## Abstract

The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics and may become advantageous for quantum technology. In noncyclic evolutions, a proposition relates the geometric phase to the area bounded by the phase-space trajectory and the shortest geodesic connecting its end points. The experimental demonstration of this geodesic rule proposition in different systems is of great interest, especially due to the potential use in quantum technology. Here, we report a previously unshown experimental confirmation of the geodesic rule for a noncyclic geometric phase by means of a spatial SU(2) matter-wave interferometer, demonstrating, with high precision, the predicted phase sign change and π jumps. We show the connection between our results and the Pancharatnam phase. Last, we point out that the geodesic rule may be applied to obtain the red shift in general relativity, enabling a new quantum tool to measure gravity.

## 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*–*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 π-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*–*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 π 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 technologically 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*–*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, * _{A}* to Ψ

*, characterized by {θ*

_{B}*, ϕ*

_{A}*} and {θ*

_{A}*, ϕ*

_{B}*}, respectively, is represented by a curve connecting points*

_{B}*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 θ

*= θ*

_{A}*= θ, and ϕ changes from ϕ*

_{B}*to ϕ*

_{A}*= ϕ*

_{B}*+ Δϕ (the curve*

_{A}*A*and

*B*. If

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 θ enabling to verify the π phase jump and the sign change, and (iii) obtaining the relative phase by allowing Ψ* _{A}* and Ψ

*to expand in free flight and overlap, different from previous atom-interferometry studies that required, for obtaining interference, an additional manipulation of the SU(2) parameters θ and Δϕ. 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.*

_{B}## 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⟩ ≡ ∣ *F* = 2, *m _{F}* = 1⟩ or ∣2⟩ ≡ ∣

*F*= 2,

*m*= 2⟩, where

_{F}*F*is the total angular momentum and

*m*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

_{F}*T*, 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 θ depends on

_{R}*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

*T*

_{R}, and Δϕ to

*T*

_{G}. ψ

*(*

_{A}*r*) and ψ

*(*

_{B}*r*) are the spatial components of the respective states. There may also be an additional global phase, identical for both Ψ

*and Ψ*

_{A}*, which plays no role in the interference between Ψ*

_{B}*and Ψ*

_{A}*. 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.*

_{B}## RESULTS

Figure 2 depicts the averaged interference patterns (raw data CCD images) averaged over all values of θ in the upper (B) or lower (C) hemispheres, for *T*_{G} = 17 μs (Δϕ ≃ π). The value of θ was independently deduced from the relative populations of states ∣1⟩ and ∣2⟩, which are given by

According to Eq. 1, the interference phase Φ, for general θ and Δϕ, is given by_{0} = arg ⟨ψ* _{A}*(

*r*) ∣ ψ

*(*

_{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 ϕ

_{0}(an overall vertical shift) and Δϕ. The excellent fit to the data allows us to determine with high precision the values of Δϕ (Fig. 3E).

The total phase (interference phase) Φ is a sum of two contributions, the GP Φ_{G} and the dynamical phase Φ_{D}. While both Φ and Φ_{D} are gauge dependent, Φ_{G} = Φ − Φ_{D} is gauge independent (*37*, *38*). Substituting for the dynamical phase (*6*, *10*, *37*, *38*), we obtain (see Methods)_{0} has dropped out.

Figure 4 displays Φ, Φ_{D}, and the resulting Φ_{G}, for two values of Δϕ, where the first term on the right-hand-side of Eq. 3 is given by Φ, the phase of the interference pattern, while the second is evaluated for the experimentally determined values of θ and Δϕ. 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 associated 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 π 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 ϕ_{0} = 0, we have arg ⟨Ψ* _{A}* ∣ 2⟩ = 0 and arg ⟨2 ∣ Ψ

*⟩ = 0, and then the states*

_{B}*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 σ of the spherical triangle defined by these three states on the Bloch sphere, namely, the area in between three geodesic lines. The area σ of the spherical triangle defined by the two arcs joining the north pole and points

*A*and

*B*is given by the relation

_{0}= 0). This geometric interpretation of Φ yields an explanation of the observed phase rigidity for Δϕ = π: 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, Φ = 0. When the two points are in the southern hemisphere, the geodesic goes through the south pole, with an area of 2π, resulting in a jump of π in the value of Φ (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 Φ

