Evolution of the fringe spacing

α2p_r

m ·T_int·x

≡cos (γ·x). (3.66) By a comparison of Eq. 3.66 and 3.60, we see thatγ =κwhich justifies our nomen-clature of the applied (π/2 - π/2) beam splitter sequence as an open Ramsey-type interferometer (ORI). We now come back to the calculation of the fringe spacing for interfering condensates based on theλ-matrices formalism

3.5.2 Evolution of the fringe spacing

By neglecting the linear contribution of the phase evolution as β = 0(i.e. no relative velocity between the wave packets) in Eq. 3.60, the periodicity of the fringe spacing (distance between two local maxima/minima) is given by

d(t) = 2π

αδx = λx(t) λ˙_x(t)· h

mδx. (3.67)

Hence, we can predict the temporal evolution of the fringe pattern for a given separa-tionδxby calculatingλ/λ. In QUANTUS-I, the confining potentials of the chip-based˙ Ioffe-Pritchard trap in ground-based experiments can be approximated as cigar-shaped (ω_x =ω_z ≡ω_rad ≫ ω_y). For a sudden opening of the trap at t = 0, the solution for the scaling factors is [130]

λ_rad(τ) =^p1 +τ², (3.68)

λ_y(τ) =1 +ǫτarctanτ −ln^p1 +τ², (3.69) with the dimensionless variableτ =ω_rad·tand the fraction of radial and axial trapping frequencyǫ=ω_y/ω_rad.

The Bragg beams are aligned along the x-direction and the interference pattern will emerge with lines of constant phase along the y-direction (see Fig. 3.5). The dynamics of the fringe spacing can thus be discussed by evaluating the temporal evolution of the fractionλ˙_rad/λ_rad (see Fig. 3.16). Here, the scaling factors are calculated for three different initial radial trapping frequencies of2π·(50,100,150)Hz.

The temporal evolution of the radial Thomas-Fermi radii of expanding BECs is pro-portional toλ_rad (see Sec. 5.2.1), which for long timescales increases linearly in time and trapping frequency (λ_rad ∼ω·t). Here, most of the mean-field energy is converted into kinetic energy which causes the expansion velocity to asymptotically reach a con-stant value (λ˙_rad ∼ω). The stepper the trap, the faster the condensate expands due

0 5 10 15 20

Figure 3.16: Calculation of the temporal evolution of the radial scaling factorλrad(t)(left), the first derivative λ˙rad(t) (center) and the fraction of λrad(t)/λ(t)˙ (right) for trapping frequencies of 2π·50 Hz (black lines), 2π·100 Hz (red lines) and 2π·150 Hz (blue lines).

to density-dependent mean-field conversion. For comparison, the steepest trap shown here (ω_rad = 2π·150 Hz), converts 95% of the mean-field within an expansion time of T_150Hz,mean≈3.25 ms, whereas the corresponding conversion time for the shallow trap (ω_rad= 2π·50 Hz) is significantly larger withT_50Hz,mean≈9.5 ms. We will review the influence of mean-field accelerated expansion on the emergence of a fringe pattern in upcoming interferometry experiments. The fraction ofλ˙_rad/λdetermines the temporal evolution of the fringe spacing and seems to converge towards a fixed slope for large timescales.

This result combined with Eq. 3.67 and t≡T_tof yields the far-field approximation of the fringe spacing

d(T_tof, δx) = h·Ttof

m·δx, (3.71)

which increases linearly with expansion time T_tof. In Fig. 3.16 (right), this would correspond to a straight line emerging from the origin with a slope of 1, which differs from the the scaling law approach due to the disregard of density dependent mean-field conversion and finite size of the condensate.

To analyze the temporal evolution of interference pattern in experiment, we first have to image it and evaluate the important properties with a suitable fit function.

