Measurements and analysis of response function of cold atoms in optical molasses
Subhajit Bhar, Maheswar Swar, Urbashi Satpathi, Supurna Sinha, Rafael D. Sorkin, Saptarishi Chaudhuri, Sanjukta Roy
MMeasurements and analysis of response function of cold atoms in optical molasses
Subhajit Bhar , ∗ Maheswar Swar , Urbashi Satpathi , † Supurna Sinha ,Rafael D. Sorkin , , Saptarishi Chaudhuri , and Sanjukta Roy ‡ Raman Research Institute, C. V. Raman Avenue, Sadashivanagar, Bangalore-560080, India. International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India and Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada (Dated: January 25, 2021)We report our experimental measurements and theoretical analysis of the response function of coldatoms in a 3D optical molasses. The response function of the atomic cloud is measured by applyinga pulsed homogeneous magnetic field which provides the perturbing force. We observe an interestingtransition from a damped oscillatory to an over-damped behaviour of the response function stemmingfrom a competition between the viscous drag provided by the 3D optical molasses and the restoringforce of the Magneto-Optical Trap (MOT). Our observations are in good qualitative and quantitativeagreement with the predictions of our theoretical model based on the Quantum Langevin equation.We also study the variation of the Diffusion coefficient of the cold atoms as the atomic cloudis cooled to lower temperatures via sub-Doppler cooling in a 3D optical molasses and compare ourexperimental data with the theoretical calculations using the Stokes-Einstein-Smoluchowski relation.
I. INTRODUCTION
The response function of a physical system in thepresence of an external driving force reveals the intrin-sic characteristics of the system e.g. electric polaris-ability, impedance of an electronic circuit and magneticsusceptibility[1–3]. In particular, the study of the diffu-sive behaviour of a system provides crucial informationregarding its transport properties. Diffusion or the Brow-nian motion of a particle in a dissipative medium in thepresence of friction and fluctuating forces is described bythe Langevin equation[1–4].In recent years, diffusion of a Brownian particle in thepresence of quantum zero-point fluctuation was analysedin [5, 6] starting from the fluctuation-dissipation theorem[1, 3]. The fluctuation-dissipation theorem is a linear re-sponse theorem according to which the linear response ofa system to an external perturbation can be expressed interms of spontaneous stochastic fluctuations of the sys-tem in thermal equilibrium and vice versa. One of thekey inputs in the analysis presented in [6] is the responsefunction characterising the system. The choice of the re-sponse function was suggested by the model of a viscousmedium. In this paper, we have a concrete experimen-tal measurement of this response function by utilising athree-dimensional configuration of laser beams known asthe ‘optical molasses’ which enables cooling as well as vis-cous confinement of the atomic cloud. In this work, wemeasure the response function of the cold atomic clouddriven by an external force in the form of a pulsed ho-mogeneous magnetic field. In the literature, there areseveral methods to measure the damping coefficient andspring-constant of the MOT, including beam imbalancemethod [7–9], oscillatory magnetic-field method [10] and ∗ [email protected] † [email protected] ‡ [email protected] parametric resonance method [11]. In our experiment,we measure both the parameters in an externally drivenMOT using a method known as the transient oscilla-tion method [12]. The work presented here gives an ex-perimental basis for the choice of the response functionin [6]. We also explore an interesting transition of thecold atomic response from damped oscillatory to over-damped monotonic behaviour stemming from a competi-tion between the reactive spring-like effect coming fromthe magneto-optical trapping and the viscous drag dueto the 3D optical molasses. This work demonstrates theutility of the Langevin equation in describing the dy-namics of cold atoms in an optical molasses effectivelyand can be extended to a variety of systems to study theresponse function of particles in a viscous medium.Spatial diffusion of cold atoms has been studied in anoptical molasses by several groups [8, 13–16] in a differ-ent context. Numerical calculations of friction and dif-fusion coefficients were done for atomic motion in laserfields with multidimensional periodicity in [17]. The dif-fusive motion of an individual Cs atom colliding with adilute gas of cold Rb atoms was experimentally ob-served and described using the Langevin equation in[18]. In this paper, we have investigated the diffusivemotion of Rb atoms in a 3D optical molasses with σ + − σ − configuration of the counter-propagating cool-ing beams. We investigate the diffusion law of particles inthe thermal fluctuation dominated classical regime pre-dicted in [6]. We study the variation of the diffusioncoefficient as the atomic cloud is cooled to lower tem-peratures via sub-Doppler cooling and observe the de-viation from the predictions of the Doppler cooling the-ory [8, 13–16]. This study opens up the possibility offuture exploration of the zero point fluctuation domi-nated quantum diffusion regime predicted in [6]. Wealso check the classical Stokes-Einstein-Smoluchowski re-lation from our measurements of the diffusion coefficient.The study of spatial