Microwave control of atomic motional states in a spin-dependent optical lattice
Noomen Belmechri, Leonid Förster, Wolfgang Alt, Artur Widera, Dieter Meschede, Andrea Alberti
MMicrowave control of atomic motional states in aspin-dependent optical lattice
Noomen Belmechri , Leonid Förster , Wolfgang Alt , ArturWidera , Dieter Meschede , and Andrea Alberti Institut für angewandte Physik, Universität Bonn, Germany Fachbereich Physik und Forschungszentrum OPTIMAS, Universität Kaiserslautern,Erwin-Schrödinger-Straße, D67663 Kaiserslautern, GermanyE-mail: [email protected]
Abstract.
Spin-dependent optical potentials allow us to use microwave radiationto manipulate the motional state of trapped neutral atoms (Förster et al. 2009
Phys. Rev. Lett. , 233001). Here, we discuss this method in greater detail,comparing it to the widely-employed Raman sideband coupling method. We provide asimplified model for sideband cooling in a spin-dependent potential, and we discuss itin terms of the generalized Lamb-Dicke parameter. Using a master equation formalism,we present a quantitative analysis of the cooling performance for our experiment, whichcan be generalized to other experimental settings. We additionally use microwavesideband transitions to engineer motional Fock states and coherent states, and wedevise a technique for measuring the population distribution of the prepared states.PACS numbers: 37.10.De: Atom cooling methods, 37.10.Jk: Atoms in optical lattices,03.65.Wj: State reconstruction (quantum mechanics), 37.10.Vz: Mechanical effects oflight on atoms, molecules, and ions.
Submitted to:
J. Phys. B: At. Mol. Opt. Phys. a r X i v : . [ qu a n t - ph ] F e b ONTENTS Contents1 Introduction 22 Microwave induced motional sideband transitions 3
Motional state control of atomic particles is achieved by the absorption and emissioncycles of a resonant or near resonant radiation, i.e., by light scattering typically at opticalfrequencies. For instance, laser Doppler cooling reduces the momentum of atoms or ionsthrough multiple recoil processes [1]. Coherent momentum transfer can be performedwith two-photon Raman processes [2] for applications in, e.g., atom interferometry [3].The quantum state of the atomic particles is composed of the internal states, e.g.,two spin states {|↑(cid:105) , |↓(cid:105)} for a two-level atom, and the external motional state. Forfree particles the simplest motional state is the momentum state | (cid:126)p (cid:105) . Trapped particlesare instead characterized by vibrational eigenstates | n (cid:105) , which in the simplest case of aharmonic oscillator of frequency ω vib have their energies equally spaced as (cid:126) ω vib ( n +1 / .In free space, the momentum state of a particle, and consequently its kineticenergy, is changed by the momentum transfer | (cid:126)p (cid:105) → | (cid:126)p (cid:48) (cid:105) in the absorption/emissioncycle of an optical photon. While the momentum transfer picture also applies approx-imately for trapped particles when the energy separation between motional states isnot spectroscopically resolved, recoil-free transitions become possible in the resolved-sideband regime (Mössbauer effect). While carrier transitions do not change thevibrational quantum state | n (cid:105) , the motional state can be controlled via sidebandtransitions | n (cid:105) → | n (cid:48) (cid:105) ( n (cid:48) (cid:54) = n ), for instance, in incoherent cooling processes | n (cid:105) →| n − (cid:105) or in coherent manipulation of vibrational states [4]. With trapped ions orneutral atoms trapped in optical lattices, the resolved-sideband regime is typicallyrealized by two-photon Raman transitions connecting two different hyperfine ground ONTENTS
2. Microwave induced motional sideband transitions
We consider a single atom with two spin states { |↑(cid:105) , |↓(cid:105) } trapped in a one-dimensionaloptical lattice. We will initially ignore the internal degree of freedom of the atom andtake the Hamiltonian governing its motion in the trap as given by ˆ H ext = ˆ p m + U ( k L ˆ x ) , (1)with U being the trap depth, k L = 2 π/λ L being the wavenumber of the twocounter propagating laser fields creating the lattice, and ˆ x , ˆ p , the atom’s position andmomentum, respectively. (b)(a) Figure 1: In a semi-classical picture, an atomic transition can affect the motional stateof an atom either (a) by a kinetic energy change caused by the momentum transferfrom an optical photon of wavevector k opt (velocity selective transition [1]), or (b) bya potential energy change when the potentials of the two internal states are displacedin space by ∆ x (position selective transition). In the two cases, the motional energy isdecreased when the detuning is set to the Doppler shift k opt v at or the potential energy m ω ∆ x / (2 (cid:126) ) , respectively. ONTENTS (cid:12)(cid:12) Ψ Bn,k (cid:11) , where n is the band index ( n = 0 for the first band) and k is thewavevector in the first Brillouin zone (BZ). In the limit of deep lattice potentials that weare considering here, the atoms remain localized for the time scales of the experimentand their spatial state is best described by the maximally localized Wannier state [13] | n, r (cid:105) = 1 √ N (cid:88) k ∈ BZ e − ikrd (cid:12)(cid:12) Ψ Bn,k (cid:11) . (2)Here N is the lattice size, r the site index, and d = λ L / the lattice spacing. In thisdeep lattice regime, we can safely view the vanishingly narrow energy bands ε n ( k ) asthe vibrational level energies ε n of the corresponding Wannier state | n, r (cid:105) at lattice site r ; in the harmonic approximation we would have ε n = (cid:126) ω vib ( n + 1 / .The Wannier states form an orthonormal basis set such that the overlaps betweentwo different states yield (cid:104) n, r | n (cid:48) , r (cid:48) (cid:105) = δ n,n (cid:48) δ r,r (cid:48) . This means that the interaction of theatomic spin with a microwave field will fail to induce motional sideband transitions, | n, r (cid:105) ↔ | n (cid:48) , r (cid:48) (cid:105) , because of the nearly negligible momentum carried by microwavephotons, five orders of magnitude smaller than that