An experimental test of the geodesic rule proposition for the noncyclic geometric phase

This paper describes geometric effects in the evolution of atoms through a double-slit type experiment on an atom chip.


INTRODUCTION
The geometric phase (GP), the phase acquired over the course of an evolution of the Hamiltonian in parameter space, is a central concept in classical and quantum physics (1)(2)(3)(4)(5)(6)(7)(8)(9). Originally, the GP was defined only for an evolution of a system in a closed trajectory in phase space, but later, it was generalized to noncyclic evolutions (7,10). For the case of a two-level system, where the evolution of the system can be described by a trajectory on the Bloch or Poincaré spheres, it has been proposed (7,10) that, using a natural definition of the phase (1), the GP is given by half the area enclosed by the trajectory and the geodesic connecting the initial and final points. A marked outcome of the proposed geodesic rule is that this noncyclic phase changes sign when the trajectory moves from the upper to the lower hemisphere, resulting in a -phase jump when the trajectory is half the circumference of a circle (7,10). While the GP for a closed trajectory has been measured experimentally in several physical systems in a fairly straightforward manner (11)(12)(13)(14)(15), the experimental verification of the GP during noncyclic evolution requires a more convoluted approach. This is so because the cyclic GP can be readily measured as the probed state is returned to its initial position in parameter space, where it can be compared with a reference state to measure the relative phase, while for noncyclic geometric evolutions, where the probed state is not returned to its initial position, one needs to project the final state onto the initial state. Using an ultracold atom spatial interferometer, we test the geodesic rule, including the predicted SU(2) phase sign change and  jumps.
Berry's original work (2) addressed a quantum system undergoing a cyclic evolution under the action of a time-dependent Hamiltonian. When the Hamiltonian returns to its initial value, the quantum state acquires an extra GP in addition to the dynamical phase. This concept has been generalized (7) to a noncyclic evolution of the system, where the parameters of the Hamiltonian do not return to their initial values. In addition to the fundamental interest in better understanding the noncyclic behavior, it may also prove to be technolog-ically advantageous. For example, as the system does not need to return to its original state, geometric operations may be done faster, e.g., geometric quantum gates (16)(17)(18). Quantum optimal control of the evolution may also benefit (19,20). In addition, metrology may be made more sensitive due to the expected phase sign change and phase jumps, e.g., in measuring a gravitational potential (21).
The geometric interpretation of this noncyclic GP takes an illuminative form for a two-level system whose state can be described by two angles,  = ( cos  _ 2 | 2⟩+ exp (i ) sin  _ 2 | 1⟩ ) , which define a point on the Poincaré or Bloch spheres. The propagation of a state under a noncyclic evolution of the Hamiltonian, from  A to  B , characterized by { A ,  A } and { B ,  B }, respectively, is represented by a curve connecting points A and B on the sphere. Using a natural definition of the phase (1), where the relative phase between two arbitrary states is zero when the visibility of their interference pattern is maximal, the GP associated with this propagation is determined by the geodesic rule: it is given by half the area on the sphere bordered by the evolution curve and the shortest geodesic connecting A and B (22). An illustration of the geodesic rule on the Bloch sphere is shown in Fig. 1, where A evolves toward B, along the latitude of fixed  A =  B = , and  changes from  A to  B =  A +  (the curve C AB ). The area corresponding to the GP, blue shaded in the figure, is enclosed by C AB and by the geodesic curve G AB joining points A and B. If C AB is on the northern hemisphere, G AB is above (toward the north pole) C AB . But if C AB is on the southern hemisphere, G AB is below C AB , leading to a sign change of the GP as C AB crosses the equator.
Since the introduction of the geodesic rule, several studies have verified it experimentally with light (23,24), neutron (25,26), and atom (27,28) interferometers [see also (29) and (30) for relevant debates and (31) for other interpretations]. In this work, we propose and realize a matter-wave experimental study using cold-atom spatial interferometry (32,33). The uniqueness of our approach includes (i) the use of a spatial interference pattern to determine the phase in a single experimental run (no need to scan any parameter to obtain the phase), (ii) the use of a common phase reference for both hemispheres and a scan of  enabling to verify the  phase jump and the sign change, and (iii) obtaining the relative phase by allowing  A and  B to expand in free flight and overlap, different from previous atominterferometry studies that required, for obtaining interference, an  additional manipulation of the SU(2) parameters  and . As a result of our novel technique, we are able to test and confirm the geodesic rule for noncyclic evolutions in a new way, including the predicted sign change and the predicted SU(2) phase jumps.

