Shimon Machluf


Coherent Stern–Gerlach momentum splitting

Interferometry is a measurement technique based on the phase difference between waves. It was used already in the 17th century in order to support the claim that light is a wave and not a stream of particles. Optical interferometry is a well-established field. It uses the phase differences to measure optical length differences. On the other hand, interferometry with matter-waves is still in its infancy. Since the phase of de-Broglie waves of massive particles is extremely sensitive to forces acting on the particles, matter-wave interferometry may serve as a powerful tool for measuring potentials and forces. Interferometry is used as a sensitive tool for investigating fundamental physics (for example, testing the equivalence principle of general relativity or measuring the phase fluctuation along a 1-dimensional Bose gas) as well as for realizing technological applications (for example, measurement of linear acceleration and rotation).

In order to produce a matter-wave interferometer, the internal and external degrees of freedom of the atoms should be controlled and manipulated. This can be done with light or magnetic fields. Interferometers based on light beam-splitters put the atoms in a superposition of momentum states. Interference experiments based on magnetic field beam-splitters have been limited so far to the splitting of trapped atoms into two trapped clouds in a double-well potential. The main advantage of splitting atoms in a trap or a waveguide is the possibility of thereby controlling the motion of both outputs of the beam-splitter and manipulating each one of them independently. The atom chip is an ideal tool for precise small-scale control and manipulation using magnetic fields.

Coherent splitting on an atom chip was first realized with light. Splitting with magnetic fields was first demonstrated by the Schmiedmayer group in Heidelberg (now located in Vienna). They used a combination of a static magnetic trap and a magnetic field alternating at radio frequencies to transform a single well into a double-well potential, and to split a Bose-Einstein condensate (BEC) into two clouds. Since then, splitting of a BEC has been demonstrated on an atom chip using microwave radiation, and first indications for coherent splitting were also shown by using only static currents for the creation of the double-well potential.

In this work we choose a different approach, which uses magnetic gradients for momentum splitting and combines some of the characteristics and advantages of both light and magnetic beam-splitters. This approach is inspired by the original Stern-Gerlach (SG) experiment. In the SG effect, an atom possessing a spin is exposed to a magnetic gradient and is thus accelerated in a direction which is dependent on its spin projection. If the atom is in a superposition of spin states, it will be split into different momentum states, which will transform into different paths. This idea forms the basis for the SG interferometer. Almost a century of attempts to realize this type of interferometer failed due to the high sensitivity and accuracy needed for this kind of splitting. The SG interferometer drew theoretical attention from physicists like Heisenberg, Wigner, Bohm, Lamb, and Scully, who estimated that this kind of interferometer will require such a high level of accuracy that it is impractical. By modifying the original idea, we split a BEC and showed that the two outputs of our new beam-splitter are coherent.

The physical mechanism that lies at the heart of the SG effect and of our beam-splitter is the interplay between internal spin dynamics and external motion of the atom, due to different interaction of the Zeeman sublevels with the external magnetic gradients. This mechanism was a key feature in a previous project, which studied the effect of noise on transitions between atomic levels with state-dependent potentials. The experimental configuration of this previous project is the basis for our implementation of the SG beam-splitter.

FGBS scheme
Full size image

General scheme of the field gradient beam-splitter (FGBS) operation. (a) Atoms in the state |2> (red) are released from a trap. (b) A π/2 pulse splits the wave-function into a superposition of internal levels |1> (blue) and |2>. (c) A state-dependent force is applied. (d) Atoms in the two internal states are accelerated to different momenta. (e) The force is turned off and another π/2 pulse produces a superposition of 4 parts with two different momenta and two different internal states. (f) After some evolution time the momentum components are spatially separated.

The field gradient beam-splitter at work: FGBS (a) input and (b-d) output images, and the corresponding schematic descriptions. (a) In the trap before release. (b) After a weak splitting of less than ħk using 5µs of interaction time and allowing 14ms time-of-flight (TOF). (c) After a strong splitting of more than 40ħk using 1ms interaction time and allowing 2ms TOF. (d) To view all four output wavepackets, we separate the two internal states by another strong gradient pulse. Images (b) and (c) demonstrates the large dynamic range of this method without requiring any complicated sequence.

FGBS demonstration
Full size image
FGBS fringes
Full size image

Interference fringes from freely falling atoms. Fringes for (a) short (T=5µs) and (b) long (T=10µs) interaction times, created by recombining the two outputs of the FGBS. Cuts of the optical density (OD) are also provided, with fits (see text for fit description). The two |2> wavepackets were recombined by an additional magnetic gradient and imaged after 14ms of TOF. The different T (5 and 10µs) gives rise to a different ∆v, and hence a different distance before recombination, leading to different fringe spacings, 33µm and 16µm, respectively.

Phase stability analysis. (a) Average over the optical densities obtained from a set of 29 consecutive single-shot images in the first half hour of an interferometric measurement session (one image per minute), and a one-dimensional cut (data and fit, see text). The visibility of the averaged fringe pattern is 0.09±0.01, reduced from the single-shot value of 0.20±0.02. The average periodicity is similar, 23.1±0.35µm in the single shots and 22.9±0.2µm in the averaged image. (b) Phase distribution of the 29 images in π/6 radian bins, with a width of 1.04 radians (rms), in accordance with that expected by our stability analysis, see text. A random distribution has a probability of FGBS phase analysis
Full size image
3·(2/3)29=2.5·10-5 of similarly occupying only 2/3 of the phase spectrum. (c) Phase of fringes obtained by averaging running sets (“windows”) of 16 images for a total set of 55 consecutive images. This shows a long-term variation of the phase which is approximated by a sine fit (solid line) with an amplitude of 1.35 radian and a period of 65 minutes. The 29 images analyzed in (a) were taken near the maximum of this long-term variation.

