Femtosecond wavepacket interferometry using the rotational dynamics of a trapped cold molecular ion
FFemtosecond wavepacket interferometryusing the rotational dynamics of a trapped cold molecular ion
J. Martin Berglund, Michael Drewsen, and Christiane P. Koch ∗ Theoretische Physik, Universit¨at Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus, Denmark (Dated: October 22, 2018)A Ramsey-type interferometer is suggested, employing a cold trapped ion and two time-delayed off-resonant femtosecond laser pulses. The laser light couples to the molecular polarization anisotropy,inducing rotational wavepacket dynamics. An interferogram is obtained from the delay dependentpopulations of the final field-free rotational states. Current experimental capabilities for cooling andpreparation of the initial state are found to yield an interferogram visibility of more than 80%. Theinterferograms can be used to determine the polarizability anisotropy with an accuracy of about ± ± I. INTRODUCTION
Interference of waves both in the form of light and mas-sive particles, such as electrons, neutrals, and atoms, hasproven to be a very sensitive method of measuring phys-ical quantities (See e.g. Ref. [1] and references therein).This includes precise measurements of electromagneticfields, gyromagnetic constants, gravitational accelerationand rotation relative to an inertial system. When itcomes to interference of quantum objects with internalenergy structure, the method of Ramsey interferometry,involving two-level objects, has been in particular suc-cessful [2]. The accuracy in determining the internal stateenergies by Ramsey interferometry scales inversely pro-portional to the square root of the interrogation time.Trapped individual atomic ions, laser cooled or sympa-thetically cooled, have led to the most precise measure-ments to date [3]. Such ions, occupying much less thana cubic micron, are ideal for controlled laser excitations,and they can exhibit coherence time in the range of hun-dreds of seconds [4].Here a Ramsey-type interferometer using the rota-tional levels of a trapped and sympathetically cooledMgH + molecular ion [5] is presented, as schematicallyillustrated in Fig. 1. Compared to the standard versionemployed in atom interferometry [6], the π/ ∗ Electronic address: E-mail: [email protected]
FIG. 1: Schematic illustration of a rotational Ramsey in-terferometer with time evolving from left to right. A sym-pathetically cooled MgH + molecular ion, initially preparedin its ground state, interacts with a first off-resonant laserpulse (the Mg + coolant ion is not shown). Then the resultingwavepacket evolves freely for a controllable time delay τ , untila second pulse is applied, generating a new wavepacket. Thisstep is sensitive to the relative individual phases of the free-field states that make up the wavepacket prior to the secondpulse and thus on the time delay. The final rotational pop-ulations are measured by a state-sensitive dissociation pulse(blue arrow). ters and the initial state. In particular, the visibility canbe enhanced by varying the intensity of the second pulse.As an application of the interferometer, the measure-ment of the molecular polarizability anisotropy is dis-cussed, taking both the experimental uncertainties ofthe initial populations as well as population measure-ment errors into account. The interferometric methodto determine the static dipole polarizabilities discussedhere represents, for molecular ions, an interesting alter-native to microwave spectroscopy previously used [8, 9].Conversely, the interferometric technique may be used toprobe local electric fields, like the radio-frequency fieldsat the position of the molecular ion in a (linear) Paultrap. The potentially very long interogation times withtrapped ions can make such an interferometer extremely a r X i v : . [ qu a n t - ph ] N ov sensitive.The article is organized as follows: Section II intro-duces the model and numerical tools. The results arepresented in Sec. III, starting with an assessment of theeffective rotor approximation in Sec. III A, followed byan analysis of the wavepacket created by the first laserpulse and the characterization of the interferograms inSecs. III B and III C. The prospects for measuring thepolarizability anisotropy are discussed in Sec. III D. Sec-tion IV concludes. II. THEORETICAL FRAMEWORK
