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In the following, we demonstrate the coherence of the splitting process by carrying out trapped-BEC interferometry with internal state labeling of the

interferometer arms. Our interferometer consists of a Ramsey(π/2)-(π/2)se- quence on the|1i ↔ |2itransition in combination with state-dependent split-

ting and recombination of the motional wave functions between the pulses. We use a non-adiabatic splitting and recombination scheme, see Figure5.3.1a, which is motivated by the sequence required for the atom chip controlled phase gate in [43]. By choosingδm= 600 kHz, we ensure that the admixture

of state|3_iis small enough so that decoherence due to magnetic field noise is not a problem on the timescale of our experiment. After the first π/2 pulse, which prepares the atoms in an equal coherent superposition of |1_i and _|2_i, the microwave on the CPW is switched on within 50µs to Pmw = 120 mW,

which corresponds to a sudden displacement of the potential minimum for state|1_iby4.3µm. After a variable delay, we switch off the microwave within 50µs, followed by the secondπ/2pulse and state-selective detection (after a time-of-flight of 4 ms) to determine the number of atoms N1 (N2) in state

|1_i (_|2_i). The time between the π/2 pulses, TR, corresponds to the overall

time the microwave was turned on. In this scheme, the switching of Vmw is

adiabatic with respect to the internal-state dynamics, but fast compared to the trap oscillation period. The wave function ψ_|1i is thus set into oscillation

in the shifted potential V_|1i. We can record these oscillations by varying TR

and imaging the atoms without applying the second π/2 pulse, see Figure

5.3.1b. The wave function ψ_|1i oscillates with a peak-to-peak amplitude of

8.5 µm and a frequency of f˜x = 116 Hz. Small deviations of the oscillation

frequency _f˜_{x from the trap frequency fx arise due to additional mean field} effects from the resting wave function ψ_|2i. Periodically ψ_|1i comes back to its initial position, whenT_R is an integer multiple of 1/f˜x = 8.6 ms. At these

times, it overlaps with the wave function ψ_|2i. Note that owing to collisions,

ψ_|2i starts to oscillate as well.

If we apply bothπ/2pulses and vary TR, we observe Ramsey interference

fringes, see Figure5.3.2. The interference contrast is modulated by the wave function overlap of the two states and thus periodically vanishes and reap- pears again owing to the oscillation of state ψ_|1i. As a measure of the wave

function overlap, we plot σ(N₂)/N¯₂ as a function of T_R, where σ(N₂) is the standard deviation and _N¯_{2 is the mean ofN2 obtained from a running aver-} age over one period of the Ramsey fringes (15 measured datapoints for each state), see Figure5.3.2a. This measure of the overlap has the advantage that it is largely insensitive to noise on the Ramsey fringes. Corresponding fringe data andin situ images of the atoms at specific timesTRare shown in Figure

5.3.2b+c. Precisely at the time when the wave function ψ_|1i has carried out

a full oscillation in V_|1i, a sharp recurrence of the contrast is observed. The

a)

x

x

c. o. m. position at time of 2nd π/2 pulse V_z |2〉 V V_|1〉

b)

|2〉

|1〉

Figure 5.3.1: Dynamical splitting and recombination scheme used for BEC

interferometry. (a) Timing sequence of the interferometer. In between the two π/2-pulses of a Ramsey sequence on the _|1_{i ↔ |}2_i transition, the mi- crowave on the CPW is pulsed on for a duration TR, resulting in a sudden

displacement of the potential minimum of V_|1i. This sets the wave function ψ_|1iinto oscillation. (b) Oscillation of the atoms, recorded with the sequence

of (a), but with the second π/2 pulse omitted. The center-of-mass (c.o.m.) position of the atoms at the end of the sequence is shown as a function of

T_R. ψ_|1i oscillates whereasψ_|2i remains initially at rest. Each time the wave

nal and motional state is coherent. The relatively high contrast of the first recurrence (Michelson contrast 50%) shows that the collisional interactions between the atoms observable in Figure5.3.1b lead only to a relatively small distortion of the wave functionsψ_|ii. Wave function distortion can be reduced

to negligible levels by optimal control of the splitting process as discussed in [43].

For the second (and subsequent) recurrences, we observe substantial phase noise on the Ramsey fringe data. In contrast, when we take Ramsey fringes without splitting the BEC, comparable noise is visible only for TR beyond

several hundred milliseconds. Fundamental as well as technical sources of this noise are discussed in Section 5.5.