diffusion and other similar trans-port properties of cold atoms [19] have important impli- a r X i v : . [ phy s i c s . a t o m - ph ] J a n cations such as exploring Anderson localisation of matterwaves in disordered potentials where the diffusion coeffi-cient of the cold atoms vanishes at the onset of Andersonlocalisation[20, 21].The paper is organised as follows: In Sec.II, we brieflydescribe the experimental setup and detection methodfor the cold atoms used in our present study. In Sec.IIIwe present the measurement of the response function ofcold atoms. In Sec.III A we use the Quantum Langevinequation as a starting point for setting up the theoreti-cal perspective. We then describe our experimental set-up and methods for the measurement of the responsefunction of the cold atoms in Sec.III B. Thereafter, wedescribe our theoretical modelling of the response func-tion using the Quantum Langevin equation in Sec.III Cand compare the analytical results with the experimen-tal observations in Sec.III D. In Sec.IV A, we present atheoretical description of the diffusion coefficient. Wepresent the theoretical expressions for the mean squaredisplacement and the velocity autocorrelation functionbased on the Quantum Langevin equation in Sec.IV B.In Sec.IV C, we discuss our experimental measurementsof the variation of the diffusion coefficient as a functionof the detuning and the beam intensity as the atomiccloud is cooled to lower temperatures and compare ourexperimental data with the theoretical calculations us-ing the Stokes-Einstein-Smoluchowski relation. Finally in Sec.V we present some concluding remarks and futureperspective. II. PREPARATION AND DETECTION OFCOLD ATOMS
In our experiment, we have used a cold atomic cloud of Rb atoms trapped in a MOT inside an ultra-high vac-uum (UHV) region ( ∼ − mbar) in a glass cell. TheMOT was vapour-loaded using a Rb Getter source. Theschematic of the experimental set-up is shown in Fig.1 .An external cavity diode laser (ECDL) has been used asthe cooling laser for laser cooling and the laser beam is 12MHz red-detuned from the 5 S / F = 2 → P / F (cid:48) = 3transition of Rb. Another ECDL is used as the re-pump laser and its frequency is tuned to the transition5 S / F = 1 → P / F (cid:48) = 2 to optically pump the atomsback to the cooling transition. This is a standard proce-dure followed in laser cooling experiments [22–25]. Thedetuning and intensity of the cooling and repump beamsare controlled using acousto-optic modulators (AOM).The restoring force needed for a MOT is provided by thepair of current carrying coils, kept in a near ideal anti-Helmholtz configuration. FIG. 1. Schematic diagram of the experimental setup where a cold atomic cloud is produced in a magneto-optical trap in aglass cell. A magnified view near the cold atomic cloud is shown in the inset. The trajectory of the cold atomic cloud is shownas a series of atomic clouds at various positions in the XY plane. The cooling beams are retro-reflected using a mirror anda λ/ The fiber coupled laser beams are expanded to havea Gaussian waist diameter of 10 mm and combined in anon-polarizing cube beam splitter. Thereafter, the com-bined cooling and repump beams are split into three pairsof beams using a combination of half wave plates andpolarizing cube beam splitters (PBS). Each of the cool-ing beams is sent through the UHV glass cell and retro-reflected back via a quarter waveplate and a mirror. Theincoming cooling beams are kept slightly converging soas to account for the losses in the optical elements andto make sure that radiation pressure imbalance of theincoming and the retro-reflected beam is eliminated.The cold atoms are detected by a time-of-flight ab-sorption imaging technique using a short ( ∼ µ sec)pulse of weak, resonant linearly polarised probe laserbeam tuned at the 5 S / F = 2 → P / F (cid:48) = 3 tran-sition. The shadow cast by the atoms is imaged onto anICCD camera with a magnification factor 0.4. In a typ-ical run of the experiment, we can trap and cool about5 × atoms at a temperature of around 150 µ K inthe magneto-optical trap. Further sub-Doppler coolingof the atoms [14, 16] in a 3D optical molasses leads toa temperature of around 20 µ K without any significantloss of atoms. Therefore, we can perform measurementsof response function as well as spatial diffusion in a widerange of temperatures (20 - 150 µ K).
III. RESPONSE FUNCTION OF THE COLDATOMSA. The Quantum Langevin equation
Our starting point is the Quantum Langevin equation(QLE) [4]. The position-operator, x ( t ), of the particleat any time t can be obtained by solving the QLE. M ¨ x + (cid:90) t −∞ dt (cid:48) α ( t − t (cid:48) ) ˙ x ( t (cid:48) ) + kx = ζ ( t ) + f ( t ) (1) where M is the mass of the particle, α ( t ) is the dissipa-tion kernel and ζ ( t ) is the noise related to the dissipa-tion kernel via the Fluctuation Dissipation Theorem[4],where, In this paper we have used the Quantum Langevin Equation asthe starting point unlike Ref [5, 6] where we use the FluctuationDissipation Theorem as the starting point. Note however that thecrucial input to the Quantum Langevin Equation is the secondquantum expectation value given in Eq.(2), and this rests entirelyon the Fluctuation Dissipation theorem. In this paper, we use theQuantum Langevin Equation as the starting point since this forceequation enables easy identification of all the forces coming intoplay in the experimental system. (cid:104) ζ ( t ) (cid:105) =0 (cid:104){ ζ ( t ) , ζ ( t (cid:48) ) }(cid:105) = 12 π (cid:90) ∞ dω (cid:126) ω coth (cid:18) (cid:126) ω k B T (cid:19) Re[˜ α ( ω )]cos( ω ( t − t (cid:48) )) (2)The last term on the L.H.S. of Eq.