by optical photons. This restrictioncan be lifted if the atom experiences a different trapping potential depending onits internal spin state as the corresponding motional eigenstates are then no longerorthogonal [14, 15]. A simple relative spatial shift of the potentials trapping each internalstate induces such a difference. A shift by a distance ∆ x is accounted for by the positionspace shift operator ˆ T ∆ x ≡ exp( − i ˆ p ∆ x/ (cid:126) ) , see figure (2a). The overlap between thetwo Wannier states then becomes (cid:68) n (cid:48) , r (cid:48) (cid:12)(cid:12)(cid:12) ˆ T ∆ x (cid:12)(cid:12)(cid:12) n, r (cid:69) ≡ I n (cid:48) ,r (cid:48) n,r (∆ x ) . (3)The resulting overlap integral, − ≤ I n (cid:48) ,r (cid:48) n,r (∆ x ) ≤ , is hence a known function of ∆ x ,see figure (2b). It is analogous to the Franck–Condon factor from molecular physics andit determines the strength of the transitions coupling different vibrational levels [16].One way to realize the shift operator ˆ T ∆ x is by two overlapped lattices which trapeach spin state separately and can be independently shifted in the longitudinal directionas shown in figure (2). The trapping potential thus becomes dependent on the spin state s = {↑ , ↓} and the shift distance ∆ x = x ↑ − x ↓ , ˆ H ext = ˆ p m + (cid:88) s = {↑ , ↓} U s [ k L (ˆ x − x s )] ⊗ | s (cid:105) (cid:104) s | (4)with x s being the position of the lattice trapping the state | s (cid:105) . The total transitionmatrix element for two spin states coupled by an interaction Hamiltonian H I , with afree-atom bare Rabi frequency Ω , is then given by (cid:126) Ω n (cid:48) ,r (cid:48) n,r (∆ x ) / (cid:68) s (cid:48) , n (cid:48) , r (cid:48) (cid:12)(cid:12)(cid:12) ˆ T ∆ x ⊗ H I (cid:12)(cid:12)(cid:12) s, n, r (cid:69) = I n (cid:48) ,r (cid:48) n,r (∆ x ) × (cid:126) Ω / . (5)The Franck-Condon factors I n (cid:48) ,r (cid:48) n,r (∆ x ) can be explicitly evaluated using equations (2)and (3). We first rewrite equation (2) using Bloch’s theorem, W n,r ( x ) = 1 √ N (cid:88) k ∈ BZ (cid:88) q ∈ Z e − ikrd e i πd q a n,q ( k ) | k (cid:105) . (6) ONTENTS Ω multiplied by the overlap between the two involvedvibrational states, the Franck-Condon factor, which is controlled by the relative shift ∆ x between the two lattices. η x is the spatial Lamb-Dicke parameter defined in section 2.1and later in 3.1. (b) Lattice shift dependence of the Franck-Condon factors for differenttransitions, denoted as n − m , calculated for typical experimental parameters (see text).with a n,q ( k ) being the Fourier coefficients of the Bloch functions and | k (cid:105) the planewavestate. These functions can be constructed using the periodic solutions of the Mathieudifferential equation [17, 18] with their phase chosen such that the resulting Wannierstates are real and have the proper parity corresponding to their respective vibrationallevels [13]. The coefficients a n,q ( k ) are numerically obtained from algorithms for thecomputation of Mathieu coefficients [19]. Inserting (6) in (3) and taking into accountthe parity of the Wannier states, or equivalently the parity of the band n , one eventuallyarrives at the following expression for the Franck-Condon factors I n (cid:48) ,r (cid:48) n,r (∆ x ) = 2 (cid:88) k ∈ BZ (cid:88) q ∈ Z F (cid:20) ( k + 2 πd q )(∆ x + r − r (cid:48) ) (cid:21) a ∗ n,q ( k ) a n (cid:48) ,q ( k ) , (7)where we have defined F ( x ) := cos( x ) if n and n (cid:48) have the same parity, and F ( x ) := sin( x ) otherwise. Numerical evaluation of (7) is shown in figure (2).Considering a single lattice site and assuming the harmonic approximation for thepotential, the shift operator takes the simple form ˆ T ∆ x = exp[ η x ( a † − a )] , where a † ( a )is the raising (lowering) operator acting on the vibrational states. Here we introducedthe spatial Lamb-Dicke parameter η x = ∆ x/ (2 x ) , (8)where x is equal to the rms width of the motional ground state. When η x (cid:28) ,taking the first order term in η x of ˆ T ∆ x allows for a simple expression of the Franck-Condon factors for transitions on the same lattice site (i.e., r = r (cid:48) =0), I n (cid:48) , n, (∆ x ) ≈ δ n,n (cid:48) + η x ( √ n (cid:48) δ n (cid:48) ,n +1 − √ nδ n (cid:48) ,n − ) . ONTENTS L a tt i c e p o t e n t i a l s Figure 3: State-dependent optical lattices relatively shifted by a distance ∆ x . The totaltrap depth difference ∆ U tot = U tot ↑ − U tot ↓ , and lattice contrast W s for spin state | s (cid:105) areshown. Unlike the spin |↑(cid:105) lattice, the contrast and total depth of the spin |↓(cid:105) latticevary with the shift distance. We load Cesium (
Cs) atoms from a magneto optical trap into a 1D optical latticeformed by two counter-propagating, far-detuned, linearly polarized laser beams. Thefilling factor is at most one atom per lattice site due to light-induced collisions [20].A weak guiding magnetic field of 3 G oriented along the lattice lifts the degeneracybetween the Zeeman sublevels of the Cesium S / ground state such that atoms canbe initialized by optical pumping beams into the hyperfine state |↑(cid:105) ≡ | F = 4 , m F = 4 (cid:105) .Microwave radiation, at around ω MW = 2 π × . GHz, couples states |↑(cid:105) and |↓(cid:105) ≡| F = 3 , m F = 3 (cid:105) with the bare Rabi frequency of Ω = 2 π × kHz [21]. The spin stateof the atom is probed using the so called “push-out” technique [22] which consists ofcounting the fraction of atoms left in |↓(cid:105) after all the atoms in |↑(cid:105) have been removedby an intense radiation pulse.An angle θ between the linear polarization vectors of the two beams forming thelattice is equivalent in the circular basis to a phase delay of θ between two collinearand independent circularly-polarized standing waves, σ + and σ − , or equivalently to astanding wave longitudinal relative shift of ∆ x sw ( θ ) = θ d/π. (9)The polarization angle θ is controlled by an electro-optical modulator (EOM) and twoquarter-wave plates in the path of one of the two lattice beams. The two in-phasecircular components of the beam are mapped by the first λ L / plate onto orthogonallinear