EXPERIMENT
Our full experimental procedure is detailed elsewhere (34-36) as well as in Methods and the Supplementary Materials. The relevant part for the determination of the GP is sketched in Fig. 1. The 87 Rb atom can be in either state |1⟩ where F is the total angular momentum and m F is the projection. We start by preparing two-atom wave packets at different positions, both in an internal state |2⟩. We first apply a uniform radio-frequency (RF) pulse, of time duration T R , which transfers population from the |2⟩ state to |1⟩, shifting both wave packets from the north pole of the Bloch sphere to a point whose latitude  depends on T R (Fig. 1A). We then apply a magnetic field gradient pulse of duration T G , which results, due to the different magnetic moments of states |1⟩ and |2⟩, in a phase difference between these states, rotating both superpositions along a constant latitude on the Bloch sphere. Because of the difference in positions, each wave packet experiences a different magnetic field and thus will rotate by a different angle, ending up at points A and B in Fig. 1A. The two states, after the application of both T R and T G , can thus be written as where  is proportional to T R , and  to T G .  A (r) and  B (r) are the spatial components of the respective states. There may also be an additional global phase, identical for both  A and  B , which plays no role in the interference between  A and  B . To measure this interference, we allow enough time of flight for the two wave packets to free fall, expand, and overlap, before taking a picture using a charge-coupled device (CCD) camera. Figure 2 depicts the averaged interference patterns (raw data CCD images) averaged over all values of  in the upper (B) or lower (C) hemispheres, for T G = 17 s ( ≃ ). The value of  was independently deduced from the relative populations of states |1⟩ and |2⟩, which are given by cos 2 ( / 2) and sin 2 ( / 2) , respectively (Figs. 2A and 5). The high visibility in both images indicates the existence of "phase rigidity," namely, that the measured phase is independent of  in each hemisphere. Moreover, the two datasets have a phase difference of , which can also be deduced from the vanishing visibility in Fig. 2D, where the two datasets in (B) and (C) are joined. Evidently, there is a sharp jump in the phase of the interference pattern as  crosses the equator. According to Eq. 1, the interference phase , for general  and , is given by where  0 = arg ⟨ A (r) |  B (r)⟩ is the phase associated with the evolution of the external degrees of freedom of the system (see Methods). Figure 3 depicts the interference phase, deduced from the raw data, as a function of T R for different values of T G . The dashed lines in this figure are a fit to Eq. 2, with the fitting parameters  0 (an overall vertical shift) and . The excellent fit to the data allows us to determine with high precision the values of  (Fig. 3E).

RESULTS
The total phase (interference phase)  is a sum of two contributions, the GP  G and the dynamical phase  D . While both  and  D are gauge dependent,  G =  −  D is gauge independent (37,38). Substituting for the dynamical phase (6,10,37,38), we obtain (see Methods) where the gauge-dependent phase  0 has dropped out. Figure 4 displays ,  D , and the resulting  G , for two values of , where the first term on the right-hand-side of Eq. 3 is given by , the phase of the interference pattern, while the second is evaluated for the experimentally determined values of  and . The dashed lines in Fig. 4 (B and D) correspond to the geodesic rule-half the area between the geodesic and the trajectory, with the correct sign. A very good agreement between data and the theoretical predictions is observed. This constitutes a complete verification of the GP asso-ciated with noncyclic evolution in an SU(2) system and accurately confirms the theoretical predictions, including a precise observation of the geodesic rule, the phase sign change, and the  phase jump.

