Pump-probe spectroscopy of two-body correlations in ultracold gases
aa r X i v : . [ qu a n t - ph ] D ec Pump-probe spectroscopy of two-body correlations in ultracold gases
Christiane P. Koch ∗ and Ronnie Kosloff Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany Institute of Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel (Dated: November 5, 2018)We suggest pump-probe spectroscopy to study pair correlations that determine the many-bodydynamics in weakly interacting, dilute ultracold gases. A suitably chosen, short laser pulse depletesthe pair density locally, creating a ’hole’ in the electronic ground state. The dynamics of this non-stationary pair density is monitored by a time-delayed probe pulse. The resulting transient signalallows to spectrally decompose the ’hole’ and to map out the pair correlation function.
PACS numbers: 03.75.Kk,32.80.Qk,82.53.Kp
Introduction
Bose-Einstein condensation in dilutegases is determined by the nature of the two-body in-teractions between the atoms [1]. These microscopic in-teractions manifest themselves in two-body correlationsand dictate the mesoscopic and macroscopic propertiesof the condensate. Formally, the dynamics of an ultra-cold gas is described in terms of field equations [1]. Fordilute gases where only two-body interactions are promi-nent, the equation of motion for the field operator thatannhilates (or creates) a particle at position ~x reads i ~ ∂ ˆ ψ ∂t ( ~x ; t ) = ˆH ˆ ψ ( ~x ; t ) + (1) Z d ~y ˆV ( ~x − ~y ) ˆ ψ † ( ~y ; t ) ˆ ψ ( ~y ; t ) ˆ ψ ( ~x ; t ) . Expectation values of the many-body system can be ex-pressed in terms of normal-ordered correlation functions.To lowest order, these are the condensate or mean fieldwavefunction, Ψ( ~x ; t ) = h ˆ ψ ( ~x ; t ) i , the one-body densitymatrix R ( ~x, ~y ; t ) = h ˆ ψ † ( ~x ; t ) ˆ ψ ( ~y ; t ) i and the pair corre-lation function, Φ( ~x, ~y ; t ) = h ˆ ψ ( ~x ; t ) ˆ ψ ( ~y ; t ) i . For practi-cal calculations, the infinite set of equations of motionfor the many-body problem needs to be truncated. Thiscan achieved by expanding the correlation functions intocumulants [2]. A separation of time or length scales, i.e.small collision time vs long free propagation time or smalleffective range of the interaction potential vs large inter-atomic distance is required to justify truncation. Fordilute Bose gases in a macroscopic trap, such an assump-tion can typically be made, and within the first-ordercumulant expansion, the dynamics of pair correlations isdecoupled from higher order terms [2]. Alternatively, onecan work with the correlation functions directly [3]. Inboth cases, the dynamics of the macroscopic pair correla-tion function is described by a Schr¨odinger-like equationwhere the mean field enters as a source term [2] or actsas an additional potential [3]. If we restrict our con-siderations to timescales that are much shorter than thetimescale of the mean field dynamics, the pair correlationdynamics are described by a standard Schr¨odinger equa-tion where the presence of the condensate only modifies the boundary conditions. The macroscopic pair correla-tion function is then given by the two-body wavefunctionof an isolated pair of atoms, Φ( r ) with r = | ~x − ~y | . Wecan thus study the many-body pair correlation dynamicsby solving the time-dependent Schr¨odinger equation fortwo colliding atoms [3].In the most simplified approach, the effect of the many-body pair correlations is captured in a single parameter,the scattering length [1]. Measuring the scattering lengthcorresponds to an indirect assessment of the pair corre-lations. If the two-body interaction is probed in a timemuch shorter than its characteristic timescale, a morecomprehensive study becomes possible. Here we suggestto employ pump-probe spectroscopy to unravel the dy-namics of pair correlations in an ultracold Bose gas. Thisrequires a combination of ultrafast and ultracold physics, interatomic distance [ Bohr radii ] -2000200 hole dynamics probe window e n e r gy [ c m - ] pump & probe pulses(time-delayed)ionization pulse(simultaneous with probe pulse) FIG. 1: (Color online) Pump-probe spectroscopy of dynami-cal pair correlations: A pump pulse excites population fromthe electronic ground, leaving the pair correlation functionin a non-stationary state, the ’hole’. A time-delayed probepulse monitors the dynamics of the ’hole’. The orange peaksindicate the action where the probe pulse measures pair am-plitude. the basic basic feasibility of which has been demonstratedin recent experiments on femtosecond photoassociation ofultracold rubidium atoms [4, 5].