A single trapped MgH + ion, translationally cooleddown to sub-Kelvin temperatures and in its electronicground state, is considered. The ion interacts with fem-tosecond laser pulses which are far off resonance fromany transition in MgH + and linearly polarized along thelaboratory fixed z -axis. The Hamiltonian describing therovibrational motion of the molecular ion and its inter-action with the off-resonant field is given by ( (cid:126) = 1) ˆH D = ˆT r + V ( ˆr )+ ˆJ m ˆr − I ( t )2 (cid:15) c (cid:16) ∆ α ( ˆr ) cos ˆ θ + α ⊥ ( ˆr ) (cid:17) , (1)where the first two terms describe the radial kinetic andpotential energy, respectively. ˆJ is the orbital angularmomentum operator, and I ( t ) the intensity profile of thelaser pulse. ∆ α ( ˆr ) denotes the molecular polarizabilityanisotropy and α ⊥ ( ˆr ) the molecular polarizability per-pendicular to the interatomic axis. θ is the angle betweenthe polarization vector of the laser pulse and the inter-atomic axis. Assuming linear and parallel pulse polariza-tions, the Hamiltonian (1) is independent of the azimutalangle φ . Therefore ∆ m = 0, and the cos ( θ )-term givesrise to the selection rule ∆ j = 0 , ±
2. The potential curveand polarizabilities are taken from Ref. [10].In Eq. (1), the leading term of the light-matter in-teraction is assumed to be via the ion’s polarizabilityanisotropy. The interaction of the light with the perma-nent dipole moment of the molecular ion has been ne-glected because it averages to zero if the laser pulses areoff-resonant. Simulations of the vibrational dynamics un-der a pulse with λ c ≈
800 nm, starting from the groundvibrational level, yield a total population of vibrationallyexcited levels of the order of 10 − to 10 − , depending onthe laser intensity. This confirms that practically no vi-brational excitations take place which is not surprisinggiven that the energy difference between the ground andfirst excited vibrational level corresponds to a wavelengthof λ ≈ v ≈
10 that vibrationaltransitions may become resonant. However, the matrixelements for these transitions is so small that they playessentially no role.Since for low-lying rovibrational levels the vibrationalenergy is much larger than the rotational energy, the twodegrees of freedom can be adiabatically separated [11]. B ν =0 (cid:104) ∆ α (cid:105) ν =0 (cid:104) α ⊥ (cid:105) ν =0 . − . · − cm . · − cm TABLE I: Parameters of MgH + , used in Hamiltonian (2). To this end, the vibrational eigenfunctions are obtainedby diagonalizing ˆH vib , given by the first two terms inEq. (1). Denoting radial expectation values by (cid:104)·(cid:105) ν forthe ν th vibrational level, the vibrational motion can beintegrated out in Eq. (1). This yields the so-called Ef-fective Rotor Approximation (ERA) [11] where all ˆr -dependent quantities in Eq. (1) are replaced by their ex-pectation values, ˆH ν = B ν ˆJ − I ( t )2 (cid:15) c (cid:16) (cid:104) ∆ α (cid:105) ν cos ˆ θ + (cid:104) α ⊥ (cid:105) ν (cid:17) , (2)with B ν = m (cid:104) r − (cid:105) ν . The ERA neglects ro-vibrationalcouplings but goes beyond the rigid rotor approximation,since its parameters are obtained by integrating over thevibrational motion instead of just replacing ˆr by the equi-librium distance. The values of the molecular parametersin Eq. (2) for MgH + , calculated using the ab initio dataof Ref. [10], are listed in Table I; they are found to be ingood agreement with the experimental values of Ref. [12].The rotational dynamics are characterized by the ro-tational period, τ rot = (cid:126) / (2 B ν =0 ), which amounts to τ rot ≈