(1) corresponds to aharmonic force characterized by a spring constant k . Inour present problem k defines the spring constant corre-sponding to the Magneto-optical trap (MOT). Here, f ( t )is a perturbing force used in this experiment in the formof an additional magneto-optical force arising due to apulsed homogeneous magnetic field.Taking the expectation value of Eq.(1), and substitut-ing (cid:104) ζ ( t ) (cid:105) = 0, we find that in the Fourier domain: − Mω ˜ x ( ω ) − iω ˜ α ( ω )˜ x ( ω ) + k ˜ x ( ω ) = ˜ f ( ω ) , (3) where ˜ x ( ω ) corresponds to the Fourier transform of themean position of the particle. Note that, here we haveused x to denote the mean position of the particle. Eq.(3)can be re-expressed as ˜ x ( ω ) = ˜ R ( ω ) ˜ f ( ω ) , (4) where ˜ R ( ω ) = 1[ − Mω − iωα + k ] , (5) corresponds to the Position Response Function of the sys-tem (in this case the cold atomic cloud) in the Fourierdomain. Here we have chosen ˜ α ( ω ) = α , which corre-sponds to the choice of an Ohmic bath to which the sys-tem is coupled. The choice of an Ohmic bath is motivatedby the optical molasses in the cold atomic experimentalsetup. In fact, the choice of the Ohmic bath is equivalentto a force proportional to the velocity with a constant ofproportionality corresponding to the friction coefficient.The present experiment serves as a test of this theoreticalmodel of the molasses .The Position Response Function in the time domaindefined as R ( t ) = i (cid:126) (cid:104) [ x (0) , x ( t )] (cid:105) θ ( t ) is given by: R ( t ) = 12 π (cid:90) ˜ R ( ω ) e − iωt dω , (6) For the Ohmic bath (for which α is a constant), the Po-sition Response Function can be obtained using Eq.(5), R ( t ) = 2 α c e − αt M sinh (cid:18) α c t M (cid:19) (7) Note that the analysis presented here is essentially classical sincewe restrict to an analysis involving the expectation value of x inwhich case we restrict to the case (cid:104) ζ (cid:105) = 0. Nevertheless we for-mulate the problem from the general point of view of a QuantumLangevin Equation, keeping the larger perspective of quantum dif-fusion in mind. In future, we intend to address the problem ofdiffusion driven by zero point fluctuations, where the full quantumformulation comes into play. In fact, we have some preliminarydiscussion of this in Sec. IV B. where, α c = √ α − kM . There are three qualita-tively distinct cases. For α > kM , one gets an over-damped behaviour of the cold atomic cloud response. For α = 4 kM the response of the cold atomic cloud is criti-cally damped and for α < kM the response of the coldatomic cloud is underdamped (Eq.(7)). For k = 0, Eq.(7)reduces to the position response function used in [6]. Wehave analytically calculated the position response func-tion R ( t ) and compared it with the experimentally ob-served oscillatory and damped motions of the cold atomiccloud in Sec.III D. Note that in the experimental setup, k is always nonzero due to the presence of the non-zeromagnetic field gradient in the MOT.Eq.(7) is the expression for the Position ResponseFunction. Taking a time derivative of R ( t ), we get theVelocity Response Function, R (cid:48) ( t ) = 1 α c M e − αt M (cid:18) α c cosh (cid:18) α c t M (cid:19) − α sinh (cid:18) α c t M (cid:19)(cid:19) (8) B. Motion of cold Rb atoms in a driven MOT
The response function of the cold atoms depends onthe damping coefficient and the spring constant of theMOT. Experimentally, both observables can be obtainedfrom the motion of the cold atoms in a MOT by exter-nally driving them. To record the motion of the coldatoms, we first prepare the laser-cooled Rb atoms in aMOT as described in Sec.II. After that, we apply a pulsedhomogeneous magnetic field (bias field, B b ). This resultsin a shift of the zero of the magnetic field and hence thetrap center. The cold atoms experience a force towardsthe new center position, and equilibrate there within ashort interval of time. After 5 sec, we turn off the biasfield and the cold atoms again come back to the initialtrap center. The path of the cold atoms depends on thedamping coefficient and spring constant of the MOT. Totrace the path, we have recorded the position of the coldatoms at regular intervals of time after turning off thebias field.Fig.2 shows a schematic diagram of the experimentalsequence for this experiment. We capture and cool theatoms in the MOT from a Rb getter source with a load-ing time of 15 sec. The MOT magnetic field gradient of18 Gauss/cm (3.5 Gauss/cm) is used for different sets ofmeasurements demonstrating the under-damped (over-damped) motion shown in Fig.4 (Fig.5 ) in Sec.III D.The cooling beams having a Gaussian cross-section witha waist size of 10 mm is red detuned by 2 . Rb D transition = 5.89 MHz)from the 5 S / F = 2 → P / F (cid:48) = 3 transition. Af-ter the preparation stage, we apply the bias field havingan amplitude of 3 Gauss (0.6 Gauss) for 5 sec, for per-forming different sets of experiments to investigate theunder-damped (over-damped) motion of the cold atoms.Thereafter, we turn off the bias field and wait for a vari-able time t. Finally, we switch off the quadrupole mag- netic field, cooling and repumper laser beams simultane-ously, and take the absorption image after allowing for aballistic motion of the cloud for τ tof =1.2 ms. The meanposition of the cold atomic cloud is obtained after fittinga Gaussian to the column density profile of the atomiccloud. w t w t g t t o f M O T Q u a d r u p o l em a g n e t i c f i e l dM O T C o o l i n g B e a mM O T R e p u m p B e a mB i a s M a g n e t i c F i e l dD e t e c t i o n G a t e &I m a g i n g B e a m A t o m L o a d i n g ( 1 5 s e c ) t
