R J Peláez

B Zurro
email: b.zurro@ciemat.es

D Rapisarda

C Clark

Peláez et al, Probing the edge ion temperature by passive Doppler spectroscopy in the TJ-II… 1 Probing the edge ion temperature by passive Doppler spectroscopy in the TJ-II stellarator

Keywords: Plasma confinement magnetic, 52.55.-s Plasma diagnostics, 52.70.-m Plasma heating, 52.50.-b Plasma impurities, 52.25

We report on the measurement of the edge ion temperature in TJ-II plasmas, heated by electron cyclotron waves and neutral beam injection, using passive spectroscopy of impurity ions in low ionization stages. The measurements, which provide both temporal and some spatial resolution, offer some advantages over more intrusive or more sophisticated methods. The capabilities and limitations of this approach are illustrated by showing typical results obtained in conditions of interest for fusion plasmas.

Introduction

Edge parameters have a strong influence on the core of magnetically confined plasmas and on the design of plasma facing components. Heating and erosion of these components generate impurities that can migrate to the plasma core and reduce the fusion gain by diluting the fuel and cooling the plasma by radiative emission. The edge ion temperature, T i , is relevant for the development of plasma-surface interaction models, for predictions of the power fraction transported to the walls and for estimation of wall erosion and impurity production [START_REF] Stangeby | The Plasma Boundary of Magnetic Fusion Devices[END_REF].

Charge eXchange Recombination Spectroscopy (CXRS) is the usual technique used to determine T i at the plasma core [START_REF] Isler | [END_REF]3], but only specialized systems can reach the plasma edge [4,5,6,7]. The CXRS system operating in TJ-II [3] has a high aperture spectrometer, but cannot measure the plasma outside ρ (normalized radius) > 0.8. Spectroscopic techniques based on a supersonic He beam have been able to obtain the relative radial temperature profile in the TJ-II plasma periphery, but it needs independent temperature data in the plasma interior in order to give absolute values [8]. Because of these shortcomings in these other techniques, we need to cover the peripheral part of the TJ-II ion temperature profile by means of passive spectroscopy.

Passive spectroscopy is an alternative technique for edge diagnosis; and although inversion of chord-integrated data is usually required in order to obtain local spatial resolution, this is not always necessary if the ion species emission is itself spatially well localized. In high magnetic field devices, the complications added to the line width analysis by the influence of the Zeeman effect have the advantage that some information on the spatial origin of the emission can be recovered [9] due to the variation of magnetic field along the line-of-sight.

Recently, the spectral line shape and T i fluctuations have been extensively modeled [10], at conditions relevant to the edge of a magnetic fusion device, but scarce effort on the experimental side of using passive spectroscopy can be found in the literature notwithstanding the excellent and detailed work performed in the TEXTOR tokamak [11]. That paper deals with all the critical issues of this kind of measurements, with the exception of the possible effect of ion mass on Doppler ion temperature measurements, which can be observed only when measurements are taken with thermalized ions of significantly different mass. This effect, first pointed out by the astrophysicists, and now routinely accounted for in their analysis, is required to understand the results of ion temperature measurements of central ions [12,13]. In contrast, retarding field analyzers have been used to monitor the edge ion temperature (see [14] for a set of references), but that is a method that is very difficult to extrapolate to thermonuclear plasmas such as ITER. Since all magnetic confinement fusion devices have high spectral resolution devices in the visible range in order to monitor the types of impurities, their concentration and their influx, it seems natural to explore the potential and difficulty of using these instruments to also deduce the edge ion temperature.

Although it is generally believed that passive spectroscopy is a standard technique for monitoring the plasma edge ion temperature, this must not be the case since it is not often used in practice. We believe that the reason why it is not standard is that the relationship between the spectral line shape and its width (FWHM), from which the ion temperature must be deduced, is complicated. It can be altered by any of several different effects, each of which has a distinct influence on different lines belonging to different ions (for instance velocity fluctuations, Zeeman effect, spatial averaging effects) that must be considered in order to understand the raw experimental data.

Taking into account all these complications, the scope of this work is to investigate which ions and particular emission lines can be used most confidently for this purpose, with the final goal of developing an experimental set-up for routinely monitoring the edge temperature with increased spectral and spatial resolution. For this purpose, measurements of T i in the edge of the TJ-II plasma taken using passive spectroscopy are reported and some of the limitations in the experimental approach and the data analysis are discussed. Spectral lines belonging to low ionization stages of C, He and Li impurities have been used for this purpose. Measurements of the ion temperature with time and chord resolution have been performed in TJ-II plasmas heated by Electron Cyclotron Resonance Heating (ECRH) alone and in discharges with Neutral Beam Injection (NBI). Typical illustrative results in all of these different scenarios will be presented.

A discussion of the different problems encountered in both the experimental approach and the data analysis is also presented. This paper is organized as follows: the experimental set-up and a brief description of the analysis methods are presented in section 2. The experimental results obtained in typical TJ-II discharges are included in section 3. Finally, conclusions are drawn in section 4.

Experimental

The spectral line shapes of intrinsically edge ions have been recorded in TJ-II plasmas using high spectral resolution spectrometers that view the plasma perpendicular to the main magnetic field through fiber guides. One of the spectrometers has spatial resolution capabilities by means of 9 equal-spaced channels, which integrate the emission along different plasma chords. The other one monitors the impurity emission lines, with higher time resolution, along a fixed chord collinear to any one of the nine spatial channels. The experimental systems and data analysis methods are described in some detail in [15] and [16]. The information provided by both spectrometers is complementary, providing an understanding of the data obtained in a complicated geometry such as that of TJ-II plasmas. The geometry of the TJ-II plasma observation is illustrated in figures 1 (a) and (b); the first one shows the toroidal position of the two spectrometers with respect to the location of the main heating sources and key diagnostics, whereas the second one gives the details of how the nine channels (1 bottom to 9 top) of the multichannel spectrometer are distributed along different plasma chords in order to cover most of the plasma cross-section; the diameter of the collection light cone is not better than 1 cm.

We must add a few comments regarding the spatial resolution that can be achieved with this approach. The spatial resolution that can be achieved in practice depends on the intensity of emission lines available, which is typically limited by the cleanliness requirements of the plasma device, the limited physical access to the plasma and the use of spectrometers with large focal length and low f-number (f/10 or f/12) to achieve the maximum spectral resolution needed for edge ion temperature measurement. The former conditions constrain the spectral system to the use of long fibers that must be 1 mm thick if you want to collect the maximum signals. A multichannel system with compact focusing optics, and with good contrast to avoid channel crosstalk, give us an optical spot not better than 1 cm at the plasma center for every channel.

Better spatial resolution could be achieved if an observation geometry where the input slit and consequently the individual pixels of the CCD array were directly focused with a single optical lens onto the plasma periphery, but this is uncommon due to space constraints. This work has been focused on the measurement of the edge ion temperature in discharges heated by only ECRH (up to 600 kW at the second harmonic) and with additional heating of two neutral beam injectors: NBI_1 and NBI_2, parallel and anti-parallel to the magnetic field (up to 400 kW from each injector).

The impurity ions selected for this work are present in TJ-II plasmas in concentrations high enough to make these measurements possible. The Li lines are very prominent because the plasma wall is covered by Li to reduce the influx of high Z (atomic number) metals from the plasma-wall interaction. The carbon lines are present because the NBI neutrals that shine through the plasma are dumped into graphite tile protection. Finally, He is present because it is used everyday during glow discharge cleaning, because it is used also to run the supersonic helium beam that diagnoses T e at the edge [17], and also because it is injected after the TJ-II discharges for Heavy-Ion Beam Probe (HIBP) calibration. Other lines of contaminants are present in much smaller proportion and it would be difficult to use them for edge ion temperature measurements with good time resolution and good statistics. In table 1, we show the ionization potential of the ions whose emission lines have been used in this work.