420 fs for MgH + . This short rotational period is aconsequence of the large difference in the atomic massesand the small hydrogen mass. Rotational wavepacket re-vivals occur when the wavepacket returns to its initialstate. They can be analysed by the correlation function, C ( t ) = (cid:104) χ (0) | χ ( t ) (cid:105) = (cid:88) j | c j | e − iE j t , (3)where E j = j ( j + 1) B is the field-free rotational eigenen-ergy with corresponding eigenfunction, χ mj ( θ ) = (cid:104) θ | j, m (cid:105) = P mj ( θ ) . (4)Here, P mj ( θ ) is the associated Legendre function of degree j and order m . Revivals occur at times T , for which C ( T ) = C (0). That is, the conditions 2 πk j = E j T with k j integer need to be fulfilled simultaneously for all j which make up the wavepacket | χ ( t ) (cid:105) .The laser pulses ε ( t ) which create the wavepackets areassumed to have a Gaussian temporal envelope such thatthe pulse fluence becomes P ( I , τ I ) = 2 (cid:15) c (cid:114) π I τ I . (5)Here I is the maximum pulse intensity and τ I is the fullwidth at half maximum (FWHM) duration of the inten-sity profile. In particular, for constant fluence P = P ,the intensity and pulse duration are inversely propor-tional.Three different initial states will be considered in theinvestigations presented below. Ideally for wavepacketinterferometry, the molecule is in its ground rotationalstate ( j = 0). In an experiment, however, a completelypure initial state cannot be fully realized, but recent ex-periments with MgH + ions trapped in a cryogenic en-vironment have led to a nearly 80% rotational groundstate population through helium buffer gas cooling [13].The ideal initial state is therefore compared to a ther-mal ensemble with a rotational temperature of 20 K andto an incoherent ensemble prepared in current room-temperature experiments by rotational cooling [14] withthe same ground state population ( P ∼ .
38) as a ther-mal ensemble at 20 K. An incoherent initial state is de-scribed by a density operator, ˆ ρ : ˆ ρ ( t = 0) = ∞ (cid:88) j =0 j (cid:88) m = − j a j | j, m (cid:105) (cid:104) j, m | . (6)For a thermal state at temperature T , a j = g j exp ( − βE j ) /Z with g j = 2 j + 1, β = 1 /k B T , and Z = (cid:80) j g j exp( − βE j ) the partition function, whereasfor the experimentally prepared initial state, the valuesof a j are taken from Ref. [14].Since the timescale of the interferometer is muchshorter than any decoherence time, the time evolutionis coherent and the density operator at time t is given by ˆ ρ ( t ) = ˆU ( t ) ˆ ρ (0) ˆU † ( t ) . Inserting Eq. (6), each m state may be considered sepa-rately, reducing the numerical effort in the calculations,since the Hamiltonian conserves m . The time-dependentpopulation of the state j (cid:48) , with all corresponding m -states taken into account, is then obtained as ρ j (cid:48) ,j (cid:48) ( t ) = ∞ (cid:88) j =0 a j j (cid:88) m = j (cid:12)(cid:12)(cid:12) (cid:104) j (cid:48) , m | ˆU ( t ) | j, m (cid:105) (cid:12)(cid:12)(cid:12) , (7)i.e., each pure state | j, m (cid:105) is propagated separately, usinga Chebychev propagator, and the resulting population isadded up incoherently with its proper weight. The sum-mation over the initially populated values of j in Eq. (7)can be truncated at j inimax = 6. The basis set expansionin the Legendre polynomials is found to be converged for j max = 20, provided I ≤ × W/cm . III. RESULTS
Before presenting the results obtained with Hamil-tonian (2), the accuracy of the ERA is checked inSec. III A. Then the interferometer is analyzed in a step-wise fashion, starting in Sec. III B with the dependenceof the wavepacket, that is created by the first pulse, onpulse intensity and duration. The complete interferom-eter time evolution is presented in Sec. III C, determin-ing pulse parameters that yield high-visibility interfero-grams. Prospects for using the interferometer to measurethe molecular polarizability are discussed in Sec. III D.
