Abstract
In the Stern–Gerlach effect, a magnetic field gradient splits particles into spatially separated paths according to their spin projection. The idea of exploiting this effect for creating coherent momentum superpositions for matterwave interferometry appeared shortly after its discovery, almost a century ago, but was judged to be far beyond practical reach. Here we demonstrate a viable version of this idea. Our scheme uses pulsed magnetic field gradients, generated by currents in an atom chip wire, and radiofrequency Rabi transitions between Zeeman sublevels. We transform an atomic Bose–Einstein condensate into a superposition of spatially separated propagating wavepackets and observe spatial interference fringes with a measurable phase repeatability. The method is versatile in its range of momentum transfer and the different available splitting geometries. These features make our method a good candidate for supporting a variety of future applications and fundamental studies.
Introduction
The development of atom interferometry over the last two decades has given rise to new insights into the tenets of quantum mechanics^{1} as well as to ultrahigh accuracy sensors for fundamental physics^{2,3,4} and technological applications^{5,6}. Examples range from the creation of momentum state superpositions by accurate momentum transfer of laser photons^{7,8} allowing high precision measurements of rotation, acceleration and gravity^{5,6,9,10}, to the splitting of trapped ultracold atoms by local potential barriers^{11,12,13} allowing the investigation of fundamental properties of quantum systems of a few or many particles, such as decoherence and entanglement^{14,15,16}.
One of the tools for atom interferometry is the atom chip^{17,18,19}. The high level of spatial and temporal control of local fields which is facilitated by the atom chip has made it an ideal tool for the splitting of a Bose–Einstein condensate (BEC) into a doublewell potential by a combination of static magnetic fields with radiofrequency (RF)^{12} or microwave^{13} fields. Pure static fields^{20} or light fields^{21,22} have also been used. However, practical atom chip schemes for interferometry with a wide dynamic range and versatile geometries are still very much sought after^{23,24,25,26,27,28}. Such schemes may enable, for example, sensitive probing of classical or quantum properties of solidstate nanoscale devices and surface physics (for example, refs 29, 30, 31, 32, 33). This is expected to enhance considerably the power of noninterferometric measurements with ultracold atoms on a chip, which have already contributed, for example, to the study of longrange order of current fluctuations in thin films^{34}, the Casimir–Polder force^{35,36,37,38} and Johnson noise from a surface^{39}. In addition, interferometry integrated on a chip is a crucial step towards the development of miniature rotation, acceleration and gravitational sensors based on guided matter waves^{23,27}.
One of the earliest attempts to envision atom interferometry was considered shortly after the discovery of the Stern–Gerlach (SG) effect almost a century ago^{40}. The SG effect, which has become a paradigm of quantum mechanics^{41}, uses a magnetic gradient to split particles into momentum states with different spin projections. SG interferometry was generally judged to be impractical because of the extreme accuracy that would be required, where fundamental issues of phase dispersion stemming from the uncertainty principle were raised (Briegel et al.^{42} and references therein), and finally coined as the HumptyDumpty effect^{43,44,45}.
Here we report, what is to the best of our knowledge, the first observation of spatial interference fringes with a measurable phase stability, originating from spatially separated paths in SG interferometry. We propose and demonstrate a field gradient beam splitter (FGBS) that may be viewed as the application of modern atom optics to the nearcentury old idea, where we take advantage of the high magnetic field gradients, fast timing and high accuracy made available on an atom chip. Although our method uses the SG effect, it is very different from previous theoretical^{42,43,44,45,46} and experimental^{47,48} schemes of SG interferometry. Although previous experiments used thermal atomic beams in which two spin states propagated on a metre scale, our experiment uses ultracold wavepackets on the micrometre scale. This allows not only for miniaturization, but also the accurate manipulation of the quantum state, enabled by the atom chip. Moreover, as we use minimal uncertainty wavepackets, the phase dispersion due to the evolution through the inhomogeneous potential is considerably smaller. Another fundamental difference is that, in contrast to previous SG interferometer schemes, the time scales in which atoms propagate while in a superposition of two different spin states are extremely short, as the output of the beam splitter demonstrated here contains different momentum states of the same spin state, a crucial advantage in noisy environments. This is why, in contrast to previous experiments, our experiment does not require magnetic shielding. In an operation time of just a few μs, the FGBS may allow for a continuous range of momentum splitting of over 100 photon recoils (100 ħk, for reference photons of 2π/k=1 μm wavelength). This may enable the advantageous largeangle interferometers^{49,50,51,52}, giving rise to highly sensitive probes. As a novel method that combines high momentum splitting with the advantages of chipscale integration, the FGBS may serve for exploring new regimes of fundamental studies and technological applications.