polarizations parallel to the EOM axes. The retardation θ induced by the EOMis proportional to the voltage signal applied to it. The last plate then converts the linearpolarizations back into the circular ones while conserving the delay.The trapping potentials resulting from the σ + and σ − standing waves for a spinstate | s (cid:105) are U s = U tot s + W s cos [ k L ( x − x s )] (10) ONTENTS W s and U tot s are the lattice contrast, taking positive values, and total trap depthfor state | s (cid:105) , respectively. Both W s and U tot s depend on the lattice lasers wavelength λ L and on the lattice shift ∆ x or equivalently the polarization angle θ , see figure (3).For alkali atoms, one can define the “magic wavelength” as the one where the state |↑(cid:105) experiences the σ + standing wave only. This occurs at λ L = λ +( λ − λ ) / (2 λ /λ +1) ≈ λ + ( λ − λ ) / , where λ ( λ ) is the wavelength of the D (D ) line [23, 24], which is λ L = 866 nm in our case. At this wavelength, for the spin |↑(cid:105) state, equation (10) reads U ↑ = − W ↑ + W ↑ cos ( k L x − θ/ , (11)while the spin |↓(cid:105) state experiences both σ + and σ − standing waves with a relativeweight of / and / , respectively. The lattice potential in this case is U ↓ = − W ↑ + (1 / W ↑ cos ( k L x − θ/
2) + (7 / W ↑ cos ( k L x + θ/ . (12)With the notation of equation (10), one finds that W ↑ = − U tot ↑ is independent fromthe angle θ , while W ↓ = [cos( θ ) + (3 / sin( θ ) ] / W ↑ and U tot ↓ = − ( W ↑ + W ↓ ) / . Inaddition, one obtains the lattice relative shift ∆ x = ( d/π ) { θ + arctan[3 tan( θ ) / } / .Equations (11) and (12) constitute the closest realization of the idealized spin-dependentlattice discussed in section 2.1. The small admixture of a σ + component in equation (12)results in a lattice depth W ↓ that depends on θ , or equivalently on the lattice shift ∆ x ,which makes the energy levels ε s,n (∆ x ) depend on the spin state and on the shift ∆ x ,see figure (3). The nonlinear position shift of the U ↓ potential, x ↓ , makes ∆ x deviatefrom the standing wave relative shift ∆ x sw in equation (9), and this has to be taken intoaccount in the calculation of the Franck-Condon factors [23].The typical total lattice depth used in our experiment is W ↑ ≈ E lattR (corresponding to µK), with E lattR = (cid:126) k L / m Cs as the lattice recoil, which amounts toan oscillation frequency along the lattice axis of ω vib ≈ π × kHz. In the transversedirection, atoms are confined only by the Gaussian profile of the lattice lasers whichresults in a transverse oscillation frequency of ω rad ≈ π × kHz. The typical initialtemperature of the atoms loaded into the lattice is T ≈ µK, which in the harmonicapproximation amounts to mean vibrational numbers of n vib ≈ . and n rad ≈ inthe axial and transverse directions, respectively. We investigate sideband transitions by recording microwave spectra for different latticeshifts. Controlling the relative distance ∆ x allows us to continuously tune the parameter η x from 0 to about 5. In order to resolve the sidebands we use Gaussian microwave pulseswith a FWHM of µs and a bare Rabi frequency of Ω / π = 36 kHz, corresponding tothe π -pulse condition for the carrier transition. Figure (4) shows a combined spectrumwhere transitions from n = 0 to levels up to n (cid:48) = 14 are well resolved [11]. Fourspectra are recorded for four different lattice shifts. With an unshifted lattice only thecarrier transition is visible, and it defines the zero of the microwave detuning δ MW .The remaining three lattice shifts were chosen such that for each shift distance ∆ x the ONTENTS r = r (cid:48) ), Ω n (cid:48) , n, (∆ x ) , is simultaneouslyclose to maximum for a small group of adjacent sideband transitions. The couplingstrength for sites r (cid:54) = r (cid:48) can be neglected at the given shifts.For each shift distance ∆ x the microwave spectra are fitted using the spectra yieldedby a numerical calculation of the time evolution based on the following Hamiltonian ˆ H = ˆ H + ˆ H MW (13)with ˆ H = (cid:88) s = ↑ , ↓ (cid:88) n (cid:16) ε s,n (∆ x ) + δ s, ↑ (cid:126) ω HS (cid:17) | s, n, r (cid:105) (cid:104) s, n, r | , (14) ˆ H MW = − (cid:126) (cid:88) r,r (cid:48) (cid:88) n,n (cid:48) I n (cid:48) ,r (cid:48) n,r (∆ x ) (cid:16) e − iω MW t |↑ , n, r (cid:105) (cid:104)↓ , n (cid:48) , r (cid:48) | + h.c. (cid:17) , (15) x=43 nmx=111nmx=176nm Microwave detuning δ (MHz) MW ΔU t o t Δx/d (E R latt ) (cid:31) (cid:31)
500 50 (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) /8– (kHz)
Δx/d (E R latt ) Figure 4: (a) Microwave spectrum of sideband transitions |↑ , n = 0 (cid:105) ↔ |↓ , n (cid:48) (cid:105) for latticeshifts ∆ x = {0nm ( • ), 43nm ( ◦ ), 111nm( (cid:4) ), 176nm ( (cid:3) )} corresponding to the parameter η x = { , . , . , . } defined in section 3.1 (data points from [11]). The microwavedetuning is given with respect to the carrier transition frequency. Data points are theaverage on about 100 atoms and they are here fitted with a model that takes intoaccount broadening mechanisms detailed in the text. The error bars, reported only forthree representative peaks, are obtained with the 68% Clopper-Pearson interval methodfor binomial statistics. The panels (b) and (c) compare the expected values (dashedlines) for the lattice contrast W ↓ and total trap depth difference ∆ U tot (see figure (3)and text in section 2.2) with the values extracted from the fits ( uncertainty). ONTENTS (arb. units) (a) (b) convolutedpeak shaperadial thermaldistributionFourier-limitedpeak shape Figure 5: (a) Inhomogeneous broadening effect due to the transverse motion of theatoms in the trap. The overall peak profile (bottom curve) is the convolution of theother two profiles. The Fourier-limited FWHM and the thermal broadening are typically kHz and kHz (for T ∼ µK and first sideband), respectively. (b) Left panel:Franck-Condon factors as a function of the radial distance ρ (here, θ = 15 ◦ ). The grayprofile shows the 2D radial distribution from equation (16) for the same temperature.Right panel: Resulting thermal distribution of Franck-Condon