Schumm, T., Hofferberth, S., Andersson, L.M., Wildermuth, S., Groth, S., Bar-Joseph, I., Schmiedmayer, J. & Krüger, P.
Matter wave interferometry in a double well on an atom chip
Nature Phys., 2005, Vol. 1, pp. 57-62
Hofferberth, S., Fischer, B., Schumm, T., Schmiedmayer, J. & Lesanovsky, I.
Ultracold atoms in radio-frequency dressed potentials beyond the rotating-wave approximation
Physical Review A, 2007, Vol. 76(1), pp. 013401
A one-dimensional Bose-Einstein condensate trapped in a tight magnetic trap can be split into two by means of an adiabatic radio frequency induced potential. By interfering the condensates we show that the splitting process preserves the relative phase between the two BECs even when they are split far enough to inhibit tunnel coupling.
At the centre of our interference experiments is the beam splitter. It is based on radio frequency induced adiabatic potentials. This technique has the advantage of allowing a smooth transition from a single trap into a double well potential and hence coherent splitting is possible. In contrast to other methods, the splitting distance can be orders of magnitude smaller than the chip structures. We precisely control the splitting of condensates over a range from 3-80 µm. Our new technique allows to construct versatile chip based atom interferometers. Our scheme is based on a combination of static and radio frequency (RF) magnetic fields. The beam splitter is fully integrated on the atom chip, as these fields are provided by microfabricated wires. By slowly changing the parameters of the RF current we can smoothly transform a tight magnetic trap into a double well and thereby dynamically split a BEC. We accurately control the splitting distance over a wide range. The potential barrier between the two wells can be raised gradually with high precision, thus enabling access to the tunnelling regime as well as to the regime of entirely isolated wells. We complete an interferometer sequence and measure the relative phase between the split BECs by recombining the clouds in time-of-flight expansion. In our experiments we find an interference pattern with a fixed phase as long as the two wells are not completely separated. The phase starts to evolve deterministically once the wells are entirely separated so that tunnelling is fully inhibited on all experimental time scales. The figure on the below side illustrates the operation principle of the beam splitter. A standard magnetic microtrap is formed by passing a current through a wire. In combination with an external bias field a static magnetic field minimum forms where atoms in low field seeking states can be trapped. An additional RF field generated by an independent wire carrying an alternating current couples internal atomicby the dressed eigenstates. In our scheme we are able to form a three dimensionally confining trap with a double well shape in one of the transverse strongly confined directions. Here it is relevant that not only intensity and frequency, but also the orientation of the RF field determine the effective potential. The angle between the RF field and the local static magnetic field varies spatially, resulting in a corresponding variation of the RF coupling strength.Our smooth micro wires enable us to create pure quasi one-dimensional condensates (aspect ratio 400) with chemical potential µ ~ hω⊥ in a trap with high transverse confinement (ω⊥ = 2 · 2.1 kHz). These BECs are located directly above an auxiliary wire. A small sinusoidally alternating current through this wire provides the RF field that splits the trap. For small splitting distances (< 6 µm) we ramp the amplitude of the RF current (typically 60-70 mA) at a constant RF frequency (~ 500 kHz) to smoothly split a BEC confined in the single well trap into two. The splitting is performed transversely to the long axis of the trap, as shown in the figure (b). The distance between the two wells can be further increased by additionally raising the frequency of the RF field (up to 4 MHz).
The split cloud is detected by in situ absorption imaging along the weak trapping direction. We are able to split BECs over distances of up to 80 µm without significant loss or heating. The measured splitting distances are in very good agreement with the theoretical expectation for different gradients of the initial single well (figure below).The splitting distances are not limited by the structure size on the chip, but rather by the ground state size of the initial single well trap that can be orders of magnitude smaller. This allows us to reach full splitting of a BEC at a double well separation of only 3.4 µm using a trapping wire of a width of 50 µm at a distance of 80 µm from the surface. Furthermore, the dynamic splitting process can be performed in a smooth (adiabatic) fashion by simply controlling the parameters of the RF field. We recombine the split clouds in time-of-flight expansion after a non-adiabatically fast extinction of the double well potential. Typical matter wave interference patterns obtained by taking absorption images 14 ms after releasing the clouds are. The integrated transverse density profile derived from these images contains information on both the distance d of the BECs in the double well potential and the relative phase Φ of the two condensates. The figure below shows the observed fringe spacing that is compared to the above approximation and to a numerical integration of the timedependent Gross-Pitaevskii equation. Again, we find excellent agreement.
The observed differential phases ΔΦ in the interference patterns are nonrandom. Hence, the splitting process is phase preserving and coherence is maintained. The differential phase spread ΔΦ is smaller than 50° for splittings of up to 3.7µm (figure below). When separating the clouds further, the phase distribution remains nonrandom
up to at least d = 6µm. An increase in phase spread and a coinciding loss of average contrast can be attributed to longitudinal phase diffusion inside the individual one-dimensional quasi BECs. Determining the phase distribution at different stages of the splitting procedure, we observe that the relative phase is locked to zero, as long as the chemical potential exceeds the potential barrier; for trap separations larger than 3.4µm, the potential barrier is sufficiently high to suppress tunnelling. We observe a deterministic phaseshift after the splitting is complete. This differential phase evolution is induced by a slight residual imbalance of the double well potential (energy difference on the order of h · 1 kHz or, equivalently, µB · 1 mG).