Results
General splitting scheme
The general scheme of the FGBS and its output is demonstrated in Fig. 1a–f. It is based on a combined manipulation of two internal states (1› and 2›) and an external potential. We perform a Ramseylike sequence of two π/2 Rabi rotations and apply a field gradient during the time interval between them. The states 1› and 2› may be any two states enabling controlled coherent transitions between them, and having a statedependent interaction with the field gradient. We start with the atoms in the internal state 2› and an external state p_{0}, x_{0}›, representing a wavepacket with central momentum p_{0} and central position x_{0}. The first π/2 rotation transfers the atoms into the superposition state . We then apply a field gradient that constitutes a stateselective force F_{j}=−V_{j} (j=1, 2) for an interaction time T, which is typically shorter than the time it takes the atoms to move in the force field. The state of the atoms after time T is then , where each level acquires a phase gradient [−V_{j}(x)T/ħ]=F_{j}T/ħ, which is equivalent to a momentum transfer p_{0} → p_{0}+F_{j}T. The second π/2 rotation transfers the atoms into the superposition state
representing two wavepackets with momenta p_{j}=p_{0}+F_{j}T entangled with the internal states , such that each of the internal states 1› and 2› is in a superposition of wavepackets with different momenta (the state in equation (1) can be viewed as spatial Ramsey fringes, see below). Equation (1) is also valid if the atomic motion during the interaction time is taken into account or when the force is not homogeneous in space and time. In this case, the states p_{j}, x_{0}› should be replaced with the more general solutions for the wavepacket evolution in the respective potentials V_{j}(x, t).
This completes the operation of momentum splitting. One may then use the 1› and 2› states to realize two parallel interferometers for noise rejection. One can also use the entangled momentum and internal state as an interferometer of two clocks (for example, ref. 53). If one wishes to stay with just one of the two internal states, one may typically find a dedicated transition to discard the redundant state.
Realization of the FGBS
To realize the FGBS, we use Zeeman sublevels of freely falling ^{87}Rb atoms and magnetic field gradients from a chip wire. We start with a BEC of ~10^{4} atoms in state F, m_{F}›=2, 2› ≡ 2› and use a RF field to perform transitions to state 2, 1› ≡ 1›. We use the same setup described in ref. 54. The trap position is at a distance of z=100 μm from the chip surface, and the radial and axial trapping frequencies of 2› are ≈2π × 100 Hz (measured) and ≈2π × 40 Hz (estimated), respectively. In order to have the 1› and 2› states form a pure twolevel system, we apply a strong homogeneous magnetic field (ΔE_{12} ≈h × 25 MHz) and push the transition to 2, 0› out of resonance by ~250 kHz due to the nonlinear Zeeman effect^{54}. Next, we release the BEC and apply two π/2 RF pulses with a Rabi frequency of Ω_{R}=20–25 kHz and with a magnetic gradient pulse of length T in between, thus forming a Ramseylike sequence. The gradient is generated by a current of 2–3 A in a 200 × 2 μm^{2} gold wire on the chip surface. The homogeneous magnetic field (in the direction of the magnetic field generated by the chip wire) is kept on during the free fall to preserve the quantization axis.
In Fig. 1g–j, we demonstrate the output of the FGBS splitting process. Figure 1g presents an image of the cloud in the trap, before it is released. In Fig. 1h1i, we present an image of the output of two events that exhibit the large dynamic range of this method, one with momentum difference of less than ħk (T=5 μs), and the other with more than 40 ħk (T=1 ms). To compare with atom interferometry based on light beam splitters, we express the momentum transfer in units of ħk, where ħk is a reference momentum of a photon with 1 μm wavelength. The only parameter that is changed is the interaction time T. Finally, in order to verify the internal state of the atoms (equation (1)), we separate between the 1› and 2› states by a second long pulse of magnetic field gradient, as shown in Fig. 1j.
Figure 2a shows the measured differential momentum transfer as a function of the interaction time T. An interaction time as short as 100 μs is required to transfer a relative velocity of 50 mm s^{−1} (equivalent to 10 ħk). The operation of the FGBS may be quantitatively understood by simple kinematics in one dimension (along the z axis). During the interaction time T, a differential acceleration between the wavepackets is induced, such that after the FGBS, each internal state is a superposition of two wavepackets which were accelerated as a 1› or 2› state. The force applied on a wavepacket of a certain m_{F} state at a distance z below the chip wire is:
where μ_{0} and μ_{B} are the magnetic permeability of free space and the Bohr magneton, g_{F} is the Landé factor for the hyperfine state F and I is the current. The equation does not include the effect of nonlinear Zeeman splitting and a geometric term 1/[1+(W/2z)^{2}], accounting for the finite width W of the wire. These terms have been taken into account in our full simulation of the FGBS which, as presented in the figure, is in good agreement with the experimental results. The linear relation in Fig. 2a is to be expected for the short interaction times during which the atoms move only slightly, and the acceleration in equation (2) is fairly constant (the kink in Fig. 2a is due to changing currents, see caption). In Fig. 2b, we present the calculated limit of our specific realization of the FGBS because of the growing distance of the atoms from the gradient source, the chip wire. In Fig. 2c, we show that for realizable chip wire parameters, momentum transfers of over 100 ħk are feasible in a few microseconds.
