)] N → ∞ .The left column [(a),(c),(e),(g)] presents spectral line shapes evaluated at a time delay of τ = 3 π/ω = 210 fs , while thetwo-dimensional spectra on the right column [(b),(d),(f),(h)] are exhibited as a function of frequency ω and time delay τ . ϑ pr (cid:28) and φ , pr = 0 , described by the interaction op-erator ˆ U pr = ϑ pr sin( α ) i ϑ pr cos( α )i ϑ pr sin( α ) 1 0i ϑ pr cos( α ) 0 1 , (56)where we have neglected terms of second or higher orderin ϑ pr . Since there is no ambiguity, in the following andin the Appendixes we drop the subscript in ϑ pu , so that ϑ always refers to the pump-pulse area. A. Appearance of light-induced states for anincreasing number of pump pulses
In Fig. 4, the time-delay-dependent absorption spectraare displayed for fixed values of the pulse area ϑ = 3 π/ and pulse-to-pulse phase shift ∆ φ = 0 , for an increasingnumber N of pump pulses. For a single δ like pump pulse1centered on t = 0 , Fig. 4(b) shows the modification of theabsorption line shapes of a probe pulse centered on t = τ .The main features in this single-pulse case were alreadythoroughly described in Refs. [21, 24]. In particular, thetwo absorption lines, centered on the atomic transitionenergies ω = 1 .
56 eV and ω = 1 .
59 eV , respectively,exhibit oscillations as a function of time delay, with aperiodicity of π/ω = 140 fs determined by the beat-ing frequency ω . At negative time delays, when theevolution of the atomic dipole between the first-arrivingprobe pulse and the subsequent pump pulse influencesthe spectrum, perturbed free-induction-decay sidebandsappear [2], which become more significant for increasingvalues of | τ | [see also the first line in Eq. (42)].Two main features emerge for increasing values of N .Firstly, a pump–probe–pump region appears for positivetime delays, where the periodic excitation due to thepump pulses, at a repetition period of T p = 2 × π/ω =280 fs , can be recognized in the time-delay dependence ofthe absorption spectral lines. Furthermore, the spectrain this positive-time-delay region also present perturbedfree-induction-decay sidebands similar to the negative-time-delay case [see also the first line in Eq. (54)], whichcan be identified in Figs. 4(d) and 4(f) for a finite num-ber of pump pulses. Secondly, for τ < ( N − T p , theperiodic excitation of the atomic dipole, resulting fromthe N − M τ pump pulses which follow the probe pulse,induces the appearance of LISs. This becomes increas-ingly significant for larger values of N , up to the limit ofinfinitely many pulses shown in Fig. 4(h). The onset andclear appearance of these additional lines is highlightedin the left column of Fig. 4, which displays absorptionspectral lines at a fixed value of the time delay.The appearance of LISs is associated with the dynam-ics of the system following the probe pulse and periodi-cally modified by a train of N or N − M τ pump pulses,for a probe–pump and pump–probe–pump setup, respec-tively. The larger the number of pump pulses followingthe probe pulse, the more defined and intense these ad-ditional lines will be. For this reason, at positive timedelays and for a fixed total number N of pump pulses,the additional spectral lines gradually fade out for in-creasing values of M τ , i.e., when one approaches the endof the pump-pulse train. This appears clearly in Fig. 4(f)for large positive values of τ . B. Dependence on laser control parameters forinfinitely many pump pulses
In this section, we explicitly focus on the case of in-finitely many pump pulses, although it is apparent fromthe above discussion that the main spectral features ex-hibited by the spectra for N → ∞ are already presentfor finite, sufficiently large numbers of pulses. We inves-tigate the information encoded in the frequency of theLISs appearing in the spectrum as a function of controlparameters such as the pulse area ϑ and pulse-to-pulse phase shift ∆ φ . As discussed in Sec. II, LISs at N → ∞ are due to the action of the infinite sequence of pumppulses following the probe pulse, reflected by the opera-tor ˆ D ∞ (¯ ω ) in Eq. (47). At the same time, we also inves-tigate the time-delay-dependent features of the spectra,especially for τ > , focusing on the influence of the M τ pump pulses preceding the probe pulse. Mathematicaldetails are presented in the Appendixes E, F, and G.
1. Frequency-dependent features of the light-induced states
In order to gain an intuitive understanding of the originof the LISs displayed in Fig. 4, we can for instance focuson the probe–pump setup ( τ < ) and look at the timeevolution of the atomic dipoles for t > , i.e., followingthe first excitation from the pump-pulse train. Withoutloss of generality, we can then write ρ k ( t, τ ) = ∞ (cid:88) j =0 ρ k ( jT +p ) e − i ω k ( t − jT p ) × { θ ( t − jT p ) − θ [ t − ( j + 1) T p ] } , (57)where ρ k ( jT +p ) is the dipole immediately following theinteraction with the j th pump pulse and where θ ( x ) isthe Heaviside step function. The spectrum in Eq. (28)will then be related to (cid:90) ∞ ρ k ( t, τ ) e − i¯ ω ( t − τ ) d t = e i¯ ωτ e − i(¯ ω − ω k ) T p / × T p sinc (cid:20) (¯ ω − ω k ) T p (cid:21) ∞ (cid:88) j =0 ρ k ( jT +p ) e − i¯ ωjT p . (58)Let us then suppose that the train of pulses, acting onthe system with repetition frequency ω r , will periodicallygenerate the same atomic state with a frequency ν , i.e., ρ k ( jT +p ) = e i νjT p is a periodic function. In such case, ∞ (cid:88) j =0 ρ k ( jT +p ) e − i¯ ωjT p = lim N →∞ N − (cid:88) j =0 e i νjT p e − i(¯ ω − ν ) jT p = lim N →∞ − e − i(¯ ω − ν ) NT p − e − i(¯ ω − ν ) T p (59)has a comb-like shape, with peaks centered on ¯ ω s = ν + sω r , s ∈ Z [see also Eq. (A15)]. Already fromthis discussion, we can expect that the spectrum willconsist of a series of lines, separated by the repetitionfrequency ω r and modulated by sinc[(¯ ω − ω k ) T p / . If ρ k ( jT +p ) contains several frequency components e i ν n jT p at frequencies ν n , then they will appear in the spectrumas groups of lines centered on the associated frequencies ¯ ω ns = ν n + sω r . This is thoroughly discussed in Ap-pendix E.For N → ∞ , Fig. 5 displays the dependence of thecentral frequencies of the absorption lines on the phaseshift ∆ φ for different values of τ and ϑ . Here, in par-ticular, we focus on the behavior around the transition2 F r e q u e n c y ( e V ) τ = - T p / ( a ) τ = T p / ( b ) τ = T p / θ = π / ( c ) ( d ) ( e ) θ = π ( f ) ( g ) ( h ) θ = π / ( i ) π / π π / π ( j ) π / π π / π Pulse - to - pulse phase shift ( rad ) ( k ) π / π π / π θ = π ( l ) - - - A b s o r p t i o n ( a r b . u . ) Figure 5. Transient-absorption spectra for an infinite number of pump pulses as a function of frequency ω and pulse-to-pulsephase shift ∆ φ , for pulse areas [(a)–(c)] ϑ = π/ , [(d)–(f)] ϑ = π , [(g)–(i)] ϑ = 3 π/ , and [(j)–(l)] ϑ = 2 π , and time delays[(a),(d),(g),(j)] τ = − T p / , [(b),(e),(h),(k)] τ = 3 T p / , and [(c),(f),(i),(l)] τ = 7 T p / . The dashed lines are centered at ∆ φ = δ r T p , the dot-dashed lines at ∆ φ = δ r T p − π . The blue boxes in panels (a)–(c) highlight the 5-level structures exhibitedby the spectra. energy ω = 1 .
