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calorimetric studies have long been valuable tools for rigorous tests of physical law , ranging from joseph black s early work on latent heat , to measurements of the metallic specific heat showing the inadequacy of drude s classical model . more recently , the high degree of control available in cold atomic gases has opened up exciting avenues for experimental verification of finite temperature theories of bose and fermi gases @xcite . to date , however , there have been few works experimentally investigating the energy - temperature relationship of a harmonically trapped bose gas . pioneering work performed by ensher  _ et al . _ relied on extracting both release energy and temperature information from time - of - flight images at different evaporation points @xcite . this work was extended by gerbier  _ et al . _ , whose measurements of the release energy were found to be in good agreement with hartree - fock theory for an interacting gas @xcite . gati  _ et al . _ have measured temperature dependent phase fluctuations of an ideal bose gas and revealed qualitatively that the system deviates from a classical gas @xcite .    to further study the energy dependence of bose - einstein condensate ( bec ) thermodynamics , two options present themselves . firstly , an improvement in the ability to extract thermodynamic information from time - of - flight images could extend the results presented in @xcite . secondly , an alternative method to study the energy - temperature relationship could be envisioned , allowing the total internal energy of the system to be measured , rather than just the release energy . new methods to study this relationship are emerging , motivated by the recent characterization of the heat capacity of a strongly interacting fermi gas @xcite . an attempt to measure the specific heat of an ultracold bose gas using a time - dependent trapping potential , as well as by heating using laser pulses , has been performed @xcite , although obtaining accurate data was found to be impractical in their system . a recent experiment has extracted information regarding the heat capacity of a bose gas using global variables @xcite .    in this paper we follow the theoretical proposal of blakie _ et al . _ to transfer a known quantity of irreversible work to a bec and measure the resulting temperature  @xcite . by utilizing two independent methods we perform known , precise amounts of work on a@xmath0rb condensate . the resulting temperature is measured after a period of thermalization , giving the transferred energy as a function of temperature . this provides a rigorous test of the energy dependence of the thermodynamics of our system , as energy and temperature measurements are performed independently . our approach contrasts that of ensher _ et al . _ , who measure their system at differing evaporation points , extracting both energy and temperature information from time - of - flight images . furthermore , our approach is not sensitive to the ground state energy of the system present at @xmath1 , allowing a more direct comparison with the specific heat . our results from the two methods compare well , both with each other and with hartree - fock numerical calculations for an interacting gas . the paper is organised as follows : section [ sec : exp ] reviews the experimental parameters and methods . [ sec : temp ] details the temperature measurements . the two methods of energy transfer are discussed in sec . [ sec : grav ] and sec . [ sec : kick ] . we close with a discussion and conclusion in sec . [ sec : dis ] and sec . [ sec : con ] respectively . [ sec : exp ] our experiment involves a bec of atoms prepared in the ground state , and held in an optical dipole trap  @xcite . the trap is formed at the intersection of two focused co@xmath2 laser beams , with wavelength 10.6  @xmath3 m , and each with a @xmath4 radius of 33  @xmath3 m . the co@xmath2 laser power is stabilised using a closed - loop feedback system to ensure long term reproducibility of the trap depth , bec atom number , and temperature . after loading atoms into the dipole trap from a magneto - optical trap operating on the 780.2  nm @xmath5 transition , a 6  second evaporative cooling sequence is used to produce a bec . we then execute the experimental sequence shown in figure [ fig:1 ] . the laser power is adiabatically ramped to a higher value over 100  ms using an exponential profile . the deeper potential resulting from this ramp prevents atom loss during the heating process . the adiabaticity of this ramp has been confirmed by ensuring that negligible non - condensed fraction exists following the ramp and a 100  ms hold time at the final laser power . we then transfer a precise amount of energy to the system , using one of two methods , before allowing the system to rethermalize for 100  ms .     - @xmath6 . work is then done on the condensate in one of two ways : ( a ) the release of the atoms for a time @xmath7 0 - 1000  @xmath3s leads to falling expansion of the cloud , resulting in increased kinetic and potential energy when the trap is subsequently reinstated ; ( b ) a 300 ns pulse of an off - resonant standing wave leads to diffraction of a fraction of the atoms . following either of these is a 100 ms period of thermalization , before the condensate is left to expand for 10 ms to allow the momentum distribution to be imaged via time - of - flight.,width=313 ]    we approximate our optical dipole trap as a harmonic potential characterized by a set of frequencies @xmath8 that define the potential in three dimensions . these frequencies are measured through a parametric heating process @xcite , where the trap depth is modulated sinusoidally for a period of 200  ms with an amplitude of @xmath9% of the total trap depth . parametric excitation along dimension @xmath10 occurs for , for integer @xmath11 . measurements of the excitation frequencies allow us to characterize our trap and calculate the critical condensation temperature for an ideal bose gas @xmath12 , given by    @xmath13^{1/3},\ ] ]    where @xmath14 is the geometric mean of the trapping frequencies , @xmath15 is the boltzmann constant , @xmath16 is the number of atoms , and is the riemann zeta function ( e.g. see @xcite ) .    we ensure that the initial system is at zero temperature by evaporating to a point where the thermal fraction can no longer be observed . we find that any further lowering of the trapping potential only leads to strong depletion of the condensate , and conclude that the zero initial temperature condition is satisfied .    in our system , the interaction energy of the initial bec far outweighs the kinetic energy , as , where @xmath17 is the s - wave scattering length , and is the characteristic harmonic oscillator length  @xcite , where @xmath18 is the mass of an atom . typically we find that , and can assume the thomas - fermi approximation applies . the temperature and number of atoms are measured using time - of - flight imaging with resonant absorption , and the properties of the atomic clouds are inferred from these images . following an experimental sequence , the dipole trap containing the atoms is rapidly switched off using an acousto - optic modulator , and the atoms are allowed to freely expand for 10  ms . after a repumping pulse , the atoms are probed with a @xmath19 pulse on resonance with the transition . the probe light has an intensity of , which is less than the saturation intensity of .    when processing time - of - flight images we first subtract the average background of all images and apply a fringe - removal algorithm @xcite , improving our signal - to - noise ratio and ability to detect low density components of the expanded atomic cloud . the images are then integrated along the @xmath20- and @xmath21-dimensions to obtain two one - dimensional density profiles , from which we extract both the temperature and the number of atoms of our sample . above the critical temperature , the expansion of an ideal bose gas evolves according to a simple scaling relation , where we can define the effective temperature after an expansion time @xmath22 in dimension @xmath23 as @xmath24 here , @xmath25 is the variance of the resulting distribution as a function of the expansion time . far above the experimentally observed critical temperature @xmath26 , this can be determined by a fit to a gaussian function . close to @xmath26 and below , the density distribution of the thermal cloud becomes predominantly the bose - distribution , and by setting the chemical potential to zero , it can be described by a bose - enhanced gaussian @xcite . in the hydrodynamic regime , there is the possibility of anisotropic expansion for a very elongated trap , which occurs when the mean free path of the atoms is less than the dimension of the trap ( e.g. see @xcite ) . for the experiments presented in this paper our trap is nearly isotropic , as we have @xmath27 , with @xmath28 and @xmath29 for the two experiments . in addition , the 10  ms expansion time used sets @xmath30 , and hence we can assume that the expansion of the ideal thermal component to be isotropic , as observed experimentally . we therefore are able to assume that @xmath31 .    below the critical temperature there exists a non - negligible condensed fraction , and as such the assumption of a ballistic expansion for the entire cloud is no longer valid . we therefore extract the temperature using the method presented in @xcite . the central interacting region is systematically excluded from our measurements by performing multiple fits of a bose - enhanced gaussian to the wings of the profile , with a varying cut - off width for the excluded central region . for each fit , the width of this region is determined by a scaling factor @xmath32 , such that the region @xmath33 is excluded from the fit , as shown in figure [ fig : splot](a ) . here @xmath34 is the thomas - fermi radius of the condensed fraction in dimension @xmath35 , with @xmath36 . @xmath32 is chosen such that the region excluded for the fit is larger than the width of the condensed fraction , as sampling this region would cause us to systematically underestimate the temperature . on the other hand , if @xmath32 is too large , we are limited by the signal - to - noise ratio of our images . there exists an intermediate region where the measured temperature depends only weakly on the width of the excluded region , which we typically find to be @xmath37 . we infer the temperature from this region , as illustrated in figure [ fig : splot](b ) .    . ( a ) the bimodal fit applied to a sample along the @xmath21-dimension , where we have inferred the temperature from a fit to the grey shaded region , here with @xmath38 . here the position @xmath21 has been scaled by the thomas - fermi radius @xmath39 . ( b ) the effect of scanning @xmath32 on the temperature measurement for a sample image . the grey shaded region indicates the typical values of @xmath32 used to infer the temperature.,width=313 ]    this technique is only valid when the extent of the thermal profile is much larger than that of the condensed fraction , and thus fails at very low temperatures . in our experiment this is more pronounced in one direction , due to higher oscillator frequency in that dimension ( @xmath27 ) . we therefore choose to extract temperature information from the @xmath40dimension , where the resulting momentum distribution of the condensate fraction is narrower , and apply a temperature cut - off at @xmath41 . an investigation of the very low temperature region ( @xmath42 ) would require a new thermometry technique to be used , such as recently presented by olf  _ et al . _ this technique involves analyzing the decoherence of a quantum superposition of spin states , which allows them to measure temperatures as low as @xmath43 @xcite .    to check the accuracy of our temperature measurement well below @xmath26 , we have simulated the mean - field effect of the condensate on the thermal cloud as the gas expands . we model the expansion of the condensate using hydrodynamic scaling @xcite . for the thermal cloud we use two approaches , monte carlo with @xmath44 test particles @xcite , and a scaling approach after @xcite . in both cases we find that the effect on the temperature measurement is less than 10% .    for higher temperatures , still below @xmath26 , the magnitude of this effect is decreased due to the smaller condensed fraction . we estimate the uncertainty on @xmath45 here to be 5% , accounting for the uncertainty in length calibration , possible collisional effects during expansion , and any variance of the temperature measurement depending on the specific choice of @xmath32 . the atom number is obtained from a bimodal fit to the absorption profile @xcite . for temperatures below @xmath26 , we perform a bimodal fit of a bose - enhanced gaussian to the thermal fraction , and a thomas - fermi profile to the condensed fraction using the method presented in @xcite . integration over the entire bimodal profile gives the total optical density @xmath46 , which is directly proportional to the number of atoms @xmath16 , given by @xmath47 . here @xmath48 is the area of a pixel , and @xmath49 is the experimental absorption cross - section . here @xmath50 , with @xmath51 being the wavelength of the probe laser beam used in the imaging process , and @xmath52 is a scaling factor that allows us to account for the clebsch - gordan coefficients combined with the experimental distribution of magnetic substates following repumping . for an even distribution across the magnetic substates , as would be the case after repumping , @xmath53 . experimentally , we independently determine @xmath52 by observing experimental images having a temperature close to @xmath26 . we then scale our measured number of atoms such that the measured temperature agrees with the theoretical critical temperature @xmath54 , which we have determined to be from hartree - fock numerical simulations . due to the nature of the interacting transition , the condensed fraction does not go abruptly to zero as the temperature crosses @xmath54 , as in the ideal case . using this method we determine that @xmath55 . we utilize two separate methods for transferring energy to our system . the first method was proposed by . here we consider an irreversible work process : a bose - einstein condensate is released from the trapping potential and allowed to expand under the influence of gravity . after a time @xmath56 ( typically @xmath57 ) , the atoms are recaptured , and allowed to rethermalize . there are three contributions to the amount of work done on the atoms during this process . one , the atoms fall under gravity ( acceleration @xmath58 ) and gain kinetic energy ; two , the displacement @xmath59 from the fall leads to a potential energy gain when the trap is reinstated ; three , the larger cloud size after the expansion results in greater potential energy when the trap potential is restored . energy from the first two contributions will be coupled to a center - of - mass oscillation in the potential in the @xmath60-direction , known as the `` kohn '' mode @xcite . although this mode will theoretically persist in a harmonic trap , we observe that these oscillations are damped in the thermalization process after @xmath61  ms . we attribute this to anharmonicities in the gaussian laser trap profile @xcite . due to the observed damping of the kohn mode , we consider this energy to be completely available for rethermalization , and to have the form    @xmath62    where @xmath63 is the trap frequency parallel to the direction of gravity .    