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PHYSICAL REVIEW A 89, 041803(R) (2014)

Population inversion in hyperfine states of Rb with a single nanosecond chirped pulsein the framework of a four-level system

G. Liu, V. Zakharov, T. Collins, P. Gould,* and S. A. MalinovskayaDepartment of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA

(Received 4 November 2013; revised manuscript received 13 February 2014; published 28 April 2014)

We implement a four-level semiclassical model of a single pulse interacting with the hyperfine structure inultracold rubidium aimed at control of population dynamics and quantum state preparation. We discuss a methodbased on pulse chirping to achieve population inversion between hyperfine states of the 5S shell. The results mayprove useful for quantum operations with ultracold atoms.

DOI: 10.1103/PhysRevA.89.041803 PACS number(s): 42.65.Dr, 32.80.Qk, 42.50.Hz

Ultracold atoms, with their minimal spectral broadeningand long interaction times, are ideal systems to explore andimplement quantum state preparation and manipulation [1]. Inparticular, Raman processes in ultracold systems have provento be a very useful tool in areas ranging from laser coolingand atom optics [2] to quantum information processing [3]and ultracold molecules [4]. The standard method of driving astimulated Raman transition between two levels is to apply twobeams of light, separated in frequency by the two-level splitting[5]. Here we propose to use instead a single pulse which isfrequency chirped through the Raman resonance. This has theadvantage of simplicity, high speed, and potentially efficienttransfer. Specifically, we examine Raman transitions betweenthe ground-state hyperfine levels of Rb upon interactionwith a single linearly polarized and linearly chirped laserpulse. With chirped light, the excited-state hyperfine structuredoes not generally allow the realization of the ideal three-level system often assumed in Raman transitions. We dealwith this complication by treating the atom as a four-levelsystem.

The linearly chirped pulse is described analytically as

E(t) = E0(t) cos[2πω0(t − T ) + α(t − T )2/2]. (1)

Here E0(t) is the Gaussian envelope of the form E0 exp[−(t −T )2/τ 2

0 ], T is the time of the pulse peak intensity, τ0 is thepulse duration related to the full width at the half maximum(FWHM) of the field E(t) as FWHM = τ02

√ln 2, ω0 is

the carrier frequency of the pulse at the peak value of thefield, and α/2π is the linear chirp (GHz/ns). The atoms areconsidered to be in an ultracold dilute gas, thus eliminatingDoppler shifting of the laser field in the atomic frame.Such ultracold gases of atoms may be readily produced andcontained in a laboratory setting using a magneto-opticaltrap (MOT) [6]. MOTs have been used to create ultracoldgases and perform, e.g., spectroscopy [7,8] and quantum statemanipulation [9].

A semiclassical model is used to describe the interactionof a linearly chirped pulse with the hyperfine (HpF) structureof the ultracold Rb. The two-photon Raman transitions areinvestigated aiming at achieving full population transfer from

*Department of Physics, University of Connecticut, Storrs,Connecticut 06269, USA.

the lower to the upper HpF state of the 5S orbital with the aidof the energetically higher 5P1/2 or 5P3/2 states. As an initialcondition, the electron resides in the ground HpF state of the5S orbital. Our goal is to determine the proper field parametersthat may accomplish population inversion within the 5S orbitalby taking advantage of the four optically allowed transitionsthat exist within both the D1 and D2 lines of 85Rb and87Rb.

Such an analysis has been done before using a model ofthe three-level � system which included two hyperfine levelsof the 5S shell and the lowest hyperfine state of the P shell,[10]. It revealed that the population inversion within the 5S

orbital was attainable using chirped, nanosecond laser pulsesof beam intensities on the order of 1 kW

cm2 and one-photondetuning on the order of the 5S hyperfine splitting. In thisRapid Communication, a model of the four-level system is usedto address all optically allowed transitions between hyperfinestates belonging to 52S1/2 and 52P1/2 or 52P3/2 states; thetransitions are optically allowed between those HpF stateswhose total angular momentum quantum number F changesas �F = {1,0, − 1}. For, e.g., 85Rb, these are two hyperfinestates having F = 2,3, separated by ω21. Note, that the splittingω43 between the F = 2 and F = 3 states in the 5P shellis an order of magnitude less than that between the stateswith the same quantum numbers in the 5S shell (see Fig. 1and [11]). Because of that, as our calculations show, a pulsehaving a duration on the order of ω−1

21 and a chirp ratesatisfying the adiabaticity condition populates the upper states|3〉 and |4〉 almost identically, demonstrating that the four-levelsystem behaves as an effective three-level � system. However,for longer pulses the resulting difference in the populationof the upper hyperfine states may be significant. Thus, themodel of the four-level system is more general and gives amore complete picture of population distribution for the fieldparameters associated with higher spectral resolution. One ofthe main points of the present work is to analyze how includingthe second excited state affects the Raman transfer. This isimportant for experimental implementations since this fourthlevel cannot in general be eliminated.