FIG. 2. Experimental timing sequence for measuring the re-sponse function of the cold Rb atoms. We have prepared thecold atomic cloud by loading the MOT for 15 sec. Thereafter,we apply a homogeneous bias magnetic field for 5 sec ( w ). Af-ter the bias field is switched off, the cold atoms come back tothe previous trap center and its motion was monitored viatime-of-flight absorption imaging. In our experiment, time-of-flight( τ tof ) is 1.2 ms, the detection gate time ( t g ) is 1 msand imaging beam pulse width is 100 µsec . The time separa-tion between two absorption images ( t w ) is 1 sec. C. Theoretical Modelling
1. Perturbing force on cold atom in a driven MOT
The temporal profile of the bias field, used in our ex-periments, is shown in Fig.3 . We fit this profile with thefollowing equation: B b ( t ) = 0 if t ≤ − w = B (cid:32) − e − t + wτ (cid:33) if − w ≤ t ≤ B (cid:32) − e − wτ (cid:33) e − tτ if t ≥ (cid:39) B e − tτ (for w >> τ ) (9) where, B is the magnitude and w is the pulse widthof the bias field. τ and τ are the rise time and falltime of the bias field. In our experiment τ and τ are912 µsec and 29 . µsec respectively. The approximationdone in the last line of Eq.(9) is due to the fact that thetime duration of the bias field ( w = 5 sec) is much largerthan the rise time of the bias field ( τ = 912 µsec ). Thevalues of τ and τ depend on the design details of the fastswitching circuit for the magnetic field coils in Helmholtzconfiguration producing the bias field [26]. It is impor-tant to have a fast ‘switching off’ of the magnetic field soas to ensure that the measurements taken after switchingoff the magnetic field are not significantly affected by thetime-constant τ . In any case, we incorporate the effectof τ and τ on the motion of the atoms in our theoreticalmodel. FIG. 3. Temporal profile of the bias field. The black solid lineis the experimental data recorded using a pick-up coil. Insets(a) and (b) show the growth and the decay, respectively, of thebias field as a function of time. After fitting the data usingEq.(9) we obtain τ = (912 ± . µsec and τ = (29 . ± . µsec . In a MOT, the component of the force along the posi-tive x -direction on the trapped atoms is given in [27] andcan be recast as follows , F MOT = − αv − g µ B λh α x ∂B m ∂x = − αv − g µ B λh α C m ∂B m ∂x (10)where g = g F (cid:48) m F (cid:48) − g F m F for transition between thehyperfine levels | F, m F (cid:105) and | F (cid:48) , m (cid:48) F (cid:105) , µ B is the Bohrmagneton, λ is the wavelength of the cooling beams, h isthe Planck’s constant and α is the damping coefficient. where, x ∂B m ∂x = B m C m ∂B m ∂x = 12 C m ∂B m ∂x with B m = C m x In the presence of an additional bias field ( B b ) along thenegative x-direction, the force on an atom is given by, F net = − αv − g µ B λh α C m ∂ ( B m − B b ) ∂x = − αv − g µ B λh α C m (cid:18) ∂B m ∂x − C m B b (cid:19) = F MOT + f ( t ) (11)where ∂B b ∂x = 0 and, f ( t ) = g µ B λh α B b (12)
2. Mean displacement of the cold atoms
The mean displacement (cid:104) x ( t ) (cid:105) is related to the PositionResponse Function R ( t ) via linear response theory [5, 6], (cid:104) x ( t ) (cid:105) = (cid:90) t −∞ R ( t − t (cid:48) ) f ( t (cid:48) ) dt (cid:48) (13) which, on differentiation, gives the expectation value (cid:104) v ( t ) (cid:105) of the velocity: (cid:104) v ( t ) (cid:105) = (cid:90) t −∞ ˙ R ( t − t (cid:48) ) f ( t (cid:48) ) dt (cid:48) (14) Here f ( t ) is the external time dependent perturbation.We consider a top hat function f ( t ) = f θ ( t + w ) θ ( − t )for the perturbing force (with f being the strength ofthe force) such that, f ( t ) = (cid:40) f , for − w < t < , for t ≤ − w, t ≥ t = 0). The exact form for f ( t ) is given byEq.(12). If we neglect the rise and fall time scales, thenthe function looks like a top hat function. Then from Eq.(14) we get: (cid:104) v ( t ) (cid:105) = f (cid:90) − w ˙ R ( t − t (cid:48) ) dt (cid:48) (16)= − f ( R ( t ) − R ( t + w )) (17) R ( t ) = − f (cid:104) v ( t ) (cid:105) + R ( t + w ) (18) For w → ∞ , R ( t + w ) → R ( ∞ ) = 0 (when the MOTis on R ( ∞ ) = 0, but it will be nonzero when only themolasses is present) we get, R ( t ) = − f (cid:104) v ( t ) (cid:105) (19) This relation enables us to directly obtain the PositionResponse Function from a measurement of the expecta-tion value of the velocity.Using the expression for the Position Response Func-tion in Eq.(7) and the perturbing force in Eq.(12), weget (cid:104) x ( t ) (cid:105) = Ae ( − α + α c ) t M + Be − ( α + α c ) t M + Ce − tτ (20) where, A = − M f α c (cid:16) e ( − α + αc ) w M − (cid:17) α − α c − τ (cid:16) e ( − α + αc ) w M − e − wτ (cid:17) ατ − M − α c τ + τ (cid:16) − e − wτ (cid:17) ατ − M − α c τ (21) B = 2 M f α c (cid:16) e − ( α + αc ) w M − (cid:17) α + α c − τ (cid:16) e − ( α + αc ) w M − e − wτ (cid:17) ατ − M + α c τ + τ (cid:16) − e − wτ (cid:17) ατ − M + α c τ (22) C = 4 M τ f (cid:16) − e − wτ (cid:17) (4 M + τ ( α − α c ) − M ατ ) (23)Here, f = g µ B λh α B and is obtained using Eq.(12).The mean velocity can also be obtained by taking atime derivative of (cid:104) x ( t ) (cid:105) in Eq.(20), (cid:104) v ( t ) (cid:105) = ( α c − α ) A M e ( − α + α c ) t M − ( α + α c ) B Me − ( α + α c ) t M − Cτ e − tτ (24)Note that for negligible τ , τ and for infinite width, i.e. w → ∞ , using Eq.(21), Eq.(22) and Eq.(23),( α c − α ) A M → − f α c , ( α + α c ) B M → − f α c , Cτ → (cid:104) v ( t ) (cid:105) → − f R ( t ) ( R ( t ) is given byEq.(7)) and thus satisfies Eq.(19). D. Experimental Results and comparisons with thetheory
1. Position of the cold atoms
In each of the experiment runs, we have allowed theballistic motion of the cloud for τ tof =1.2 ms after switch-ing off the MOT light beams and the quadrupole field fol-lowing the switching off of the bias field. τ tof is required to record the images in a magnetic field-free environment.Depending on the velocity, v ( t ), of the atoms the meanposition of the cloud is given by, (cid:104) x observed (cid:105) = (cid:104) x ( t ) (cid:105) + τ tof (cid:104) v ( t ) (cid:105) (25)The graphs in Fig.4 and Fig.5 show the time varia-tion of this position ( (cid:104) x observed (cid:105) ) of the cold atomic cloudafter the bias field is turned off. In Fig.4 , we observe anunderdamped oscillatory motion of the cold atomic cloudwhere the MOT magnetic field gradient is 18 Gauss/cmand the magnitude of the bias field is 3 Gauss along the x -direction as in Fig.1 . In Fig.5 , we observe an over-damped motion of the cold atomic cloud where the MOTmagnetic field gradient is 3 . . x -direction.The experimental data in Fig.4 and Fig.5 are fittedwith the analytical expression for the mean displacementof the cold atomic cloud given by Eq.(25), where (cid:104) x ( t ) (cid:105) corresponds to the mean position of the cloud as definedin Eq.(20). In this fitting algorithm, only the dampingcoefficient ( α ) is used as a free parameter. Each experi-mental data point shown in Fig.4 and Fig.5 is the av-erage of three experimental runs and the error bar is thestandard deviation of the mean position of the cloud asdescribed in Sec.III B. In the insets of Fig.4 and Fig.5 ,we have fitted the experimental data points with Eq.(25),where (cid:104) x ( t ) (cid:105) is the solution of a damped-harmonic oscilla-tor, given in Eq.(26). In this particular fitting procedure,three parameters (A, B and α ) are kept as free param-eters. From these fits, we obtain initial estimates of thedamping coefficient. (cid:104) x ( t ) (cid:105) = A e ( − α + α c ) t M + B e − ( α + α c ) t M (26)Such an approach to explain the motion of the atom in aMOT was presented in [7] and [12], however, while beingapproximately correct, it does not capture the details ofthe external driving force in its entirety. In contrast, ourtheoretical model based on the QLE captures the detailsof the external driving force and can be used for anyform of the driving force. Hence it makes this theoreticalmodel versatile and widely applicable for this class ofexperiments.As discussed in Sec.III A, the cold atomic cloud showsan underdamped oscillatory motion or an over-dampedmotion in response to the applied bias field dependingon whether α < kM or α > kM respectively, where α is the damping coefficient, k is the spring constantcorresponding to the MOT and can be calculated fromthe 1D Doppler cooling theory[8, 13–15, 28] as, α = 4 (cid:126) κ s | δ | / Γ (cid:18) s + δ Γ (cid:19) (27) k = g µ B λh α ∂B m ∂x (28)where λ is the wavelength and κ = 2 π/λ is the wavenum-ber of the cooling beams, δ is the detuning of the coolingbeams from the atomic transition, µ B is the Bohr mag-neton, ∂B m ∂x is the MOT magnetic field gradient and s is the the saturation parameter of the cooling beams.Hence, the damping coefficient α depends on the detun-ing and intensity of the cooling beams of the MOT. Thebalance between the rate of cooling of the atomic clouddue to the viscous damping force of the optical molassesand the rate of heating of atoms due to spontaneous emis-sions leads to the equilibrium temperature in the MOT.The spring constant k of the MOT has an additional de-pendence on the MOT magnetic field gradient apart fromits dependence on the intensity and detuning of the cool-ing beams. Hence the manifestation of the underdampedoscillatory motion and the over-damped motion of thecold atomic cloud results from a competition betweenthe viscous damping force due to the optical molassesand the restoring spring force due to the magneto-opticaltrapping. We have repeated the experiment to study thevariation of the damping coefficient ( α ) on the intensityand detuning of the cooling laser beams.
2. Estimation of the damping coefficient ( α ) in the MOT In Fig.6 , the damping coefficients ( α ) obtained fromfitting the experimental data with the analytical expres-sion given in Eq.(20) and Eq.(24) are plotted against thelight shift, where the light shift (∆) is given by: ∆ = (cid:126) | δ | I/I sat δ / Γ (29) where δ is the detuning of the cooling beam from theatomic transition, I is the total intensity of the cool-ing beams and I sat is the saturation intensity ( I sat =1 . mW/cm for Rb 5 S / F = 2 → P / F (cid:48) = 3transition and σ ± polarised light) and Γ is the naturallinewidth of the atomic transition.We have obtained the value of α in a range of light-shiftbetween 0.3 to 2 (cid:126) Γ and note an excellent agreement ofthe values of α with those independently obtained fromEq.(27). This justifies our theoretical model and its gen-eral applicability.