The FGBS output as a spatial Ramsey fringe pattern
In Fig. 3, we present a Gross–Pitaevskii simulation of the atomic density for state 2› just after the second π/2 pulse (‘near field’). The density shows a sinusoidal pattern. This pattern may be viewed as spatial Ramsey fringes: an atom at a specific point x is subjected to a Ramsey sequence in which the phase accumulated between the two π/2 pulses is (1/ħ) where ΔV=V_{1}−V_{2} is the potential difference between the two levels. As the potential approximately varies linearly along the axis, ΔV ≈−ΔFz, and at short times the atom stays stationary, the Ramsey phase and consequently the population of the two states 1› and 2› is modulated with a wavelength λ=h/2ΔFT (where the factor 2 is due to the fact that we observe intensity modulations and not amplitude). As a perfect sine function Fourier transforms into two identical and counter propagating k components (with momentum difference Δp=h/2λ), this ‘nearfield’ density distribution transforms after a sufficient timeofflight (TOF) into a pair of spatially separated wavepackets. The ‘nearfield’ fringes shown in the figure are not resolvable by our imaging system, and what we observe is only the four wavepackets (two for the state 1› and two for the state 2›) that evolve after a long TOF, as shown in Fig. 1.
Observation of interference
In order to examine the coherence of the FGBS output, we have applied a simple procedure to stop the relative motion of the two output wavepackets of internal state 2›. We then let them freely expand and overlap to create spatial interference fringes, as shown in Fig. 4 (the state is out of the field of view). This method is suitable for the observation of interference between wavepackets with small spatial separation. If the spatial separation and the momentum difference is large, the wavelength of the interference pattern may be smaller than the resolution limit of the imaging system. In this case, interferometric methods based on internal state population should be applied (see Discussion).
The simple procedure we use to stop the relative motion consists of applying a second gradient pulse giving a stronger kick to the wavepacket with smaller momentum, which is at this time closer to the chip wire relative to the wavepacket with larger momentum. The duration of the second momentum kick is tuned such that after this kick the two wavepackets have the same momentum (with a spatial separation that we denote by 2d).
In order to understand the formation of the interference pattern, we use a Gaussian model in which the two interfering state 2› wavepackets p_{1}, z_{1}› and p_{2}, z_{2}› have a Gaussian shape of initial width σ_{0} and centre trajectories z_{1}(t) and z_{2}(t), corresponding to atoms that have been in the internal states 1› and 2›, respectively, during the FGBS gradient pulse (see Methods for more details). Given that the final momentum difference between the two interfering wavepackets is smaller than the momentum spread of each one of them, after a long enough time they overlap and an interference pattern appears having the approximate form:
where A is a constant, z_{CM}=(z_{1}+z_{2})/2 is the centreofmass (CM) position of the combined wavepacket at the time of imaging, σ_{z}(t) ≈ħt/2mσ_{0} is the final Gaussian width, λ=ht/2md is the fringe periodicity (2d=z_{1}−z_{2}), v is the visibility and φ=φ_{2}−φ_{1} is the global phase difference. The phases φ_{1} and φ_{2} are determined by an integral over the trajectories of the two wavepacket centres. We emphasize that equation (3) is not a phenomenological equation, but rather an outcome of our analytical model (see Methods).
Equation (3) is used for fitting the interference patterns as those shown in Fig. 4. For a pure superposition state, as in our model, the fringe visibility v should be 1. The observed visibility is reduced because of various possible effects, such as unequal amplitudes of the two wavepackets or partial overlap between them, the finite temperature of the atomic cloud, effects of atomic collisions and limited imaging resolution. We note that some of the manybody collisional effects, such as phase diffusion, would not lead to a reduction of the singleshot visibility, but may cause the randomization of shottoshot phase. As the singleshot visibility does not imply that the interferometric process is coherent, one needs to examine the stability of the phase along many experimental realizations of the interference.
Phase and momentum stability
An analysis of a sequence of interference patterns (Fig. 4a–c) reveals shortterm phase fluctuations of δφ~1 radians and longterm drifts over a time scale of an hour. The coherence of the underlying interferometric process is clearly proven by this analysis. In particular, we directly observe statistically significant phase repeatability in a time window of about half an hour where the drift is slow, such that an image obtained by averaging over the 29 singleshot images shows a pronounced visibility of only a factor of ~2 less than the singleshot visibility.
To identify the sources of instability and to suggest ways to reduce it, we analyse the propagation of the wavepackets with the help of our Gaussian model. This analysis shows that the major source of phase instability in our experiment is the difference in magnetic field energy during the time between the two π/2 pulses of the FGBS, in which the two wavepackets occupy two different spin states. As the magnetic energy is linearly proportional to the pulse current, the phase fluctuation at a given reference point z_{0} is
where the relative current fluctuations during the pulse δI/I and the timing uncertainty δT/T are both independently estimated for our electronics to have a rootmeansquare (r.m.s.) value of ~10^{−3}. As the field applied by the chip wire at z_{0}=100 μm is about 27 G, corresponding to a Zeeman potential of ~19 MHz, we expect, for T=5 μs, phase fluctuations of δφ~1 radian, similar to the observed shortterm phase fluctuations (width of the phase distribution in Fig. 4b). Note that during the 100 μs time interval between the two π/2 pulses, a bias field of about 40 G (in the same direction as the wire field) is on. Changes in the distant coils responsible for this field are most likely the source of the observed longterm drift of the phase.