59 eV . Some general features can berecognized in the first row of Fig. 5 [panels (a)–(c)] for ϑ = π/ . Firstly, we notice that, for values of the phaseshift ∆ φ > π/ , the absorption line present at ω is nowshaped into five LISs, as highlighted in the blue boxes.As introduced in the above discussion, the frequencies ofthese five lines are associated with the frequency com-ponents of the evolution of ρ ( jT +p ) . In particular, andas discussed thoroughly in Appendix E 1, when ω o = δ r ,i.e., at ∆ φ = δ r T p (dashed lines), the five lines are equallyspaced, separated by a frequency gap of ∆ ω = ϑ/ (2 T p ) which here is equal to π/ (4 T p ) = 2 meV . Several five-line structures appear in the spectrum, as expected fromthe above discussion: the structures are separated by therepetition frequency ω r = 2 π/T p = 15 meV , with the s thstructure thus centered on ω = ω + sω r , s ∈ Z . Wenotice that the pulse-to-pulse phase shift and the timedelay both affect the shape of the lines, which turn froma Lorentzian to a Fano-like shape depending on the valueof ∆ φ and τ .As a second general feature of the spectra, we notice that the spacing between the five lines changes with ∆ φ ,with the lines forming groups as shown in Figs. 5(a)–(c). In particular, when ω o = δ r − π/T p , such that ∆ φ = δ r T p − π (dot-dashed line), the lines merge intosingle lines centered on ω = ω or ω = ω ± ω r / , as dis-cussed in Appendix E 2. When decreasing ∆ φ even fur-ther, the lines ungroup again, to newly approach a five-line structure—the results are periodic in ∆ φ mod 2 π .This line merging takes place also for higher values ofthe pulse area, as one can see by comparing Figs. 5(a),5(d), and 5(g) [for the π -area case of Fig.5(j), no merg-ing takes place, as we will discuss afterwards]. In partic-ular, the frequencies at which the lines merge, ω = ω or ω = ω ± ω r / , do not depend on ϑ , as shown inAppendix E 2. Other lines tend to group towards singlelines centered on ω = ω ± ω r , but their intensities de-crease for ∆ φ → δ r T p − π so that no spectral line appearsat ω ± ω r when ∆ φ is exactly equal to δ r T p − π .With the increase in the pulse area, the frequency gap ∆ ω = ϑ/ (2 T p ) between individual lines in each five-linestructure also grows. This can lead to the intersection or3merging of lines belonging to different structures. ϑ = π is the smallest pulse area for which such intersectionstake place: in this case and for ∆ φ = δ r T p − π , thecentral frequency of the top line in the s th structure, ω + sω r + 2∆ ω , and that of the bottom line in the ( s + 1) th structure, ω + ( s + 1) ω r − ω , coincide andare equal to ω + 2 sπ/T p + π/T p .This can be recognized in Figs. 5(d)–(f) for ϑ = π .The behavior of the spectrum and the position of theLISs for ϑ = π are described in detail in Appendix E 3.Firstly, we notice that the position of all absorption linesdepends linearly upon ∆ φ for this value of the pulse area.Furthermore, it is now more difficult than in the previous π/ -area case to identify groups consisting of five lines inthe spectrum, because two lines belonging to differentgroups are here completely merged. It is interesting tosee in Fig. 5(e) how some of the above lines do not appearat all when t < τ < t . This dependence is a directresult of the action of the π -area pump pulses precedingthe arrival of the probe pulse. These M τ pulses preparethe system in the state which is then encountered by theprobe pulse, and which determines the shapes of the linesin the spectrum, as explained in Appendix F. This is afirst example of the dependence of the spectra on timedelay, which will be more clearly visible in Figs. 7 and 8.While intersections of different lines appear only at ∆ φ = δ r T p − π for ϑ ≤ π , lines will intersect also atadditional values of ∆ φ for larger pulse areas. This isexhibited in Figs. 5(g)–(i) for ϑ = 3 π/ . However, theseintersections render it also more difficult to distinguishfive-line structures in the spectrum, although it wouldstill be possible to formally group the lines as in the caseof ϑ = π/ . Finally, when ϑ = 2 π , as in Figs. 5(j)–(l),only three lines can be distinguished, whose positions andshapes do not depend on the pulse-to-pulse phase shift ∆ φ . The lines are centered on ω and ω ± ϑ/ (2 T p ) . For ϑ = 2 π , these frequencies are equal to ω = ω and ω = ω ± ω r / , and thus correspond to the above-mentioned ϑ -independent frequencies at which the spectral lines arecentered when ∆ φ = δ r T p − π . Appendix E 4 presentsthe details of this π -area case.Also the absorption spectral line centered at ω = ω is shaped into several five-line structures when N → ∞ .In order to render this apparent, in Fig. 6 we displaytransient-absorption spectra as a function of frequencyand pulse area, evaluated at the two values of the pulse-to-pulse phase shift ∆ φ which were recognized to be im-portant in the above discussion, and for the same dis-crete values of the time delay τ already used in Fig. 5.The left column [Figs. 6(a)–(c)] displays results evalu-ated at ∆ φ = δ r T p . Here, the expansion of the five-levelstructures as a function of ϑ , with already described in-tersections for values of the pulse area larger than π ,can be clearly recognized (see also Appendix E 1). Theright column [Figs. 6(d)–(f)], with the results evaluatedat ∆ φ = δ r T p − π , shows once more that the position ofthe lines is not influenced by the value of ϑ for this par-ticular choice of the pulse-to-pulse phase shift (see also Appendix E 2).The central frequencies of the LISs appearing in thespectrum are related to the action of the intense pumppulses: this is immediate for ∆ φ = δ r T p , where the spac-ing between the lines in the same five-level structure isgiven by ∆ ω = ϑ/ (2 T p ) and is thus due to the amplitudeand phase action of each single pump pulse. For N = 1 , alight-imposed amplitude and phase change would modifythe dipole decay, and would therefore lead to a changeof the absorption line shapes from Lorentzian to Fano-like. By acting on the system several times, however, therepeated amplitude and phase changes imposed by thepulses lead to the appearance of several LIS structures.Information on the action of the pulses can therefore bedirectly extracted from the central frequency of the LISs,complementing the information which could be obtainedby a detailed analysis of the absorption line shapes.