Pump-probe spectroscopy
Our scheme involves threeshort laser pulses and is sketched in Fig. 1 for the exampleof an ultracold gas of Rb atoms. The pump pulse ex-cites population from the electronic ground to an excitedstate, leaving a ’hole’ in the initial pair correlation func-tion. The ’hole’ represents a non-stationary state thatmoves under the influence of the ground state potential,cf. Fig. 1. The pump pulse thus induces the dynamicsof the pair correlations. A time-delayed probe monitorsthese dynamics by measuring the amount of probabilityamplitude in a range of internuclear distances. The mea-surement consists of applying simultaneously a photoas-sociation and an ionization pulse (combination of red andgreen arrows in Fig. 1). The pair density on the groundstate is thus photoassociated and immediately ionized fordetection. The largest probe signal is obtained when theprobe pulse is identical to the pump pulse. The dynam-ics are then monitored at the position where the ’hole’was created. The spatial region where the probe pulsedetects pair density is indicated in orange in Fig. 1.For alkali atoms, the initial pair correlation functionconsists of a superposition of singlet and triplet com-ponents. The corresponding interaction potentials areshown in solid (singlet) and dashed (triplet) lines. Forclarity’s sake, only triplet pair wave functions are de-picted in Fig. 1. Since the respective excited state poten-tials differ, the probe pulse is resonant at different inter-atomic distances for singlet and triplet. This is indicatedby the orange peaks representing the probe windows inFig. 1 (striped for singlet, dotted for triplet).Detection of the pair correlation dynamics proceedsvia the creation of molecular ions and is inspired byRefs. [4, 5]. In those experiments, the pump pulsephotoassociates ultracold atoms and the ensuing excitedstate molecular dynamics are detected by the ionizationpulse. In contrast to that, it is the ground state dynam-ics of the many-body pair correlations that are probedin our proposal; photoassociation just serves as a meansfor detection. Hence, while the experimental setups envi-sioned in this work and realized in Refs. [4, 5] are similar,the probed physics is rather different.
Modelling the two-body interaction
We consider twocolliding Rb atoms. Hyperfine interaction couples theground state singlet and lowest triplet scattering chan-nels. However, this interaction cannot be resolved on thetimescales considered below. We therefore assume a su-perposition of singlet and triplet components, but neglectthe effect of hyperfine interaction on binding energies anddynamics. The two-body Hamiltonian is represented ona grid large enough to faithfully represent the scatteringatoms. The interaction of the atom pair with the pumppulse is treated within the dipole and rotating wave ap-proximations. Excitation is considered exemplarily into the 0 + u (5 s +5 p / ) and 0 − g (5 s +5 p / ) excited states. Thepulses are taken to be transform-limited Gaussian pulseswith a full-width at half-maximum (FWHM) of 10 ps.This corresponds to a spectral bandwidth of roughly1 . − or 45 GHz. Details on the potentials and theemployed methods are found in Ref. [6]. For a Bose-Einstein condensate, a single low energy scattering stateneeds to be considered. The collision energy of this initialstate is chosen to correspond to 20 µ K with 75% (25%)triplet (singlet) character. At higher temperatures, thebosonic nature of the atoms can be neglected; and theultracold thermal ensemble is described by a Boltzmannaverage over all thermally populated two-body scatteringstates [7].