Although the FGBS intrinsic phase instability was found to be the main source of the interferometric phase instability in our experiment, it is important to analyse the FGBS momentum instability, which may become the dominant factor in interferometers with a larger spacetime area. In Fig. 4d, we show a few images and fits of singleinterference patterns that reveal an instability in the CM momentum. The source of this instability may be understood on the basis of equation (2), indicating that the momentum kick fluctuations of our FGBS are given by
where z_{i} is the initial distance from the wire responsible for the momentum kick. As the trapping potential was generated by a wire at a distance of more than 1 mm (with its own relative current fluctuations δI/I~10^{−3}) rather than a chip wire (for technical reasons), we estimate the uncertainty of this position to be δz_{i}~1 mm × 10^{−3}=1 μm. For z_{i}=100 μm, we have δz_{i}/z_{i}~10^{−2}, making it the dominant source of momentum instability. Indeed, the observed width of the final CM position distribution is δz_{CM}/z_{CM}~0.02.
However, perhaps surprisingly, the observed momentum difference between the two wavepackets after the second momentum kick is much more stable and gives rise to a good overlap at each experimental shot, as observed in Fig. 4. Let us elaborate. We have approximated the 1/z potential in our experiment by a quadratic form
where represents the inhomogeneity of the force =−(z, t)/∂z, acting as a harmonic force when <0. The observed stability, namely the small relative momentum fluctuations, is due to the fact that the second momentum pulse reverses the effect of momentum fluctuations due to the first pulse. The second pulse applies a differential force F_{2}(z_{1})−F_{2}(z_{2})=F′_{2}(z_{0})(z_{1}−z_{2})∝−(p_{1}−p_{2}), which acts against initial momentum fluctuations. In the interferometric scheme used in our experiment, the second pulse introduces its own fluctuations of the momentum difference through current fluctuations δI and δT, but as noted, this contribution is an order of magnitude smaller than that of the fluctuations introduced by the initial position fluctuations during the first pulse.
Finally, our Gaussian model shows that initial position fluctuations contribute very little to the fluctuations of the accumulated phase difference after a long TOF. As demonstrated in Fig. 4e, initial position fluctuations give rise to fringe patterns whose Gaussian envelopes appear at a different position in each shot, but their phase is the same in the lab frame. This decoupling between the wavepacket position and its phase is also observed in the experiment, as demonstrated in Fig. 4d. Significant phase fluctuations that reduce the visibility of the averaged pattern are mainly caused by current strength and timing fluctuations during the gradient pulse (Fig. 4f), as discussed above.
Discussion
Here we have used the old idea of SG interferometry as a basis for a novel method in atom chip interferometry, and observed stable interference fringes from spatially separated paths. This achievement is due to three main differences compared with previous SG schemes. First, the use of minimal uncertainty wavepackets (a BEC) rather than thermal beams. Second, in most of the interferometric cycle, the two interferometer arms are in the same spin state. Finally, chipscale temporal and spatial control allows the cancellation of path difference fluctuations.
The FGBS has two main advantages in comparison to previously demonstrated matterwave beam splitters: its tuneable high dynamic range of momentum transfer and its natural integrability with an atom chip. Compared with previous atom chip experiments with doublewell potentials, which are limited to relatively slow splitting to prevent higher mode excitations, the FGBS allows a wide range of splitting times that will enable the investigation of manybody effects of entanglement and squeezing over new parameter regimes. For example, in the presence of atom–atom interactions, generation of a coherent manybody state is possible only by fast splitting^{28}. On the other hand, the FGBS that uses magnetic field gradients for splitting is more naturally and easily suited for integration with an atom chip compared with laser light splitting methods. These advantages are expected to make FGBSbased interferometry suitable for high sensitivity measurements on the micron scale.
We have gained an accurate understanding of the sources of instability in our experimental system that was not dedicated to atom interferometry, and used a simple wire configuration as well as electronics with significant technical noise. We have shown that our Gaussian model correctly predicts the position and phase fluctuations of the observed fringes, as well as the decoupling between the position and phase. We may thus attempt to suggest prospects for the stability in future experiments. A straight forward way to improve phase stability (equation (4)) is to improve the stability of current amplitude and timing. In addition, one can use a configuration with decreased ratio between the magnetic field at the trapping position and its gradient (see equation (2)), such that the momentum kick may be increased while keeping the same phase fluctuations, or the phase fluctuations may be reduced while keeping the same momentum kick. One way to achieve this is to set the initial trapping position z_{0} closer to a narrower chip wire. An alternative way is to use three parallel wires with alternating currents, such that a quadrupole field is formed near the initial position of the atoms. Such a field provides a high gradient and a small absolute value of the magnetic field. Momentum kick variations may also be reduced by using chipbased initial trapping for a better control of the initial position.