factors.where ω HS denotes the hyperfine splitting frequency of the ground state. With thisnotation, the microwave detuning reads δ MW = ω MW − ω HS .Given the deep lattice regime considered here, in the numerical solution of equation(13) the maximum number of vibrational levels per site can be restricted with a goodapproximation to n max = 15 , before atoms start to behave like free particles tunnelingbetween sites or directly coupling to the continuum. In this regime, the coupling strengthfor a sideband transition between two lattice sites separated by a distance x > d aretwo orders of magnitude lower than the typical time scales of our experiment; therefore,we limit the site indices to r = r (cid:48) .In the fitting of the sideband spectra, the energy levels ε s,n and Franck-Condonfactors I n (cid:48) , n, depend on the fitting parameters ∆ x , U tot s and W s . In particular, in theharmonic approximation the spacing between two adjacent peaks is equal to the trapfrequency of the U ↓ potential, which therefore determines the lattice contrast W ↓ ; theabsolute offset of each spectrum is mainly determined by the difference of the total trapdepths, ∆ U tot = U tot ↑ − U tot ↓ , expressed in frequency units. Additionally, an averageover the thermal motion of the atoms in the transverse direction of the one-dimensionaloptical lattice has to be performed. In fact, the lattice parameters U tot s and W s dependon the transverse position of the atom, and to take this dependence into accountwe assume that during the microwave dynamics an atom has a “frozen” transverseposition ρ . This assumption is justified by the slow transverse motion of the atoms, ω rad / π ≈ kHz, compared to the lowest bare Rabi frequency used for the microwave ONTENTS Ω / π ≈ kHz. The transverse positions of the atoms are then assumed to bedistributed according to a two-dimensional Boltzmann distribution, shown in figure (5b)and given in the harmonic approximation by P ( ρ ) = ρσ exp( − ρ σ ) , with σ = (cid:115) k B T m Cs ω rad (16)with T the transverse temperature. The thermal transverse position distributionresults in an inhomogeneous distribution of microwave sideband resonance frequenciesand of Franck-Condon factors, shown qualitatively in figure (5). Both distributionsare used to weight the calculated spectra, with T as an additional fitting parameter.The figure shows that thermal broadening effect becomes larger for higher sidebands,exhibiting a more pronounced asymmetric peak shapes. This behavior has a clearexplanation: in the harmonic approximation, for instance, one expects the thermalbroadening to increase linearly with band index n while the Fourier-limited FWHMremains constant.The best-fit results for W ↓ and ∆ U tot are shown in figures (4b) and (4c). Thismethod allows us to spectroscopically determine the parameters of the spin-dependentpotentials seen by the atoms with a relative uncertainty of about . The smalldeviations from the expected values (dashed curves in the figure) can be attributed inpart to measurement uncertainty and to polarization imperfections in the standing wavebeams, resulting in slightly distorted potentials. For instance, polarization distortioncan be responsible for the non monotonic behavior of the data points in figure (4c).From the fit, we obtain a temperature of T = (2 . ± . µK. Without axial groundstate cooling, we measure a three-dimensional temperature of µK by means of theadiabatic lowering technique [25]. This discrepancy requires further investigations.
3. Microwave sideband cooling
The general principle of resolved sideband cooling, depicted in figure (6), relies onthe repetition of cooling cycles where each cycle starts by a sideband transition |↑ , n (cid:105) → |↓ , n − (cid:105) removing a vibrational energy quantum (cid:126) ω vib . The cycle is thenclosed by an optical repumping process with a transition to an optically excited state | e (cid:105) followed by a spontaneous decay to the initial spin state. Because of the opticalrepumping, the motional energy of the atom in each cycle increases on average, whichcorresponds to heating. Therefore, in order to achieve cooling the overall energy gainedby an atom after one cycle must be negative. In general, heating is caused by themomentum recoil from the optical repumping photons, i.e. recoil heating. In themicrowave-based scheme however, shown in figure (6b), an additional source of heating,called hereafter “projection heating,” is present. It is due to the difference between thetrapping potentials of the internal states, in this case it is a spatial shift. This differencemakes the projection of a vibrational state |↓ , n (cid:105) on an arbitrary state |↑ , m (cid:105) appreciablein contrast to the case of identical potentials where transitions beyond m = n, n ± are ONTENTS |↑ , n (cid:105) → |↓ , n − (cid:105) . Thewavefunction is shifted in momentum space by (cid:126) ∆ k . (b) Microwave sideband coolingscheme: a microwave transition between two shifted trapping potentials reduces thevibrational state. Note that we use here a blue sideband transition to reduce thevibrational state, instead of the typical usage of a red sideband transition [7].negligible. In the standard Raman-based sideband cooling schemes [6, 26] the sideband is inducedby a two-photon transition where the coupling is given by the matrix element Ω Raman n − ,n = (cid:104)↓ , n − | ˆ T ∆ k | ↑ , n (cid:105) × Ω Raman , (17)where ˆ T ∆ k ≡ exp( i ˆ x ∆ k ) is the momentum shift operator and ∆ k ≈ k opt is thewavevector difference between the two optical photons for counterpropagating beams.From here on, it is understood that all transitions occur on the same site, r = r (cid:48) . Inthe microwave-based scheme we neglect the microwave photon recoil, and the sidebandcoupling corresponding to a lattice shift ∆ x between nearest neighboring sites is thengiven by Ω n − ,n = (cid:104)↓ , n − | ˆ T ∆ x | ↑ , n (cid:105) × Ω . (18)Using the harmonic approximation, the Raman and microwave sideband couplings canbe expanded to the first order in the parameters η k = (cid:126) k opt / (2 p ) and η x = ∆ x/ (2 x ) ,as shown in table (1), where p = (cid:112) m Cs (cid:126) ω vib / and x = (cid:112) (cid:126) / (2 m Cs ω vib ) are themomentum and spatial rms width of the ground-state wavefunction, respectively [27].From table (1) we can note a clear duality between momentum and spatial shifts in thetwo sideband cooling methods. The duality is better emphasized by using the generalcomplex Lamb-Dicke parameter η = η k + iη x = (cid:126) k opt / (2 p ) + i ∆ x/ (2 x ) = k opt x + ip ∆ x/ (cid:126) , (19) ONTENTS η k and η x in the Lamb-Dicke regimedefined by | η | = | η k + iη x | (cid:28) , under the harmonic approximation.Raman MicrowaveSideband coupling strength Ω n − ,n / Ω i η k √ n − η x √ n Recoil heating per cycle (cid:126) ω vib η k (cid:126) ω vib η k Projection heating per cycle — (cid:126) ω vib η x Overall heating per cycle (cid:126) ω vib η k (cid:126) ω vib ( η x + 2 η k ) which accounts for both degrees of freedom via the momentum and spatial Lamb-Dickeparameters, η k and η x , respectively. This generalized approach was introduced first inion systems to describe microwave-induced sidebands in the presence of spin-dependentforces [14].In the Raman-based cooling schemes with identical trapping potentials, the spatialLamb-Dicke parameter η x vanishes and the heating comes from the recoil of the opticalrepumping photons, as depicted in the figure (6a). In the microwave-based schemehowever, the generalized Lamb-Dicke parameter is complex and the heating is causedby a combination of recoil and projection heating.The energy gained by an atom from recoil heating after one cycle results from tworecoils, one from absorption and one from spontaneous emission, and is therefore givenby ∆ E rec = 2 E R , (20)where E R = (cid:126) k opt / m Cs is the optical photon recoil energy [28]. This quantity doesnot depend on the details of the potentials but only on the atom’s properties, and itexpresses the overall three-dimensional recoil heating.In the shifted potentials shown in figure (6b), in addition to the recoil heating,the atom’s motional energy increases on average by the projection heating energy ∆ E proj . This is due to the non-vanishing projection of the atom’s initial vibrational state |↓ , n (cid:105) onto the vibrational basis |↑ , m (cid:105) of the final spin state in the optical repumpingprocess. In the harmonic approximation, with H ext = (cid:126) ω vib ( n +1 / , and after adiabaticelimination of the excited state | e (cid:105) , the projection heating contribution for a relativeshift ∆ x can be derived as ∆ E proj = (cid:126) ω vib (cid:88) m ( m − n ) (cid:12)(cid:12)(cid:12)(cid:68) m (cid:12)(cid:12)(cid:12) ˆ T ∆ x (cid:12)(cid:12)(cid:12) n (cid:69)(cid:12)(cid:12)(cid:12) == (cid:88) m (cid:68) m (cid:12)(cid:12)(cid:12) ˆ[ H ˆ T ∆ x − ˆ T ∆ x ˆ H ] (cid:12)(cid:12)(cid:12) n (cid:69) (cid:68) n (cid:12)(cid:12)(cid:12) ˆ T † ∆ x (cid:12)(cid:12)(cid:12) m (cid:69) == (cid:68) n (cid:12)(cid:12)(cid:12) ( ˆ H ext (∆ x ) − ˆ H ext ) (cid:12)(cid:12)(cid:12) n (cid:69) , (21) ONTENTS ˆ H ext (∆ x ) = ˆ T † ∆ x ˆ H ext ˆ T ∆ x = ˆ p m Cs + 12 m Cs ω vib (ˆ x + ∆ x ) . (22)The result of equation (21) applies in general for any potential profile, and in theharmonic approximation it results in a quantity which is independent of n , ∆ E proj = 12 m Cs ω vib ∆ x , (23)which is nothing but the potential energy difference as expected from the semi-classicalpicture in figure (1).Using the same method, one can generally show that in the microwave sidebandcooling scheme the total average heating energy gained by an atom in one cooling cycleis the sum of the recoil and projection contributions. The total energy balance per cyclethen becomes ∆ E tot = ∆ E proj + ∆ E rec − (cid:126) ω vib = (cid:126) ω vib ( η x + 2 η k − . (24)Similarly to the usual definition of the Lamb-Dicke regime [29], the condition for cooling ∆ E tot < defines a generalized Lamb-Dicke regime as the range where | η | < . The general theory of sideband cooling is very well known and has been extensivelystudied in the literature [5, 27, 30, 31]. Here, we discuss a quantitative model based onthe Lindblad master equation formalism. To provide a concrete example, we apply themodel to the level scheme of our specific system, though the model can be adapted toother similar systems.In the cooling cycle depicted in figure (7), microwave radiation resonant with thefirst blue sideband transfers atoms from states |↑ , n (cid:105) to states |↓ , n − (cid:105) . Concurrentwith the microwave, a σ + -polarized repumper laser beam couples state |↓(cid:105) to state (cid:12)(cid:12) P / , F (cid:48) = 4 (cid:11) ≡ | e (cid:105) , from where the atoms close the cooling cycle by spontaneouslydecaying back to state |↑(cid:105) . Due to the appreciable probability of atoms decaying fromstate | e (cid:105) to state | F = 4 , m F = 3 (cid:105) ≡ | a (cid:105) , a second equally polarized pumping lasercouples the two states and brings the atoms which have decayed to state | a (cid:105) back intothe cooling cycle. In each cycle, an atom loses energy on average until it reaches the“dark state” |↑ , n = 0 (cid:105) where it is no longer affected by the microwave or the repumpinglasers. Nevertheless, a small probability remains that the dark state is depopulateddue to photon scattering from the lattice lasers or an off-resonant microwave carriertransition.To describe the cooling dynamics, we reduce the problem at hand to an effectivemodel with three spin states with the set of motional states associated with each one ofthem. The considered Hilbert space is then the one spanned by the states | s, n (cid:105) , with n being the vibrational level and | s (cid:105) being one of the three internal states |↑(cid:105) , |↓(cid:105) or | a (cid:105) .The optically excited state | e (cid:105) is adiabatically eliminated due to its very short lifetime, ONTENTS τ = 30 ns, compared to the motional time scale. We use the Lindblad master equationformalism to write the time evolution of the effective model’s density matrix as [31] dρdt = − i (cid:126) [ ˆ H (cid:48) + ˆ H MW , ρ ] + L [ ρ ] , (25)where ˆ H (cid:48) is the extension of the Hamiltonian from equation (14) to the states | a, n (cid:105) ˆ H (cid:48) = (cid:88) s = {↑ , ↓ ,a } (cid:88) n ε s,n (∆ x ) | s, n, r (cid:105) (cid:104) s, n, r | , (26)and L is the Lindblad superoperator with the projectors L n (cid:48) ,r (cid:48) ,s (cid:48) n,r,s = | s, n, r (cid:105) (cid:104) s (cid:48) , n (cid:48) , r (cid:48) | , (27)and the effective decay rates γ s (cid:48) ,n (cid:48) ,r (cid:48) s,n,r for the transitions | s (cid:48) , n (cid:48) , r (cid:48) (cid:105) → | s, n, r (cid:105) which aregiven by γ s (cid:48) ,n (cid:48) ,r (cid:48) s,n,r = α s R s (cid:48) (cid:68) | M s (cid:48) ,n (cid:48) ,r (cid:48) s,n,r | (cid:69) (cid:126)k sp , (28)with M s (cid:48) ,n (cid:48) ,r (cid:48) s,n,r = (cid:104) n, r, s | ˆ T ∆ k s,s (cid:48) ˆ T ∆ x | s (cid:48) , n (cid:48) , r (cid:48) (cid:105) . (29)Here, α s is the branching ratio for the spontaneous emission from state | e (cid:105) to state | s (cid:105) ,and R a , R ↓ are the pumping and repumping rate, respectively, as shown in figure (7). Inaddition, we account for the possibility that an atom in |↑(cid:105) state scatters a photon fromthe lattice with the rate R ↑ . The matrix element M s (cid:48) ,n (cid:48) ,r (cid:48) s,n,r accounts for the relative spatialshift between the two involved vibrational states and for the transferred momentum ofboth optical photons ∆ k s,s (cid:48) = k opt + (cid:126)k sp · (cid:126)e x in the optical repumping process, withFigure 7: Microwave sideband cooling scheme in a realistic physical system using Cs atoms. (i) Microwave radiation tuned to the first blue sideband induces a |↑ , n (cid:105) → |↓ , n − (cid:105) transition decreasing the motional quantum number by one. (ii)The cooling cycle is closed by an optical repumping transition |↓(cid:105) → | F (cid:48) = 4 (cid:105) , withrate R ↓ , and (iii) a spontaneous decay back to state |↑(cid:105) . In (iv) an additional pumpinglaser brings the atoms which have decayed to state | a (cid:105) back into the cooling cycle, withrate R a . Atoms reaching the dark state |↑ , n = 0 (cid:105) are out of the cooling cycle unlessoff-resonantly excited or heated externally. ONTENTS (cid:126)k sp being the wavevector of the spontaneously emitted photon and (cid:126)e x being the unitvector along the lattice direction. Additionally, one has to perform an average over (cid:126)k sp ,indicated by the angle brackets in equation (28) .Given our experimental parameters, we compute the steady-state solution toequation (25) numerically, using the same approximations as in section 2.3. In thecomputation, the microwave is resonant with the |↑ , (cid:105) → |↓ , (cid:105) transition. The rates R a and R ↓ are set by the experimental values, which are chosen comparable to Ω , andsmaller than the vibrational level separations to avoid off-resonant transitions by powerbroadening of the vibrational levels of the F = 3 ground state. Figure (8) shows acontour plot of the ground state population P | n =0 (cid:105) ≡ (cid:80) s,r P | s, ,r (cid:105) in the steady state as afunction of the bare microwave Rabi frequency and the relative shift distance expressedin terms of η x . When projection heating dominates, η x (cid:38) η k , the energy balance inequation (24) just requires η x < for cooling; for instance, figure (8) shows that aground state population P | (cid:105) > can be reached with η x < . . For very small latticeshifts however, with η x (cid:28) , the microwave coupling for the blue sideband transitionbecomes small compared to that of the carrier transition, which renders the microwaveaction of removing an energy (cid:126) ω vib per cycle inefficient compared to the recoil heating,which is the dominant heating source for such small shifts. Weak microwave sidebandcoupling and hence inefficient microwave cooling will also be present at very low bareRabi frequency, namely at the Rabi frequencies where the sideband coupling becomeslower than the rate of depopulation of the dark state. For high Rabi frequencies of thesame order of magnitude as the vibrational level spacing, the carrier coupling becomescomparable to the blue sideband coupling, and the microwave cooling action is againreduced. Figure 8: Steady state population in the motional ground state P | n =0 (cid:105) as a function of η x and the bare microwave Rabi frequency Ω . ONTENTS Microwave cooling is obtained by applying microwave radiation on resonance with thefirst blue sideband, |↑ , (cid:105) → |↓ , (cid:105) , for a certain duration τ cooling , at a certain lattice shift ∆ x , concurrently with the two optical pumping lasers as shown in figure (7). In orderto probe the final vibrational state distribution, we record a spectrum of the first ordersideband transitions using a Gaussian microwave pulse satisfying the π -pulse conditionfor the first red sideband, figure (9a). In the low temperature limit, the height of the firstblue sideband peak provides a good measure of the motional ground state population, P |↑ , (cid:105) , and thus of the cooling efficiency. For instance, for atoms in the ground state oneexpects to detect no blue sideband. Figures (9a) and (9b) show two microwave spectrarecorded before and after cooling, clearly indicating a reduction of temperature by thecooling process.In order to determine the optimum cooling parameters, the blue sideband height isremeasured while scanning different variables, namely the optical pumping intensities,the cooling microwave power and frequency, the lattice shift distance and the durationof the cooling pulse. Figure (9c) shows a scan of the cooling microwave frequency. Asindicated by a nearly zero detected signal from the blue sideband, the optimum frequencyfor cooling lies evidently in the vicinity of the first blue sideband, while a less pronouncedcooling is also present at the position of the second blue sideband. Furthermore, themeasurement reveals the absence of the blue sideband signal in a broad range extendingto negative detunings in addition to a weak dip at the position of the carrier. These twoobservations are correlated