The main assumptions implicit in Doppler broadening measurements of ion temperature are that the impurity ions are thermalized with protons and that the Zeeman effect is negligible or can be corrected. The thermalization time between protons and other ions (C and Li) has been estimated, assuming solely Coulomb collisions, using the well-known Spitzer formula [START_REF] Spitzer | Physics of Fully Ionized Gases[END_REF]. For typical edge densities of 10 18 m -3 and a proton temperature of 50 eV, the thermalization time is about 50 µs with C + and 100 µs for Li + . On the other hand, the ionization rate by electronic collisions is strongly influenced by the electron density and temperature. If we assume the same values as before, the ionization times of C + and C 2+ , estimated using the analytical expressions for the rate coefficients given by Suno and Kato [START_REF] Suno | [END_REF], are comparable to the thermalization time between C + and protons. The ionization time for Li + is about 1 ms [START_REF] Fang | Atomic Data and Nuclear Data Tables[END_REF], so this is the most probable ion, among those herein chosen, to reach thermal equilibrium with protons. These time estimations are shown in the figure 2(a). These times should also be compared with the global particle confinement time, which has been estimated to be about 5-6 ms for lithium-coated walls in TJ-II [START_REF] Tabarés | [END_REF]. Similar results have been obtained for impurity confinement time in TJ-II as determined by measuring the time relaxation of global radiation signals after injecting a small amount of either a light impurity such as Boron (B) or a heavy impurity like Iron (Fe) by laser blow-off technique. These measurements have been performed for pure H plasma and with freshly Li wall deposition [START_REF] Zurro | Proc. 36th EPS Conference on Controlled Fusion and Plasma Physics Prague[END_REF]. We must emphasize that other processes, such as fluctuations, could enhance the thermalization times but it is difficult to estimate their contribution to this problem. Regarding the spectral line shape, under some considerations of thermodynamic equilibrium for the ions, it has been shown that Doppler line width for the edge conditions, apart from mass motion, is not influenced by temperature fluctuations, although these variations might theoretically affect the spectral line wings [START_REF] Marandet | [END_REF].

In figure 2(b) we depict the typical electron temperature profiles in the TJ-II periphery at three selected times during an ECRH+NBI discharge, obtained from the line ratio method by means of the periodic supersonic He beam [17]. Edge electron temperature decreases as the chord-averaged density increases (n e = 0.6, 1 and 3x10 19 m -3 ). The lines correspond to exponential fits of the data and guide the reader through the experimental points. This approach achieves its moderate spatial resolution by assuming that in the ideal case the plasma edge should be symmetric: the electron density and temperature would be solely a function of the flux surface. Under that assumption, impurity ions with different ionization potentials are located in distinct radial shells. The lower the ionization potential, the more external the layer from which its line emission comes from. Therefore, the ion temperature obtained in a plasma with a sufficiently large electron temperature gradient should be distinctive of the charge state and type of ion whose emission line is selected by the Doppler spectrometer.

The influence of the Zeeman effect on the two carbon ions used in this work has been estimated by a synthetic method previously described in [24]. The method consists of a simulation of the line width, as a function of the magnetic field. This process has been performed and illustrated with the results shown in figure 2(c), where we have estimated the influence of the Zeeman effect on the temperature estimated from the line width of the carbon lines in the lowest ionization stage considered in this work. For the transition of the Li + line selected the Zeeman effect is zero in the approximation of low magnetic field, which is valid in TJ-II. We have considered the transition of the He + line in old calculations [25].

In figure 3, we display traces of a typical TJ-II discharge. Figure 3

Results and discussion

In order to illustrate both the power and the limitations of using passive spectroscopy to monitor the ion temperature of peripheral ions, we will focus on a selected set of results relevant to several important physical issues. We will consider the behavior of edge ion temperature in low frequency ECRH modulation experiments. We will also perform a comparison between the behavior of edge ion temperatures monitored with different impurity ions, in low density ECRH discharges and high density NBI shots in both spatial and temporal evolution. Finally, we will compare results provided by different impurity ions with different ionization potentials since they provide information about different radial portions of the plasma periphery even if it can be extremely complex to determine those position and extents with high accuracy. The influence of Ion Cyclotron Resonance Heating (ICRH) on edge ion temperatures has been studied in great detail [26][27][28]. Although these plasma heating methods are theoretically well tuned to the plasma center, some unexpected mechanism might allow some power coupling to the plasma edge and these effects can be assessed by measuring ion kinetic properties at the plasma periphery.

In order to illustrate the results provided by this passive spectroscopy technique, we have selected a TJ-II discharge heated solely by ECRH where one of the gyrotrons was modulated in its full power (300 kW) at a rather low frequency (30/60 Hz). The ion temperature evolution of Li + , as determined from passive spectroscopy, is shown in figure 4(a), while its line intensity is shown in figure 4(b). The very prominent line at 548.4 nm emitted by this ion has been crucial for measuring the temperature with good time resolution. It is difficult to achieve such results in TJ-II with other, weaker, peripheral ion emission lines. We must emphasise that when TJ-II walls are coated with Li, oxygen lines are too weak for performing this type of measurement and we must wait for several weeks after a deposition before we can use carbon lines. In addition, NBI operation is needed for providing enough carbon for these measurements. The modulation timing provided by one central Electron Cyclotron Emission (ECE) trace is also displayed in both plots. As can be seen, the modulation is echoed in the line intensity evolution (figure 4(b)), but with a slower edge rate than ECE, which is proportional to the electron temperature. The effect on the ion temperature is delayed and less pronounced. m -3 ). Notice that in the first discharge the ECRH power was modulated at a lower rate than the second one. The delay between changes in ECRH power, Li + density and temperature is longer for the lower electron density case (approximately a cycle), which is the behaviour expected for an electron-collision heating mechanism in which the collision rate scales with density. This delay between the microwave modulation and the ion temperature and density response at the edge shows that the ion temperature modulation is not a direct consequence of the radiofrequency source as it crosses the edge, but rather is a consequence of the central T e modulating in response to the heating source and the subsequent ion heating modulation due to electron-ion heat transfer, which propagates towards the edge due to transport. The electron-ion heating time, as estimated for typical TJ-II edge plasma conditions (T e = 50 -100 eV and n e = 1-2 x10 18 m -3 ), predicts values (3-6 ms) compatible with the delays measured in the temporal traces depicted in figure 4. This data rules out the need for any anomalous effect to explain the observed behavior of Li + temperature modulation at the edge. As can be easily seen, in figure 4 because the edge electron temperature is reduced and the density is augmented. The carbon intensity increases because the part of the beam not absorbed by the plasma hits the graphite tiles that protect the stainless steel and prevent the production of heavy impurities. The discharge #21733, where a low modulation in the Li + ion temperature is observed, correspond to a TJ-II experiment where an electrode inserted into the plasma edge, was dynamically polarized with a higher frequency than the modulation seen in the temperature.

The lowest density discharge taken while viewing the He + line (#20077) is a measurement in the ECRH phase of the discharge. In the intermediate density discharge (#20076), only the NBI1 injector was operating, while in the highest density case (#20072) both injectors were used to heat the discharge. For the selected Li + data, similar operating modes were used, (#21176) is a measurement in the ECRH phase of the discharge, (#20041) with one injector and (#20044) was heated by two NBI injectors. Therefore, for similar operation modes the Li + and He + ion temperatures behave differently because of the different penetration of both ions. Radial line intensity measurements, discussed when commenting figures 7 and 8, show that in discharges heated by NBI, a significant He + population exist at the plasma centre, whereas Li + is confined at the edge. In addition, Li + is more probably thermalized with the protons at all densities, because of its higher ionization potential. This may explain why the Li + temperatures are higher than the He + , even for the peripheral channel [START_REF] Stangeby | The Plasma Boundary of Magnetic Fusion Devices[END_REF][START_REF] Isler | [END_REF].

Based on a broader set of data than that presented here for illustration, the He + ion temperature typically responds to the NBI by approximately doubling its temperature, whereas C 2+ temperature either does not change or even decreases during the NBI phase and little change in Li + temperature is observed. As we will see later, to understand the distinct behavior of He + under NBI we must consider how its observed penetration differs from the other ions. That rules out a simple argument based on the pure ordering of ionization potential. In addition, you can observe that the ion temperature of Li + is less prone to change with both density and NBI heating. In contrast, the temperature of C 2+ exhibits high "apparent" temperature values that decrease as the density increases (see figure 5(c)). This might be due to the influence of plasma velocity turbulence on different ions, already reported by us for highly ionized ions [12,13], that is more important for the higher the mass ions. In general, C 2+ and C + exhibit more significant temporal temperature variations than other ions. These fluctuations have a dominant frequency of 50-100 Hz and have been observed in both NBI and ECRH discharges. We attribute this to the more external radial location of these ions where temperature fluctuations are enhanced.

Finally, in figure 6 we present results of edge ion temperatures provided by the single shot multichannel system measured along different chords. In the labels displayed on each plot we show the type of ion, wavelength of the emission spectral line used, the discharge number and the mean chord-averaged density in the time window where the data collection took place.

