Bragg spectroscopy of expanding Bose-Einstein condensates

We can analyze the momentum distribution of the Bose-Einstein condensate using Bragg spectroscopy [168]. Atoms are diffracted by interacting with Bragg beams whose mutual detuningδdetermines the momentum component which is diffracted out of the

condensate (e.g. in resonance). A typical measurement performed for a condensate released from a steep trap (I_bias = 0.7 A,ω_x ≈2π·150 Hz) is shown in Fig. 3.11.

After T₀ = 10 ms of free expansion, we probe the condensate with a τ = 5 ms lasting first-order Bragg pulse (rectangular shaped). The intensities are chosen to be in resonance for first-order processes only, moreover, they have been reduced in order ensure pulse areas of Φ·τ ≪ π. The beams are ∆ ≈ 640MHz red detuned below the cooling laser frequency. After a separation time of T_sep = 17 ms, we detect the diffracted and un-diffracted cloud of atoms in dependence of the frequency detuningδ between the Bragg beams.

Typical absorption images for the different detunings (δ = 6−17 kHz) and corre-sponding column densities are given in Fig. 3.11. In contrast to the expected resonance for first-order Bragg diffraction of resting atoms at δ ≈15 kHz, the center is shifted by about ∆δ ≈ 3 kHz. Large crossing angles ϑ between the Bragg beams poten-tially shift the resonance (see Sec. 3.1), but are prohibited due to the geometry of the vacuum chamber and the beam telescopes. The observed deviation is related to the fact, that the drop capsule itself (e.g., in-between of two drop campaigns) is not per-fectly aligned w.r.t gravity while standing in the lab. This causes a non-zero velocity component vx of the freely falling atoms along the wave vectors of the Bragg beams.

In our case, the Doppler shift ∆f_d of the Bragg laser light can be approximated to

∆f_d ≈ v_x·1.258 kHz/(mm/s) [150], leading to a fractional error of the effective de-tuning ∆δ/δ0 ≈ 3/15 for velocities vx on the order of 1.2 mm/s. For example, such velocities already occur at an expansion time ofT₀= 10 ms, if the angle between wave vector of the Bragg lattice and gravity exceeds only α_~k,~g ≈1^◦.

The sensitivity is limited by the stability of δ during the pulse duration time τ = 5 ms. As long as the Doppler-shift is constant, this would only absolutely shift the spectrum but has no influence on the width. Another systematic is given by the spatial dependence of the Rabi frequency due to the Gaussian beam profile, which we neglect sinceΦ·τ ≪π.

Influence of the trap steepness

From each picture in Fig. 3.11, the Bragg diffraction efficiency can be calculated and displayed versus the difference to the center resonance ∆δ to achieve a spectrum in momentum space. This has been done to evaluate the momentum width of condensates expanding from four differently steep holding traps (see open symbols in Fig. 6.3.1, left). The corresponding trapping frequencies in the beam splitter direction wereω_x ≈ 2π·(50,100,150,350)Hz.

A Gaussian was fitted to the central peak of each spectrum (solid lines) to approxi-mate the Bragg resonance rms widthσ_∆δ of the evolving condensates⁵. The obtained rms width of the four data sets are then used to calculate the momentum width of the atomic distribution as

σkx = m·σ_∆δ

2k , (3.51)

5This was done by either fitting a double-Gaussian distribution to the spectrum or by manually subtracting the thermal background.

d= 6 kHz

integrated column density [a.u.]

d= 7 kHz

d= 8 kHz

d= 9 kHz

d= 10 kHz

d= 11 kHz

d= 12 kHz

d= 13 kHz

d= 14 kHz

d= 15 kHz

d= 16 kHz

d= 17 kHz

100 µm position in x-direction [a.u.]

Figure 3.11:Bragg spectroscopy as a tool to measure the momentum distribution of a freely expanding Bose-Einstein condensate. In each picture from top to bottom, the pulse duration of the box-shaped Bragg pulses is kept constant at τ = 5 ms and the detuning is scanned fromδ= 6 kHz−17 kHz. After application of the Bragg pulse, we let the distribution expand for 22 ms before taking the image.

Diffracted atoms are indicated with the green box, the resulting hole in the initial distribution is marked red. More details in text.

-8 -6 -4 -2 0 2 4 6 8 .