Modelling the absorption of the probe pulse
The dy-namics of the non-stationary ’hole’ is monitored by acombination of a probe pulse and an ionization pulse,cf. Fig. 1. This two-color scheme converts the absorp-tion of the probe pulse into detection of molecular ions.Assuming the probe pulse to be weak and the ionizationstep to be saturated, absorption of the probe pulse can bemodelled within first order perturbation theory [9, 10].The transient absorption signal is then represented by thetime-dependent expectation value of a window operator, ˆW ( ˆr ) = π ( τ p E p, ) e − ˆ∆ ( ˆr ) τ p · ˆ µ , (2)where τ p and E p, denote duration (FWHM) and peakamplitude of the probe pulse. ˆ µ p is the transition dipolemoment between ground and first excited state. Thecentral frequency of the probe pulse, ω p , determines thedifference potential, ˆ∆ ( ˆr ) = V e ( ˆr ) − V g ( ˆr ) − ~ ω p . Aposition measurement becomes possible if the differencebetween the ground and excited state potential is suffi-ciently large, and, moreover, the spectral bandwidth suf-ficiently small to probe only non-zero ˆ∆ ( ˆr ). Since thedifference potential vanishes for r → ∞ , this implies suffi-ciently detuned, narrow-band probe pulses. Fig. 1 showstriplet and singlet window operators (orange peaks) as-suming identical parameters for pump and probe pulses. Characterization of the ’hole’
As sketched in Fig. 1,the pump pulse carves a ’hole’ into the ground state paircorrelation function. The resulting non-stationary wavepacket is a superposition of a few weakly bound vibra-tional wave functions and many scattering states [8]. Thedetuning of the pump pulse from the atomic resonancefrequency, ∆ L = ω L − ω at , determines the position wherethe ’hole’ is created: For larger detuning, excitation oc-curs at shorter distance and populates deeper bound lev-els. A pulse energy of E P = 1 . time delay [ ps ] w i ndo w e xp ec t a ti on v a l u e < W ( t ) > × -6 × -5 × -6 × -6 (b) ∆ L = -4.0 cm -1 ∆ L = -14 cm -1 (a) FIG. 2: (Color online) Probing the two-body correlation dy-namics: Absorption of the probe pulse as a function of thedelay between pump and probe pulses. The pump pulse de-tunings are ∆ L = − . − (a) and ∆ L = −
14 cm − (b),and three different pump pulse energies are shown. Dynamics of the ’hole’
Fig. 1 also illustrates the timeevolution of the pair correlation amplitude after a weakpump pulse ( E P = 1 . t = 24 ps (taking t = 0 to be the time of the pumppulse maximum). A probe measurement at that timewill find no amplitude within the probe window. Dueto the attractive interaction potential, the ’hole’ movestoward shorter distances, cf. the brown wave function( t = 126 ps). This brings amplitude that is initially atlarger r into the probe window. Eventually the motion ofthe ’hole’ will be reflected at the repulsive barrier of thepotential. The bound part of the wave packet will remainat short distance and oscillate, while the scattering partwill pass through the probe window once not to return. Pump-probe spectra
The dynamics of the ’hole’ is re-flected in the transient probe absorption, i.e. the time-dependent expectation value of the window operator (reddotted curve in Fig. 2 a): A depletion of the signal due tothe creation of the ’hole’ by the pump pulse, referred toas ’bleach’ in traditional pump-probe spectroscopy, is fol-lowed by a recovery that peaks at 550 ps. At later timesoscillations due to partial recurrence are observed but afull recovery does not occur. Rabi cycling induced bylarger pulse energies may partially (green solid curve) orcompletely (blue dashed curve) refill the ’hole’. For largerdetuning, cf. Fig. 2b, the ’hole’ is created at shorter in-teratomic distance, r ∼
48 a for ∆ L = −
14 cm − vs r ∼
76 a for ∆ L = − − . Obviously, the time tomove to the repulsive barrier and back is then shorter.A faster recovery of the bleach is hence observed – at t = 110 ps in Fig. 2b. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 020406080 r ec on s t r u c t e d s p ec t r u m o f < W ( t ) > [ a r b . un it s ] -1 -0.5 0 0.5 1 0246810-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.41 × -4 × -3 × -2 r e l a ti v e s p ec t r a l w e i gh t s -1 -0.5 0 0.5 1 frequency [ cm -1 ] × -7 × -6 × -5 ∆ L = -4.0 cm -1 ∆ L = -14 cm -1 (a)(b) scaleddown by 2 scaleddown by 2 FIG. 3: (Color online) Spectra of the transient absorptionsignals shown in Fig. 2. The eigenergies of the interactionpotential are recovered: Black arrows indicate the positionof the binding energies of the triplet (upper row) and singlet(lower row) levels, grey arrows half-multiples of the bindingenergies. The spectra for 23 . The spectrum of the transient absorption signals shownin Fig. 2 can be obtained by filter-diagonalization [11], amethod allowing to accurately extract frequencies fromjust a few oscillation periods. The spectra are shownin Fig. 3. The vibrational energies of the two-body in-teraction potential are recovered. Fig. 3 shows further-more that Rabi cycling during the pump pulse leads to alarger bound part in the ’hole’, cf. the increase of spectralweights with pump pulse energy. While a direct measure-ment of the vibrational populations would be difficult toimplement experimentally, wave packet spectral analysisvia probe absorption is fairly straightforward.