DISCUSSION
Last, we make a fundamental connection between our experiment and the Pancharatnam phase (1). We begin by noting that in the case  0 = 0, we have arg ⟨ A | 2⟩ = 0 and arg ⟨2 |  B ⟩ = 0, and then the states A, |2⟩, and B fulfill the Pancharatnam consecutive in-phase criterion (1,22). It then follows that arg ⟨A | B⟩ is given by half the area  of the spherical triangle defined by these three states on the Bloch sphere, namely, the area in between three geodesic lines. The area  of the spherical triangle defined by the two arcs joining the north pole and points A and B is given by the relation tan ( / 2 ) = tan 2 ( / 2 ) sin ( ) / [1 + tan 2 ( / 2 ) cos ] , which is identical to Eq. 2 with  = /2 (for  0 = 0). This geometric interpretation of  yields an explanation of the observed phase rigidity for  = : When the two points are in the northern hemisphere, the geodesic between the two points goes through the north pole. The enclosed area is zero; hence,  = 0. When the two points are in the southern hemisphere, the geodesic goes through the south pole, with an area of 2, resulting in a jump of  in the value of  (Fig. 2). The geometric interpretation of our experiment is now evident, namely, what is measured in the experiment (the interference-pattern phase) is the Pancharatnam phase  P . The difference between the areas associated with  P and  D gives the light blue area in Fig. 1, associated

Fig. 4. Geometric SU(2) phase jump and sign flip, experiment (dots) versus theory (Eq. 3, dashed lines). (A) Total phase and dynamical phase for  =  as a function
of T R (). The total phase is directly measured from the imaged interference pattern (Fig. 3), and the dynamical phase  _ 2 (1 − cos ) is deduced from the independently measured values of  and . (B) GP  G determined as the difference between the two sets of points appearing in (A). The predicted sign change as the latitude crosses the equator is clearly visible. The evident phase jump is due to the geodesic rule. When  = , the geodesic must go through the Bloch sphere pole for any  ≠ /2. As the latitude approaches the equator (i.e., increasing ), the blue area in Fig. 1 (twice  G ) continuously grows to reach a maximum of  in the limit of  = /2. As the latitude crosses the equator, the geodesic jumps from one pole to the other pole, resulting in an instantaneous change of sign of this large area and a phase jump of . This plot exactly confirms the prediction in (10). (C) Total phase and dynamical phase for  = 2.24 rad (0.71). (D)  G , determined as the difference between the two sets of points appearing in (C). The predicted sign change is again visible. However, in the case of  = 2.24 rad (0.71), the geodesic line does not go through the pole, and as the latitude approaches the equator,  G actually reduces (after reaching its maximum for an intermediate ), so no abrupt phase jump is expected.
with  G . This now naturally explains both the sign change of  G as the latitude crosses the equator, as well as the phase jump for  =  (Fig. 4).

