Stern-Gerlach Interferometry with the Atom Chip
Mark Keil, Shimon Machluf, Yair Margalit, Zhifan Zhou, Omer Amit, Or Dobkowski, Yonathan Japha, Samuel Moukouri, Daniel Rohrlich, Zina Binstock, Yaniv Bar-Haim, Menachem Givon, David Groswasser, Yigal Meir, Ron Folman
SStern-Gerlach Interferometry with the Atom Chip
Mark Keil, ∗ Shimon Machluf, † Yair Margalit, ‡ Zhifan Zhou, § Omer Amit,Or Dobkowski, Yonathan Japha, Samuel Moukouri, Daniel Rohrlich, Zina Binstock,Yaniv Bar-Haim, Menachem Givon, David Groswasser, Yigal Meir, and Ron Folman ∗ Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel (Dated: November 9, 2020)In this invited review in honor of 100 years since the Stern-Gerlach (SG) experiments, we describe a decadeof SG interferometry on the atom chip. The SG effect has been a paradigm of quantum mechanics throughoutthe last century, but there has been surprisingly little evidence that the original scheme, with freely propagatingatoms exposed to gradients from macroscopic magnets, is a fully coherent quantum process. Specifically, nofull-loop SG interferometer (SGI) has been realized with the scheme as envisioned decades ago. Furthermore,several theoretical studies have explained why it is a formidable challenge. Here we provide a review of our SGexperiments over the last decade. We describe several novel configurations such as that giving rise to thefirst SG spatial interference fringes, and the first full-loop SGI realization. These devices are based on highlyaccurate magnetic fields, originating from an atom chip, that ensure coherent operation within strict constraintsdescribed by previous theoretical analyses. Achieving this high level of control over magnetic gradients isexpected to facilitate technological applications such as probing of surfaces and currents, as well as metrology.Fundamental applications include the probing of the foundations of quantum theory, gravity, and the interfaceof quantum mechanics and gravity. We end with an outlook describing possible future experiments.
I. INTRODUCTION
This review follows the centennial conference held inFrankfurt in the same building housing the original Stern-Gerlach (SG) experiments. Here we describe the SG in-terferometry performed in our laboratories at Ben-GurionUniversity of the Negev (BGU) over the last decade.The trail-blazing experiments of Otto Stern and WaltherGerlach one hundred years ago [1–4] required a few basicingredients: a source of isolated atoms with well-specifiedmomentum components (provided by their atomic beam),an inhomogeneous magnetic field and, if we follow thehistorical account of events in [5], also a smoky cigar. Inthis review, we present our approach to these first twoingredients, with our sincere apologies that we will notbe able to adequately address the third.As Dudley Herschbach notes [4], the SG experimentsformed the basis for a “symbiotic entwining of molecularbeams with quantum theory” and, as shown in many ofthe papers at this centennial conference, this symbioticrelationship remains vigorous to the present day. In thisreview, our source of isolated atoms is instead provided bythe new world of ultra-cold atomic physics, to which wecouple inhomogeneous magnetic fields that are providednaturally by an atom chip [6]. Current-carrying wires ∗ Corresponding authors: [email protected] , [email protected] † Present address: Analytics Lab, Amsterdam, The Netherlands ‡ Present address: Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,Massachusetts Institute of Technology, Cambridge, MA 02139,USA § Present address: Joint Quantum Institute, National Instituteof Standards and Technology and the University of Maryland,College Park, Maryland 20742 USA on such chips were first realized as magnetic traps forultra-cold atoms at the turn of the (21 st ) century [7–9]and reviewed extensively since [6, 10–14]. We are usingthe atom chip as our basis for coherently manipulatingatoms in a way that is complementary to the atomic andmolecular beam techniques pioneered by Otto Stern andpracticed so energetically and creatively by his scientificdescendants.The work presented here is performed with high-qualityatom chips fabricated by our nano-fabrication facility [15].The atom chip is advantageous for Stern-Gerlach interfer-ometry (SGI) for 4 main reasons. First, the source (Bose-Einstein condensates, BEC) is a minimal-uncertaintywavepacket so it is very well defined in position andmomentum. Second, the source of the magnetic gradi-ents (current-carrying wires on the atom chip) is verywell