Typically, each plot includes one case corresponding to the ECRH phase, the lowest density, and two cases corresponding to the NBI phase, corresponding to densities higher than 1.5x10 19 m -3 . A few features must be highlighted from these figures. Notice that Li + ion temperature does not change significantly along the different parallel set of chords, see figure 6(a), and also note the low sensitivity to density changes. A more significant and unexpected change is observed in the spatially resolved results of He + temperatures, shown in figure 6(b), where we observe that its temperature increases significantly when the electron density decreases. It is much higher for central chords (4-7 channels) and lower for the bottom channels (1,2) than for the top edge channels (8,9). The first effect can be qualitatively understood, because of the deeper penetration of He + ion, as we will see when we present and comment on the results in the next figure. The second effect might be understood because of the B x grad B drifts, which tend to concentrate the hottest ions in the top part of the plasma. Therefore, as the temperature is one of the first moments of the distribution function, that effect leaves a track in this parameter.

In contrast with Li + and He + , we can see from figures 6(c) and (d) the behavior of similar data for C 2+ and C + ions as a function of density. On the one hand, C 2+ exhibits a greater temperature change with density than Li + , but its temperature decreases as the density increases, consistent with the density behavior of edge T e presented in figure 2(b). A similar trend is also expected for plasma turbulence. This behavior can be qualitatively explained because, on the one hand, the C + and C 2+ ions are more external than Li + , consistent with their lower ionization potential (see table 1), and they do not exhibit the same penetration as the He + , as we will see when we discuss next figure. In addition, due to the higher mass of carbon it is more prone to velocity turbulence influence on Doppler line width than lighter ions [12,13].

Regarding to the behaviour of C 2+ temperatures, we must point out that our data set using this ion is small because its emission is often too weak to perform these measurements meaningfully. However, we feel obligated to show its temperature behaviour, even if we do not fully understand it. In fact, the C 2+ temperature data, shown in figure 6 (c), exhibits a trend with density similar to that shown by the edge electron temperature profiles selected and presented in figure 2 (b); it tends to decrease when the density rises and the edge electron temperature decreases. That behaviour would be very difficult to explain from a classical mechanism such as electron-ion heating through Coulomb collisions. However, it is generally observed that some types of plasma turbulence tend to decrease, when increasing the electron plasma density, see ref. [12]. That might be a hint to look more deeply into this effect in the future, to figure out whether this behaviour of C 2+ temperature could be caused by turbulent velocities.

Regarding the low frequency modulation of C 2+ ion temperature for discharge #21187, we suspect that the presence of an 8/5 resonance at the very edge, for the particular magnetic configuration of the former discharge (which is not the standard one), might explain that modulation. It is not seen in neighbouring discharges, which were performed with the standard magnetic configuration. However, in order to corroborate that theory we would need a deeper analysis involving other diagnostics, which is outside the scope of the present work. More puzzling, at first sight, is the fact that the C 2+ temperature is higher than that of the other ions in the data shown in figure 5 (c). We must remark that although we cannot compare data for different ions in the same discharge, and this could be the reason for some of the differences, but we do not believe that it is the only explanation. Since carbon mass is significantly higher than that of Li and He, carbon temperatures might be more influenced by the effect of turbulent velocities on the kinetic temperature deduced form line width, see our references [12,13] for a more detailed discussion of this effect.

The decrease of ion temperature in channel 7 of discharge #21326 was correlated with a short pulse injection of gas, which was done because that discharge was part of an experiment where the T e profile was forcibly transitioned from "bell" to "dome" shape by timed gas injection. This result illustrates how sensitive these measurements are to anything that happens at the plasma edge, even if it is observed only in one channel.

In order to quantify the influence of the finite radial extent of the emission region for each ion charge state, we have carried out an analysis using the method described in [15] and references therein. In summary, we use a simple emissivity localization function, of the type I(ρ) = I 0 exp(-[(ρ-ρ 0 )/λ] 2 ), to account for the chord-integrated results, where I 0 is the maximum emissivity, ρ 0 the radius of the magnetic surface where that value occurs and λ the decay length of the selected line emission of the corresponding ion. The local ion temperature profile is simulated by this analytical expression: T i (ρ)= (T i0 -T edge ) (1-ρ α ) β +T edge , where the central value is taken from Doppler measurements of inner ions. The TJ-II magnetic configuration, needed to choose the shell structure seen by any observational cone, is taken from the TJ-II data base. Some results from this type of simulation are shown in figure 7 In contrast, for the Li + case selected, which is representative of many cases, the edge temperature assumed to account for the multichannel system results is between 60-70 eV. Much lower values of around 25 eV are found for C 2+ . These cases represent typical behavior of results obtained for these different ions. The edge temperature deduced from the simulation is, within the experimental uncertainties very close to the temperature provided by the more peripheral channels, although slightly higher. This fact suggests that a single channel looking at the plasma edge tangentially would be a good ion temperature monitor. However, simulation of chord integrated values either of line emissions or of ion temperatures does not fully clarify some difficult to interpret points with local or chord integrated values. For instance, the fact that the Li + temperatures tend to be a factor 2 higher than that found with any other emission lines. We think that a further and deeper study of the emission mechanism of this line, such as role of electron excitation versus charge exchange recombination might help to understand better these results.

Conclusions

Typical peripheral T i obtained with the 548.4 nm Li + spectral line is in the range of 60-100 eV. Due to the prominence of this line, high temporal resolution has been achieved in these measurements, as illustrated by data obtained in an ECRH power modulation experiment. He + ions seems to penetrate deeper into the plasma, as indicated by its emissivity profile, which explains the higher temperature values measured in central chords in the NBI regime. The temperature of C 2+ and C + ions exhibit more pronounced temporal variations in standard discharges without any input power modulation. We believe this is because they are the most external ions and so they must be sensitive to what occurs at the very plasma edge. Edge temperatures estimated by the 464.7 nm C 2+ exhibit a behaviour that suggests that its spectral line width might be influenced, more than the other lighter ions, by the turbulence at the very plasma edge. A dedicated system monitoring the plasma edge with spatial resolution of a few millimetres would help to understand these results better, and would be worth setting up in a fusion plasma in order to compare passive spectroscopic results with other methods that are more popular nowadays. In fusion devices with much higher magnetic field it would be essential to explicitly incorporate the Zeeman effect into the analysis. However, the line of Li + (548.4 nm) used in this work, which is not affected by the Zeeman effect and which will be very prominent in Li coated-wall machines, would be the right choice for monitoring in the very edge of fusion plasmas. In addition, since the ionization potential of this Li + ion is higher than that of the other studied ions, it can reach easier the thermal equilibrium with protons at the edge.

  (a) depicts the traces of the chord-averaged density, diamagnetic energy and the time evolution of C 4+ emission.

Figure 3 (

 3 Figure 3(b) shows typical Thomson scattering profiles of electron temperature and density.

Figure 4 (

 4 Figure 4(c) shows the effect of ECRH modulation on Li + intensity and temperature when the spectral line detector was run at its highest sampling rate, and where the modulation is seen reasonably well in the temperature and line intensity traces. A peak-to-peak modulation of 15 eV is easily seen in some time cycles. The two selected discharges with ECRH modulation correspond to two different density cases (#21108, n e =0.6x10 19 m -3 ) and (#22361, n e =0.3x1019

  (a) and 4(c), the oscillations in ion temperature from the Li + measurement are almost as large (~20 eV) as those during the ECRH modulation, but Li + intensity is a clearer indication of the modulation timing. Therefore, to obtain the maximum information in dedicated ECRH modulation experiments, we should choose discharges with minimum temperature fluctuations and perform the modulation at a lower frequency. In any case, a time correlation between both modulations would help to disentangle variations associated to natural fluctuations from forced modulations due to ECRH source modulation. The plots of figure 5(a) and (b) show the time evolution of ion temperature and intensity deduced from the emission lines of other two ions used in this work, He + and C 2+ . They correspond to discharges with NBI. The combination of temperature and line intensity temporal evolution allows us to understand the effect of the NBI, or the rise of density and decrease of temperature due to them, on different ions. In figure 5(c) and (d) we display only the temperature evolution of representative peripheral ions (Li + , He + , C 2+ and C + ); figure 5(d) for discharges heated solely by ECRH, and figure 5(c) for NBI discharges. Notice the different behavior of the line intensity time evolution of impurities He and C shown in figures 5(a) and (b). Whereas the intensity of He emission decreases, with the entrance of NBI, presumably

  (a) and (b) for He + in an NBI discharge and figure 7 (c) and (d) for an ECRH discharge; in figure 8 (a) and (b) we present a similar simulation for Li + and figure 8 (c) and (d) for C 2+ . Both the chord integrated intensity and temperatures are shown. The edge temperatures assumed in the simulations are 40 and 20 eV for the two He cases; one corresponds to the NBI phase (#20072), whereas the accompanying case corresponds to ECRH alone (#20077).