with a decrease in the atom survival given in the same figure.This shows that, instead of being due to cooling, the absence of the blue sideband hereis due to increased atom losses. In fact, for zero and negative microwave detunings,that is if the microwave is resonant with the carrier |↑ , n (cid:105) → |↓ , n (cid:105) or red sideband |↑ , n (cid:105) → |↓ , n + m (cid:105) transitions respectively, the energy of the atom increases on averagein each cooling cycle. In the case of zero detuning the increase is due mainly to recoiland projection heating in the absence of microwave cooling, while for negative detuningsmicrowave sideband heating occurs in addition to the recoil and projection heating.Once the optimum cooling parameters have been determined, we extract theachieved steady state temperature assuming a thermal Boltzmann distribution andneglecting the anharmonic spacing of the vibrational levels. The ratio between thered and blue sideband heights is proportional to the Boltzmann factor which is relatedto the average motional quantum number (cid:104) n (cid:105) by P ↑ , P ↑ , = exp( − (cid:126) ω x k B T ) = (cid:104) n (cid:105)(cid:104) n (cid:105) + 1 (30)Using the fitted sideband heights from figure (9b), we calculate (cid:104) n (cid:105) = 0 . ± . , anda ground state population of P ↑ , (cid:39) , corresponding to a temperature T (cid:39) . µK.Table (2) summarizes the optimum cooling parameters for our setup. ONTENTS ω vib = 2 π × kHz . Ω / π η x η k R ↓ R a R ↑ τ cooling kHz . . ms − ms − s − ms
4. Motional state control
We have developed a vibrational state detection scheme which allows us to determinethe vibrational state distribution of any given motional state. It relies on removing allatoms above a selected vibrational state n from the trap and counting the remainingatoms, as illustrated in figure (10a). The distribution is then reconstructed from thedifferences of subsequent measurements.Atoms are first transferred to state |↓(cid:105) by means of an adiabatic passage microwavepulse that is resonant with the carrier transition in unshifted lattices, which preservesvibrational states’ populations. A microwave pulse resonant with the red sideband |↓ , n (cid:105) → |↑ , (cid:105) transfers atoms from states |↓ , m (cid:105) with m (cid:62) n to states | ↑ , m − n (cid:105) .The transferred atoms are eventually pushed out of the trap (see section 2.2). However,since the sideband transition rates depend on the initial vibrational state |↓ , n (cid:105) (dueto, e.g., trap anharmonicity and Franck-Condon factor differences) the microwave pulsedoes not achieve full transfer efficiency for all transitions. To overcome this limitation,the procedure of microwave pulse plus push-out is repeated several times to deplete allvibrational states |↓ , m (cid:105) with m (cid:62) n . If f is the population transfer efficiency for agiven n , then after N repetitions the effective population transfer efficiency becomes f (cid:48) = 1 − (1 − f ) N . For instance, an initial efficiency of f = 70% is thus increased to f (cid:48) ∼ with N = 3 repetitions. Measuring the fraction of remaining atoms as afunction of the microwave frequency, we obtain a sequence of plateaus at the successiveFigure 9: Microwave spectroscopy performed (a) before cooling and (b) after msof microwave cooling, with optimal experimental parameters (see Table 2). (c) Bluesideband height vs. detuning of the microwave cooling frequency ( ◦ ) and atom survivalprobability measured after the cooling ( (cid:3) ), (data points in (a) and (b) are from [11]). ONTENTS n th sideband indicates the integrated population of states m < n , that is, the cumulativedistribution function F n = (cid:80) n − m =0 p m , from which the individual populations of thevibrational states are then derived.Figure 10: (a) Method for measuring vibrational state population distributions: (i) aninitial microwave pulse resonant with the n th red sideband transfers all atoms in states |↓ , m (cid:105) , with m (cid:62) n , to state |↑(cid:105) ; (ii) the transferred atoms are pushed out of the lattice;(i) and (ii) are repeated N times to overcome low pulse efficiency. (b) Surviving fractionof atoms for a thermal state, with the dotted lines indicating a thermal distribution of T ≈ . µK; this temperature is compatible with the independently measured one ofabout µK. For each sideband n , after N = 3 repetitions of the microwave pulse pluspush-out, only the atoms in states m < n survive. The horizontal dashed line indicatesthe maximum survival probability, limited by the off-resonant transitions during therepeated pulses. For the sake of clarity, error bars have been displayed for the carriertransition only. With 97% of the atoms cooled to state |↑ , n = 0 (cid:105) (see section 3.3), controlled preparationof different motional states is possible using a combination of microwave pulses andselected lattice shifts.The simplest state that can be prepared is the Fock state |↓ , m (cid:105) . It requiresaddressing the m -th red sideband transition at the lattice shift ∆ x chosen to maximizethe coupling |↑ , (cid:105) ↔ |↓ , m (cid:105) . The fidelity for preparing this state is limited by the coolingefficiency, the population transfer efficiency and the selectivity of the microwave pulse.Using an adiabatic passage pulse [32], a state preparation fidelity close to 98% has beenachieved for states up to m = 6 .A superposition of two Fock states is created by a two-pulse sequence as shownin the inset of figure (11b). An initial microwave pulse resonant with the transition ONTENTS Vibrational state n Vibrational state n=0.3=0.6=1.4=1.7 P opu l a t i on P , n P opu l a t i on P , n Figure 11: Motional state preparation and analysis. Shown are the populations of thevibrational states n = 0 , .., after (a) creating superposition states of | n = 0 (cid:105) and | n = 2 (cid:105) with different weights (from top to bottom, area of the first MW pulse 0.30, 0.40, 0.55,0.70 in units of π ) and (b) coherent vibrational states for different amplitudes α , wherethe left bars (brighter red) indicate the theoretically