to calculate the energy acquired from expansion of the cloud , we assume that the atoms will undergo a self - similar expansion , with widths evolving according to @xcite . here @xmath64 are the thomas - fermi profile condensate widths in directions @xmath65 , and the evolution of @xmath66 is given by    @xmath67    with @xmath68 . et al . _ @xcite have shown that the energy transferred to the system due to a symmetric expansion about the trap minimum is given by @xmath69 where @xmath70 is the thomas - fermi chemical potential , is the ratio of trapping frequencies before and after @xmath56 , and . in our experiment we constrain the trapping frequencies both before and after @xmath56 to be identical , such that @xmath71 , reducing this expression to @xmath72 the various contributions to @xmath73 are shown in figure [ fig : simtest](a ) . the total amount of energy transferred to the system and available for rethermalization is then the sum of the contributions @xmath74 and @xmath73 . we wish to reduce the dependence of our energy calculation on the absolute number of atoms @xmath16 , which has an uncertainty of 15% due to the error in @xmath52 . we therefore choose to calculate the energy per particle , rather than the total energy transferred to the atoms . in this scenario , only @xmath75 maintains a dependence on @xmath16 , with @xmath70 proportional to @xmath76 . this term accounts for less than 20% of the total energy in our experiment , as shown by a numerical simulation in figure [ fig : simtest ] . here we have calculated the ground state of our harmonic potential in three dimensions by solving for the ground state of the gross - pitaevskii equation for our trap parameters . this initial condition is then allowed to expand under gravity for up to 1000  @xmath3s using a split - step fourier method . the uncertainty in @xmath77 is mainly due to the uncertainty in the measurement of the trap frequency @xmath63 , used for determining the potential energy contribution in equation [ eq : edrop ] , giving an uncertainty of 4% for this component . the accuracy of the kinetic component of @xmath77 is due to our accuracy in determining the local value of @xmath58 , as well as the timing accuracy in our experiment . this component accounts for more than half of the total energy transferred to the system , and has an uncertainty . experimental results are shown in figure [ fig:2 ] , and have been plotted with a hartree - fock calculation for this system . as we are only interested in the energy transferred to the system , the ground state energy present at @xmath1 is subtracted from the theoretical curve . this allows us to make close comparisons with the specific heat of the system , which is defined as the temperature derivative of the energy per particle . the theoretical curve for an ideal gas with finite - size effects includes a shaded region representing the uncertainty in our determination of the critical temperature @xmath12 , which is mainly due to the uncertainty in the absolute measurement of @xmath16 . notwithstanding the 15% error in @xmath16 , the hartree - fock numerical simulation gives a better description of the behavior of our data than an ideal gas having finite - size effects . to support our previous evidence , which relies on a calculation of the work done on the bec , we utilize a separate method which allows for a direct measurement of the transferred energy . here , we transfer energy to the system using a single pulse of an optical standing wave , as indicated in figure [ fig:1](b ) , before allowing the system to rethermalize . using a setup analogous to the atom - optics kicked rotor pioneered by raizen and co - workers @xcite , we apply a short 300 ns pulse to the atoms , diffracting the system into quantized momentum orders . an additional advantage of this method is that the kohn mode is naturally not present , as the diffraction is symmetric about zero momentum . we use a pair of counter - propagating laser beams to form our standing wave , with the beams red - detuned by @xmath78  ghz from the resonant transition , such that the effects of spontaneous emission are negligible . the atoms are diffracted into quantized momentum states , with the @xmath11th momentum state having momenta @xmath79 , where @xmath80 is the wavenumber of the laser and @xmath11 is integer .    