In order to estimate the bounds on phase and momentum stability in future realizations, one should consider the available technology. Let us assume a 10μs pulse. Then, for a 2A current (containing ~10^{14} electrons), the shot noise leads to δ(IT)/IT~10^{−7}. Power sources with subshot noise are being developed (for example, at Jet Propulsion Laboratory^{55}) and may enable an even better stability. Stable current pulses may be driven by ultra stable capacitors, which reach stability of δC/C=10^{−7} at mK temperature stabilities. For picosecond switching electronics, one similarly finds δT/T~10^{−7}. Taking these limits and assuming that the momentum pulse could be performed in a medium magnetic field of 1 G (splitting of 0.7 MHz), the limit on the phase uncertainty of the FGBS becomes δφ~6 × 10^{−6} radian, while momentum stability is bound by δp/p~10^{−7}. Phase and momentum stability may improve even more for longer and larger current pulses giving rise to higher momentum transfer, as the relative timing instability and shot noise reduces. A more careful estimation would require taking into account the specific structure of the FGBS and the whole interferometer, as well as environmental factors such as thermal expansion of chip elements.
As an outlook for guided interferometry, an alternative scheme using trapped atoms was implemented, though currently without fringe repeatability (S.M., Y.J., R.F., manuscript in preparation). Other possible alternatives for the operation of the FGBS include different level schemes. One example is the possible use of magnetically insensitive atomic levels such as 2, 0›. A superposition of two momentum states of 2, 0› can be easily achieved by a Rabi rotation with a RF pulse tuned to the transition 2, 1› →2, 0›. Another example is a FGBS using a microwave transition between two hyperfine states with opposite magnetic moments, such as the states 2, 1› and 1, 1›, enabling symmetric splitting with opposite momenta. Through nonlinear Zeeman effect, the FGBS may also split directly the magnetic noise immune ‘clock’ states 1, 0› and 2, 0› (see Methods).
Finally, we propose two interferometer schemes, each based on two FGBSs. The first scheme creates spatial fringes and replaces the second momentum kick used in our experiment with a second FGBS when the spatial separation between the two wavepackets is 2d, as described in Fig. 5a–c. Unlike the method used in the current experiment, which is based on a long inhomogeneous gradient pulse, the proposed method uses a short pulse with the price of reducing the signal intensity by a factor of 2. The second scheme realizes internal state population fringes and requires reversing the velocity of the wavepackets. When the two parts of the wavefunction spatially overlap, another FGBS is applied, as described in Fig. 5d–f. See Methods for more details. In view of the versatility of the FGBS, we expect it to enable a wide range of fundamental as well as technological applications.
To conclude, we have demonstrated a method for splitting atoms into momentum states by using local magnetic field gradients and have observed repeatable spatial interference fringes, which indicate the coherence of the splitting process. The method has a wide dynamic range of momentum transfer, it is versatile in geometry and utilized states, and is easily integrable with an atom chip. We have analysed in detail the causes for phase and momentum instabilities. Our analysis exhibits a good fit to the experimental observations. Based on this analysis, we propose practical ways to exploit the potential of the wide dynamic range of momentum transfer and to increase interferometric stability. This enables us to extrapolate and predict the ultimate accuracy of such a device—which we find to be high.
Methods
Gaussian wavepacket model for interferometry
Our model assumes that the atomic state at each stage of the interferometric process is a superposition of wavepackets as in equation (1). The spatial representation can generally be written as
where w_{j}› represent internal state trajectories, such that at time t, two states w_{j}› and w_{k}› may either represent two different internal states or the same internal atomic state with different internal state histories. In our case, for example, the state w_{1}› will represent atoms that were initially at the state m_{F}=2›, then transformed into m_{F}=1› during the first π/2 pulse and then back to m_{F}=2› during the second π/2 pulse, and w_{2}› will represent a trajectory where the atoms stayed at m_{F}=2› throughout this process. In what follows, we omit the ket symbols w_{j}› whenever they represent the same internal state at time t. In equation (7), ψ_{j}(z, t) represent spatial wavefunctions that we take as Gaussian wavepackets
where a_{j}, b_{j} and c_{j} are complex. This is equivalent to the form
where Z_{j}(t) is the central position and P_{j}(t) is the central momentum of the j'th wavepacket, whereas φ_{j} is a real phase of the wavepacket at the centre.
We assume that the potential is smooth enough on the scale of the wavepacket, such that it can be approximated by a quadratic form as in equation (6) with the force F_{j}=−∂_{z}V_{j} and the potential curvature (with m_{F}→j). With this approximation and neglecting atom–atom interactions, the Gaussian ansatz is an exact solution for the propagation problem. By substituting the Gaussian form (8) in the Schrödinger equation and equating terms proportional to z^{2}, z and 1, we obtain the equations for the coefficients
By comparing the forms (8) and (9) we find b_{j}=2a_{j}Z_{j}+iP_{j}/ħ and , where the equations for the centre coordinates are given by the Newtonian equations of motion
where the solution for the phase in the wavepacket CM frame is
An analytical solution for a_{j} is possible for constant coefficients F_{j} and F′_{j}
where
with and
Let us now take a superposition of two wavepackets of the form (9) with equal amplitudes C_{j} and widths (a_{1}=a_{2}=a). We obtain
where Z_{CM}=(Z_{1}+Z_{2})/2 is the position of the CM of the two wavepackets and P_{CM}=(P_{1}+P_{2})/2 is the CM momentum, whereas Δz=Z_{1}−Z_{2} and Δp=P_{1}−P_{2} are the corresponding position and momentum differences. ψ_{CM}(z) is the wavefunction of the form of equation (9) with Z_{CM}, P_{CM} and φ_{CM}≡(φ_{1}+φ_{2})/2 replacing the corresponding singlewavepacket coordinates and phase. The exponential arguments are
In freespace propagation we have a(t)=a(0)(1+2ia(0)ħt/m)^{−1}. By substituting Z_{j}(t)=Z_{j}(0)+P_{j}t/m in the expression for ξ_{j} we obtain
After a time t, such that tm/2ħa(0), we have a(t)~−im/2ħt such that the term containing the momentum vanishes.