Pure state vs thermally averaged dynamics
Pumpprobe spectroscopy of the pair correlations can be appliedto a condensate as well as a thermal ultracold gas. For thetimescales considered here, this translates into compar-ing the dynamics of a pure state to that of an incoherentensemble. Fig. 4 shows the transient probe absorption forthe two cases. Thermal averaging markedly smears outthe recovery of the bleach at 550 ps. In principle, ther-mal averaging introduces two effects – the finite width inscattering energies which is too small to be resolved ona nanosecond timescale, and the contribution of higherpartial waves. The latter becomes particularly prominentin the presence of resonances. Fig. 4 compares calcula-tions including both singlet and triplet channels (a) tothose for the triplet component only (b) in order to high-light the role of shape resonances. For Rb, shape reso-nances are observed at ∼ µ K ( J = 2) and at ∼ µ K( J = 6) in the singlet, and for J = 2 at ∼ µ K in thetriplet channel. The second peak observed in Fig. 4a at1100 ps corresponds to the singlet recovery of the bleach. time delay [ ps ]
T = 25 µ KT = 100 µ KT = 500 µ K no r m a li ze d w i ndo w e xp ec t a ti on v a l u e < W ( t ) > E coll /k B = 25 µ K (b)(a) FIG. 4: (Color online) Absorption of the probe pulse as afunction of the delay between pump and probe pulses com-paring pure state dynamics (black solid lines) to those of athermal ensemble (colored broken lines). (a) Calculation in-cluding two scattering channels (singlet and triplet): Thethermal dynamics are dominated by a singlet shape reso-nance while the pure dynamics shows features of both singletand triplet dynamics. (b) Calculation for a single channel(triplet): The thermal and pure state dynamics are similarwith the recovery of the bleach smeared out at higher tem-peratures. (∆ L = − . − , E P = 3 . The shape resonances lead to a much larger weight of thesinglet contribution in the thermal averages than in thepure state s -wave calculation. Since probe absorption inthe singlet channel occurs at larger distances than for thetriplet channel, cf. Fig. 1, the recovery of the bleach isobserved at later times. This observation opens up theperspective of analysing pair correlation functions in cou-pled channels scattering near a resonance where tuningan external field through the resonance will modify therespective weight of the channels. We emphasize that thisnovel pair correlation spectroscopy is possible even in thepresence of, e.g., three-body losses as long as the decayoccurs on a timescale larger than a few nanoseconds. Mapping out the pair correlation function
Pump-probe spectroscopy, a well established tool in chemicalphysics, allows for retrieving the amplitude and phase ofa wave function [12]. In the present context we can re-trieve the pair correlation density operator ρ ( r, r ′ , t ) with ρ ( r, r ′ ; t ) = Φ( r ; t )Φ ∗ ( r ′ ; t ) for a BEC. This is based onEq. (2), where probe absorption corresponds to a posi-tion measurement with finite resolution: Different cen-tral frequencies, ω P , define the position that is mea-sured, and the difference potential, ˆ∆ ( ˆr ), together withthe pulse duration, τ p , control the resolution. These mea-surements resolve the amplitude of the pair correlation, | Φ ∗ ( r ′ ; t ) | . The phase information is obtained by chirp-ing the probe pulse which corresponds to a momentum measurement [10]. The window operator then defines afinite resolution measurement in phase space [13]. Col-lecting the expectation values for a sufficiently large setof window operators with different positions/frequenciesand momenta/chirps corresponds to quantum state to-mography of ρ ( r, r ′ ; t ) [14]. Conclusions
Pump-probe spectroscopy unravels di-rectly many-body pair correlations in dilute Bose gas.Existing experimental setups [4, 5] need to be onlyslightly modified to implement our proposal. In particu-lar, transform-limited pulses of about 1 cm − bandwidthare required for detection of the probe absorption viamolecular ions. Spectral features on a scale of less than1 cm − can be resolved for pump-probe delays of a fewnanoseconds. Pump-probe spectroscopy of the pair cor-relation dynamics allows to capture transient states of ul-tracold gases such as collapsing condensates. Moreover,it can be combined with static external field control. Forexample, tuning a magnetic field close to a Feshbach res-onance may enhance the pair density at short and inter-mediate distances [15]. The resulting coupled channelspair correlation function can be mapped out despite thefinite lifetime of the resonance. Future work will considershaped pulses. Once picosecond pulse shaping becomesavailable, the full power of coherent control can be em-ployed to study pair correlation dynamics. Acknowledgements
We are grateful to F. Masnou-Seeuws and P. Naidon for many fruitful discussions, toour referees for helpful comments and to the DeutscheForschungsgemeinschaft for financial support. ∗ Electronic address: [email protected][1] C. J. Pethick and H. Smith,
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