expected populations. The analysistechnique used here, see figure (10a), can only measure vibrational states’ populationsbut not coherences. |↑ , (cid:105) → |↓ , (cid:105) , performed at the lattice shift which maximizes the coupling for thetransition, generates the state | ψ (cid:105) = c ↑ , |↑ , (cid:105) + c ↓ , |↓ , (cid:105) (31)with variable coefficients c ↑ , and c ↓ , determined by the pulse duration. The latticeshift ∆ x is then changed to the distance at which the Franck-Condon factor for thetransition |↑ , (cid:105) ↔ |↓ , (cid:105) is zero. The shifting is precisely timed so that the probabilityof changing the vibrational state by the acceleration of the lattices is zero [33]. At thenew lattice shift, a second microwave pulse resonant with the carrier transition mapsthe population | c ↑ , | onto |↓ , (cid:105) . One would expect as a result a coherent superpositionbetween |↓ , (cid:105) and |↓ , (cid:105) . However, because of the appreciable duration of the sequence of µs (two sideband-resolved pulses plus lattice shift operation) compared to the totalspin coherence time of ∼ µs in our setup, the coherence between the two vibrationalstates is partially lost during the preparation procedure. This is a technical limitationwhich can be overcome by improving the coherence time, for instance in our setup, bycooling the transverse motion of the atoms to the three-dimensional ground state [34]. ONTENTS |↓ , n = 0 (cid:105) ontoa shifted potential to create the state | α (cid:105) = ˆ T ∆ x |↑ , n = 0 (cid:105) = e αa † − α ∗ a |↑ , n = 0 (cid:105) (32)with α = η x . We realize this by applying an optical repumping pulse while the latticeis displaced by ∆ x . This corresponds to exciting the transition |↓(cid:105) → | e (cid:105) followedby a spontaneous decay to state |↑(cid:105) , which occur on a time scale much shorter thanthe oscillation period of the atom in the trap. We also neglect the recoil transferredby the optical repumping photons, which is equivalent to assuming η k = 0 . Becausethe decay process additionally involves transitions to states | a (cid:105) and |↓(cid:105) , the resultingstate is a statistical mixture of the three internal states; our analysis scheme howevermeasures the vibrational population of the state projected on |↑(cid:105) state shown in equation(32). The statistical mixture can be avoided by replacing the optical repumpingpulse and spontaneous decay by a fast two-photon Raman transition. Measuring thepopulation distribution of the created state reveals a clear agreement with the theoreticalexpectation, as shown in figure (11b). With the state detection scheme presented insection 4.1, so far we can only measure populations, while coherences could be accessedin the future through interferometric schemes.
5. Conclusions and outlook
We have shown that microwave sideband transitions in spin-dependent optical latticesare a favorable alternative to Raman transitions for sideband cooling and motionalstate engineering. The effective Lamb-Dicke parameter can be continuously adjustedfrom zero to above one, giving the possibility to address directly higher-order sidebandwith coupling frequencies comparable to the bare Rabi frequency. We investigated theperformance of microwave sideband cooling in the generalized Lamb-Dicke regime, andwe compared it to the Raman sideband cooling; our analysis can be easily extended tothe three-dimensional case [15].Quantum engineering of motional states represents one of the most attractive uses ofmicrowave-induced sidebands. We demonstrated here a first step towards the creationof superposition between Fock states, and the preparation of coherent states. In thefuture, the interest resides in proving the coherence properties of these states throughinterferometric schemes, for instance, by measuring the accumulated phase between twodistinct Fock states, or through quantum beat experiments [35].Along the same line, spin-dependent shift operations can be employed to transfer astate-dependent momentum kick, allowing the realization of a superposition of oppositecoherent states, producing Schrödinger-cat-like states as has been realized with ions [36].Microwave control of atomic motion in a spin-dependent optical lattice can be ofinterest for storing and processing quantum information via the motional states [37]. For
ONTENTS ∆ x is close to d/ , i.e., close to half the lattice spacing [38].Finally, it is worth noting that the microwave cooling technique studied here doesnot strictly require the use of the “magic” wavelength for the lattice potential, but canstill be operated with the same efficiency at other wavelengths, e.g., at λ L = 1064 nmas we have tested. In fact, the optimal cooling efficiency occurring at around η x ∼ . ,see figure (8), can be reached by adjusting the polarization angle θ . Acknowledgments
We thank Andreas Steffen and Tobias Kampschulte for fruitful discussions. Commentsby an anonymous referee helped improve the manuscript. We acknowledge financialsupport by the DFG Research Unit (FOR 635), NRW-Nachwuchsforschergruppe“Quantenkontrolle auf der Nanoskala”, AQUTE project, and Studienstiftung desdeutschen Volkes. AA acknowledges also support by Alexander von HumboldtFoundation.
References [1] Wineland D J, Drullinger R E and Walls F L 1978 Radiation-Pressure Cooling of Bound ResonantAbsorbers
Phys. Rev. Lett. Phys. Rev.Lett. Rev. Mod. Phys. Rev. Mod. Phys. Phys. Rev. A Phys. Rev. Lett. Europhys. Lett. Phys.Rev. Lett.
Phys. Rev. A Proc.SPIE
Phys. Rev. Lett.
Nature
ONTENTS [13] Kohn W 1959 Analytic Properties of Bloch Waves and Wannier Functions Phys. Rev.
Phys. Rev, Lett. Phys. Rev. Lett.
Spectra of Atoms and Molecules (Topics in Physical Chemistry)
Handbook of Mathematical Functions
Ninth printing (NewYork: Dover)[18] Slater J C 1952 A Soluble Problem in Energy Bands
Phys. Rev. ACM T. Math. Software Nature
New J. Phys. Phys. Rev. Lett. Phys. Rev. A Phys. Rev. Lett. Phys. Rev. A Phys. Rev. Lett. Rev. Mod. Phys. Phys. Rev. A Phys. Rev. Lett. Phys. Rev. A Phys. Rev. A Phys. Rev. A Science
Phys. Rev. X Nature Phys. Science
Phys. Rep.
Phys. Rev. A81