Asymmetric transition rates

Coupling between internal and external (motional) degrees of freedom plays a major role in cooling atoms, ions and molecules and in manipulating their quantum states, e.g., in logic gates for quantum computation. The state of these particles is usually controlled by monochromatic (or transform limited) electromagnetic fields, while incoherent fluctuations (noise) must be suppressed in order to prevent decoherence, heating and loss. The latter are usually studied under the “white noise” assumption. Little is known about what happens between the monochromatic and white noise limits (“colored noise”) with respect to control and hindering effects.

To understand how the noise spectrum can affect the rate of transitions between internal states, consider a system of two levels representing electronic configurations of an atom, ion or molecule, coupled to external (translational, rotational or vibrational) degrees of freedom and to a weak homogeneous field inducing transitions between the internal levels. If this field imposes a monochromatic perturbation, then the rate of transitions from an initial state to a final state is given by Fermi's golden rule. The transition rates are proportional to the density of states in their respective final level, and are therefore asymmetric between two levels that experience different external potentials. At the other extreme, one has a fluctuating random field with a spectral density which is flat over a large bandwidth (“white noise”). This now allows transitions to all states such that the external degrees of freedom decouple from the transition dynamics and the transition rates between the two levels become symmetric.

In this work we study the dynamics of transitions between two internal atomic states with different external potentials in the form of two magnetically trapped Zeeman levels (|F,mF> =|2,2> and |2,1>) in the presence of colored noise which is neither monochromatic nor white. We experimentally observe that the relative transition rates between the levels strongly depend on the spectral shape of the noise. While a clear understanding of this dependence is important in order to find effective ways to combat uncontrolled noise, we demonstrate that this dependence also allows steering the transitions in the desired direction by utilizing engineered noise.

Spin flips ratios
Full size image
Measured ratio R(t) and the applied noise. (a) The measured ratio R(t) between the number of atoms in mF=1 and the number of atoms in both trapped levels. Different curves correspond to different values of Δf, the detuning between the noise peak frequency f0 and the Zeeman splitting E120/h between the two trapped levels at the magnetic field minimum. The measured R(t) goes beyond the band 0≤R≤1/2 representing asymptotic values of R(t) in a model where the external degrees of freedom are decoupled from the transition dynamics
(the dashed line at R=1/3 is for white noise). Data variance gives an error of ±0.01 (not shown). (b) The noise spectrum we introduce into the system relative to f0=18MHz (red line is the fit used in our calculations).

Asymptotic values R, as a function of Δf, compared with theory with no fitting parameters (T=0.5-1.5µK). Error bars are rms values for the data variance and mean fit error in measurements of R(t), except for open-circle data points, for which only one measurement is available and the error is the average of the entire data set. The horizontal error is the uncertainty in the magnetic field minimum.

Spin flips asymtotic values
Full size image
FGBS fringes
Full size image
A simplified model showing the asymmetry in the transitions. (a) A simplified model showing the asymmetry in the transitions between the two trapped levels. We assume that the atoms are in thermal equilibrium at temperature T, in both levels. We plot the transitions at the mean atom distances from the trap center. A spin flip from mF=2 to mF=1 requires photon energy E2→1, which is smaller than the photon energy E1→2 needed for the reverse transition. Eif corresponds to the difference between the potentials since the atomic recoil is
negligible. (b) The two transitions sample different frequencies of the noise spectrum (plotted twice for opposite detuning), giving rise to the asymmetric transition rates. The two pairs of arrows (blue and red) point to the sampled noise intensities for each detuning.

Cold atoms on atom chips

The phase transition to BEC occurs when the phase space density (the density of the atoms multiplied by their thermal wavelength cubed) becomes larger than one. The RF evaporation technique is able to bring the atomic cloud to the needed parameters. However, this technique requires a trap with a controllable depth, namely, that the height of the trap walls may be varied so hot atoms may be expelled. In our experiment, we use for this purpose a magnetic trap. These traps are typically not deep enough to trap atoms at room temperature and pre-cooling, to ~100µK, is necessary. In our system, we use a magneto-optical trap to cool and trap an atomic sample (from the background gas), as a preparatory step for the magnetic trap.

After cooling the atomic cloud through the phase transition, ~300nK, one should observe the differences between a Bose gas and a gas in a Maxwell-Boltzmann distribution. As long as the thermal gas does not reach the phase transition, or is very close to it, it is still considered an ideal gas that obeys the Maxwell-Boltzmann distribution. The velocity distribution has a Gaussian shape, and after release it expands isotropically.

BEC transition
Full size image

When the atomic gas condenses, both the shape and the expansion after release change. In a non-interacting BEC, the shape of the condensed atoms is still a Gaussian, but much sharper than the thermal distribution. If the atoms are interacting, the condensate gets the trap shape - an inverted parabola. This parabola is much wider than the ground state Gaussian, but it is still much narrower than the broad thermal distribution.

In addition, when releasing a BEC from the trap, its expansion depends significantly on the trap shape, setting the initial conditions. The cloud expands faster in the axis which was narrower in the trap, i.e. the tighter axis. When the atoms are non-interacting, this effect can be explained as a result of the uncertainty principle, Δx Δp > ħ. In the tighter axis, Δx - the uncertainty in position, is smaller, and hence Δp - the uncertainty in momentum, is larger. If the atoms are interacting this anisotropic expansion is mostly due to the interaction energy converted into kinetic energy.

BEC transition
Full size image