A complete treatment would require inclusion of allthe magnetic sublevels of each hyperfine level, but in anexperiment, the system could be restricted to four states bypreparing a single state via optical pumping and using selectionrules based on polarization of the chirped light to restrict thenumber of states involved.

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LIU, ZAKHAROV, COLLINS, GOULD, AND MALINOVSKAYA PHYSICAL REVIEW A 89, 041803(R) (2014)

The field-interaction Hamiltonian used in the time-dependent Schrodinger equation reads

H = h

⎡⎢⎢⎢⎣

� + ω43 + α(t − T ) 0 �R(t)/2 �R(t)/2

0 � + ω43 + ω21 + α(t − T ) �R(t)/2 �R(t)/2

�R(t)/2 �R(t)/2 0 0

�R(t)/2 �R(t)/2 0 ω43

⎤⎥⎥⎥⎦ (2)

Here �R(t) ≡ −μE0(t)/h is the Rabi frequency with the peakvalue �R . If the one-photon detuning � is equal to zero, thelaser carrier frequency becomes equal to the ω41 transitionfrequency at the peak value of the field amplitude. The valueof the transition dipole moment μ for 87Rb and 85Rb is 3.58 ×10−29 C m for the D2 line, and 2.54 × 10−29 C m for theD1 line. In our calculations, transition probabilities betweenhyperfine levels within these lines are considered to be equal.In addition, we ignore spontaneous emission because of ourfast time scales.

We investigated numerically the population transfer withinthe four-level system as a function of the pulse duration τ0,the linear chirp rate α/2π , the one-photon detuning � (where� = ω0 − ω41), and the peak Rabi frequency �R . Intuitively,we expect that a negative chirp should be implemented inorder to perform the two-photon population transfer witha narrow-band pulse by sweeping through the one-photonresonances beginning from the high-frequency resonance withthe |1〉 - |4〉 transition. The peak Rabi frequency was chosento be 30.35 GHz (10ω21) and 3.035 GHz (ω21) correspondingto the peak value of the field intensity about 83 kW/cm2 and833.8 W/cm2, respectively.

The results of calculations for �R = 30.35 GHz (10ω21)are shown in Fig. 2. An informative region is presented bythe FWHM from 1.0 ns to 5.0 ns and the α/2π from zeroto ±5 GHz/ns. These ranges of parameters may be reachedusing a setup similar to that described in [12]. The figureshows that for pulses having τ0 > ω−1

21 (FWHM = 0.56 ns)and a chirp rate such that |α/(2π )|τ0 � ω21, the solution is aRabi-type oscillation as a function of τ0 and α/(2π ) resultingin a periodic population transfer to the upper hyperfine state

FIG. 1. (Color online) Four optically attainable hyperfine statesof 5S and 5P shells with the energy differences that correspond tothe D1 line, modeled by a four-level system. Initially, the populationis in the ground state |1〉. Note that the hyperfine splitting of the5S1/2 orbital is approximately an order of magnitude greater than thesplitting of the 5P1/2 orbital.

|2〉 of the 5S shell. For |α/(2π )|τ0 > ω21, the dynamics entersthe adiabatic regime.

For small values of FWHM, such that the pulse spectralwidth is 1/τ0 > ω21, the population dynamics follows thepulse area solution. For such broad pulses, chirping is ofsecondary importance since population inversion may beachieved following the pulse area solution.

When the condition |α/(2π )|τ0 > ω21 is satisfied, a smooth,adiabatic regime of population transfer to state |2〉 may beachieved by implementing a lower intensity field satisfying theLandau-Zener adiabaticity condition |α/(2π )| � �2

R . If weconsider the peak intensity value of 833.8 W/cm2, which gives�R = ω21 = 3.035 GHz, a significant population transfer tothe final state |2〉 is manifested starting from FWHM ≈ 2.5 ns(τ0 = 1.5 ns) and α/(2π ) = −2 GHz/ns, giving α/(2π )τ0 ≈3 GHz; and the adiabatic regime is reached for FWHM ≈ 3 nsand α/(2π ) ≈ −3 GHz/ns (see Fig. 3). In the time-dependentpicture, Fig. 4(a), as the pulse instantaneous carrier frequencygradually decreases toward the first one-photon resonance withω41, the population adiabatically moves from the initial state|1〉 to the final state |2〉 through the excited-state manifold.In the vicinity of the peak value of the field intensity andω0 ≈ ω41, almost half of the population has been transferredfrom the initial to the final state and about 1% resides instates |3〉 and |4〉. As the instantaneous frequency furtherdecreases toward the second one-photon resonance with thetransition frequency ω42, more population is accumulated inthe final state |2〉, resulting in almost complete populationtransfer to this state by the end of the pulse. The mechanismrelies on two-photon adiabatic passage: As the frequencymoves toward the first resonance, the excited states workas mediators in the off-resonant two-photon transition to thefinal state |2〉. When a gradual decrease in the instantaneousfrequency leads to the second one-photon resonance withω42, it accomplishes the two-photon resonance condition andeffectively moves population from the excited states to thefinal state. The effective detuning switches sign at this point.The rest of the pulse completes population transfer to the finalstate off-resonantly. A one-photon detuning on the order of thepeak Rabi frequency leads to similar population dynamics withadiabatic passage to the final state, Fig. 4(b). Further increaseof the one-photon detuning diminishes the contribution of theexcited-state manifold and prevents the adiabatic populationflow to the final state for � � �R .