The atomic density per unit length is given by Nψ(z)^{2}, where N is the total atom number. In the long time limit, the coefficient a becomes imaginary, such that ξ_{j} and θ_{j} in equation (17) become imaginary as well. The last line of equation (17) becomes cos(Δξz+Δθ/2)=cos(mdz/ħt+φ/2), where 2d=Z_{2}−Z_{1} and φ=θ_{2}−θ_{1}. To obtain equation (3), we take the square absolute value of equation (17), and use Re{a(t)}=1/4σ_{z}(t)^{2} and . The visibility v is ideally equal to 1 and was included as a parameter in equation (3) in order to account for the real interference patterns whose visibility is lower than the ideal one.
Splitting magnetically insensitive states
Another interferometric scheme that is based on the same idea of the FGBS involves the two firstorder magnetically insensitive substates 1,0› and 2,0› of the two hyperfine states, which are analogous to those used in presentday precision interferometers. A microwave π/2 pulse may be used to create an equally populated superposition of the two states and then they may be split into two momentum states using a magnetic gradient at a high magnetic field. The nonlinear Zeeman shift of the transition energy between the states is ΔE ≈ αB^{2}, where α=2πħ × 575 Hz G^{−2}. At a distance of 10 μm from a wire carrying 2A of current, the atoms are exposed to a magnetic field of B=400 G and a magnetic gradient of ∂_{z}B=40 kG mm^{−1}. It then follows that F=∂_{z}ΔE(z)=2πħ × 575 × 2B∂_{z}B=1.22 × 10^{−20} N, and consequently atoms receive a differential velocity of 84.7 mm s^{−1} for a 1μs pulse, equivalent to about 18 ħk. As the two states are relatively magnetically insensitive, the second π/2 pulse of the FGBS would not be needed and the two output beams of this beam splitter could be used for interferometry in a completely analogous way to existing interferometers based on light beam splitters. Recombining the two wavepackets similarly requires only a gradient and one π/2 pulse, and the internal state population can now be measured.
Future interferometry schemes
In order to construct a full interferometer based on spatial or internal state interference, one needs to recombine the two momentum outputs of the FGBS. Here we present two schemes to recombine the two momentum outputs of the FGBS and create a full interferometer, with spatial or internal state interference signals.
In the CM frame, an atom that was accelerated by the force F_{2} (F_{1}) is at time t after the FGBS in the state p, d› (−p,−d›) where p=(p_{2}−p_{1})/2 and d=p t/m represent the external degrees of freedom of the centre of the wavepackets, m being the mass. At this point, their relative motion can be stopped and after sufficient free expansion time (or TOF) t_{TOF} they overlap and create a spatial fringe pattern with periodicity λ=ht_{TOF}/2md.
The simplest way to stop the relative wavepacket motion is to apply a gradient (for example, an harmonic potential), which will accelerate each part of the wavefunction in an opposite direction. This method was followed in our experiment, as described in the text.
Another way to stop the relative motion of the two wavepackets is to apply the FGBS again, as shown in Fig. 5a–c. Considering only the state 2›, the spatial wavefunction in the CM momentum and position coordinates is , where φ is the relative phase accumulated between the two paths during the propagation. After the second FGBS, which applies a momentum difference p′, the new wavefunction in the CM frame is
such that if p′=±p we are left with two wavepackets with the same momentum at ±d, giving rise to spatial interference after expansion.
If one wishes to use the internal state population as a signal, one needs to overlap the two parts of the wavefunction spatially and then apply another FGBS, as shown in Fig. 5d–f. Once the wavepackets overlap, the second FGBS is operated, giving rise to a wavefunction of the same form as in equation (20) with d→0. If the magnitude of the momentum kick p′ of the second FGBS is equal to that of the first FGBS, we are left with two wavepackets with a zero CM momentum in an internal state which depends on the propagation phase φ. For p′= ±p in equation (20), the internal state with zero CM momentum is
Additional information
How to cite this article: Machluf, S. et al. Coherent Stern–Gerlach momentum splitting on an atom chip. Nat. Commun. 4:2424 doi: 10.1038/ncomms3424 (2013).
References
Berman P. R. (ed.) Atom Interferometry Academic Press (1997).
Müller, H., Chiow, S.w., Herrmann, S., Chu, S. & Chung, K.Y. Atominterferometry tests of the isotropy of postNewtonian gravity. Phys. Rev. Lett. 100, 031101 (2008).