_{P}. The difference between the areas associated with Φ

_{P}and Φ

_{D}gives the light blue area in Fig. 1, associated with Φ

_{G}. This now naturally explains both the sign change of Φ

_{G}as the latitude crosses the equator, as well as the phase jump for Δϕ = π (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 θ = π/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 Δϕ = Δ(*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., Δ(*E*_{1} − *E*_{2}) is the difference of energy splitting between two wave packets Ψ* _{A}* and Ψ

*], 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*

_{B}*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 Δ

*t*). In this case, we have Δϕ = (

*E*

_{1}−

*E*

_{2}) × Δ

*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 θ around π/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 (ΔΦ = π; 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 Δϕ = π, so when dealing with proper time differences that give rise to a substantially different Δϕ, a bias would have to be introduced to keep the system at this optimal point, and this bias may introduce its own errors.

## METHODS

### 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 μm 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*= 1⟩) via the nonlinear Zeeman effect with

_{F}*E*=

_{ij}*E*

_{21}≈

*h*× 25 MHz (where

*i*and

*j*are the

*m*numbers, all in the

_{F}*F*= 2 manifold), and

*E*

_{21}−

*E*

_{10}≈

*h*× 180 kHz. We then applied an RF pulse (duration

*TR*

_{1}, where typically 10 μs gives rise to a θ = π/2 rotation) to prepare a spin superposition

*B*/∂

*z*of duration

*TG*

_{1}= 4 μs, 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 π/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 θ 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 Δϕ 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 reciprocal 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 μm, and a thickness of 2 μm. 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 μm (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 μm 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 θ

In Fig. 5, we explained how the values of θ 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*= 2 parts, and absorption imaging was performed to evaluate the atom number. See the details in the figure caption.

_{F}### The CCD image of the interference pattern while θ 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 θ) and in Fig. 3B (where the phase for different values of θ is presented), when TG_{3} (*T*_{G} in the main text) equals 17 μs (Δϕ ≃ π). The whole scanning range of *T _{R}* is 40 μs, corresponding to one full cycle (2π) 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 π phase jump in between.

The interference pattern was fitted with the function *A*exp*A* is a constant 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, σ* _{z}* is the Gaussian width of the combined wave packet obtained after time of flight,

*v*is the visibility,

*z*

_{ref}is a fixed reference point,

*c*is the background optical density from the absorption imaging, and Φ is the phase of the interference pattern that appears in Eq. 2. In the fringe periodicity

*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 Φ on θ (

*T*

_{R}) for a fixed

*T*

_{G}and then fit the data to Eq. 2, returning values for both ϕ

_{0}and Δϕ.

### Geometrical phases for different values of ∆ϕ

Here, we describe the approach used to derive the expression of Φ* _{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 ψ belonging to a Hilbert space *s*) should be differentiable, and the non-orthogonality of the initial and final states. The GP is given by_{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 Φ during an evolution along C is given by

Taking *s*_{1}, *s*_{2}} = [0,1], we found for the total phase_{0} arising from the evolution of the spatial part. This yields Eq. 2.

The dynamical phase Φ_{D} can be calculated from the integral of the evolution curve C (*7*)

We find_{0} should also be added. Subtracting Φ_{D} from Φ yields the expression for Φ_{G} of Eq. 3. Φ_{G} is more suitable to use for analysis because gauge-dependent phases in Φ and Φ_{D} mutually cancel.

In fig. S2, we presented the detailed theoretical behavior of Φ_{G} (Eq. 3) as a function of θ and Δϕ. The characteristics of Φ_{G} are the singularity at Δϕ = π and θ = (*n* + 1/2)π (where *n* is an integer), and the change of sign when θ 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 Φ_{G} versus θ for different values of Δϕ.

This is an open-access article distributed under the terms of the Creative Commons Attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We thank the BGU nano-fabrication facility for the high-quality chip.

**Funding:**This work was funded, in part, by the Israel Science Foundation (grants no. 292/15 and 856/18), the German-Israeli DIP project (Hybrid devices: FO 703/2-1) supported by the DFG, and the European ROCIT consortium for stable optical clocks. We also acknowledge support from the Israeli Council for Higher Education and from the Ministry of Immigrant Absorption (Israel).

**Author contributions:**Z.Z. and Y.Ma. did the experiment. Z.Z., S.M., and Y.Me. did the theory. All authors discussed the results and participated in the writing of the manuscript. R.F. supervised the experiment, the theory, and the writing of the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and Materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution License 4.0 (CC BY).