3. Position Response Functions and Velocity
In Fig.7a and Fig.7b , we show comparisons betweenthe theoretically obtained position response functions a (10-22 kg/sec) D ( (cid:1)
G )
FIG. 6. Damping coefficient( α ) as a function of light shift.MOT magnetic field gradient: 3.5 G/cm. Here, all measure-ments are taken in over damped regime and Fig.5 shows oneof the representative data. given in Eq. 7 (solid lines) and the experimentally ob-tained scaled velocities - f (cid:104) v ( t ) (cid:105) (circle with error bars)for the motion of the atomic clouds given in Fig.4 (oscil-latory motion) and Fig.5 (damped motion) respectively.Note that the scaled velocity data agrees very well withthe curves for the response functions, confirming Eq. 19,which is indeed a very good approximation to the exactresponse function (in other words, the top hat functionapproximates the exact bias field and in turn the per-turbing force well).In the experiment, we observe both oscillatory andmonotonic motions of the centroid of the cloud (cid:104) x ( t ) (cid:105) ) de-pending on the molasses parameters and the MOT mag-netic field gradient corresponding to a transition froman underdamped to an over-damped regime. However,in the experiment it is difficult to vary all the param-eters (intensity, detuning, magnetic field gradient) in awide range to demonstrate a transition point between tworegimes. Nevertheless, in a reasonable parameter spacethat we have explored, we do observe a clear variation ofthe response function with respect to the experimentalparameters exhibiting the existence of both regimes inmagneto-optical traps. IV. SPATIAL DIFFUSION OF COLD ATOMS
After the measurement of the response function of thecold atoms in the magneto-optical trap and compari-son with theoretical modelling using Quantum Langevinequation[4], we investigate the diffusion law in varioustemperature regimes as predicted in [6]. We study thediffusive behaviour of the cold atoms in the viscousmedium provided by the 3D optical molasses at differ-
R(t) (1021 sec/kg) t ( m s ) (a)
R(t) (1021 sec/kg) t ( m s ) (b)
FIG. 7. Position Response function for (a) oscillatory motionwith α = (1 . ± . × − kg/sec and (b) damped motionwith α = (1 . ± . × − kg/sec. In both the graphs,solid lines represent the theoretical prediction of R ( t ) givenin Eq.(7) and the shaded region shows the 68% confidenceband. The experimental data points correspond to the scaledvelocities - f (cid:104) v ( t ) (cid:105) of the cold atoms. ent temperature regimes as the atomic cloud is cooled tolower temperatures via sub-Doppler cooling. We presenta theoretical analysis of the mean-square displacementand the velocity auto-correlation function before describ-ing our experimental measurements of the spatial Diffu-sion coefficients. A. Theoretical Background
The force on an atom in the 3D optical molasses canbe written as, F OM = − αv (30)where α is the friction coefficient, which is same as thedamping coefficient introduced earlier and v is the veloc-ity of the atoms, Since the cold atoms in a 3D optical mo-lasses act like Brownian particles, the diffusion coefficient D can be determined by the Fokker-Planck equation[29]leading to the Stokes-Einstein-Smoluchowski relationgiven by D = k B Tα (31)where k B is the Boltzmann constant, T is the tempera-ture of the cold atomic cloud. The thermal energy k B T can be expressed as [13] k B T = (cid:126) Γ4 1 + 2 d s + δ Γ | δ | / Γ (32)where d is the relevant spatial dimension. Hence thediffusion coefficient is , D = Γ16 κ s (cid:18) d s + δ Γ (cid:19) δ / Γ (33)This expression shows the explicit dependence of thediffusion coefficient on the detuning and the intensity ofthe cooling beam. B. Mean square displacement and velocityautocorrelation function
In this section we highlight the relevant quantities : themean square displacement and the velocity autocorrela-tion function, that stem out of the Quantum LangevinEquation presented in Sec III. The mean square displace-ment had already been derived in [5, 6] via the Fluctua-tion Dissipation Theorem. Here we present it again viathe Quantum Langevin equation for the sake of complete-ness.The mean square displacement is defined as (cid:104) ∆ r (cid:105) = (cid:104) [ r ( t ) − r (0)] (cid:105) = (cid:104) ∆ x (cid:105) + (cid:104) ∆ y (cid:105) (34)Using the noise properties discussed in Eq.(2) we get, (cid:104) ∆ x (cid:105) = (cid:104) x ( t ) (cid:105) + (cid:104) x (0) (cid:105) − (cid:104){ x ( t ) , x (0) }(cid:105) = 2( C x (0) − C x ( t ))= 2 π (cid:90) ∞ dω (cid:126) ω coth (cid:18) (cid:126) ω k B T (cid:19) Im[ ˜ R ( ω )] (1 − cos( ωt )) ω = (cid:104) ∆ y (cid:105) (35)Here, C x ( t ) is the position autocorrelation function andis given by, C x ( t ) = 12 (cid:104){ x ( t ) , x (0) }(cid:105) Then the mean square displacement is (cid:104) ∆ r (cid:105) = 4 (cid:126) π (cid:90) ∞ dω coth (cid:18) (cid:126) ω k B T (cid:19) Im[ ˜ R ( ω )] (1 − cos( ωt )) (36)Using Eq.(5) we get,Im[ ˜ R ( ω )] = ωα [( − M ω + k ) + ( ωα ) ] (37)As will be further discussed in Section IV C, for the mea-surement of the mean square displacement the trap po-tential is switched off and the cloud is allowed to expandpurely in an optical molasses. We can therefore set k = 0for the calculation of the mean square displacement andget: (cid:104) ∆ r (cid:105) = 4 (cid:126) π (cid:90) ∞ dω coth (cid:18) (cid:126) ω k B T (cid:19) (1 − cos( ωt )) ωαω [ M ω + α ] (38)In the higher temperature regime ( T > µK ) probedin our experiments, we get: (cid:104) ∆ r (cid:105) = 4 (cid:126) π (cid:90) ∞ dω k B T (cid:126) ω (1 − cos( ωt )) ω ωαM ω + α = 4 Dt , for t >> Mα (39)where D = k B Tα is the Stokes-Einstein-Smoluchowski re-lation displayed in Eq.(31).The velocity autocorrelation function is obtained bytaking the second time derivative of the mean square dis-placement [30]: C v ( t ) = 12 (cid:104){ v ( t ) , v (0) }(cid:105) (40)= 12 d (cid:104) ∆ r (cid:105) dt = 2 (cid:126) π (cid:90) ∞ dω coth (cid:18) (cid:126) ω k B T (cid:19) ω Im[ ˜ R ( ω )]cos( ωt )= 2 (cid:126) π (cid:90) ∞ dω coth (cid:18) (cid:126) ω k B T (cid:19) cos( ωt ) ωαM ω + α (41)0In the higher temperature regime ( T > µK ) probedin our experiments, we get [30] : C v ( t ) = 2 k B TM e − αM t (42)Note, as expected, we get C v (0) = (cid:104) v (cid:105) = k B TM . C. Measurement of the Diffusion Coefficient