Detection of the fringe pattern

As described earlier (see Sec. 2.4.5), we can calculate the optical densityDof a camera image to

In QUANTUS-I, the atomic clouds are detected destructively and spatial information is obtained by fitting different distributions to the obtained column densities. With the resonant cross section σ, the atomic column density follow as n = D/σ. For analyzing the spatial interference pattern, we integrate the density along the dimension perpendicular to the interferometer beams (z-direction) and evaluate the 1D density profile (x-direction) by fitting the following distribution to the column densities

n^1D =n^1D_max With constant background n^1D₀ and amplitude n^1D_max, this function describes the center of the output ports at positionsx₁ and x₂ with a Gaussian density profile. To consider interference, it is multiplied by a periodic function with fringe spacing dand phasesφ₁ andφ₂. Most important for interferometry studies of the condensates phase coherence and related systematics are fringe spacingdand the contrastC.

In Fig. 3.15 (right), a typical absorption image and the integrated 1D column density are given for ORI performed on ground. The positions x₁ and x₂ correspond to the output ports with momentum classesp₁= 0¯hkand p₂ = 2¯hk at the time of detection.

In a first set of experiments, we start by analyzing the fringe pattern and discussing the temporal evolution by scanning either T_tof or the separation distance δx of the interfering matter waves in an ORI.

Linear scaling of the fringe pattern with time-of-flight

In Fig. 3.17, the linear scaling of a fringe pattern with increasingT_tof is shown. There, the condensate is released from a steep potential withω_rad ≈2π·350 Hz, to be able to observe several fringes within the overlapping region and to minimize mean-field effects.

After switching-off the trapping potential, it evolves freely for a total expansion time ofT₀= 10 ms until the first beam splitter (π/2) is applied with durations of typically τ ≈100µs. During an interrogation time ofT_int= 240µs, the two coupled parts of the wave function separate to a maximum distance of onlyδx= 2v_r·T_int≈2.6µm. The second beam splitter (π/2) thus recombines still overlapping clouds and generates two complementary output ports. As already mentioned, we use rectangular-shaped pulses in the time-domain and the interrogation time T_int is approximated by the distance between the temporal centers of each pulse.

In the case of a Ramsey-type interferometer with first-order Bragg pulses, the fringe spacingd_ori in the far field can be re-written as

d_ori= hT_tof

mδx = π·T_tof k·Tint

, (3.74)

with wave vector k = 2π/λ of the Bragg beams. An absorption image is taken after an additional separation timeT_sep which was scanned to realize different total time-of-flight between release of the matter waves and detection ofT_tof =T₀+T_int+T_sep. Error bars in Fig. 3.17 depict1σ confidence bounds of the fitted parameters. Since the interrogation time T_int is kept constant, the fringe spacing (blue circles) evolves

0 5 10 15 20 25 30 35

Figure 3.17: Temporal fringe spacing evolution of a ground-based ORI with constant interro-gation timeTint= 240µs. The interferometer is appliedT0= 10 msafter release of the condensate from a steep trap (ωrad≈2π·350 Hz). Scaling law calcula-tions for the fringe spacing of differently steep radial-symmetric traps (dashed lines) and the far-field approximation (solid line) are given for comparison.

linearly in time which fits well to the scaling law calculation using the corresponding trapping frequency (dashed orange line). The far-field approximation coincides with the numerical calculation after a few ms (solid blue line), which confirms our mea-surement to be operated in the linear regime. A gallery of absorption images of this measurement is shown as a temporal sequence in Fig. 3.20 (left).

For comparison, the scaling law calculations of d(t) for more shallow initial traps are given (dashed red and black lines). These correspond the the most shallow trap operable in ground-based measurements (ω_rad ≈2π·50 Hz) and the holding trap used in microgravity experiments (ω_rad ≈ 2π·25 Hz, see Sec. 5.2.2). As expected, the presence of mean-field acceleration would shiftd(t) to larger values.