A. Accuracy of the effective rotor approximation
The ERA, Eq. (2), is tested against the full rovibra-tional dynamics, generated by Hamiltonian (1), consider-ing the interaction with one pulse of 100 fs duration. Tothis end, the radial coordinate in Eq. (1) is represented ona Fourier grid, and the time-dependent Schr¨odinger equa-tion with Hamiltonian (1) is solved. The absolute differ-ence in the final-time population is found to be within0.01 for j = 0 , . . . , I ≤ × W/cm .The relative error amounts to less than one percent for j up to j = 4 and intensities up to 1 × W/cm .For intensities 1 × W/cm ≤ I ≤ × W/cm ,the absolute error due to the ERA is within 0.015 for j = 0 , . . . ,
6, whereas the relative error reaches 10% to15%. While both absolute and relative errors becomelarger for higher j states, the ERA is applicable for ourpurposes since low-lying j states ( j ≤
6) are most rele-vant for interferometry and only moderate pulse intensi-ties will be considered to ease experimental feasibility.
B. Creating a rotational wavepacket by a singlefemtosecond laser pulse
Starting from a pure initial state ( j = 0, m = 0 ), afemtosecond laser pulse creates a rotational wavepacketwhich, due to the selection rules, is made up of states j = 0 , , , . . . , all with m = 0. Since the laser-moleculeinteraction is off-resonant and Gaussian pulse envelopesare assumed, the composition of the wavepacket is onlydetermined by the intensity and duration of the pulse.The dependence of the final rotational state populationson pulse intensity and duration is shown in Fig. 2 for j = 0 , , ,
6. Curves of constant pulse duration (inten-sity) correspond to horizontal (vertical) cuts in Fig. 2.The black line indicates pulses of τ I = 100 fs duration.Constant pulse fluences correspond to hyperbolas in thelandscape, cf. Eq. (5). Hyperbolas are clearly visiblein Fig. 2, indicating that it is the pulse fluence that de-termines the population transfer. For the pulse parame-ters corresponding to the upper left part of each panel inFig. 2, the molecule can approximately be described as atwo-level system, consisting of the states j = 0 and j = 2.This should allow for the closest analogy to a Ramsey in-terferometer as used with atoms [6]. For short pulses, τ I (cid:46)
300 fs, the final populations show a more compli-cated behavior, with more states being significantly pop-ulated, save for very small I . A wide range of pulseparameters gives rise to significant population of j = 2,see the lower left corner of Fig. 2 (b). Significant popu-lation of the j = 4 state is obtained for intensities largerthan 0 . × W/cm and pulses shorter than 150 fs.However, the j = 4 state starts to be populated alreadyat smaller pulse intensities. This inhibits a perfect 50%-50% superposition of the states j = 0 and j = 2.In a typical experimental setup, the transform-limitedpulse duration is fixed, i.e., to 100 fs, whereas the inten- FIG. 2: Final population of field-free rotational states (a: j = 0, b: j = 2, c: j = 4, d: j = 6) after interaction with a singlelaser pulse as function of pulse intensity and duration. The initial state is j = 0. The black line indicates pulses of 100 fsFWHM. I (10 W/cm ) | c j (I ) | j = 0j = 2j = 4j = 6 I *0 FIG. 3: Final population of field-free rotational states afterinteraction with a single 100 fs pulse as a function of pulseintensity. The arrow at I ∗ indicates the laser intensity forwhich equal population of the states j = 0 and j = 2 isobtained, corresponding to the black line in Fig. 2. sity is more easily varied. The dependence of the finalstate populations on pulse intensity for a 100 fs pulse ispresented in Fig. 3, as marked by the black lines in Fig. 2.For low intensity, the behaviour is similar to a two-levelsystem, as to be expected from the nearest neighbourcoupling in Hamiltonian (2). As the intensity increases,more levels are populated and the result deviates moreand more from the simple two-level picture. For evenhigher pulse intensities, recurring peaks of low j statesappear. Note that only a small number of levels can besuperimposed at a given field intensity. Intensities forwhich population curves cross, indicating equal popula-tion, are particularly interesting for interferometry sincethey should yield good contrast. The first such occur-rence, at I ≈ . · W/cm for j = 0 and j = 2, ismarked by I ∗ in the figure. Another crossing occurs at I ≈ . · W/cm for j = 2 and j = 4. (10 W/cm )00.10.20.30.4 | c j (I ) | j = 0, Exp. dist.j = 0, 20 K j = 2, Exp. dist.j = 2, 20 K j = 4, Exp. dist.j = 4, 20 K j = 6, Exp. dist.j = 6, 20 K FIG. 4: Same as Fig. 3 but for incoherent initial states, cor-responding to the experimental distribution of Ref. [14] (solidlines) and a thermal ensemble at 20 K (dashed lines).