Braggdiffractionefficiency [a.u.]

difference to resonant detuning [kHz]

x

Figure 3.12: Bragg spectroscopy of Bose-Einstein condensates released from differently steep magnetic traps withωx ≈2π·(50,100,150,350) Hz. The observed spectra in momentum space are fitted with a Gaussian function (left). The extracted rms width is plotted versus the trapping frequency and fitted withA·√ωx(right).

with mass of the atoms m and wave vector of the Bragg beams k. The resulting width in units of¯hkare separately displayed as a function of the trapping frequency of the holding trap (full symbols in Fig. 6.3.1, right). In a classical gas, the momentum width is proportional to the square root of temperatureT of the expanding cloud (see Sec. 2.6.1). In a harmonic trap, the temperature dependents linearly on the trapping frequency, we thus expect the Bragg resonance to evolve asσ_kx ∼A·√ω_x, of which a corresponding function is fitted to the data (solid red line).

Mean-field expansion from the shallow trap

Bragg spectroscopy can also be used to analyze the mean-field driven expansion of a condensate. Therefore, we prepared our BEC in the shallow trap (ω_x≈2π·50 Hz) and probed the momentum distribution after different free expansion times (see Fig. 3.13).

Here, pulse durations of 3 ms (blue squares) and 5 ms (red triangles) were applied.

Directly after release, the width of the Bragg resonance increases immediately due to mean-field acceleration and asymptotically reaches a finite value in the far-field, which is expected for entirely converted mean-field energy. The resonance width σ_k can be translated into a velocity width σ_v. The dashed red line is a theoretical prediction of the latter using the measured trapping frequencies of Sec. 2.6.2 in a scaling law approximation for cigarre-shaped condensates in the Thomas-Fermi regime [168, 130].

This data shows, that mean-field contributions are still present for expansion times of T_tof ≤10 ms, which is expected for shallow traps (see Sec. 3.5.2). The non-linear expansion of matter wave packets will play an important role as a systematic error in interference experiments and therefore addressed in detail in Sec. 3.5.

Scattering in the Raman-Nath regime

Scattering in the Bragg regime occurs if the interaction time of the atoms with the optical lattice beam is long enough, to fulfill the criteria of a thick grating. The atomic

0 2 4 6 8 10 12 14 16 18 20 22

Figure 3.13:Mean-field acceleration of a Bose-Einstein condensates released from a shallow magnetic trap (ωx≈2π·50 Hz), depicted by the increase of the Bragg resonance width during free expansion of the sample. The dashed red line corresponds to the calculated velocity evolutionvx∼vinfωxt/p

(1 + (ωxt))² [130, 168].

waves scatter from all layers of the diffraction plane and add constructively in a single order diffraction process, as previously used in first-order Bragg diffraction and Bragg spectroscopy experiments.

Very short pulse durations instead broaden the resonance and lead to scattering in neighboring momentum states, which is called Raman-Nath scattering [182, 172, 158].

Due to the comparably short interaction time, the optical lattice can be seen as a stationary wave, imprinting a sinusoidal phase distribution at the wave function due to the AC Stark shift, which develops into a Bessel-type distribution in the far field [161].

To find an approximation for the transition to the KD regime, we assume a cloud of atoms which is exposed to a standing wave (δ= 0) and adapt the description of power broadening of the Rabi amplitude for multi-order scattering (see Eq. 3.4.2). If the Rabi amplitude is on the order of the effective detuning to the neighboring momentum states as

Ωeff≈ ¯h m

4nk²−2kσ_kx, (3.52)

multi-order scattering will occur [184]. For example, simultaneous diffraction of a Gaussian wave packet (released from the shallow trap) in±2momentum states would for our parameters results inπ−pulse durations of about

τ±2 = π

Ωeff ≈ m·π

¯

h(8k²−2k(0.13¯hk)) ≈8.6µs, (3.53) In Fig. 3.14 (left), the realization of multi-order Bragg scattering is shown. The con-densate is adiabatically released from the shallow trap (ωx≈2π·50 Hz) and expands forT_tof = 13 ms. Then, a short pulse is applied with a duration ofτ = 8µs and an

in-Figure 3.14: Diffraction in the Raman-Nath regime. The initial condensate (0¯hk) is simul-taneously scattered in± 3 momentum states after application of a pulse with τ = 8µs (left). The application of two successive multi-order beam splitters, separated by about100µs, leads to a multi-order interference pattern (right).

tensity of IKD ≈10 W/m². The sub-recoil momentum distribution of the condensate allows to distinguish between the different momentum states in the absorption images, where atoms are mainly scattered from 0¯hk order into ±4¯hk with a less significant occupancy of ±2¯hk and±6¯hk.

If we after application of the first Raman-Nath beam splitter wait for an interrogation time of about100µsand subsequently apply a second beam splitter, we observe spatial interference patterns in each of the outputs associated with the diffraction order (see Fig. 3.14, right). The interference pattern can be clearly seen in the absorption image as well as in the integrated column densities⁶. We will discuss the reason for the formation of such interference patterns in the next section.