Dimopoulos, S., Graham, P. W., Hogan, J. M. & Kasevich, M. A. Testing general relativity with atom interferometry. Phys. Rev. Lett. 98, 111102 (2007).
Lan, S. Y. et al. A clock directly linking time to a particle's mass. Science 339, 554–557 (2013).
Canuel, B. et al. Sixaxis inertial sensor using coldatom interferometry. Phys. Rev. Lett. 97, 010402 (2006).
Müller, T. et al. A compact dual atom interferometer gyroscope based on lasercooled rubidium. Eur. Phys. J. D 53, 273–281 (2009).
Kasevich, M. & Chu, S. Atomic interferometry using stimulated Raman transitions. Phys. Rev. Lett. 67, 181–184 (1991).
Rasel, E. M., Oberthaler, M. K., Batelaan, H., Schmiedmayer, J. & Zeilinger, A. Atom wave interferometry with diffraction gratings of light. Phys. Rev. Lett. 75, 2633–2637 (1995).
Peters, A., Chung, K. & Chu, S. Measurement of gravitational acceleration by dropping atoms. Nature 400, 849–852 (1999).
Fixler, J. B., Foster, G. T., McGuirk, J. M. & Kasevich, M. A. Atom interferometer measurement of the Newtonian constant of gravity. Science 315, 74–77 (2007).
Shin, Y. et al. Atom interferometry with BoseEinstein condensates in a doublewell potential. Phys. Rev. Lett. 92, 050405 (2004).
Schumm, T. et al. Matterwave interferometry in a double well on an atom chip. Nat. Phys. 1, 57–62 (2005).
Boehi, P. et al. Coherent manipulation of BoseEinstein condensates with statedependent microwave potentials on an atom chip. Nat. Phys. 5, 592–597 (2009).
Esteve, J., Gross, C., Weller, A., Giovanazzi, S. & Oberthaler, M. K. Squeezing and entanglement in a BoseEinstein condensate. Nature 455, 1216–1219 (2008).
Hofferberth, S., Lesanovsky, I., Fischer, B., Schumm, T. & Schmiedmayer, J. Nonequilibrium coherence dynamics in onedimensional Bose gases. Nature 449, 324–327 (2007).
Japha, Y. & Band, Y. B. Ground state and excitations of a Bose gas: from a harmonic trap to a double well. Phys. Rev. A 84, 033630 (2011).
Folman, R., Krüger, P., Schmiedmayer, J., Denschlag, J. & Henkel, C. Microscopic atom optics: from wires to an atom chip. Adv. At. Mol. Opt. Phys. 48, 263–356 (2002).
Reichel, J. Microchip traps and BoseEinstein condensation. Appl. Phys. B 74, 469–487 (2002).
Fortágh, J. & Zimmermann, C. Magnetic microtraps for ultracold atoms. Rev. Mod. Phys. 79, 235–289 (2007).
Maussang, K. et al. Enhanced and reduced atom number fluctuations in a BEC splitter. Phys. Rev. Lett. 105, 080403 (2010).
Wang, Y.J. et al. Atom Michelson interferometer on a chip using a BoseEinstein condensate. Phys. Rev. Lett. 94, 090405 (2005).
Müntinga, H. et al. Interferometry with BoseEinstein condensates in microgravity. Phys. Rev. Lett. 110, 093602 (2013).
Baker, P. M. et al. Adjustable microchip ring trap for cold atoms and molecules. Phys. Rev. A 80, 063615 (2009).
Sherlock, B. E., Gildemeister, M., Owen, E., Nugent, E. & Foot, C. J. Timeaveraged adiabatic ring potential for ultracold atoms. Phys. Rev. A 83, 043408 (2011).
Zhang, B. et al. Magnetic confinement of neutral atoms based on patterned vortex distributions in superconducting disks and rings. Phys. Rev. A 85, 013404 (2012).
Griffin, P. F., Riis, E. & Arnold, A. S. Smooth inductively coupled ring trap for atoms. Phys. Rev. A 77, 051402 (2008).
Japha, Y., Arzouan, O., Avishai, Y. & Folman, R. Using timereversal symmetry for sensitive incoherent matterwave Sagnac interferometry. Phys. Rev. Lett. 99, 060402 (2007).
Jo, G.B. et al. Long phase coherence time and number squeezing of two BoseEinstein condensates on an atom chip. Phys. Rev. Lett. 98, 030407 (2007).
Nishio, T. et al. Scanning SQUID microscopy of vortex clusters in multiband superconductors. Phys. Rev. B 81, 020506 (2010).
Hicks, C. W. et al. Limits on superconductivityrelated magnetization in Sr2RuO4 and PrOs4Sb12 from scanning SQUID microscopy. Phys. Rev. B 81, 214501 (2010).
Heiblum, M. Fractional charge determination via quantum shot noise measurements. In Perspectives of Mesoscopic Physics. 115 (World Scientific, (2010).
Blanter, Y. & Buttiker, M. Shot noise in mesoscopic conductors. Phys. Rep. 336, 1–166 (2000).
Milton, K. A. et al. Repulsive Casimir effects. Int. J. Mod. Phys. A 27, 1260014 (2012).