Figure 1 .Figure 2 .

 12 Figure 1. Illustration of experiment geometry: (a) the location of the spectrometers with respect to the TJ-II heating systems and a key diagnostic; (b) the observation geometry of the multichannel system and the vacuum chamber showing a typical TJ-II set of contours corresponding to the magnetic configuration 100_44_64.

Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 3456 Figure 3. (a) Display of the most relevant continuous traces in a typical TJ-II discharge, and (b) typical Thomson scattering profiles in a TJ-II discharge, taken at 1106 ms.

Figure 7 .

 7 Figure 7. Simulations of chord integrated results by means of a local model for two cases of He + : (a) and (b) A case with NBI, where ρ 0 , λ and T edge assumed are: 0.95, 0.6 and 40 eV, respectively and (c) and (d) A case with solely ECRH, where these parameters are: 0.7, 0.45 and 20 eV.

Figure 8 .

 8 Figure 8. Simulations of chord integrated results by means of a local model: (a) and (b) A case of Li + , where ρ 0 , λ and T edge assumed are: 0.7, 0.16 and 70 eV, respectively and (c) and (d) A case of C 2+ , where these parameters are: 0.75, 0.12 and 25 eV.

Table 1 .

 1 Ionization potentials (eV) of the lowest three ionization stages of the intrinsic edge ions used in this work. The specific ions whose emission lines have been used are in bold and

			are shaded.
				Ionization potential (eV)
	Element	Atomic weight			Ion charge
			0	+1	+2	+3
	He	4.003	24.588	54.418
	Li	6.939	5.39	75.64	122.4
	C	12.011	11.26	24.384	47.89	64.49

Acknowledgements

This work was partially funded by the Spanish "Ministerio de Educación y Ciencia" under Grant No. ENE2007-65007 and access of external researchers to the national Fusion Laboratory facility "Stellarator TJ-II".


J J P Van Es

P Wicke

A H Van Amerongen

C Rétif

S Whitlock

N J Van Druten

Box traps on an atom chip for one-dimensional quantum gases

Keywords: numbers: 03.75.Be, 05.30.Jp, 67.85.-d

We present the implementation of tailored trapping potentials for ultracold gases on an atom chip.

We realize highly elongated traps with box-like confinement along the long, axial direction combined with conventional harmonic confinement along the two radial directions. The design, fabrication and characterization of the atom chip and the box traps is described. We load ultracold ( 1 µK) clouds of 87 Rb in a box trap, and demonstrate Bose-gas focusing as a means to characterize these atomic clouds in arbitrarily shaped potentials. Our results show that box-like axial potentials on atom chips are very promising for studies of onedimensional quantum gases.

Introduction

Micro-electromagnetic traps produced using atom chips allow for extremely tight confinement and precise control of ultracold atoms and quantum degenerate gases [1,2,3]. These chips typically consist of lithographically defined wire patterns on a substrate which carry modest currents to create stable and tailor-made magnetic potentials for controlling atomic motion on the micrometer scale. The success of atom chips is due to the close proximity of the atoms to the field-producing elements which allow for high field gradients and tight confinement. For this reason, atom chips are also versatile tools for fundamental studies of quantum gases in one dimension [4,5,6,7,8,9,[START_REF] Bouchoule | Atom chips and one-dimensional Bose gases[END_REF].

The one-dimensional Bose gas exhibits remarkable phenomena that are not present in either 2D or 3D [START_REF] Petrov | [END_REF]12]. Further interest stems from the availability of exact solutions for the many-body eigenstates and corresponding thermodynamics [13,14,[START_REF] Korepin | Quantum Inverse Scattering Method and Correlation Functions[END_REF][START_REF] Takahashi | Thermodynamics of One-Dimensional Solvable Models[END_REF][START_REF] Kheruntsyan | [END_REF]18]. Almost all experimental work in this direction has employed highaspect-ratio harmonic potentials. The strong radial harmonic confinement allows one to treat the gas as being effectively one-dimensional [19] along the axial direction, as long as the relevant energy scales are below the radial excitation energy, ω ⊥ . However, the axial harmonic confinement can be disadvantageous, since it leads to a spatially inhomogeneous density distribution, for which a comparison to the above exact theoretical treatments necessarily relies on the local-density approximation. In contrast, exact solutions do exist for the limiting case of a true box potential with infinitely steep walls [20,21]. The ideal Bose gas also behaves rather differently in a box when compared to a harmonic trap [22,23,24]. Furthermore, some interesting features of interacting 1D gases, such as the quantum decoherent regime [START_REF] Kheruntsyan | [END_REF]25], are limited to a relatively narrow range of temperatures and densities. In an axially harmonic trap, this will typically limit the fraction of the cloud that is in the regime of interest. A boxlike optical trap in one dimension was recently realized for Bose-Einstein condensates (BECs) [26]. Atom chips hold the promise of simpler and more robust access to tailored axial potentials by employing specifically designed wire patterns [4,5].

In this paper we demonstrate the experimental implementation of simple boxshaped axial potentials for studying one-dimensional gases on an atom chip. We describe in some detail the design, fabrication and characterization of our atom chips. These chips have already been successfully used in experiments on one-dimensional Bose gases [9], and on the character of radio-frequency-dressed potentials [27]. We extend the technique of Bose-gas focusing [28,29,9] to box-like axial traps, and show that it provides an accurate measure of the energy distribution of atoms in these traps.

The outline of this paper is as follows. In section 2 we discuss the basic trapping geometry used to produce the box-shaped axial potentials. Typical experimental conditions allow access to the one-dimensional regime. In Sec. 3 we describe in detail the fabrication process used to realize our atom chips. This section also includes a description of the various trapping geometries on the chip, including several box traps. In Sec. 4, we describe experiments with ultracold atoms loaded into one of the box- The numbers at the right identify the atom chip wires and correspond to the numbers in Table 1.

shaped traps. This allows for the resulting magnetic potentials to be characterized. We analyze the technique of Bose-gas focusing from box-shaped trapping potentials and demonstrate its experimental use. We discuss some implications of our results and conclude in Sec. 5.

Trapping geometry

One of the simplest trapping geometries for atoms consists of the field of a straight current-carrying wire combined with a uniform bias field oriented perpendicular to the wire direction [1,2,3]. The fields cancel at a position determined by the bias field, producing a quadrupolar field in the plane orthogonal to the wire, and a minimum field line parallel to the wire which can be used to guide atoms. The high field gradients possible in close proximity to the wire (which scale as r -2 ) allow for tight confinement in the transverse directions. In order to produce a trap, full three dimensional confinement is required. This is typically achieved by bending the wire into a Z-shape, with a long central segment and two segments at right angles, the latter being referred to as end wires. The end wires add a spatially varying field component oriented along the central segment of the wire which provides longitudinal confinement to the atoms. Typically the longitudinal potential is weak compared to the transverse confinement resulting in a highly elongated trap with a trap frequency ratio (ω ⊥ /ω ) of a few hundred.

The Z-wire geometry has been extremely powerful for preparing ultracold atoms and Bose-Einstein condensates on atom chips in various experiments [1,2,3]. An important application is in generating high-aspect-ratio traps to investigate quantum gases in the 1D regime [6,7,8,9,[START_REF] Bouchoule | Atom chips and one-dimensional Bose gases[END_REF]. The drawback of the Z-wire for this application, however, is the inherent harmonic confinement in the longitudinal direction which limits the possible length of the studied systems and results in a spatially inhomogeneous atomic density requiring approximate theoretical treatments. Alternative wire geometries have been proposed to create long box-shaped axial potentials with nearly constant potential energy at the trap minimum, combined with tight harmonic confinement in the transverse directions [4,5]. The wire patterns we have produced to create box-trap geometries of various dimensions are shown in Fig. 1. The basic trap is created by a long, straight and thin wire combined with a perpendicular bias field for transverse confinement, and two 'wiggles' positioned at both ends of the wire. Each wiggle consists of two opposing notches in the trapping wire which generate end caps for the trapping potential, eliminating the need for end wires.