2. Time-delay-dependent featuresand periodicity of the spectra
In order to focus on the time-delay-dependent fea-tures of the spectrum, especially in the pump–probe–pump region at τ > , in Fig. 7 we display transient-absorption spectra as a function of frequency and timedelay for given values of the pulse-to-pulse phase shift ∆ φ and pulse area ϑ . The left column [Figs. 7(a)–(d)]presents spectra at ∆ φ = δ r T p for increasing values ofthe pulse area ϑ . Several five-level structures are rec-ognizable in Fig. 7(a), separated by the repetition fre-quency ω r . Furthermore, Figs. 7(a)–(c) highlight the in-crease in the frequency spacing between lines belongingto the same structure for growing values of ϑ , with theabove-described intersections and merging for pulse ar-eas ϑ ≥ π . For π -area pulses, as shown in Fig. 7(d)and already discussed for Figs. 5(j)–(l), a lower numberof spectral lines appear.The results displayed in the right column [Figs. 7(e)–(h)] are obtained for ∆ φ = δ r T p − π . In this case, thecentral frequencies of the lines appearing in the spectrumdo not depend on ϑ , and are the same in all four panels.They also coincide with the ∆ φ -independent central fre-quencies of the spectra evaluated at ϑ = 2 π . We noticethat the two spectra in Figs. 7(d) and 7(h), evaluatedat different values of ∆ φ and for ϑ = 2 π , show identi-cal frequency- and time-delay-dependent features: this isa general feature of the spectra for ϑ = 2 π , which areindependent of ∆ φ as we show in Appendix G 4.Figure 7 also allows one to focus on the time-delay-dependent features of the spectrum in the pump–probe–pump region at positive delays. Figures 7(a)–(c) showthat certain lines, otherwise present in the spectrum, aresuppressed for given time-delay intervals. While the po-sition of the lines is, in general, determined by the pe-riodic action of the pump-pulse sequence following theprobe pulse [and in particular by the poles of the oper-ator ˆ D ∞ (¯ ω ) in Eq. (47)], the shape of the spectral lines4 F r e q u e n c y ( e V ) Δϕ = δ r T p ( a ) Δϕ = δ r T p - π τ = - T p / ( d ) ( b ) τ = T p / ( e ) π / π π / π ( c ) π / π π / π τ = T p / ( f ) - - A b s o r p t i o n ( a r b . u . ) Pulse area ( rad ) Figure 6. Transient-absorption spectra for an infinite number of pump pulses as a function of frequency ω and pulse area ϑ ,for pulse-to-pulse phase shifts [(a)–(c)] ∆ φ = δ r T p and [(d)–(f)] ∆ φ = δ r T p − π , and time delays [(a),(d)] τ = − T p / , [(b),(e)] τ = 3 T p / , and [(c),(f)] τ = 7 T p / . is determined by the state encountered by the probepulse, resulting from the action of the sequence of M τ pump pulses which precede it. A change of M τ causesa modification in the resulting prepared state, and forgiven values of ϑ there exists a number of pulses M τ forwhich some of the lines in the spectrum are suppressed.Although this is a general property of the time-delay-dependent spectra displayed here, in Appendix F we ex-plain the disappearance of the spectral lines in the par-ticular case exhibited in Fig. 7(b), i.e., for ϑ = π and foran odd number M τ of pump pulses preceding the probepulse.Figure 7 exhibits the periodic features of the spectrumas a function of time delay for τ > . For instance, for ∆ φ = δ r T p , one can recognize a periodicity of T p at ϑ = π/ and ϑ = 3 π/ [Figs. 7(a) and 7(c), respectively], T p at ϑ = π [Fig. 7(c)], and T p at ϑ = 2 π [Fig. 7(d)].In contrast, all the spectra evaluated at ∆ φ = δ r T p − π [Figs. 7(e)–(h)] have a periodicity of T p , including theparticular case of the spectrum at ϑ = 2 π with a period- icity of T p . This is discussed in detail in Appendixes G 1and G 2.These periodic features are further highlighted inFig. 8, showing time-delay-dependent spectra as a func-tion of the pulse area ϑ for given values of pulse-to-pulsephase shift ∆ φ = δ r T p and ∆ φ = δ r T p − π , and evalu-ated at frequencies equal to the transition energies ω and ω . In previous works of transient-absorption spec-troscopy in the presence of a single intense pump pulse[21], it was shown that the line shapes encode amplitudeand phase information about the action of the pulse onthe atomic system. In particular, the spectra feature,both at positive and negative time delays, oscillationsin τ at the beating frequency ω , whose phases wereshown to be directly related to the intensity-dependentatomic-phase change imposed the pump pulse. In thecase investigated here for δ pulses, however, the phasesof the matrix elements of the operator ˆ A ( ϑ, α ) in Eq. (26)are not affected by the intensity of the pulse, i.e., by thevalue of ϑ —only a change of amplitude is possible, in-5 F r e q u e n c y ( e V ) Δϕ = δ r T p ( a ) Δϕ = δ r T p - π θ = π / ( e ) ( b ) θ = π ( f ) ( c ) θ = π / ( g ) - -
500 0 500 1000 1500 20001.541.561.581.601.62 ( d ) - -
500 0 500 1000 1500 2000 θ = π ( h ) - - A b s o r p t i o n ( a r b . u . ) Time delay ( fs ) Figure 7. Transient-absorption spectra for an infinite number of pump pulses as a function of frequency ω and time delay τ , forpulse-to-pulse phase shifts [(a)–(d)] ∆ φ = δ r T p and [(e)–(h)] ∆ φ = δ r T p − π , and pulse areas [(a),(e)] ϑ = π/ , [(b),(f)] ϑ = π ,[(c),(g)] ϑ = 3 π/ , and [(d),(h)] ϑ = 2 π . cluding a change of sign. As a result, the phase of thetime-delay-dependent oscillations exhibited by the spec-trum at the beating frequency ω is independent of ϑ .This clearly appears in Fig. 8.At positive time delays, the spectra display a modula-tion of their intensity as a function of ϑ . This modulationreflects the action of the M τ pump pulses preceding theprobe pulse, and therefore strongly depends on τ as well.This is further discussed in Appendix G. The propertiesof this modulation can be more precisely investigated forthe two values of ∆ φ used in Fig. 8, as discussed in Ap- pendixes G 1 and G 2 and as shown below.For ∆ φ = δ r T p as in Figs. 8(a) and 8(b), one can showthat the dipoles generated by M τ pulses of area ϑ and M τ pulses of area ϑ are equal if there exists an integer K for which M τ ϑ = M τ ϑ + 2 πK. (60)When this condition is fulfilled and the generated stateis the same, then also the associated spectra coincide.This can be recognized by inspecting the position of theminima in Figs. 8(a) and 8(b) at positive time delays,6 P u l s e a r e a ( r a d ) π / π π / π Δϕ = δ r T p ( a ) Δϕ = δ r T p - π ω = ω ( c ) - -
500 0 500 1000 1500 20000 π / π π / π ( b ) - -