The dynamics becomes more involved for incoherentinitial states since in this case also states with m (cid:54) = 0are initially populated. Figure 4 shows the final field-free state populations as a function of laser intensity fora 100 fs pulse, comparing two different initial states, theexperimental distribution of Ref. [14] and a thermal en-semble at 20 K. The final j = 0 population of the twoensembles behaves very similarly, with only a small offsetin initial population. Also, both ensembles qualitativelylead to the same dynamics as the pure initial state inFig. 3, confirming that the ensemble dynamics is dom-inated by j = 0, at least for the intensities examinedin Fig. 4. The final j = 2 population on the other handshows some differences between the two initial ensembles,with the peak occurring slightly earlier and the maxi-mum population difference being slightly smaller for thethermal ensemble. For j = 4 and j = 6, the initial popu- τ (ps)00.20.40.60.81 | c j ( τ ) | S j ( E ) j = 0 j = 2 j = 4 j = 6T rev (a)(b) FIG. 5: a: Interferogram, i.e., the final populations of thefield-free rotational states, obtained after interaction of themolecule with two laser pulses, as a function of time delay.Both pulses have I ≈ . × W/cm (as indicated bythe arrow in Fig. 3) and τ I = 100 fs. b: Spectrum S j of thepopulations shown in panel (a). lations are negligible in the thermal ensemble and small,but non-zero in the experimental distribution of Ref. [14].Therefore the resulting final populations after interactionwith the pulse in Fig. 4 are similar to those obtained forthe pure initial state in Fig. 3. Although the initial popu-lation in j = 4 and j = 6 is non-negliglible for the exper-imental distribution, these states do not take part in thedynamics for pulse intensities up to 0 . × W/cm .This is promising in view of obtaining high-visibility in-terferograms even with incoherent initial states. C. Creating and probing rotational wavepacketsusing a sequence of two femtosecond laser pulses
Interferograms are obtained when the molecule inter-acts with two laser pulses, separated by a time delay, τ .For simplicity, the parameters I and τ I are chosen to bethe same for both pulses except where indicated. First,consider the pure initial state with j = 0. The laserintensity and pulse duration were chosen such that thefirst pulse yields equal populations for j = 0 and j = 2,as marked by I ∗ in Fig. 3. Since very little population istransferred to states other than j = 0 and j = 2 by thefirst pulse ( | c | = | c | ≈ . | c | ≈ . j = 0 and j = 2 for all timedelays. The condition fo revivals of the wavepacket cre-ated by the first pulse becomes T rev = π/B ≈ . j states. The visibility is defined | c j ( τ ) | τ (ps)00.20.40.60.81 | c j ( τ ) | j = 0 Exp. dist. 20 K (a)(b) FIG. 6: Interferograms for incoherent initial states, corre-sponding to the experimental distribution of Ref. [14] (red)and a thermal ensemble at 20 K (black), compared to that ofa pure initial state with j = 0 (blue): Final populations of thefield-free rotational states (a: j = 0, b: j = 2) as a functionof pulse delay. Pulse parameters as in Fig. 5. as V j = | c j,max | − | c j,min | | c j,max | + | c j,min | , (8)where | c j,max | ( | c j,min | ) is the maximum (minimum)population of the j th state. The j = 0 state reaches analmost perfect visibility of one and the j = 2 state around0.9, reflecting the almost perfect 50%-50% superpositionof these two states.The Fourier spectra of the delay-dependent final pop-ulations in Fig. 5(a) are shown in Fig. 5(b), S j = (cid:112) F [ f j ]with F denoting the Fourier transform and f j ( τ ) = | c j ( τ ) | . As is evident from Fig. 5(b), the spectrum isuseful to visualize the components of the wave packetcreated by the first pulse, since it displays peaks at theeigenenergies ( E = 0, E = 6 B , E = 20 B ) as well asat the quantum beats ( E / = 14 B ). The similar peakheights of S and S at E = 0, E = 6 B reflect the al-most identical population of the states j = 0 and j = 2.They differ at higher energies, since population from the j = 2 state is further excited into the j = 4 state.Next, Fig. 6 examines the potentially detrimental effectof incoherence in the initial states on the interferogram.It compares the interference patterns for the pure initialstate of Fig. 5 with those obtained for the experimentaldistribution of Ref. [14] and a 20 K thermal ensemble.When measuring the final population of j = 0, the twoincoherent ensembles give practically identical results, cf.Fig. 6(a). Their delay-dependence is qualitatively thesame as that of the pure initial state, but the maximumamplitudes are reduced from close to one down to about0 .