We first describe the field produced by such a box wire geometry. Current through a single wiggle produces a rapidly decaying magnetic field component oriented along the wire which produces a good approximation of a hard-wall potential for the atoms. The result can be accurately modelled by the field of two anti-aligned magnetic dipoles separated by a distance

√

2b and with an effective magnetic moment Ib 2 each, where I is the wire current and b is the characteristic dimension (extension) of the wiggle. For small b the two dipoles combine to produce a field with quadrupole character:

B w =    B x B y B z    ≈ 3µ 0 Ib 3 4πr 7    z(4x 2 -5xy -y 2 -z 2 ) z(x 2 + 5xy -4y 2 + z 2 ) (y -x)(x 2 + y 2 -4z 2 )    , (1)
where µ 0 = 4π ×10 -7 Hm -1 is the magnetic permeability of vacuum, r = x 2 + y 2 + z 2 the distance from the wiggle, x the distance from the wiggle in the direction of the wire, y is along the chip surface orthogonal to the wire, and z is the distance from the chip surface. The potential experienced by the atoms trapped above the wire (y = 0) is determined predominately by the B x field component which decays as b 3 z/x 5 , where b and z are typically 20 µm. This is rapid compared to the end-wire of a Z-trap (which decays relatively slowly, as z/x 2 ), therefore the wiggle results in a harder potential wall. In a trap the transverse field components B y , B z produced by the wiggle are much smaller than the applied bias field and simply cause small displacements of the quadrupolar field in the y, z plane; these can be neglected in most applications. The box-shaped potential is created by positioning two wiggles in a long straight wire. In this case the trapping field can be written as:

B box = B w (x -L/2, y, z) + B w (x + L/2, y, z) + B l (x, y, z) + B b ( 2
)
where L is the length of the central segment between the wiggles, B l is the field of the long wire (excluding the wiggles), and B b is the bias field. Near x = 0, the longitudinal potential consists of a vanishingly small curvature which scales as b 3 z/L 7 . This should be compared with the longitudinal potential of a Z-trap with a central segment of length L which results in a harmonic potential with larger axial curvature ∝ z/L 4 . The full form of the magnetic field [Eqs. (1) and ( 2)] is relevant for experiments involving rf-dressed potentials, where the coupling of the rf field depends on the local orientation of the static field. Such box traps hold great promise for studying quantum gases in the onedimensional regime. To illustrate this we discuss the numbers involved in a typical experiment. Using 87 Rb atoms in the state (F, m F ) = (2, 2), (scattering length a = 5.24 nm) trapped 20 µm above a 10-µm-wide wire carrying 0.5 A of current, the radial trapping frequency would be ω ⊥ = 2π × 30 kHz, and the radial size of the ground state l ⊥ = ( /mω ⊥ ) 1/2 would be 62 nm. The Lieb-Liniger parameter can be expressed as γ = 2a/l 2 ⊥ n 1 , with n 1 the linear density [19]. This parameter describes the cross-over from the weakly interacting regime γ < 1 to the strongly interacting regime γ > 1 and would have a value of 1 at a linear density n 1 = 2.7 µm -1 . The maximum length of a cloud in the 1D regime (temperature T , and chemical potential µ smaller than ω ⊥ ) can now be estimated by calculating the length over which the axial trapping potential increases at most ω ⊥ with respect to its minimum. For wiggles with b = 20 µm separated by 1 mm, as in wire 4 in Fig. 1, this leads to a length of 0.73 mm (≈ 1800 atoms at γ = 1). Further increasing the distance between the wiggles would allow a proportional increase in the length of the cloud and in the total number of atoms trapped. In comparison, for a Z-shaped wire with a central segment of 1 mm length, the higher harmonic axial confinement leads to an axial trapping frequency of 17.6 Hz, and a resulting cloud length of 100 µm, limiting the number of atoms to below 180 for a cloud with n 1 = 2.7 µm -1 at its center.

Chip fabrication

This section describes the fabrication of our chip, focusing on the production of long, straight and high-quality micrometer-sized wires for producing magnetic potentials. The chips were produced and characterized using the facilities of the Amsterdam nanoCenter. We describe in some detail the fabrication recipe, the wire pattern design, characterization of the produced chips, and the assembly used to support the chip. Further details can be found in Refs. [30,31].

Two key factors played an important role in choosing the fabrication process. Firstly, we require well-defined and smooth magnetic potentials which can be accurately calculated from the corresponding conductor geometry to determine relevant trap parameters. The techniques chosen for the fabrication to a large extent determine the quality of the current-carrying wires and subsequently the smoothness of the resulting magnetic potentials [1,2,3,32,33]. It is also important that the wires are relatively thin and that the atoms can be trapped close to the surface where large field gradients provide maximum radial confinement. Second, we desire a simple fabrication process in order to save time mastering the procedure and to allow many chip designs to be implemented in a short time. For this reason we designed a simple single-layer chip with only a limited number of wire patterns tailored for experiments with quantum gases in the one-dimensional regime and with box potentials. The equipment used to produce these chips is commonly available in any cleanroom facility.

We chose a 1.8 µm-thick evaporated gold layer for the conductors and a 300 µmthick silicon substrate. Evaporated gold films on silicon substrates patterned using photolithographic lift-off techniques are known to produced high quality wires with smoother surfaces as compared with electroplated wires [32,34,35,33,36]. Additionally, these materials have good thermal conductivity and the advantage that ample experience processing them is often available. At the center of the chip we placed a 125-µm broad Zwire with length L = 3 mm (Fig. 1-wire 5), which is used to produce a deep magnetic trap for initial trapping and evaporative cooling of atoms to the BEC regime. This wire has a resistance of 0.72 Ω and is capable of carrying ∼ 2.5 A continuous current (maximum current density ≥ 10 6 A/cm 2 ). The associated power dissipation of 4.5 W is sufficiently low so that conduction through the silicon substrate keeps the temperature below 80 • C. As the evaporated gold layer is relatively thick by microfabrication standards, we use two layers of resist. The top layer is a photo-sensitive resist patterned using optical lithography. The bottom layer is a lift-off resist used to create the insulating channels and as a spacer to prevent the top layer from being buried under the gold. During development the lower resist layer dissolves faster than the top layer, naturally generating an undercut (Fig. 2a). A layer of gold is then deposited using metal vapor deposition. By virtue of the overhang the edges of the deposited gold are well-defined, while the open structure allows exposure to a solvent. The solvent dissolves the resist, effectively removing the gold lying on top of it, leaving only the patterned gold layer on the silicon substrate. Electrical insulation between the chip wires is through the resistivity of the silicon and the native silicon-oxide layer produced by exposure to air. We chose to keep the chip surface area large (25×16 mm 2 limited by the CF40 vacuum flange) and gold coated to allow for the reflection of laser beams. The reflective surface is used for a mirror magneto optical trap (MOT) to collect atoms from background vapor. The small insulating channels between the wires produce negligible scattering of the reflected light and do not affect normal MOT operation.

Microfabrication recipe

1. Substrate preparation. Substrates are cut from 3-inch silicon wafers with a (1-1-1) crystal orientation, a thickness of 300 µm and a resistivity of > 4000 Ωcm. Atom chip substrates were produced from the wafers by laser cutting to dimensions of 24.63 × 16.00 mm 2 . We clean the substrate prior to microfabrication in a heated mixture of ammonia and hydrogen-peroxide solution to remove possible Si debris and organic contamination from the surface.

2. Resist. The substrate is dehydrated on a hotplate at 150 • C for 5 minutes prior to applying the resist. After the substrate has cooled down we spin coat (1 minute at 3000 rpm) the bottom resist layer (LOR 20B) to a thickness of 2.0-2.5 µm. The substrate is then baked for 5 minutes at 150 • C. The top resist layer is a positive photoresist (Microposit S1813), spin coated (1 minute at 5000 rpm) to a layer of 1.2 µm thickness. It is then baked for another 30 minutes at 90 • C.

3. Optical lithography. The resist is exposed with a dose of 74 mJ/cm 2 of 365-nm near-ultra-violet (NUV) light from a Karl Suss MJB3 mask aligner. A 4-inch Cr mask was made to specification by Deltamask (Enschede, The Netherlands) using a 442-nm HeCd laser with a resolution of ∼1 µm, based on an AutoCAD design drawing. We have found that the minimum feature size of our chips of about 3 µm is not limited by the lithography mask, but rather by scattering and divergence of the light inside the thick bilayer resist. After exposure we submerge the substrate in Microposit MF319 developer for 1 minute then rinse in water to stop development. As the lower layer dissolves faster than the top layer an undercut is produced, extending ∼1 µm. We then expose the chip surface to a low-energy oxygen plasma for 5 minutes to remove remaining resist residue from the exposed silicon. This may also enhance the thin native oxide layer present (1-2 nm thick), which provides electrical isolation between the gold conductor layer and the silicon substrate but allows high thermal conductivity.