500 0 500 1000 1500 2000 ω = ω ( d ) - - A b s o r p t i o n ( a r b . u . ) Time delay ( fs ) Figure 8. Transient-absorption spectra for an infinite number of pump pulses as a function of pulse area ϑ and time delay τ ,for pulse-to-pulse phase shifts [(a),(b)] ∆ φ = δ r T p and [(c),(d)] ∆ φ = δ r T p − π , and frequencies [(a),(c)] ω = ω and [(b),(d)] ω = ω . which lie on the hyperbolic curves ϑ = (2 K + 1) π/M τ inagreement with Eq. (60). For < ϑ < π , as exhibitedin the figure, there exist exactly M τ possible integers K for which the above condition is satisfied. This explainswhy the number of minima increases with τ and matchesthe associated value of M τ = (cid:98) τ /T p (cid:99) + 1 . Furthermore,by applying Eq. (60) with ϑ = ϑ = ϑ , one obtains thattwo sequences of identically intense pulses prepare thesystem in the same state if ∆ M τ = 2 πK/ϑ , where ∆ M τ and K are both integers. This explains the periodicity ofthe spectra as a function of time delay, which we alreadynoticed in Figs. 7(a)–(d). For a given pulse area ϑ andat positive time delays, the spectra have namely a peri-odicity of X ϑ T p , where X ϑ is the smallest integer whichis also a multiple of π/ϑ . This agrees with the valueswe have already identified while discussing the spectra inFigs. 7(a)–(d) for the pulse areas used therein.In contrast, when ∆ φ = δ r T p − π as in Figs. 8(c) and8(d), the spectra have a periodicity of T p , as alreadyidentified in Figs. 7(e)–(h). Also in this case, this reflectsthe action of the M τ preparatory pump pulses precedingthe probe pulse, and in particular the fact that, for thisvalue of the pulse-to-pulse phase shift, an even number ofpulses acting on the ground state brings the system backto it, independent of the pulse area ϑ . Consequently,any odd number of pulses will prepare the system in thesame excited state. As a result, the ensuing spectra havea periodicity given by T p for any value of the pulse area.By using a train of pump pulses, the evolution of thetransient-absorption line shapes as a function of time de- lay thus exhibits periodic features, with a periodicitywhich can be directly related to the properties of thepump pulses used. The time-delay-dependent featuresof the spectra, as well as the frequency of the LISs, cantherefore be used to access the intensity-dependent actionof each pump pulse on the atomic system. IV. CONCLUSION
In conclusion, we have investigated the dynamics andthe transient-absorption spectrum of a V -type three-levelsystem excited by a train of δ -like pulses and probed bya short pulse at different delays. We have shown that theperiodic modification of the dipole response induces theappearance of LISs in the absorption spectrum of theprobe pulse, in spite of the fact that each δ -like pumppulse is as short as the probe pulse. We have also shownthat the LIS frequencies are directly related to the ac-tion of each single intense pump pulse. Furthermore, wehave shown that the spectrum exhibits periodic featuresas a function of time delay for τ > , which are relatedto the action of the pump pulses preceding the probepulse. In the presence of a periodically pumped system,these frequency- and time-delay-dependent features pro-vide further variables, in addition to the shape of theabsorption lines, which can be experimentally measuredin order to access and reconstruct the quantum dynamicsof a strong-field-excited system.While the dynamics and spectra presented in this pa-7per were calculated assuming a fixed ratio between therepetition frequency ω r = ω / and the beating fre-quency ω , further studies could investigate the depen-dence of the transient-absorption spectra on ω r . Fur-thermore, by considering pump and probe pulses of finiteduration, instead of the δ -like pulses assumed here, onewould expect intensity-dependent phase effects analogousto those already reported in Refs. [21, 24]: understand-ing how these atomic phases are encoded in the spectrumof a periodically pumped system would be an interestingextension of the work presented here.Towards an experimental realization of the schemewith Rb atoms, an atomic-system description couldbe considered beyond the three-level model used here.Control schemes in Rb [51], also with shaped optical-frequency combs [38], have considered a closed-loop four-level model, including the coupling of the two excitedstates | (cid:105) and | (cid:105) to the more highly excited state | (cid:105) =5 d D / . However, this coupling is weaker than that tothe ground state, and these studies explicitly aimed atshaping the pulses in order to optimize population trans-fer to this more highly excited state. This is not the casefor the TAS experiments considered here, and studies ofTAS with Rb atoms for a single pump pulse have alreadyshown that a V -type three-level model well describes thefrequency- and time-delay-dependent features of the ab-sorption spectra for different pump-pulse intensities [21].Finally, one could further study the influence of propaga-tion effects on the resulting transient-absorption spectrabeyond the single-atom response [52], e.g., towards theexperimental investigation of media which are not opti-cally thin due to large densities or medium lengths. ACKNOWLEDGMENTS
The authors acknowledge valuable discussions withChristoph H. Keitel and Thomas Pfeifer.
Appendix A: Spectral features of a train of pumppulses
In order to study the spectral features of the train ofequally spaced pump pulses in Eq. (2), we first introducethe positive-frequency part of the field E (+)pu ( t ) = e i φ , pu N − (cid:88) n =0 E , pu ( t − nT p ) e i ω c ( t − nT p ) e i n ∆ φ , (A1)such that E pu ( t ) = E (+)pu ( t ) + (cid:2) E (+)pu ( t ) (cid:3) ∗ (A2)and ˜ E pu ( ω ) = ˜ E (+)pu ( ω ) + (cid:2) ˜ E (+)pu ( − ω ) (cid:3) ∗ . (A3) By defining the convolution of two functions f ( t ) ∗ g ( t ) = (cid:90) f ( t − t (cid:48) ) g ( t (cid:48) ) d t (cid:48) , (A4)whose Fourier transform is given by (cid:90) ∞−∞ f ( t ) ∗ g ( t ) e − i ωt d t = ˜ f ( ω ) ˜ g ( ω ) , (A5)the positive-frequency part of the pump field can be writ-ten as E (+)pu ( t ) = e i φ , pu (cid:2) E , pu ( t ) e i ω c t (cid:3) ∗ N − (cid:88) n =0 δ ( t − nT p ) e i n ∆ φ , (A6)whose Fourier transform is given by ˜ E (+)pu ( ω ) = e i φ , pu E , pu ( ω − ω c ) N − (cid:88) n =0 e − i( ω − ω o ) nT p = e i φ , pu E , pu ( ω − ω c ) 1 − e − i( ω − ω o ) NT p − e − i( ω − ω o ) T p . (A7)In order to render the peak structure of ˜ E (+)pu ( ω ) moreapparent, one can write E (+)pu ( t ) as E (+)pu ( t ) = e i φ , pu (cid:2) E , pu ( t ) e i ω c t (cid:3) ∗ (cid:34) ∞ (cid:88) n = −∞ δ ( t − nT p ) e i n ∆ φ × { θ ( t − aT p ) − θ [ t − ( N − a ) T p ] } (cid:35) , (A8)with the Heaviside step function θ ( x ) and with < a < .Notice that the field is independent of the explicit value of a . Thereby, the field can be written in terms of an infinitetrain of δ pulses, whose Fourier transform is given by aninfinite comb of δ peaks (cid:90) ∞−∞ ∞ (cid:88) n = −∞ δ ( t − nT p ) e i n ∆ φ e − i ωt d t = ω r ∞ (cid:88) m = −∞ δ ( ω − ω m ) , (A9)where we have used the definitions in Eqs. (4) and (5).By recalling that (cid:90) ∞−∞ f ( t ) g ( t ) e − i ωt d t = 12 π ˜ f ( ω ) ∗ ˜ g ( ω ) , (A10)the Fourier transform of E (+)pu ( t ) is given by ˜ E (+)pu ( ω ) = e i φ , pu E , pu ( ω − ω c ) × (cid:34) ω r π (cid:18) e i ωaT p − e − i ωNT p i ω (cid:19) ∗ ∞ (cid:88) m = −∞ δ ( ω − ω m ) (cid:35) = e i φ , pu E , pu ( ω − ω c ) N ∞ (cid:88) m = −∞ e i( ω − ω m ) aT p × e − i( ω − ω m ) NT p / sinc (cid:20) N T p ω − ω m ) (cid:21) . (A11)8One can therefore recognize that the Fourier transformof a train of N pulses is given by peaks centered at thefrequency ω m = mω r + ω o . The strength of the peaks ismodulated by the Fourier transform ˜ E , pu ( ω − ω c ) of asingle pulse, while the width of each peak is associatedwith the width of sinc[ N T p ( ω − ω m ) / , which is muchsmaller than the separation frequency ω r if N (cid:29) .The Fourier transform ˜ E (+)pu ( ω ) is independent of a .The second line in Eq. (A11) can namely be written as ω r π (cid:18) e i ωaT p − e − i ωNT p i ω (cid:19) ∗ ∞ (cid:88) m = −∞ δ ( ω − ω m )= ω r π ∞ (cid:88) m = −∞ e i( ω − ω m ) aT p − e − i( ω − ω m ) NT p i ( ω − ω m )= e i( ω − ω o ) aT p (1 − e − i( ω − ω o ) NT p ) ω r π ∞ (cid:88) m = −∞ e − i2 πma i ( ω − ω m ) , (A12)due to the fact that − e − i( ω − ω m ) NT p = 1 − e − i( ω − ω o ) NT p (A13)is independent of m . By recognizing that [53] ω r π ∞ (cid:88) m = −∞ e − i2 πma i ( ω − ω m ) = 12 π i ∞ (cid:88) m = −∞ e − i2 πmaω − ω o ω r − m = 12 π i (cid:32) ω − ω o ω r + 2i ∞ (cid:88) m =1 m sin(2 πam ) m − (cid:16) ω − ω o ω r (cid:17) − ω − ω o ω r ∞ (cid:88) m =1 cos(2 πam ) m − (cid:16) ω − ω o ω r (cid:17) (cid:33) = e − i( ω − ω o ) aT p − e − i( ω − ω o ) T p (A14)for < a < , one can conclude that ω r π (cid:18) e i ωaT p − e − i ωNT p i ω (cid:19) ∗ ∞ (cid:88) m = −∞ δ ( ω − ω m )= 1 − e − i( ω − ω o ) NT p − e − i( ω − ω o ) T p , (A15)which is independent of a and coincides with the result in Eq. (A7). Appendix B: Evolution of the system between twogeneric pump pulses a and b The interaction with two or more consecutive pumppulses explicitly depends on their position in the train ofpulses as a result of the phase-dependent term [ ˆ F (∆ φ )] n .We will show this here explicitly, by considering the evo-lution of (cid:126)R ( t ) between t − a and t + b , where a and b are twointegers, ≤ a ≤ b ≤ N − , associated with the a th and b th pump pulses, respectively, and where t − n ( t + n ) denotesthe time t n approached from the left (right), preceding(following) the interaction with the n th pump pulse. Weassume that τ (cid:54)∈ [ t a , t b ] , such that the evolution of thesystem results from the interaction with ( b − a + 1) pumppulses, separated by ( b − a ) intervals of free evolution.The state reached by the system is then given by (cid:126)R ( t + b ) = ˆ U pu ,b ˆ V p ˆ U pu ,b − · · · ˆ V p ˆ U pu ,a (cid:126)R ( t − a )= ˆ F † , pu ( ˆ F † ∆ ) b ˆ A pu × ( ˆ F ∆ ) b ˆ F , pu ˆ V p ˆ F † , pu ( ˆ F † ∆ ) b − ˆ A pu × ( ˆ F ∆ ) b − ˆ F , pu · · · ˆ V p ˆ F † , pu ( ˆ F † ∆ ) a ˆ A pu × ( ˆ F ∆ ) a ˆ F , pu (cid:126)R ( t − a )= ˆ F † , pu ( ˆ F † ∆ ) b ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) b − a × ( ˆ F ∆ ) a ˆ F , pu (cid:126)R ( t − a ) , (B1)where we have used the fact that the diagonal matrices ˆ V ( t ) and ˆ F ( φ ) commute. Appendix C: Evolution of the dipole response ρ k ( t ) The off-diagonal matrix elements ρ k = R k ( t ) = (cid:126)v k (cid:126)R ( t ) used for the calculation of the absorption spec-trum are displayed below for the probe–pump [Eq. (C1)],pump–probe [Eq. (C2)], and pump–probe–pump setup[Eq. (C3)]. R k ( t ) = , t < τ, e i ω k ( t − τ ) (cid:126)v k ˆ U pr (cid:126)R , τ < t < , e i ω k ( t − lT p ) e i φ , pu (cid:126)v k ˆ A pu (e i∆ φ ˆ F ∆ ˆ V p ˆ A pu ) l ˆ F , pu ˆ V ( − τ ) ˆ U pr (cid:126)R , lT p < t < ( l + 1) T p , e i ω k [ t − ( N − T p ] e i φ , pu (cid:126)v k ˆ A pu (e i∆ φ ˆ F ∆ ˆ V p ˆ A pu ) N − ˆ F , pu ˆ V ( − τ ) ˆ U pr (cid:126)R , t > ( N − T p . (C1) R k ( t ) = , t < , e i ω k ( t − lT p ) e i φ , pu e i l ∆ φ (cid:126)v k ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) l (cid:126)R , lT p < t < ( l + 1) T p , e i ω k [ t − ( N − T p ] e i φ , pu e i( N − φ (cid:126)v k ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − (cid:126)R , ( N − T p < t < τ, e i ω k ( t − τ ) (cid:126)v k ˆ U pr ˆ V ( τ − ( N − T p ) ˆ F † , pu ( ˆ F † ∆ ) N − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) N − (cid:126)R , t > τ, (C2)9 R k ( t ) = , t < , e i ω k ( t − pT p ) e i φ , pu (cid:126)v k ˆ A pu (e i∆ φ ˆ F ∆ ˆ V p ˆ A pu ) p (cid:126)R , pT p < t < ( p + 1) T p , e i ω k [ t − ( M τ − T p ] e i φ , pu (cid:126)v k ˆ A pu (e i∆ φ ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R , ( M τ − T p < t < τ, e i ω k ( t − τ ) (cid:126)v k ˆ U pr ˆ V ( τ − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R , τ < t < M τ T p , e i ω k ( t − qT p ) e i φ , pu (cid:126)v k ˆ A pu (e i∆ φ ˆ F ∆ ˆ V p ˆ A pu ) q − M τ e i M τ ∆ φ ( ˆ F ∆ ) M τ ˆ F , pu × ˆ V ( M τ T p − τ ) ˆ U pr ˆ V ( τ − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R , (cid:41) qT p < t < ( q + 1) T p , e i ω k [ t − ( N − T p ] e i φ , pu (cid:126)v k ˆ A pu (e i∆ φ ˆ F ∆ ˆ V p ˆ A pu ) N − M τ − e i M τ ∆ φ ( ˆ F ∆ ) M τ ˆ F , pu × ˆ V ( M τ T p − τ ) ˆ U pr ˆ V ( τ − ( M τ − T p ) ˆ F † , pu ( ˆ F † ∆ ) M τ − ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R , (cid:41) t > ( N − T p , (C3) Appendix D: The operator ˆ Z ( τ ) By averaging over the fast time-delay-dependent oscillations in Eq. (53), several matrix elements of ˆ Z ( τ (cid:48) ) vanish.The × matrix ˆ Z ( τ (cid:48) ) can then be written in terms of a sum of Kronecker products ˆ Z = ˆ Z ⊗ ˆ Z ∗ + ˆ Z ⊗ ˆ Z ∗ + ˆ Z ⊗ ˆ Z ∗ + ˆ Z ⊗ ˆ Z ∗ , (D1)involving the operator ˆ Z ( τ (cid:48) ) = ˆ V ( T p − τ (cid:48) ) ˆ U pr ( ϑ pr , α, φ , pr ) ˆ V ( τ (cid:48) ) . (D2)Written explicitly, the operator reads ˆ Z = Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z , (D3)with nonvanishing elements equal to Z ij ( τ (cid:48) ) = V ii ( T p − τ (cid:48) ) U pr ,ij V jj ( τ (cid:48) ) . Notice that some of the nonvanishingmatrix elements of ˆ Z ( τ (cid:48) ) may be negligibly small com-pared to others for small intensities of the probe pulse,since they are of different orders in ϑ pr , and may thusvanish if we use the probe-pulse interaction operatorgiven in Eq. (56). Appendix E: Central frequencies of thelight-induced states appearing in the spectrum