4. Notwithstanding, the visibility, V , is only reducedto 0.96 for the experimental distribution and 0.91 for thethermal ensemble. For j = 2 (Fig. 6(b)), the maximumpopulation still amounts to about 0 .
4, at least for the | c ( τ ) | τ (ps)0.10.20.30.4 | c ( τ ) | τ (ps) ∆α = 16.20 ∆α = 17.01 ∆α = 15.39(a) (b)(c) (d) FIG. 7: Interferograms, i.e., final j = 0 populations asfunction of delay, for different values of the polarizabilityanisotropy ∆ α and increasing intensity of the second pulse(a: identical pulse intensities, b: I = 1 . I , c: I = 1 . I , d: I = 1 . I ). The pulse duration is τ I = 100 fs for both pulses,and the intensity of the first pulse is I = 0 . × W/cm .The initial state corresponds to the experimental distributionof Ref. [14]. experimental distribution. The interferograms of the in-coherent ensembles differ significantly from each other,and they also differ qualitatively from the interferogramof the pure initial state. This simply reflects the fact thatthe dynamics of the j = 2 state is affected by more statesin the initial ensemble. If one wants to test the composi-tion of the initial state interferometrically, measurementof j = 2 is therefore preferred to j = 0. The visibilityof the interferograms for j = 2 is reduced, compared tothe pure state, to 0.85 for the experimental distributionof Ref. [14] and to 0.76 for the thermal state. Thesenumbers are very encouraging in view of the feasibilityof a rotational interferometer. In summary, incoherencein the initial state does not preclude interferometry, inparticular if one measures the j = 0 population. D. Prospect of measuring the polarizabilityanisotropy
The high visibility of the interferograms presented inSec. III C suggests that the rotational interferometer canbe employed to determine molecular parameters such asthe polarizability anisotropy. Figure 7 shows interfero-grams obtained by measuring the final j = 0 popula-tion for several values of ∆ α – in atomic units: 16.20 a (the ab initio value), 17.01 a (5% larger) and 15.39 a (5% smaller), increasing the peak intensity of the secondpulse. Whereas the interferograms are essential identicalfor the three values of the polarizability anitropy if thepulses have the same intensity (Fig. 7a), the curves be- FIG. 8: Interferograms, i.e., final j = 0 populations as a func-tion of time delay, for different values of the polarizabilityanisotropy ∆ α , accounting for 2% (a) and 5% (b) measure-ment uncertainty in the final populations. The initial statecorresponds to the experimental distribution of Ref. [14] with2% uncertainty in the initial populations taken into account( τ I = 100 fs for both pulses, I = 0 . × W/cm for thefirst pulse, I = 1 . · . × W/cm for the second pulse).FIG. 9: Same as Fig. 8 for a smaller range of pulse delays, τ ∈ [3 .
45 ps , .