Scanning the wave packet separation

Another method to analyze the fringe spacing evolution and to verify the validity of Eq. 3.74 has been demonstrated by scanning the interrogation time T_int for a fixed time-of-flight T_tof. This changed the separation of the interfering wave packets δx.

In Fig. 3.18, two experimental data sets of condensates released from a steep trap (ω_rad ≈ 2π·350 Hz) are shown for which the initial expansion time T₀ is different.

Red circles show the fringe spacing for an ORI immediately applied after release of the condensate (T₀= 0 ms), whereas blue circles correspond to a sequence applied during the linear expansion phase (T₀ = 10 ms).

The total time-of-flight T_tof between both data sets is slightly different, which is why two dashed lines (red and blue) are calculated for the expected fringe spacing in

100 150 200 250 300 350 400

Figure 3.18:Fringe spacing for increasing interrogation timeTint(b=wave packet separation) in an ORI.

bias current of the holding trap I bias

Figure 3.19: Influence of the trap steepness on the relative far-field approx-imation error in an ORI.

far-field approximation. Error bars are on the same order as compared to Fig. 3.17, but not given here for better visibility. A gallery of absorption images for the measurement withT0 = 10 ms is shown as a temporal sequence in Fig. 3.20 (right).

The fringe spacing is expected to decrease as1/δx∼1/T_int, which could be verified forT₀ = 10 ms. Large discrepancies between experiment and far-field approximation occur for the data withT0 = 0 ms. During the first400µsafter release, 73% of mean-field energy is converted into kinetic energy, given the dynamics of λ˙_rad for the used trapping potential. The acceleration of the condensate during the ORI is therefore strongly mean-field driven. If now a beam splitter is applied, the wave function is split into a coherent superposition of two momentum states with bisected density. The wave packets expansion rateλ˙ will therefore reach its asymptotic value earlier and changes the slope ofα∼λ/λ. The observed discrepancy may thus be a consequence of slowing˙ down the mean-field conversion process.

Based on this assumption, the relative error w.r.t the far-field approximation should decrease for even steeper trapping potentials which lead to a faster mean-field conver-sion.

Influence of the trap steepness

If we prepare the condensate in more steeper traps, they expand much faster leading to almost entirely converted mean-field energy even at timescales of a few hundreds of µs. An ORI with a fixed interrogation time of Tint= 400µs is applied to condensates directly after release. After T_tof = 33.1 ms, the fringe pattern is detected and the spacing evaluated for different I_bias, which determines the steepness of the holding trap (the trapping frequency roughly scales asω_rad∼I_bias^3/2, see Sec. 2.4.2).

The relative error between the measured fringe spacings and the far-field approxi-mation

d_meas−d_ori dori

(3.75)

Figure 3.20: Temporal sequences of absorption images for an ORI with Bose-Einstein con-densates. Every picture corresponds to a single measurement from the data sets evaluated in Fig. 3.17 and 3.18, respectively. For a fixed interrogation time (=b distanceδxbetween the interfering wave packets), the fringe spacing increases linear with the time-of-flightTtof (left). If theTtof is constant and the interro-gation timeTint is increased (right), the fringe spacing decreases as1/Tint. In these pictures, the condensate widths exceed the applied separationδxby about a factor of∼25, thus leading to a fringe pattern over the whole envelope. More details in text.

is given in Fig. 3.19. The error bars correspond to the standard deviation of two independent measurement cycles. At the beginning, the dominating mean-field accel-eration of dense ensembles emerging from shallow traps (lower values of I_bias) yields comparably high discrepancy of the measured fringe spacingd_exp w.r.t. to the far-field approximation. For steeper traps (I_bias →1.3 A), most of the mean-field is converted and the measured fringe spacing slowly converges to the corresponding values given by the far-field approximation.

Until now, we implemented an ORI with an atom-chip-based BEC and investigated the fringe spacing evolution under influence of mean-field driven acceleration. With the same setup, a coherence length measurement of the condensate will be presented next.