In order to quantify the central frequencies of the LISsappearing in the spectrum, we show that they are deter-mined by the poles of the operator [ (cid:126)v k ˆ D N (¯ ω )] / [i(¯ ω − ω k )] in Eq. (46). The same can be used to explain thespectra at positive time delays, determined by the term [ (cid:126)v k ˆ D N − M τ (¯ ω )] / [i(¯ ω − ω k )] in Eq. (55). It is important tonotice that the poles are real, so that a divergence in thespectrum would appear if ¯ ω were real. Since we evaluatethe spectrum at the complex frequency ¯ ω = ω − i γ/ ,no divergences appear in the spectrum, as these reduceto peaks with a width of γ/ and centered on the corre-sponding real poles.For N = 1 , [ (cid:126)v k ˆ D N (¯ ω )] / [i(¯ ω − ω k )] reduces to [ (cid:126)v k ˆ A pu ] / [i(¯ ω − ω k )] , whose only poles are ¯ ω = ω k . How-ever, when N → ∞ , this operator reads (cid:126)v k ˆ D ∞ (¯ ω )i(¯ ω − ω k ) = − i T p − i(¯ ω − ω k ) T p2 sinc (cid:20) (¯ ω − ω k ) T p (cid:21) × (cid:126)v k ˆ A pu (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − . (E1)Firstly, due to the presence of sinc[(¯ ω − ω k ) T p / in thefirst line, the pole at ω k present for a finite number ofpump pulses is here removed, unless it appears explicitlyas a pole of the inverse operator in the second line. Wealso notice that this operator has zeros at ¯ ω = ¯ ω zero r = ω k + rω r , r (cid:54) = 0 (E2)for any r ∈ Z other than 0. In order to identify the polesof Eq. (E1), we need to focus on the inverse operator inthe second line. In particular, we notice that ˆ F ∆ ˆ V p ˆ A pu = [ ˆ F ∆ ˆ V p ˆ A pu ] ⊗ [ ˆ F ∆ ˆ V p ˆ A pu ] ∗ , ˆ F ∆ = ˆ F (∆ φ ) , ˆ V p = ˆ V ( T p ) ,and ˆ A pu = ˆ A ( ϑ, α ) . The product ˆ F ∆ ˆ V p = − i( δ r − ω o ) T p
00 0 e − i( δ r − ω o ) T p (E3)is a diagonal matrix describing the change in the atomicphases of the two excited states during one period. Since ˆ F ∆ ˆ V p ˆ A pu is a unitary operator, its eigenvalues e i λ j , j ∈{ , , } , lie on the unit circle. After introducing thephases β = ( δ r − ω o ) T p (E4)and ε = arccos (cid:20) cos (cid:18) ϑ (cid:19) cos (cid:18) β (cid:19)(cid:21) , (E5)the eigenvalues e i λ j and associated eigenvectors (cid:126)P j canbe calculated exactly as e i λ = e − i β , e i λ = e − i β/ ε , e i λ = e − i β/ − i ε , (E6)and (cid:126)P = − cos( α )sin( α ) ,(cid:126)P = sin( ϑ/ (cid:113) sin ( ϑ/ | cos( ϑ/ − e i β/ − i ε | i cos( ϑ/ − e − i β/ ε (cid:113) sin ( ϑ/ | cos( ϑ/ − e i β/ − i ε | sin( α )i cos( ϑ/ − e − i β/ ε (cid:113) sin ( ϑ/ | cos( ϑ/ − e i β/ − i ε | cos( α ) ,(cid:126)P = i cos( ϑ/ − e i β/ − i ε (cid:113) sin ( ϑ/ | cos( ϑ/ − e i β/ − i ε | sin( ϑ/ (cid:113) sin ( ϑ/ | cos( ϑ/ − e i β/ − i ε | sin( α ) sin( ϑ/ (cid:113) sin ( ϑ/ | cos( ϑ/ − e i β/ − i ε | cos( α ) . (E7)By introducing the diagonal matrix ˆ Λ =diag(e i λ , e i λ , e i λ ) , and the matrix ˆ P = ( (cid:126)P , (cid:126)P , (cid:126)P ) ,whose j th column is the eigenvectors (cid:126)P j of ˆ F ∆ ˆ V p ˆ A pu ,we obtain that ˆ F ∆ ˆ V p ˆ A pu = ˆ P ˆ Λ ˆ P − . (E8)Notice that the eigenvectors in Eq. (E7) have been de-termined such that ˆ P − = ˆ P † . As a result, the inverseoperator in the second line in Eq. (E1) reduces to ( ˆ P ⊗ ˆ P ∗ ) (cid:2) ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ ) (cid:3) − ( ˆ P ⊗ ˆ P ∗ ) − , where [ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ )] − is a diagonal matrix ofelements (ˆ I − e − i(¯ ωT p − ∆ φ ) e i( λ j − λ j (cid:48) ) )) − , j, j (cid:48) ∈ { , , } , with poles at ¯ ω = ω o +( λ j − λ j (cid:48) ) /T p + s (cid:48) ω r , for any s (cid:48) ∈ Z .However, from Eq. (E8), we also notice that ˆ A pu ˆ P = ( ˆ F ∆ ˆ V p ) − ˆ P ˆ Λ, (E9)such that (cid:126)v k ˆ A pu ( ˆ P ⊗ ˆ P ∗ )= [(1 , ,
0) ˆ A pu ˆ P ] ⊗ [(0 , δ k , δ k ) ˆ A ∗ pu ˆ P ∗ ]= (cid:88) k (cid:48) ,k (cid:48)(cid:48) =2 3 (cid:88) j (cid:48) =1 δ kk (cid:48)(cid:48) P k (cid:48) P ∗ k (cid:48)(cid:48) j × [(0 , δ k (cid:48) , δ k (cid:48) ) ⊗ ( δ j (cid:48) , δ j (cid:48) , δ j (cid:48) )] [e − i β ˆ Λ ⊗ ˆ Λ ∗ ] , (E10)where we have explicitly used the fact that P = 0 . Asa result, the second line in Eq. (E1) can be written as (cid:126)v k ˆ A pu (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − = (cid:88) k (cid:48) ,k (cid:48)(cid:48) =2 3 (cid:88) j (cid:48) =1 δ kk (cid:48)(cid:48) P k (cid:48) P ∗ k (cid:48)(cid:48) j × [(0 , δ k (cid:48) , δ k (cid:48) ) ⊗ ( δ j (cid:48) , δ j (cid:48) , δ j (cid:48) )] × (cid:2) ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ ) (cid:3) − × [e − i β ˆ Λ ⊗ ˆ Λ ∗ ] ( ˆ P ⊗ ˆ P ∗ ) − . (E11)Due to the term [(0 , δ k (cid:48) , δ k (cid:48) ) ⊗ ( δ j , δ j , δ j )] , notall matrix elements of the diagonal operator [ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ )] − contribute to the spectrum, butonly (ˆ I− e − i(¯ ωT p − ∆ φ ) e i( λ k (cid:48) − λ j (cid:48) ) )) − , with k (cid:48) ∈ { , } and j (cid:48) ∈ { , , } . The only poles determining the peaks inthe spectrum are thus ¯ ω = ¯ ω pole k (cid:48) j (cid:48) s k = ω o + λ k (cid:48) − λ j (cid:48) T p + s (cid:48) ω r , = ω k + λ k (cid:48) − λ j (cid:48) − βT p + s k ω r (E12)for k, k (cid:48) ∈ { , } , j (cid:48) ∈ { , , } , and for any s k ∈ Z (inthe above equality, s k = s (cid:48) − (cid:98) ω k /ω r (cid:99) , with s = s − since ω = 2 ω r ).For a fixed value of s k , this provides the central fre-quencies of the five-line structures appearing in the spec-trum for N → ∞ and discussed in Sec. III: ¯ ω pole k (cid:48) k (cid:48) s k = ω k − βT p + s k ω r , ¯ ω pole k (cid:48) s k = ω k − β T p ± εT p + s k ω r , ¯ ω pole k (cid:48) k (cid:48)(cid:48) s k = ω k − βT p ± εT p + s k ω r , (E13)with k (cid:48) (cid:54) = k (cid:48)(cid:48) . Different values of the index s k are as-sociated with different five-level structures. The term sinc[(¯ ω − ω k ) T p / in the first line of Eq. (E1) modulatesthe intensity of the lines, such that structures in proxim-ity of the transition energies ω k are stronger than the1remaining ones. Furthermore, whenever the frequencies ¯ ω pole k (cid:48) j (cid:48) s k in Eq. (E13) coincide with the frequencies ¯ ω zero r inEq. (E2), the corresponding lines are suppressed in thespectrum.The dependence