OUTLOOK
As an outlook, we consider a situation in which the two wave packets are viewed as a split wave packet of a single clock, where  = /2 for a perfect two-level clock (35,36). When we place the two wave packets along a vertical line parallel to gravity at different distances from earth, they are exposed to different proper times. In the experiment described in this paper, the relative phase  = (E 1 − E 2 ) × t/ħ between the wave packets is determined by a magnetic gradient, which changes the energy splitting E 1 − E 2 between states |1⟩ and |2⟩ [i.e., (E 1 − E 2 ) is the difference of energy splitting between two where A is the pixel area in the object plane,  0 = 3 2 /2 is the cross section for resonant atom-light scattering, and  ≈ 780 nm is the optical transition wavelength. The total atom number is equal to ∫N(x, z)dxdz. We can then reliably determine the relation between population transfer and T R as presented in Fig. 2A, e.g., 10 s corresponds to  = /2, 20 s corresponds to  = , and 40 s corresponds to  = 2. When the Bloch vectors are in the northern hemisphere, the interference phase is seen to be rigid (fixed). When the Bloch vectors cross the equator at T R = 10 s, there is a  phase jump. The interference phase will jump by another  when the vectors cross the equator again at T R = 30 s. Namely, phase rigidity appears when the Bloch vectors are located in either the northern or southern hemisphere, with a  phase jump in between, as presented in Fig. 2 (B to D) and Fig. 3B. The fluctuations in the interference pattern's location are due to fluctuations in the initial conditions from shot to shot, while the inferred phase is stable, as explained in (34).
wave packets  A and  B ], while time (from the moment the two wave packets were allowed to free fall) is the same for both wave packets. However, the same GP situation occurs when the magnetic gradient is zero and consequently the splitting E 1 − E 2 is identical for the two wave packets, but time elapsed is different for the two wave packets due to the different red shift (with time difference t).
In this case, we have  = (E 1 − E 2 ) × t/ħ, and the same theory presented in this paper may be used to analyze via the GP an experimental situation on the interface between quantum mechanics and general relativity. Moreover, by scanning  around /2 (i.e., change the relative populations of the |1⟩ and |2⟩ states from below to above half), one should observe a sign change that may allow the construction of a novel type of gravitational sensor. A main limitation that will have to be examined is the sharpness of the slope. Even when working at the best point ( = ; see Fig. 4B), the practical slope will never be the theoretical infinite slope, as the visibility at this exact point is zero (as may be seen from the large error bar). However, the visibility quickly recovers and numerical and experimental studies are needed to reveal the ultimate realizable slope. An additional limitation has to do with the fact that the sharp slope appears when  = , so when dealing with proper time differences that give rise to a substantially different , a bias would have to be introduced to keep the system at this optimal point, and this bias may introduce its own errors.

Detailed experimental scheme
The experiment was realized in an atom chip setup (39). We present the detailed experimental scheme in fig. S1, which includes the two-level system preparation. We first prepared a Bose-Einstein condensate (BEC) of about 10 4 87 Rb atoms in the state |2⟩ ≡ | F = 2, m F = 2⟩ in a magnetic trap located 90 m below the chip surface. After the BEC atoms were released from the trap, the entire experimental sequence took place in the presence of a homogeneous magnetic bias field of 36.7 G in the y direction (z is the direction of gravity), which created an effective two-level system (with |1⟩ ≡ | F = 2, m F = 1⟩) via the nonlinear Zeeman effect with E ij = E 21 ≈ h × 25 MHz (where i and j are the m F numbers, all in the F = 2 manifold), and E 21 − E 10 ≈ h × 180 kHz. We then applied an RF pulse (duration TR 1 , where typically 10 s gives rise to a  = /2 rotation) to prepare a spin superposition (| 1⟩+ | 2⟩) / √ _ 2 between the |2⟩ and |1⟩ states. A magnetic gradient pulse ∂B/∂z of duration TG 1 = 4 s, generated by currents in the atom chip wires, was applied to create the Stern-Gerlach splitting, in which the different spins are exposed to differential forces. To enable interference between the two wave packets (|2⟩ and |1⟩ are orthogonal), a second /2 pulse (TR 2 ) was applied to mix the spins. To stop the relative velocity of the wave packets, a second magnetic gradient pulse (TG 2 ) was applied to yield differential forces for the same-spin states at different locations. A spatial superposition of two wave packets in state |2⟩ now exists (separated along the z axis, with zero relative velocity). Note that during TG 2 , the |1⟩ state from the two wave packets was pushed outside the experimental zone. The control of  introduced in Fig. 1A is realized by a third RF pulse of duration TR 3 (T R in the main text). The relative rotation between the two wave packets  may be changed by applying a third magnetic field gradient of duration TG 3 (T G in the main text). The wave packets were then allowed to expand (during time of flight of ∼10 ms, much larger than the re-ciprocal of the trap frequency ∼500 Hz) and overlap to form the interference pattern. An image based on the absorption imaging was taken in the end.
The magnetic gradient pulses were generated by three parallel gold wires located on the chip surface with a length of 10 mm, a width of 40 m, and a thickness of 2 m. The chip wire current was driven using a simple 12.5-V battery and modulated using a homemade current shutter. The three parallel gold wires were separated by 100 m (center to center), and the same current runs through them in alternating directions. The benefit of using this three-wire configuration instead of a single gold wire is that a two-dimensional quadrupole field was created at z = 100 m below the atom chip. As the magnetic instability is proportional to the field strength, and as the main instability originates in the gradient pulses (the bias fields from external coils are very stable), positioning the atoms near the middle (zero) of the quadrupole field significantly reduces the magnetic noise while maintaining the strength of the magnetic gradients.
Determination of the population transfer and the value of  In Fig. 5, we explained how the values of  are obtained from the measurement of population transfer when we apply TR 3 (T R in the main text). Stern-Gerlach splitting was used to separate the m F = 1 and m F = 2 parts, and absorption imaging was performed to evaluate the atom number. See the details in the figure caption.