When a pulse of much longer duration and small negativechirp rate is applied, such that |α/(2π )|τ0 � ω21, no transferto state |2〉 is observed regardless of the field intensity[see Figs. 5(a) and 5(b) where the dependence of the statepopulation is shown as a function of time for a smallnegative chirp rate α/2π = −0.026 GHz/ns, FWHM = 50 ns,and �R = 3.035 GHz (a), and �R = 30.35 GHz (b)]. Here

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POPULATION INVERSION IN HYPERFINE STATES OF . . . PHYSICAL REVIEW A 89, 041803(R) (2014)

FIG. 2. (Color online) Final population distribution in the four-level system as a function of α/2π and FWHM for �R = 30.35 GHz and� = 0. The values of the system parameters are ω21 = 3.035 GHz, ω43 = 0.362 GHz, characteristic for 85Rb [11].

adiabatic passage takes place to the excited-state manifoldleading to differently populated states |3〉 and |4〉.

For positive values of the chirp rate and τ0 > ω−121 , the

excited-state manifold is populated adiabatically in Figs. 2and 3 owing to the resonance condition being satisfied with

these closely spaced states at the peak field intensity andsubsequent moving away from all resonances in the four-levelsystem.

In summary, we have demonstrated that a single chirpednanosecond pulse of intensity on the order of 800 W/cm2 may

FIG. 3. (Color online) Final population distribution in the four-level system as a function of α/2π and FWHM for �R = 3.035 GHz and� = 0. The values of the system parameters are ω21 = 3.035 GHz and ω43 = 0.362 GHz, characteristic for 85Rb [11].

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LIU, ZAKHAROV, COLLINS, GOULD, AND MALINOVSKAYA PHYSICAL REVIEW A 89, 041803(R) (2014)

FIG. 4. (Color online) (a) Time-dependent picture of the population dynamics of four hyperfine states leading to population transfer to thefinal state |2〉. The field parameters are �R = 3.035 GHz, FWHM = 2.993 53 ns, and α/(2π ) = −2.947 52 GHz/ns. (b) The end-of-pulsepopulation of four hyperfine states as a function of the one-photon detuning �. The field parameters are �R = 3.035 GHz, α/2π = −1 GHz/ns,and FWHM = 17 ns.

be successfully implemented to manipulate valence electrondynamics in alkali-metal atoms at ultracold temperaturesaiming at the population transfer within the hyperfine structure.The results are based on the developed model of a classicalpulse interaction with the four-level system representing opti-cally accessible hyperfine states in ultracold 85Rb. Numericalanalysis is performed revealing the dependence of populationdynamics and the quantum yield on the key field parameters:the pulse duration, the linear chirp rate, the peak amplitude, andthe one-photon detuning. We analyzed a possibility to performthe population inversion within the hyperfine states of the 5S

shell. The adiabatic region of light-matter interaction leading topopulation inversion was found for parameters that satisfy the

adiabaticity conditions |α/(2π )|τ0 > ω21 and |α/(2π )| < �2R .

The physical values applicable to 85Rb are the peak Rabifrequency �R = ω21 (3.035 GHz), the chirp rate α/2π =−0.3[ω2

21] (−3 GHz/ns) or faster, and the pulse durationτ0 � 5.5ω−1

21 (�1.8 ns). The one-photon detuning � on theorder of the peak Rabi frequency does not diminish adiabaticpassage and still leads to the inversion between hyperfinestates. The four-level system may be approximated by aneffective three-level � system giving a qualitatively similarquantum yield when the pulse duration and the chirp ratesatisfy |α/(2π )|τ0 � ω43 and �R � ω43. If these conditionsare not met, the validity of the three-level system breaksdown.

FIG. 5. (Color online) (a), (b) Time-dependent picture of adiabatic passage to states |3〉 and |4〉 achieved by a pulse having FWHM = 50 ns,α/2π = −0.026 GHz/ns and (a) �R = 3.035 GHz, (b) �R = 30.35 GHz.

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POPULATION INVERSION IN HYPERFINE STATES OF . . . PHYSICAL REVIEW A 89, 041803(R) (2014)

This research was supported in part by the National ScienceFoundation under Grants No. NSF PHY11-25915 and No. NSFPHY12-05454. The work at the University of Connecticut

was supported by the Chemical Sciences, Geosciences andBiosciences Division, Office of Basic Energy Sciences, Officeof Science, US Department of Energy.

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