of the poles upon the pulse-to-pulsephase shift ∆ φ . = ω o T p is in general complex due to thepresence of ε in Eq. (E13). We notice, however, that oneof the spectral peaks is always centered on ¯ ω pole k (cid:48) k (cid:48) s k , inde-pendent of the pulse area ϑ . This central frequency hasa linear dependence on ∆ φ , and the corresponding peakcan be recognized in Fig. 5 for all values of ϑ except π .As we will discuss later, the contribution to the spectrumdue to this line is suppressed for ϑ = 2 π .In the following, we investigate in detail a few particu-lar cases on which we have focused during the discussionof the results in Sec. III. ω o = δ r Whenever the offset frequency ω o is equal to the effec-tive detuning δ r (see also Fig. 3), then ˆ F ∆ ˆ V p = ˆ I , i.e.,the pulse-to-pulse phase shift ∆ φ perfectly balances thedifference in the phase of the two excited states ω T p accumulated during the interval T p in between the twopump pulses. The operator ˆ F ∆ ˆ V p ˆ A pu then reduces tothe symmetric operator ˆ A pu in Eq. (26), β = 0 , ε = ϑ/ ,such that Eq. (E13) gives ˆ Λ = diag(1 , e i ϑ/ , e − i ϑ/ ) , ˆ P = √ − √ − cos( α ) sin( α ) √ α ) √ sin( α ) cos( α ) √ α ) √ , (E14)with ϑ -independent eigenvectors. The central frequen-cies of the spectral lines from Eq. (E13) can therefore bewritten as ¯ ω pole k (cid:48) k (cid:48) s k = ω k + s k ω r , ¯ ω pole k (cid:48) s k = ω k ± ϑ T p + s k ω r , ¯ ω pole k (cid:48) k (cid:48)(cid:48) s k = ω k ± ϑT p + s k ω r . (E15)These frequencies correspond to the central frequenciesof the five-level structures identified in Sec. III for ∆ φ = δ r T p , separated by the frequency gap ∆ ω = ϑ/ (2 T p ) .Notice that ¯ ω pole k (cid:48) k (cid:48) s k in Eq. (E15) corresponds to the posi-tion ¯ ω zero r of the zeros of sinc[(¯ ω − ω k ) T p / except when s k = 0 . Hence, while the five-level structures centered on ω k do show the associated central line, this is suppressedin the additional structures appearing above and below,as apparent in Figs. 5, 6(a)–(c), and 7(a)–(d) in Sec. III. ω o = δ r − π/T p When offset frequency and effective detuning differ by π/T p , it follows that ˆ F ∆ ˆ V p = diag(1 , − , − , β = π ,and ε = π/ . As a result, the Hermitian operator ˆ F ∆ ˆ V p ˆ A pu has eigenvalues and eigenvectors given by ˆ Λ = diag( − , , − , ˆ P = (cid:0) ϑ (cid:1) − i sin (cid:0) ϑ (cid:1) − cos( α ) − i sin (cid:0) ϑ (cid:1) sin( α ) cos (cid:0) ϑ (cid:1) sin( α )sin( α ) − i sin (cid:0) ϑ (cid:1) cos( α ) cos (cid:0) ϑ (cid:1) cos( α ) , (E16)such that all poles in Eq. (E13) are given by ¯ ω pole22 s k = ¯ ω pole33 s k = ¯ ω pole31 s k = ω k − πT p + s k ω r , ¯ ω pole21 s k = ¯ ω pole23 s k = ω k − πT p + s k ω r , ¯ ω pole32 s k = ω k + s k ω r , (E17)which can be summarized as the ϑ -independent frequen-cies ¯ ω pole s = ω k + s ω r , (E18)with s ∈ Z . Notice that the corresponding spectral lineswill be suppressed whenever their central frequencies areequal to the zeros in Eq. (E2). This is apparent in Figs. 5,6(d)–(f), and 7(e)–(h) in Sec. III. π -area pulses When ϑ = π , such that ε = π/ , then the diagonaliza-tion of the operator ˆ F ∆ ˆ V p ˆ A pu leads to ˆ Λ = diag(e − i( δ r − ω o ) T p , i e − i( δ r − ω o ) T p / , − i e − i( δ r − ω o ) T p / ) , ˆ P = √ − √ e i( δ r − ω o ) T p / − cos( α ) e − i( δ r − ω o ) T p / α ) √ α ) √ sin( α ) e − i( δ r − ω o ) T p / α ) √ α ) √ . (E19)Equation (E13) then provides the equations for the cen-tral frequencies of the peaks as a function of both offsetfrequency and effective detuning ¯ ω pole k (cid:48) k (cid:48) s k = ω k + ( ω o − δ r ) + s k ω r , ¯ ω pole k (cid:48) k (cid:48)(cid:48) s k = ω k + (cid:20) ω o − (cid:18) δ r ± πT p (cid:19)(cid:21) + s k ω r , ¯ ω pole k (cid:48) s k = ω k + ω o − δ r ± ω r s k ω r . (E20)Notice that the ± sign in ¯ ω pole k (cid:48) k (cid:48)(cid:48) s k is superfluous, since the + solution associated with the index s k coincides with the − solution for the index ( s k + 1) . This leads to the levelstructures shown in Figs. 5(d)–(f), explaining the linear2dependence of the position of the absorption lines uponthe pulse-to-pulse phase shift. Two parallel lines, givenby ¯ ω pole k (cid:48) k (cid:48) s k and ¯ ω pole k (cid:48) k (cid:48)(cid:48) s k , have the same unitary slope andare spaced by π/T p = ω r / . The remaining two lines,given by ¯ ω pole k (cid:48) s k , are also parallel and separated by ω r / ,but with a slope equal to / . These two couples of linesintersect at ω o = δ r − π/T p , as confirmed in Figs. 5(d)–(f). π -area pulses A pulse with area ϑ = 2 π will not mix the subspaceformed by the ground state with that associated withthe two excited states, since its action is given by theblock-diagonal operator ˆ A pu = − α ) − sin(2 α )0 − sin(2 α ) − cos(2 α ) . (E21)Multiplying it by ˆ F ∆ ˆ V p still preserves its block-diagonalform. In this case, ε = π − β/ , such that eigenvaluesand eigenvectors of ˆ F ∆ ˆ V p ˆ A pu can be written as ˆ Λ = diag(e − i( δ r − ω o ) T p , − e − i( δ r − ω o ) T p , − , ˆ P = − cos( α ) sin( α ) 0sin( α ) cos( α ) 0 . (E22)Owing to the many vanishing elements of ˆ P , not all 5peaks in Eq. (E13) contribute to the spectrum. To seethis, one can refer to Eq. (E11), which for a π -area pulsereads (cid:126)v k ˆ A pu (cid:0) ˆ I − e − i(¯ ωT p − ∆ φ ) ˆ F ∆ ˆ V p ˆ A pu (cid:1) − = (cid:88) k (cid:48)(cid:48) =2 2 (cid:88) j (cid:48) =1 δ kk (cid:48)(cid:48) P ∗ k (cid:48)(cid:48) j [(0 , , ⊗ ( δ j (cid:48) , δ j (cid:48) , × [e − i β ˆ Λ ⊗ ˆ Λ ∗ ] (cid:2) ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ ) (cid:3) − × ( ˆ P ⊗ ˆ P ∗ ) − , (E23)where we have used the fact that P k (cid:48) = δ k (cid:48) and that P k (cid:48)(cid:48) j (cid:48) vanishes for j (cid:48) = 3 . As a result, the only linesappearing in the spectrum are due to the poles of e − i β e i( λ − λ j (cid:48) ) − e − i(¯ ω − ω k ) T p ) e − i β e i( λ − λ j (cid:48) ) = ( − j (cid:48) +1 − e − i(¯ ω − ω k ) T p ) ( − j (cid:48) +1 , (E24)which are given by ¯ ω pole31 s k = ω k + ω r s k ω r , ¯ ω pole32 s k = ω k + ( s k + 1) ω r , (E25) are spaced by ω r / , and independent of ω o , as shownin Figs. 5(j)–(l). They are equal to the ϑ -independentfrequencies in Eq. (E18) for ω o = δ r − π/T p . Alsohere, if these central frequencies are equal to the zerosin Eq. (E2), then the corresponding spectral lines aresuppressed, as shown in Figs. 7(d) and 7(h). We finallynotice that Eq. (E23) is independent of β as a conse-quence of Eqs. (E22) and (E24). This will be used inAppendix G 4. Appendix F: Spectral features in apump–probe–pump setup determined by the pumppulses preceding the probe pulse