in a separate calibration experiment , we quantitatively measure the amount of energy transferred as a function of kick - strength , given by @xmath81 , where @xmath82 is the pulse length , @xmath83 is the rabi frequency of a single beam and @xmath84 is the detuning . experimentally , @xmath85 is varied by controlling the intensity of the laser for each pulse using an acousto - optic modulator , and we utilise @xmath86 . we measure the resulting momentum distribution by turning off the trap immediately after the kick and observing the atoms after a 10  ms expansion time , as shown in figure [ fig : kicktest ] . the total energy transferred to the system is computed as @xmath87 where @xmath88 is the fraction of atoms in the @xmath11th momentum state , for integer @xmath11 . we again scale this energy measurement by @xmath16 to remove the dependence of our energy measurement on our determination of the absolute number of atoms . the uncertainty in @xmath89 is then due to shot - to - shot variation in intensity of the kicking laser and the temporal width of the pulse . we perform multiple calibration runs to obtain an average transferred energy , and allow the variation to be experimentally represented as a variation in the resulting temperature after thermalization . we find that the shot - to - shot variation in @xmath89 can be up to 10% . once @xmath89 has been calibrated we repeat the experiment , but allow the atoms to rethermalize in the trap for 100  ms following the kick , before imaging the system via time - of - flight . the experimental data are shown in figure [ fig:3 ] . figure [ fig:3 ] shows both sets of experimental data on a logarithmic axis . here we scale the temperature by the ideal gas critical temperature @xmath12 , and the energy per particle by the characteristic energy of the transition @xmath90 . the data clearly deviates from the classical prediction of @xmath91 , where we would expect @xmath92 in each of the three potential and three kinetic degrees of freedom from the equipartition theorem . there is also a deviation from the prediction for an ideal bose gas . this deviation is beyond finite - size corrections @xcite , and is consistent with hartree - fock simulations of an interacting gas . corrections to the hartree - fock approximation , such as in hartree - fock - bogoliubov - popov theory @xcite , are very small for our system , and impractical to measure . theoretical studies have shown that the hartree - fock approximation can accurately reproduce the thermodynamic properties of a trapped bose gas @xcite , and we have confirmed through simulation that these two theories give very similar results for our system . the specific heat is defined as the temperature derivative of the energy per particle , in our case with the external potential held constant . taking numerical derivatives of our experimental data is impractical . we can instead make comparisons with the specific heat extracted from derivatives of the theoretical curves , shown in figure [ fig : hc ] . we find that our experiments support the notion that the presence of interactions will tend to increase the specific heat at low temperatures when compared to an ideal bose gas . this can can be understood as a consequence of the repulsion of the thermal atoms by a large condensate fraction . the effective potential seen by the thermal atoms is modified to a `` mexican hat '' type potential @xcite , increasing the volume occupied by the thermal atoms , thereby increasing the density of states . consequently , this allows the otherwise `` saturated '' thermal cloud to hold more atoms , and hence more energy . we have directly measured the energy - temperature relationship of an interacting , harmonically trapped , ultracold bose gas . two separate calorimetric techniques have produced similar results ; namely , that interactions lead to an increased specific heat from the ideal gas prediction , which is proportional to @xmath93 below @xmath12 . we have performed quantitative measurements , utilising independent determinations of the energy and temperature , that are well described by hartree - fock theory . future research could involve a thorough investigation of the effect of interactions on the specific heat by employing feshbach resonances , an investigation into ways to reduce error in the experiment , and a detailed investigation of the thermalization process . 28ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) _ _ ,  @noop , ( ) @noop * * ,   ( ) @noop * * , ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * , ( ) @noop * * ,   ( ) in  @noop _ _ ,  ( ,  ) link:\doibase    10.1063/1.3125051 [ * * ,   ( ) ] @noop * * ,   ( ) @noop * * ,   ( ) link:\doibase 10.1103/physreva.66.033606 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.67.063614 [ * * ,   ( ) ] @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop  ,  * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( )
we experimentally study the energy - temperature relationship of a harmonically trapped bose - einstein condensate by transferring a known quantity of energy to the condensate and measuring the resulting temperature change . 
 we consider two methods of heat transfer , the first using a free expansion under gravity and the second using an optical standing wave to diffract the atoms in the potential . 
 we investigate the effect of interactions on the thermodynamics and compare our results to various finite temperature theories .