75 ps]. come more and more distinguishable when the intensityof the second pulse is increased (Fig. 7b-d). The betterdistinguishability comes at the price of a slightly dete-riorated visibility of 0.79 for I = 1 . I (Fig. 7c) com-pared to 0.97 for equal intensities (Fig. 7a). Particularlypromising features are observed for I = 1 . I in Fig. 7cfor delays around 1 . . . j = 0 population of2% and 5% are assumed in Fig. 8(a) and (b), respec-tively. In the case of a 2% measurement error, the inter-ferograms are easily distinguishable from each other, i.e.,the curves including error bars do not overlap, in variousranges of time delays, for example for τ ∈ [1 . . . ± α ,16.52 a and 15.88 a , larger by ±
2% than the ab initio value, have also been included. It is seen that for a 2%measurement error the interferometer is readily sensitiveto ±
2% shifts in ∆ α , whereas for a 5% measurementerror, only shifts of ±
5% can unequivocally be distin-guished.Our predictions for the sensitivity of the rota-tional interferometer are based on averaging over manywavepacket calculations to account for inevitable exper-imental inaccuracies. One might argue that the corre-sponding noise effects may come into play differently inthe ERA and the full rovibrational dynamics. In order tobe sure that our conclusions are not compromised by abreak-down of the ERA, Figs. 8 and 9 compare the inter-ferogram obtained within the ERA for the ab initio valueof ∆ α with that obtained from full two-dimensional cal-culations, using Hamiltonian (1). While slight deviationsin the error bars between ERA and 2D model are visible,in particular in Fig. 8(b) and 9(b), they are sufficientlysmall not to affect the confidence levels stated above.That is, a sensitivity of the interferometer to changes inthe polarizability anisotropy of ±
2% ( ± IV. CONCLUSIONS
A Ramsey-type interferometer, employing off-resonantfemtosecond laser pulses to induce rotational wavepacketdynamics in a trapped, cooled MgH + molecular ion, canbe implemented using current experimental capabilities.Unlike in atom interferometry, where it is comparativelystraight-forward to pick two isolated levels for Rabi cy-cling, application of the second pulse leads to rotationalladder climbing in the molecule. Perfect visibility ofthe interferogram can thus only be obtained for j = 0,whereas measuring j = 2 leads to 90%. It also requires apure initial state with j = 0, m = 0.Preparing a molecule perfectly in its rovibrationalground state is a very challenging task. However, aground state population of 38%, as prepared in Ref. [14], is found to decrease the visibility for j = 0 to only 96%and that for j = 2 to only 85%. Even for a thermaldistribution with a rotational temperature of 20 K, high-visibility interferograms are predicted. The required in-tensities are moderate for the case of the MgH + molecu-lar ion, of the order of 10 W/cm for 100 fs laser pulses.This suggests feasibility of a rotational Ramsey interfer-ometer, combining standard trapping and cooling tech-niques for molecular ions with 800 nm femtosecond laserpulses.Such an interferometer could be used for example todetermine the molecular polarizability anisotropy in thevibrational ground state by comparing an experimentalinterferogram to theoretical predictions for various valuesof (cid:104) ∆ α (cid:105) . Taking experimental uncertainties in the ini-tial populations as well as population measurement errorsinto account, the interferometer is found to be sensitiveto changes in the polarizability anisotropy of ± ±
5% assuming the same level of experimentalinaccuracy.It will be interesting to see whether the rotationalRamsey interferometer can also be used to determine thedependence of the molecular polarizability anisotropy onthe internuclear separation. A possible route could beprovided by recording rotational interferograms for sev-eral vibrational states. Alternatively, one could exploitthe full rovibrational dynamics. In both cases, however,it might turn out to be difficult to disentangle effectsthat are due to the shape of the potential energy curve(which is also known only approximately) from thosethat are caused by the r -dependence of the polarizabilityanisotropy.Additionally, the interferometric technique may be uti-lized to probe local electric fields, such as the radio-frequency fields at different positions within a (linear)Paul trap. Obviously, the rotational Ramsey interfer-ometer will work as well for other molecular ion species,provided the laser light does not drive resonant transi-tions. Acknowledgments
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