The area of the pump pulses preceding the probe pulsedetermines the state in which the system is preparedand encountered by the probe pulse. This influences thefrequency-dependent features of the spectrum in a pump–probe–pump setup, causing, e.g., the disappearance ofsome of the spectral lines identified in Appendix E. Thisis clearly visible in Figs. 6 and 7, displaying the depen-dence of the spectral lines upon pulse area and time delay:one can see that lines otherwise present in the spectrumare suppressed for given values of ϑ and τ .This feature is a result of the state in which the systemis prepared by the M τ pump pulses preceding the probepulse. In order to provide an example for this generalproperty, we focus on the case of ϑ = π , and show how thepreparation of the system determines the disappearanceof given lines. This is clearly apparent in Fig. 5(e) for M τ = 1 : in this figure, half of the spectral lines identifiedin Appendix E 3 for ϑ = π are suppressed, whereas theyappear in Fig. 5(f) for M τ = 2 .To show this, we notice that, for ϑ = π , a train of M τ pulses prepares the system in the state ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T = (i e − i β/ ) M τ − M τ e i β/ sin( α ) [1 − ( − M τ ]e i β/ cos( α ) [1 − ( − M τ ] , (F1)see also Eq. (G1). Therefore, whenever M τ is odd, onlythe two excited states are occupied. In such case, thespectrum from Eq. (55) contains only the last addendappearing in Eq. (D1), e − i¯ ω ( T p − τ (cid:48) ) ˆ Z ( τ (cid:48) ) ˆ F ∆ ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R = e − i¯ ω ( T p − τ (cid:48) ) ˆ Z α )cos( α ) ⊗ ˆ Z ∗ α )cos( α ) , (F2)and the central frequencies of the lines appearing in the3spectrum can be determined by inspecting (cid:2) ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ ) (cid:3) − ( ˆ P ⊗ ˆ P ∗ ) − × e − i¯ ω ( T p − τ (cid:48) ) ˆ Z ( τ (cid:48) ) ˆ F ∆ ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R = e − i¯ ω ( T p − τ (cid:48) ) (cid:2) ˆ I − e − i(¯ ωT p − ∆ φ ) ( ˆ Λ ⊗ ˆ Λ ∗ ) (cid:3) − (cid:126)x ⊗ (cid:126)y, (F3)with the 3-dimensional vectors (cid:126)x = x x x = ˆ P † ˆ Z α )cos( α ) (F4)and (cid:126)y = y y y = ˆ P T ˆ Z ∗ α )cos( α ) . (F5)By noticing that the components x and y vanish for ϑ = π and for the weak probe pulses ( ϑ pr (cid:28) ) describedby Eq. (56), then one can conclude from Eq. (F3) that thepoles ¯ ω pole k (cid:48) s k identified in Eq. (E20) do not correspond topeaks in the pump–probe–pump spectrum for ϑ = π andfor an odd number M τ of pulses preceding the weak probepulse. This is in agreement with the results exhibited inFig. 5(e). Appendix G: Periodicity of the spectra as a functionof time delay
The periodicity of the pump–probe–pump spectrumin Eq. (55) is exclusively determined by the operator ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R , which prepares the system inthe state encountered by the probe pulse. All remainingterms in the spectrum depend on T p − τ (cid:48) = M τ T p − τ and are thus periodic in τ with period T p . Whenevertwo sequences of pump pulses M τ and M τ prepare thesystem in the same state, also the associated spectra willexhibit the same features.In order to investigate the properties of the state pre-pared by the pump pulses preceding the probe pulse, weobserve that ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T = ( ˆ F ∆ ˆ V p ) − ˆ P ˆ Λ M τ ˆ P − (1 , , T = e − i βM τ / sin ( ϑ/
2) + (cid:12)(cid:12) cos ( ϑ/ − e i β/ − i ε (cid:12)(cid:12) × sin ( ϑ/
2) e i εM τ + (cid:12)(cid:12) cos ( ϑ/ − e i β/ − i ε (cid:12)(cid:12) e − i εM τ − ϑ/ (cid:2) cos ( ϑ/ − e − i β/ ε (cid:3) e i β sin( εM τ ) sin( α ) − ϑ/ (cid:2) cos ( ϑ/ − e − i β/ ε (cid:3) e i β sin( εM τ ) cos( α ) . (G1)Since the spectrum depends on ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R = ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T ⊗ ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T , (G2) we observe that (i) it does not depend on the commonphase term e − i βM τ / in Eq. (G1), and (ii) its dependenceupon M τ is only via terms of the form e ± i2 εM τ . In otherwords, the dipoles generated by M τ pulses associatedwith ε and M τ pulses associated with ε are equal—and the corresponding spectra coincide—if there existsan integer K for which M τ ε = M τ ε + πK. (G3)For fixed pulse parameters ϑ and β , the spectrum is pe-riodic with respect to the number of preparatory pumppulses, with period ∆ M τ = πK/ε , where ∆ M τ and K are both integers.We analyze this in depth for the same particular casesalready discussed in Appendix E. ω o = δ r In this case, with β = 0 and ε = ϑ/ , the state pre-pared by the initial M τ pump pulses is given by ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T = cos (cid:0) ϑM τ (cid:1) i sin (cid:0) ϑM τ (cid:1) sin( α )i sin (cid:0) ϑM τ (cid:1) cos( α ) (G4)and Eq. (G3) leads to Eq. (60), thus explaining the peri-odic features in Figs. 8(a) and 8(b) and their dependenceon ϑ . ω o = δ r − π/T p With β = π and ε = π/ , the state encountered by theprobe pulse is given by ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T = − Mτ + − ( − Mτ cos (cid:0) ϑ (cid:1) i − ( − Mτ sin (cid:0) ϑ (cid:1) sin( α )i − ( − Mτ sin (cid:0) ϑ (cid:1) cos( α ) = cos (cid:0) ϑ (cid:1) i sin (cid:0) ϑ (cid:1) sin( α )i sin (cid:0) ϑ (cid:1) cos( α ) , if M τ odd, (1 , , T , if M τ even, (G5)explaining the results in Figs. 8(c) and 8(d) and the pe-riodicity of the spectra as a function of τ , with period T p .4 π -area pulses As shown in Eq. (F1), a sequence of M τ π -area pulsesprepares the system in the state ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T = (cid:40) (i e − i β/ ) M τ e i β/ (0 , sin( α ) , cos( α )) T , if M τ odd, (i e − i β/ ) M τ (1 , , T , if M τ even,(G6)so that the associated spectra are periodic in τ , withperiod T p for any β . π -area pulses A sequence of M τ π -area pulses prepares the systemin the state ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (1 , , T = (( − M τ , , T , (G7) and the time-delay-dependent spectra have period T p —the spectra are not sensitive to the absolute phase ofthe state associated with ( − M τ . In Appendix E 4, wealready noticed that Eq. (E23) is independent of β . Dueto Eq. (G7) and therefore as a result of ˆ F ∆ ˆ A pu ( ˆ F ∆ ˆ V p ˆ A pu ) M τ − (cid:126)R = (cid:126)R , (G8)the spectrum in Eq. (55) contains only the second addendappearing in Eq. 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