To measure the diffusion coefficient of an atom in the coldatomic cloud, the magneto-optical trap was first loadedfrom the background Rb vapour to around 10 atomsas described in Sec.II. Thereafter, the MOT magneticfield is switched off and the atomic cloud was cooled tolower temperatures ranging from 150 µK to 20 µK viasub-Doppler cooling. The cooling lasers are detuned fur-ther away from the cooling transition and its intensityis decreased by a variable amount to obtain lower tem-peratures of the atomic cloud in the sub-Doppler regime.Thereafter, the cold atomic cloud was allowed to diffusein the presence of the cooling laser beams forming the 3Doptical molasses. For an initial atomic cloud size of σ expanding to a size of σ t in time t , the spatial Diffusioncoefficient is measured using the relation : D = σ ( t ) − σ (0)4 t (43) The atomic cloud size σ ( t ) is the Gaussian width ob-tained from the Gaussian fit to the atomic column den-sity profile of the absorption image at time t. If one approximates the integrand of (41) under the assumption, k B T >> (cid:126) ω , one gets the quoted answer for C v (0), consistentwith the classical Equipartition Theorem. Strictly speaking, how-ever, (41) implies that C v (0) is infinite at all temperatures. Thisdivergence stems from the ω → ∞ part of the integral and is aparticularity of an Ohmic bath model without an upper frequencycutoff. This is tied to the fact that our model uses the instanta-neous relation, F = − αv , connecting the force F to the velocity v . This unphysical feature can be removed by introducing a non-trivial memory in the relation connecting the force to the velocity(for instance by choosing F = − (cid:82) α ( t − t (cid:48) ) v ( t (cid:48) ) dt (cid:48) as is done in theDrude bath model [30]). In Sec
IV B the Diffusion constant is associated with the meansquare displacement of a single particle, that we calculated usingthe Langevin equation as a starting point. In contrast, in the ex-perimental setup, the measurement of the Diffusion constant hasbeen done by measuring the size of the cloud and by taking the dif-ference of the sizes at time t and at time 0. Thus, in the experimentwe are essentially measuring the growth of the width of the prob-ability distribution associated with the diffusion of the particle,which is studied via the Fokker-Planck equation or the diffusionequation.The Langevin and the Fokker-Planck descriptions, are,however, equivalent. D. Results and Discussions
The mean square displacement of the cold atomiccloud in the 3D optical molasses as a function of time isshown in Fig.8 which shows a growth law consistent withEq.(39) and gives an estimate of the diffusion coefficientof the cold atomic cloud which we get from Eq.(43).The variation of the diffusion coefficient versus lightshift is shown in Fig.9 where the light shift ∆ is given inEq.(29). As shown in Fig.9 , we observe a sharp variationof the Diffusion coefficient as a function of the light shiftwith a minimum of the Diffusion coefficient observed ata light shift of around 0 . (cid:126) Γ. This corresponds to thevalue of the light shift below which the Diffusion coeffi-cient diverges which is evident from the steep increase inthe Diffusion coefficient as the light shift is decreased be-low 0 . (cid:126) Γ. This happens because at very low light shifts,the molasses cooling beams are too weak to provide anyviscous confinement to the atomic cloud thereby caus-ing the atomic cloud to expand faster than the diffusiveexpansion. The Diffusion coefficient reaches a minimumat the optimum values of the intensity and detuning ofthe cooling laser which enables an efficient sub-Dopplercooling mechanism. This occurs typically at smaller in-tensities and larger detunings of the cooling beams ascompared to the Doppler-cooling regime. At larger lightshifts, which corresponds to larger intensities and smallerdetunings, the sub-Doppler cooling mechanism becomesless efficient and the Diffusion coefficient increases as thelight shift is increased to higher values. At large values oflight shifts, the Diffusion coefficient matches the theoret-ical prediction from the Doppler cooling theory as shownin Fig.9 .The spatial diffusion coefficient as a function of thedetuning of the cooling laser beams is shown in the in-set of Fig.9 . At small detuning of the cooling beams ofthe 3D optical molasses from the cooling atomic transi-tion, the experimental data points for the Diffusion coef-ficient show good agreement with the theoretical predic-tions Eq.(33) obtained from a two-level Doppler coolingtheory as shown in the inset of Fig.9 . According to theDoppler cooling theory, the lowest temperature obtainedvia Doppler cooling defined as the Doppler temperature T D is given by T D = (cid:126) Γ2 k B . For Rb , the Doppler theorypredicts a Doppler temperature of around 140 µK . How-ever, in experiments temperatures much lower than theDoppler temperature were observed for cold atoms dueto sub-Doppler cooling mechanism [7, 14, 24, 31, 32].At larger detunings of the cooling beams, the deviationof the experimental data from the theoretical predictionsfrom the Doppler cooling theory manifests the onset ofsub-Doppler cooling [14, 16]. Sub-Doppler cooling occursdue to the existence of multiple sub-levels in the loweratomic ground state in contrast to the two-level atomicsystem considered in the Doppler cooling theory. Thishappens due to the appearance of different time-scales,the optical pumping times τ p between the ground-statesub-levels. At larger detunings and lower intensities of1 s (t) (10-6 m2) t ( m s ) FIG. 8. Measurement of the atomic cloud size expanding ina 3D optical molasses at different expansion times. Spatialdiffusion coefficient can be calculated from the slope by usingEq.