4. Gold deposition. The gold layer is deposited in a PVD (Physical Vapor Deposition) system at ≤ 10 -5 mbar. A 10 nm-thick titanium bonding layer is first deposited to provide adhesion between the silicon surface and the gold layer. Gold is then evaporated at a rate of 1 nm/s to a total thickness of 1.25 µm, the maximum possible given the crucible used can only hold 3 g of gold at a time. After a refill we continue the evaporation to 1.8 µm. The achievable thickness is ultimately limited by the thickness of the lift-off resist which must prevent filling of the insulating channels.

5. Lift off. We rest the chip in a beaker containing Microposit Remover 1165 at a temperature of 68 • C for ∼1 hour to lift off the excess gold. It was helpful to create a bit of flow in the beaker with a syringe. We do not use ultrasonic agitation as that can damage the remaining gold wires. After lift off we rinse the chip several times in acetone and iso-propanol then use the oxygen plasma again to remove any remaining resist or solvent.

Wire pattern design

The current chip design (Fig 1, Table 1) includes 8 wires, all adjacent to the broad 125-µm-wide Z-shaped wire which is used for initial trapping and evaporative cooling. Due to the shape of this wire, all chip wires have an unavoidable Z shape to prevent crossings in the single conductor layer. Wires 3 and 7 are regular Z-shaped wires with 3-mm-long straight segments and widths of 50 and 30 µm respectively, placed almost symmetrically on either side of the central Z-wire at a distance of ∼100 µm. These wires have been used with rf-currents to produce radio-frequency-dressed potentials for manipulating Bose-Einstein condensates [27]. The remaining wires have been designed to produce one-dimensional box potentials with various lengths defined by the distances between wiggle segments (Table 1). Each wire is separated by 10-µm-wide insulating channels in the gold layer. At the center of the chip the exact shape and quality of the wires is crucial because of the close proximity to the trapped atoms, however outside the central sections we allow the wires to fan out as much as possible to reduce wire resistance and ohmic heating. The long edges of the chip host 2.1-mm-wide contact pads which provide ample space to connect each wire to a contact pin on the chip mount (4.4mm-wide pads for the 125-µm Z-wire). The edges of the chip running parallel to the wires do not have contacts in order to maximize optical access for imaging.

Chip characterization

Surface and edge roughness. The quality of the magnetic potentials produced by microfabricated conducting wires depends critically on both edge and surface wire w (µm) R (Ω) function Characteristics of the wires included on the atom chip shown in Fig. 1.

Widths w and resistances R of the wires are listed, along with their intended functions for on-going experiments roughness [32,35,3,5,33]. To characterize these properties we have used a combination of optical microscopy, atomic force microscopy (AFM), scanning electron microscopy (SEM) and surface profilometry. We summarize our findings below. Figure 2 shows images of the resulting wire structures obtained with a SEM. From this and AFM measurements we find the top surface and the sides of the wires are smooth down to the grain size of the gold (typically 50 nm). This is comparable to some of the best results reported so far in terms of potential smoothness [34,33,36]. The observed rounding of the wire corners seen in Figure 2b is the result of the finite resolution of optical lithography. The two deposition layers of the gold film are also visible as a slight change in structure along the side of the wire, however, this does not appear to increase the wire surface roughness. It is generally difficult to quantify tiny fluctuations of the wire edge over large length scales which are relevant for experiments with atoms. Using an optical microscope the wires appear perfectly straight within the optical resolution of ∼1 µm over length scales of hundreds of micrometers. We have also measured the flatness of the top surface of a chip wire over a distance of 800 µm using a surface profiler (KLA Tencor Alphastep 500). We find a roughness of 1.8 nm rms with correlations on the 300 µm length scale. The measured profiles are similar to those obtained on a bare Si substrate suggesting the evaporated gold layer accurately follows the substrate surface and has an approximately uniform thickness. From these measurements we conclude that the wire patterns are fabricated with high quality and should produce very smooth potentials for atoms. Ultimately the quality of the wires is inferred from the magnetic field roughness measured using ultracold atoms as a probe (Sec. IVb).

Chip wire resistance. We have also measured the resistance of the chip wires under typical experimental conditions. Placed in a vacuum of 10 -6 mbar to suppress cooling through convection, we performed a four-terminal measurement of the resistance with a current of about 100 mA (current density 5 × 10 5 A/cm 2 ), which is sufficiently low to neglect the increase in resistance due to ohmic heating. The measured resistances are ∼ 10% larger than those expected from a simple calculation, which we attribute to the additional resistance of the in-vacuum leads and wire bonds not included in the calculation.

Additional measurements of the resistance between different chip wires were performed to ensure the insulating channels are sufficient to prevent current leakage under typical operational conditions. Immediately after fabrication the inter-wire resistances are around 200-600 kΩ, approximately 10 6 times larger than the wire resistance. These values are too large to be explained by the resistivity of the silicon alone, hence we conclude that the resistance is mainly determined by the thin native silicon-oxide surface layer, possibly enhanced by the exposure to the oxygen plasma during fabrication [START_REF] Nicollian | MOS (Metal Oxide Semiconductor) Physics and Technology[END_REF]. After three months of use we found that the inter-wire resistances had decreased significantly to between 1 kΩ and 60 kΩ. This is most likely attributable to diffusion of gold into the silicon-oxide layer. With the new resistances we anticipate current leakage between wires on the level of 10 -4 , possibly causing deviations from the nominal magnetic fields at the 10 mG level. This effect appears to be accelerated by regular heating of the wires during experiments, as the lowest inter-wire resistances are measured between those wires used most frequently. After the initial decrease, the resistances appear to have stabilized at their current values. We expect this effect may be reduced in the future by incorporating a dedicated insulating top layer on the Si substrate.

Assembly

The atom chip is mounted on a macroscopic support structure which includes six additional wires capable of carrying currents of up to 20 A for loading atoms to the chipbased microtraps (Fig. 3). The 300-µm-diameter Kapton-coated copper "miniwires" run in two isolated layers oriented along x and y (each with three parallel wires) positioned at z = -0.5 mm and z = -0.8 mm underneath the chip surface respectively. The separation between the wires within one layer is 0.65 mm (top layer) and 3 mm (bottom layer), for the wires along x and y respectively. These wires fit in grooves in a boronnitride ceramic disc which has been machined by a computer-controlled milling machine. Boron-nitride is easily machined and has 20 times higher thermal conductivity than Macor. The miniwires are electrically connected using standard vacuum-compatible sub-D-type gold-plated connector pins. We strip the Kapton from the end of the miniwire and press it, along with a piece of bare copper, into the male pin. The resistance of the miniwires (including the connection to the male pins) is 10 mΩ. Operated at a current of 10 A, the power dissipation is 1 W per wire over its whole length, resulting in negligible heating (10 K/W). The Boron-nitride disc is bonded using Epo-tek H77 epoxy to a copper heat sink mount made of oxygen-free high-conductance (OFHC) copper. This is bolted to the end cap of a hollow stainless-steel (type 316L) rod with an outer (inner) diameter of 16 mm (8 mm) which is welded to a CF40 flange for insertion into the vacuum system. The chip can be water cooled by means of a polyvinyl chloride (PVC) tube of 6 mm diameter which runs coaxially inside the stainless-steel rod. We typically run 0.1 l/min of cold tap water through the system during operation. At night this supply is replaced by heated water at 40 • C intended to limit the amount of Rb collected on the chip surface.

In the final assembly the microchip is carefully glued to the ceramic disc using Epotek 377 epoxy. The alignment error between the on-chip Z-wire and the miniwires was found to be smaller than 50 µm. The eight chip wires are connected to the contact pins with 20 µm-diameter aluminium wires with a wire bonding technique. Each contact pad was bonded with 10 wires except for the Z-wire where we have used 14 wire bonds to sustain higher currents. The chip wires are then connected to a set of sub-D-type vacuum feedthroughs. The maximum bake-out temperature for the chip mount in the vacuum system is 180 • C, limited by the Kapton coated wires and the epoxy. The entire chip setup is compatible with ultra high vacuum, and after bakeout operates at a pressure of 10 -10 mbar.

Ultracold atoms

To demonstrate the suitability of the produced potentials for studying ultracold atoms and quantum gases we have loaded the L = 1 mm box-shaped trap (wire 4 in Fig. 1). This allows the resulting magnetic potentials to be characterized using the atoms as a probe. We demonstrate evaporative cooling of atoms confined in the box trap and apply the method of Bose-gas focusing [28,29,9] to perform thermometry of atom clouds.