The CCD image of the interference pattern while  is scanned
In Fig. 6, we showed the raw data of the interference patterns, which are displayed in Fig. 2 (averaged over numerous values of ) and in Fig. 3B (where the phase for different values of  is presented), when TG 3 (T G in the main text) equals 17 s ( ≃ ). The whole scanning range of T R is 40 s, corresponding to one full cycle (2) of the Rabi oscillation. The phase of the interference pattern was found to be rigid when the Bloch vector is located in the northern hemisphere or in the southern hemisphere, with a  phase jump in between.
The interference pattern was fitted with the function Aexp related the optical density in the system, z CM is the center-of-mass (CM) position of the combined wave packet at the time of imaging,  z is the Gaussian width of the combined wave packet obtained after time of flight,  = ht _ md is the fringe periodicity, v is the visibility, z ref is a fixed reference point, c is the background optical density from the absorption imaging, and  is the phase of the interference pattern that appears in Eq. 2. In the fringe periodicity  = ht _ md , h is the Planck constant, t is the duration of time of flight, m is the mass of 87 Rb atom, and d is the distance between the two wave packets. In Fig. 3, we measure the dependence of  on  (T R ) for a fixed T G and then fit the data to Eq. 2, returning values for both  0 and .

Geometrical phases for different values of ∆
Here, we describe the approach used to derive the expression of  G in Eq. 3. Mukunda and Simon (37) developed a general formalism called the quantum kinematic approach for the GP in quantum systems.
In the formalism of Mukunda and Simon, a one-parameter smooth curve was defined from a vector  belonging to a Hilbert space ℋ, C = {(s ) ∈ N 0 , s ∈ [ s 1 , s 2 ] } . N 0 is the subset of unit vectors of ℋ . Note that the curve C is not necessarily closed. The only requirements of the theory are the smoothness of C , i.e., (s) should be differentiable, and the non-orthogonality of the initial and final states. The GP is given by where  is the total phase.  D is the dynamical phase arising from the energy dependence on s during the evolution. This general formalism naturally reduces to the evolution under the time-dependent Schrödinger equation if the parameter s is time. The curve C is the trajectory of the wave function during the propagation time 0 ≤ t ≤ T. The total phase  during an evolution along C is given by Taking (s ) = ( cos  _ 2 | 2⟩ + exp (is ) sin  _ 2 | 1⟩ ) and {s 1 , s 2 } = [0,1], we found for the total phase  = arctan { sin 2 ( / 2 ) sin  ─────────────── cos 2 ( / 2 ) + sin 2 ( / 2 ) cos  } where we should add to  the phase  0 arising from the evolution of the spatial part. This yields Eq. 2. The dynamical phase  D can be calculated from the integral of the evolution curve C (7)  D = Im ∫ s 1 s 2 〈(s ) |  ̇ (s ) 〉ds (7) We find to which phase  0 should also be added. Subtracting  D from  yields the expression for  G of Eq. 3.  G is more suitable to use for analysis because gauge-dependent phases in  and  D mutually cancel. In fig. S2, we presented the detailed theoretical behavior of  G (Eq. 3) as a function of  and . The characteristics of  G are the singularity at  =  and  = (n + 1/2) (where n is an integer), and the change of sign when  goes across these values. This result was originally obtained from (10) (see figure 4 in this reference).

SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ content/full/6/9/eaay8345/DC1 Fig. S1. Detailed scheme of the spatial SU(2) interferometer. Fig. S2. Theoretical curves of the GP G versus  for different values of .