(43) which is derived from Eq.(39). For this measurement,the cooling beams forming the 3D optical molasses beam weredetuned by δ = -2.2 Γ from the cooling transition and the totalintensity: 16.4 I sat . The estimated α from this measurementis (1 . ± . × − kg/sec. the cooling beams, the optical pumping times τ p becomesmuch longer than the radiative life-times of the excitedstates τ R . In the 3D optical molasses, the superpositionof three pairs of counter-propagating cooling laser beamsalong the three orthogonal directions causes the variationof the resultant electromagnetic field over a length-scaleof the wavelength of the laser beams. This creation oflarge polarisation gradients gives rise to the sub-Dopplercooling effect depending on the polarisation configura-tion of the cooling laser beams. Due to the ground-statesub-levels, the atoms have long internal time-scales andhence the internal variables cannot follow adiabaticallythe variations of the cooling laser fields as seen by themoving atom. This time lag which appears between theinternal and translational degrees of freedom of the atomis the origin of the large frictional force which gives rise tosub-Doppler cooling. A detailed description of the sub-Doppler cooling mechanism is provided in the seminalpapers [31, 32].Our experimental measurements of the Diffusion coef-ficient as a function of the cooling beam detuning are inqualitative agreement with the results of the numericalcalculations for the variation of the Diffusion coefficientwith the detuning of the cooling beams of the 3D opti-cal molasses in the σ + − σ − configuration in [17]. Thenumerical calculations were done using a semi-classicaltreatment of the atomic motion in the laser fields. - 6 - 5 - 4 - 2 - 3 - 4 - 5 - 6 - 75 x 1 0 -6 -5 -4 Diffusion Coefficient (m2/sec)
D e t u n in g ( in G ) Diffusion Coefficient (m2/sec)
D ( (cid:1) G ) FIG. 9. The data points show the experimental observation ofthe variation of the spatial diffusion coefficient with the lightshift (Eq.(29)). The inset shows the experimental observationof the variation of the diffusion coefficient with the detuningof the cooling beams. The red solid line shows the theoreticalprediction calculated from the Doppler cooling theory in boththe plots.
V. CONCLUSION AND OUTLOOK
In this work we have performed measurements to ob-tain the position response function of the cold atoms ina driven MOT. We have tested theoretical predictionsregarding the nature of the response function using ourmeasurements and have done extensive theoretical anal-ysis and numerical modelling of our experimental obser-vations. One of the significant outcomes of the studyhas been the verification of the position response func-tion which was used as an input to a recent theoreti-cal study[6] for predicting diffusion in the thermal fluc-tuation dominated classical domain as well as the zeropoint fluctuation dominated quantum domain [33]. Ourstudy has led to an interesting experimental observationof a transition from an oscillatory to an over-dampedbehaviour of the response function as a result of a com-petition between elastic and dissipative effects which isconsistent with our theoretical analysis. These measure-ments can be readily extended to lighter atomic speciescompared to Rb such as Na and K so as to access a largerrange of parameter space to observe a smooth transitionof the response function from an under-damped to anover-damped behaviour. Our study provides a generalframework to analyse the motion of a particle in opticalmolasses combined with a restoring force e.g. MOT, ion-traps in the presence of cooling laser beams, ultra-coldatoms in optical lattices in the presence of additional op-tical molasses etc. These and other similar experimentalsystems are of interest in the context of quantum tech-nology devices.2We have also presented a theoretical analysis of themean square displacement and velocity autocorrelationfunction of cold atoms as well as measurements of thediffusion coefficient of the cold atoms in the dissipativemedium of a 3D optical molasses. We study the growthof the mean square displacement with time in the ther-mal fluctuation dominated classical regime which is oneof the regimes studied in [6]. These measurements canbe further explored in a heterogenous mixture of atomicspecies to study the diffusive behaviour of one in thepresence of the other atomic species and investigate therole of inter-species collisional interaction in the diffu-sive transport. We have made a theoretical predictionfor the velocity autocorrelation function which we intendto experimentally verify in future work. We studied thevariation of the Diffusion coefficient as a function of thedetuning of the cooling laser beams forming the opti-cal molasses as well as light shifts as the atomic cloudwas cooled to lower temperatures from 150 µK to 20 µK .Thus, this study also paves the way for exploring spa-tial diffusion of ultra-cold atoms in the quantum regimein a dissipative medium driven by quantum zero point fluctuation[5, 6]. ACKNOWLEDGMENTS
This work was partially supported by the Ministry ofElectronics and Information Technology (MeitY), Gov-ernment of India, through the Center for Excellence inQuantum Technology, under Grant4(7)/2020-ITEA. S.Racknowledges funding from the Department of Scienceand Technology, India, via the WOS-A project grant no.SR/WOS-A/PM-59/2019. This research was supportedin part by Perimeter Institute for Theoretical Physics.Research at Perimeter Institute is supported in part bythe Government of Canada through the Department ofInnovation, Science and Economic Development Canadaand by the Province of Ontario through the Ministry ofColleges and Universities. We thank Hema Ramachan-dran, Meena M. S., Priyanka G. L. and RRI mechanicalworkshop for the instruments and assistance with the ex-periments. [1] R. Kubo, Reports on Progress in Physics , 306 (1966).[2] G. F. Mazenko, Nonequilibrium Statistical Mechanics (Wiley, 2006) p. 478.[3] R. Balescu,
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