Loading ultracold atoms in the box trap

A cloud of 87 Rb atoms is initially collected from the background vapour by a mirror magneto-optical trap positioned below the chip surface. We then apply a polarization gradient cooling stage and optical pumping to the atoms in preparation for magnetic trapping. The spin-polarized atoms are first trapped in an Ioffe-Pritchard magnetic trap formed by the miniwires underneath our chip (Fig. 3). For best mode-matching we make this trap as weak as possible while maintaining sufficient confinement to hold the atoms against gravity. We then immediately transfer the atoms to a trap formed by the Z-wire on the chip by ramping down the miniwire currents within 30 ms. Approximately 2 × 10 7 atoms in the F = 2, m F = 2 state are caught in a magnetic trap at z ≈ 400 µm produced by the central Z-shaped wire (I z =2.25 A) and an external field (B 0y = 13 G). This field is then ramped up to B 0y = 48 G over 200 ms to compress the trap and move to ∼ 100 µm from the surface. The central miniwire remains on at 2.7 A which produces a dimple in the longitudinal potential to improve evaporative cooling efficiency. Next, a logarithmic radio frequency sweep from 27 MHz to 2.1 MHz is applied over 750 ms to evaporatively cool the atoms to a few microkelvin. The radio frequency is then increased again to 2.3 MHz prior to transferring the cloud to the box potential.

To transfer, we first weaken the confinement of the Z-wire trap and increase the trap-surface distance to ∼ 200 µm by reducing the bias field to B 0y = 20 G within 50 ms. In 25 ms the Z-wire current is ramped down to 1 A while the current in the adjacent box-wire is increased to 0.8 A to position the cloud between the two wires. Finally, the Z-wire current is ramped off completely and the box wire current is reduced to 0.37 A with another 25 ms ramp.

A final cooling stage is applied to the atoms in the box trap using a two-part rf sweep, first from 2.30 MHz to 1.95 MHz in 60 ms and next from 1.95 MHz to a final frequency between 1.80 MHz and 1.41 MHz in 50 ms. The trap bottom as measured using rf spectroscopy is at 1.40 MHz. After evaporation the rf amplitude is switched off completely. Atoms are imaged by turning off the trap, waiting for a variable time of flight, and then applying a resonant probe laser pulse to the atoms. The shadow created by the atomic distribution is then imaged to a CCD camera for analysis. The effective pixel size in the object plane is 4.3 µm and the optical resolution is 4 µm.

Box potential

Figure 4 shows an optical density image of the atom cloud after transfer to the box potential for a final radio frequency of 1.80 MHz. We observe an elongated square-top profile for the atomic distribution as expected for a Boltzmann gas at thermal equilibrium in the box-shaped potential defined by Eq. ( 2). We note that the distributions are slightly asymmetric for final radio frequencies above 1.50 MHz (corresponding to higher cloud temperatures), but become increasingly asymmetric as the radio frequency is reduced (to lower cloud temperatures). We attribute this to a small potential gradient, possibly a tilt of the chip surface with respect to gravity. Additionally, we observe time-independent optical density modulations in the axial profiles. We have estimated the magnetic field inhomogeneity that could cause such modulations from the data presented in Fig. 4 and with additional data for various temperatures between 0.28 and 4.16 µK. This was performed using a procedure similar to previous experiments in the 1D regime [START_REF] Wildermuth | [END_REF]39], but for thermal atom distributions [32,35,40]. The observed density modulations do not scale with cloud temperature as would be expected for potential corrugations; their amplitude is roughly 20% of the peak density and independent of temperature. Secondly, the modulations are correlated on length-scales comparable to intensity variations present in the raw absorption (and reference) images (≈ 100 µm). Based on these observations we attribute the modulations to imaging artifacts due to the nearby chip surface rather than potential corrugations. The observed modulations do however provide an upper limit for the possible potential variations experienced by atoms in the box-trap. From the amplitude of the density modulations for the hottest temperature (4.16 µK) we infer an upper bound on the potential roughness of ∆V /k B ≤ 130 nK (with k B Boltzmann's constant) or ∆B ≤ 1.9 mG rms for a wire current of 0.37 A and a height of z = 40-50 µm. The figure of merit for the box trap is thus ∆B/B 0y ≈ 1 × 10 -4 which compares well to other chip traps [32,33]. The measured value is consistent with previous measurements on the broader Z-shaped wire for which we determine ∆B/B 0y < 5 × 10 -5 for a height of z = 120 µm [27,30].

The following calculation shows that this trap can support quantum gases in the 1D regime. To support an unfragmented 1D quasi-condensate (with µ < ω ⊥ ) requires that m F g F µ B ∆B < ω ⊥ . We estimate the transverse confinement for the parameters used in experiments as ω ⊥ /2π ≈ 4 kHz. This corresponds to an thermal excitation energy of ≈ 200 nK, which is greater than the upper bound for the potential corrugations of 130 nK and satisfies the above constraints. A 1D zero-temperature condensate in this trap would fragment for linear densities below n 1 = 100 µm -1 . Lower linear densities can be reached without significant modifications using time-dependent potentials to smooth the corrugated potential [41,27,33]. Alternatively, strong transverse confinement while minimizing corrugation could be provided by an auxiliary wire located far from the atoms [START_REF] Wildermuth | [END_REF], with the walls of the box-trap produced by a small current in the boxshaped wire. Due to the steep walls provided by the wiggle wires, currents less than ∼ 10 mA would be sufficient to provide good longitudinal confinement.

Bose-gas focusing

Bose-gas focusing is a powerful experimental approach for measuring the axial momentum distribution of harmonically trapped ultracold gases. In particular it has been applied to study non-equilibrium dynamics of Bose-Einstein condensates [28] and to equilibrium systems in the cross-over from the three-dimensional to the one-dimensional regimes [9,31]. Here, we extend the description to arbitrarily shaped axial potentials, and apply this to extract temperatures of atom clouds in box-shaped potentials. This technique is particularly relevant for one-dimensional gases in box-shaped traps as the spatial distribution of atoms depends weakly on temperature and conventional thermometry methods fail.

The concept of atom focusing of an elongated cloud is as follows: initially, the atoms are in thermal equilibrium in the confining potential. A relatively strong parabolic (harmonic) potential is switched on in the axial direction. This is applied for a short time, typically smaller than the oscillation period, before being switched off. This pulse gives a kick to the atoms which is proportional to their distance from the trap center and is analogous to the action of a lens in optics [29]. Then the atoms are released from the trap and allowed to evolve during a period of free propagation. The effect of the kick brings the atoms to an axial focus at a time t focus . At this point the distribution has a minimum axial width proportional to the cloud temperature and the in-focus distribution reflects the axial momentum distribution prior to focusing.

In a classical description, suitable for a non-degenerate gas, the time evolution of the atomic distribution induced by the focusing action can be expressed by

g(t) = a(t)x 0 + b(t)v 0 h(t) = c(t)x 0 + d(t)v 0 . (3)
where the final position g(t) and velocity h(t) of an atom depends on the initial position and velocity x 0 , v 0 through the focus parameters a, b, c and d. These parameters directly correspond to those in the ABCD matrix formalism known from conventional (ray-and gaussian-beam) optics [START_REF] Siegman | Lasers[END_REF]. Transformation of the phase space distribution f (x, v, t) by this evolution is given by

f (x, v, t) = f (x 0 , v 0 , 0)δ(x-g(t))δ(v-h(t))dx 0 dv 0 (4)
with δ() the Dirac delta function. From this it is possible to determine the real space atomic density distribution n(x, t) = f (x, v, t)dv. This is conveniently expressed in terms of the initial distribution

n(x, t) = 1 |b(t)| f x 0 , x -a(t)x 0 b(t) , 0 dx 0 (5)
Thus, we are left with a single integral over the modified initial phase-space distribution.

We assume a classical gas in thermal equilibrium given by the Boltzmann law:

f (x 0 , v 0 , 0) = C exp - 1 k B T 1 2 mv 2 0 + V (x 0 ) (6)
where V (x 0 ) is the confining potential energy and C a normalization constant which yields the total number of atoms. Inserting this initial distribution in Eq. 5 gives

n(x, t) = 1 |b(t)|v T √ π n(x 0 ) exp - x -a(t)x 0 b(t)v T 2 dx 0 (7)
with the thermal velocity v T = 2k B T /m. The density distribution during time of flight depends on the initial density distribution n(x 0 ) = n 0 exp[-(V (x 0 ) -V 0 )/kT ] where n 0 is the peak atomic density and V 0 is the potential energy at the trap minimum. For a harmonic focusing pulse with a duration t p and strength ω followed by a period of free evolution t, the relevant evolution parameters are given by

a(t) = cos(ωt p ) -ωt sin(ωt p ) b(t) = ω -1 sin(ωt p ) + t cos(ωt p ) ( 8 )
Equations ( 7) and ( 8) describe the one-dimensional spatial distribution of ultracold atoms during and after the application of a parabolic focusing pulse for an arbitrary shaped initial potential V (x 0 ). In the following, we apply this model to analyze our experimental results for focusing from the box-shaped potential of Eq. ( 2).