Aigner, S. et al. Longrange order in electronic transport through disordered metal films. Science 319, 1226–1229 (2008).
Lin, Y., Teper, I., Chin, C. & Vuletić, V. Impact of the CasimirPolder potential and Johnson noise on BoseEinstein condensate stability near surfaces. Phys. Rev. Lett. 92, 050404 (2004).
Pasquini, T. A. et al. Quantum reflection from a solid surface at normal incidence. Phys. Rev. Lett. 93, 223201 (2004).
Pasquini, T. A. et al. Low velocity quantum reflection of BoseEinstein condensates. Phys. Rev. Lett. 97, 093201 (2006).
Obrecht, J. M. et al. Measurement of the temperature dependence of the CasimirPolder force. Phys. Rev. Lett. 98, 063201 (2007).
Jones, M. P. A., Vale, C. J., Sahagun, D., Hall, B. V. & Hinds, E. A. Spin coupling between cold atoms and the thermal fluctuations of a metal surface. Phys. Rev. Lett. 91, 080401 (2003).
Gerlach, W. & Stern, O. Das magnetische moment des silberatoms. Z. Phys. 9, 353–355 (1922).
Scully, M. O., Lamb, W. E. & Barut, A. On the theory of the sterngerlach apparatus. Found. Phys. 17, 575–583 (1987).
Briegel, H. J. et al. inAtom Interferometry (ed. Berman P. R.) 240, (Academic Press (1997).
Englert, B. G., Schwinger, J. & Scully, M. O. Is spin coherence like HumptyDumpty? I. Simplified treatment. Found. Phys. 18, 1045–1056 (1988).
Schwinger, J., Scully, M. O. & Englert, B. G. Is spin coherence like HumptyDumpty? II. General theory. Z. Phys. D 10, 135–144 (1988).
Scully, M. O., Englert, B. G. & Schwinger, J. Spin coherence and HumptyDumpty. III. The effects of observation. Phys. Rev. A 40, 1775 (1989).
Dubetsky, B. & Raithel, G. Nanometer scale period sinusoidal atom gratings produced by a SternGerlach beam splitter. Preprint at http://arxiv.org/abs/physics/0206029 (2002).
Boustimi, M. et al. Atomic interference patterns in the transverse plane. Phys. Rev. A 61, 033602 (2000).
Viaris de Lesegno, B. et al. Stern Gerlach interferometry with metastable argon atoms: an immaterial mask modulating the profile of a supersonic beam. Eur. Phys. J. D 23, 25–34 (2003).
Müller, H., Chiow, S., Long, Q., Herrmann, S. & Chu, S. Atom interferometry with up to 24photonmomentumtransfer beam splitters. Phys. Rev. Lett. 100, 180405 (2008).
Müller, H., Chiow, S.W., Herrmann, S. & Chu, S. Atom interferometers with scalable enclosed area. Phys. Rev. Lett. 102, 240403 (2009).
Cladé, P., GuellatiKhélifa, S., Nez, F. & Biraben, F. Large momentum beam splitter using Bloch oscillations. Phys. Rev. Lett. 102, 240402 (2009).
Chiow, S., Kovachy, T., Chien, H. C. & Kasevich, M. A. 102ħk large area atom interferometers. Phys. Rev. Lett. 107, 130403 (2011).
Zych, M., Costa, F., Pikovski, I. & Brukner, C. Quantum interferometric visibility as a witness of general relativistic proper time. Nat. Commun. 2, 505 (2011).
Machluf, S., Coslovsky, J., Petrov, P. G., Japha, Y. & Folman, R. Coupling between internal spin dynamics and external degrees of freedom in the presence of colored noise. Phys. Rev. Lett. 105, 203002 (2010).
Strekalov, D. V., Yu, N. & Mansour, K. Subshot noise power source for microelectronics. NASA Tech Briefshttp://www.techbriefs.com/component/content/article/11341 (2011).
Acknowledgements
We are most grateful to David Groswasser, Zina Binshtok, Shuyu Zhou, Mark Keil, Daniel Rohrlich and the rest of the AtomChip group for their assistance. We are grateful to the Ben–Gurion University fabrication facility for providing the chip. We acknowledge support from the FP7 European consortium ‘matterwave interferometry’ (601180).
Author information
Authors and Affiliations
Contributions
S.M. performed the experiment and analysed the data. Y.J. developed the theory. All authors contributed to the conceptual design of the experiment, the interpretation of the results and writing of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Machluf, S., Japha, Y. & Folman, R. Coherent Stern–Gerlach momentum splitting on an atom chip. Nat Commun 4, 2424 (2013). https://doi.org/10.1038/ncomms3424
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms3424
This article is cited by

Observation of the alloptical Stern–Gerlach effect in nonlinear optics
Nature Photonics (2022)

Phaselocking matterwave interferometer of vortex states
npj Quantum Information (2022)

Present status and future challenges of noninterferometric tests of collapse models
Nature Physics (2022)

Nonlocal single particle steering generated through single particle entanglement
Scientific Reports (2021)

Using a quantum work meter to test nonequilibrium fluctuation theorems
Nature Communications (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.