Results of the focusing experiments are shown in Fig. 5. To perform the measurements we first increase the axial offset magnetic field by 4.25 G within 50 ms. To push the atoms away from the chip surface the box-wire current is increased to 0.95 A in 2 ms. This is followed by a short trapezoidal current pulse applied to the miniwires beneath and perpendicular to the box-wire to temporarily introduce a parabolic axial potential. This pulse consists of a 0.8 ms linear ramp to 0.45 A and 9.8 A for the central and outer-miniwires respectively, held constant for 4 ms, then finally turned off again in 0.8 ms. The pulse amplitude corresponds to a peak axial oscillation frequency of ω = 2π × 25 Hz. Finally all the fields are switched off (t = 0), and the atomic distribution is allowed to freely evolve while falling under gravity. During the pulse (t < 0) or after a variable time-of-flight (t > 0) between t = -4.8 ms and t = 17.2 ms an optical density image of the distribution is recorded. The image is then integrated over the radial coordinate to produce a profile of the linear atomic density. Profiles for each time-of-flight are then compiled into a two-dimensional data set n 1 (x, t). An example of one such data set is shown in Fig. 5(a). In total we obtain seven data sets, each for a different value of the final radio frequency corresponding to a range of trap depths and final cloud temperatures.

To analyze the data we numerically perform least-squares minimization using the Nelder-Mead method to the model of Eq. ( 7) in order to determine the temperature T for each value of the final trap depth. The shape of the initial confining potential is calculated from the known wire geometry. We use the approximation of Eq. ( 2) with the box length L = 1040 µm, wiggle size b = 20 µm and trap height z = 50 µm. The residual harmonic confinement due to the ends of the box-wire corresponding to a curvature of 1.5 µK/mm 2 or ω = 2π × 2.7 Hz. One additional parameter specifying an additional gradient to the potential is required to fully reproduce the observed density distributions and was determined to be ∆ = -1.0 µK/mm. This gradient would be consistent with a tilt of the chip surface with respect to gravity of less than 11 mrad, highlighting the sensitivity of such systems to small forces [START_REF] Hall | [END_REF].

The focus model provides the expected spatial distribution of atoms before the focus pulse and agrees well with the measured distribution for each value of the final radio frequency, see for example Fig. 5(a,b). The focus pulse is modelled with two parameters, the pulse amplitude [ω/2π = 25 Hz in Eq. ( 8)] and a small displacement between the center of the parabolic potential and the box potential which causes a small kick to the cloud visible in Figs. 5(a,b) as an upward slope in the cloud center of mass. Finally fitting is performed with temperature T for each data set as the only free parameter. The measurements and the result of this fitting procedure are shown in Fig. 5(a,b) for the 1.50 MHz data set. Good agreement is found for the full range of time-of-flights and the model accurately reproduces the focus region around t = 6 ms. The results for all trap depths are shown in Fig. 5(c), where the cloud width defined as the second central moment of the distribution is plotted. In this definition a perfect square-top distribution with length L = 1 mm equates to a width of L/2 √ 3 = 0.29 mm. The widths of the observed density profiles are shown with symbols, while the widths of the model distribution are shown with connected lines. For all data between 1.80 MHz and 1.46 MHz we obtain excellent agreement between the measured widths and the model distribution. Some deviation is visible for the lower temperature data sets which may be attributed to deviations from Boltzmann statistics resulting in a tighter focus than predicted by the model. For the measurements we determine the temperatures indicated in the caption of Fig. 5. The fitted temperatures are also consistent with measurements obtained by fitting the radial expansion rate in order to directly determine the thermal velocity.

Discussion and conclusion

We have implemented and experimentally investigated simple box-shaped potentials for trapping ultracold quantum gases on an atom chip. Our chip hosts a series of long, thin and straight current-carrying wires which create extremely elongated traps, presently used in studies of Bose gases in one dimension and in radio-frequency-dressed potentials [9,27]. The described fabrication procedure is tailored for relatively thick wires for high current capacities and with high surface and edge quality to produce smooth trapping potentials. We investigate the quality and suitability of the resulting potentials by loading ultracold atoms into a 1-mm-long box-shaped trap at a distance of z ≈ 50 µm. The observed linear density profile has a square-top shape, as expected for a thermal gas in the box-shaped potential, with small density modulations from which we infer a potential roughness of ≤ 130 nK rms. Finally, we have extended the technique of Bose-gas focusing to arbitrarily shaped axial potentials and applied the method to measure the energy distribution of thermal clouds prepared in the boxshaped trap between 0.28 µK and 4.16 µK. We expect this thermometry technique to be particularly valuable for one-dimensional systems, as the radial expansion velocity becomes independent of temperature. Furthermore, in an ideal box trap the spatial distribution of atoms is independent of the cloud temperature thereby limiting the application of conventional thermometry techniques. Our results show that box-shaped potentials implemented on an atom chip are very promising for the study of onedimensional quantum gases.

Figure 1 .

 1 Figure 1. (Colour online) Optical microscope image of the central section of an atom chip. The chip hosts a series of micro-wires of various geometries for creating onedimensional traps and box-shaped potentials. The length of the scale bar is 500 µm.The numbers at the right identify the atom chip wires and correspond to the numbers in Table1.

Figure 2 .

 2 Figure 2. Scanning Electron Microscope (SEM) images of (a) the cross-section of the resist during fabrication of the atom chip. The undercut is clearly visible. The image was made immediately after gold evaporation and a ∼ 0.5-µm gold layer can be seen lying on top of the resist structure. (b) a 5-µm-wide chip wire. The corner displayed is part of a wiggle used to form a box potential. The scale bars in both figures have a length of 5 µm.

Figure 3 .

 3 Figure 3. (Colour online) (a) Chip mount showing the arrangement of the two layers of miniwires. The orientation of the Z-wire is indicated at the center of the drawing. (b) Photograph of the completed atom chip construction.

Figure 4 .

 4 Figure 4. (Colour online) In-situ atomic density distribution after loading the boxshaped potential for a final evaporation radio frequency of 1.80 MHz. (a) Optical density image showing a 70 × 465 pixel region (4.3 µm pixel size in the object plane). The dashed line indicates the approximate position of the chip surface. (b) Integrated profile showing the corresponding linear atomic density n 1 as a function of longitudinal position x. The solid line is a result from the analysis of Sec. IVc for the expected in-trap density distribution corresponding to a temperature of T = 4.16 µK.

Figure 5 .

 5 Figure 5. (Colour online) Focusing ultracold atoms from a box-shaped potential. (a) Measured optical density profiles during the focus sequence. (b) Calculated profiles based on the focusing model of Eq. (7). (c) Cloud width during the focus sequence for several values of the final evaporation radio frequency and corresponding fits used to determine the cloud temperature. Symbols (curves) correspond to final radio frequencies (fitted temperatures) of 1.80 MHz (4.16 µK), ∇ 1.70 MHz (3.44 µK), 1.64 MHz (2.95 µK), 1.50 MHz (1.51 µK), 1.46 MHz (1.02 µK), • 1.42 MHz (0.42 µK) and * 1.41 MHz (0.28 µK). The shape and duration of the focus pulse is indicated by the shaded area.

Table 1 .

 1

	1	10	3.15 double box structure
	2	5	3.70 wiggle test wire
	3	50	1.47 Z wire, radio-frequency field
	4	10	3.89 1.00 mm long box
	5	125	0.72 Z wire, initial trapping and cooling
	6	10	2.42 0.20 mm long box, rf evaporation
	7	30	2.06 Z wire, radio-frequency field
	8	20	2.30 0.90 mm long box

Acknowledgments

We gratefully acknowledge R. J. C. Spreeuw, T. Gregorkiewicz, G. V. Shlyapnikov and J. T. M. Walraven for helpful discussions. The atom chip was produced and characterized using the facilities of the Amsterdam nanoCenter and with the help of J. Rövekamp. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek van de Materie (Foundation for the Fundamental Research on Matter), and was made possible by financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Research) and by the European Union through contract MRTN-CT-2003-505032 ("Atom Chips"), and a Marie Curie fellowship (SW, grant number PIIF-GA-2008-220794).