aligned relative to the atomic source. Third, due tothe very small atom-chip distance, the gradients are verystrong, and significant Stern-Gerlach splitting can be re-alized in very short times. Fourth, the gradients are verywell defined in time since there are no coils and the induc-tance of the chip wires is negligible. We will describe howthese advantages have overcome long-standing difficultiesand have enabled different SG configurations to be real-ized at BGU ( e.g., spatial interference patterns [16, 17]and a “full-loop” SGI [18, 19]) alongside several applica-tions, such as spatially splitting a clock [20, 21]. Finally,let us mention that while the interferometers presentedhere are of a new type, it is worthwhile noting decades ofprogress in matter-wave interferometry [22].The discovery of the Stern-Gerlach (SG) effect [1] wasfollowed by ideas concerning a full-loop SGI that wouldconsist of freely propagating atoms exposed to magneticgradients from macroscopic magnets. However, startingwith Heisenberg [23], Bohm [24] and Wigner [25] consid- a r X i v : . [ phy s i c s . a t o m - ph ] N ov ered a coherent SGI impractical because it was thoughtthat the macroscopic device could not be made accu-rate enough to ensure a reversible splitting process [26].Bohm, for example, noted that the magnet would needto have “fantastic” accuracy [24]. Englert, Schwinger andScully analyzed the problem in more detail and coinedit the Humpty-Dumpty (HD) effect [28–31]. They tooconcluded that for significant coherence to be observed,exceptional accuracy in controlling magnetic fields wouldbe required. Indeed, while atom interferometers basedon light beam-splitters enjoy the quantum accuracy ofthe photon momentum transfer, the SGI magnets notonly have no such quantum discreteness, but they alsosuffer from inherent lack of flatness due to Maxwell’sequations [32]. Later work added the effect of dissipa-tion and suggested that only low-temperature magneticfield sources would enable an operational SGI [33]. Claimshave even been made that no coherent splitting is possibleat all [34].Undeterred, we utilize the novel capabilities of the atomchip to address these significant hurdles. Let us brieflypreview our most recent and most challenging realization,the full-loop SGI, in which magnetic field gradients act onthe atom during its flight through the interferometer, firstsplitting, and then re-combining, the atomic wavepacket.We obtain a high full-loop SGI visibility of 95% with a spininterference signal [18, 19] by utilizing the highly accuratemagnetic fields of an atom chip [6]. Notwithstandingthe impressive endeavors of [35–45] this is, to the bestof our knowledge, the first realization of a complete SGinterferometer analogous to that originally envisioned acentury ago.Achieving this high level of control over magnetic gradi-ents may facilitate fundamental research. Stern-Gerlachinterferometry with mesoscopic objects has been sug-gested as a compact detector for space-time metric andcurvature [46], possibly enabling detection of gravitationalwaves. It has also been suggested as a probe for the quan-tum nature of gravity [47]. Such SG capabilities may alsoenable searches for exotic effects like the fifth force or thehypothesized self-gravitation interaction [48]. We notethat the realization presented here has already enabledthe construction of a unique matter-wave interferometerwhose phase scales with the cube of the time the atomspends in the interferometer [19], a configuration thathas been suggested as an experimental test for Einstein’sequivalence principle when extended to the quantum do-main [49]. Can a fragile item be taken apart and be re-assembled perfectly?. . . another tough problem, according to the popular Englishrhyme [27]Humpty Dumpty sat on a wall,Humpty Dumpty had a great fall.All the king’s horsesAnd all the king’s menCouldn’t put Humpty together again.
High magnetic stability and accuracy may also bene-fit technological applications such as large-momentum-transfer beam splitting for metrology with atom interfer-ometry [50–52], sensitive probing of electron transport, e.g., squeezed currents [53], as well as nuclear magneticresonance and compact accelerators [54]. We note thatsince the SGI makes no use of light, it may serve as ahigh-precision surface probe at short distances for whichadministering light is difficult.For the purpose of this review, it is especially importantto also realize that the atom chip allows our atoms tobe completely isolated from their environment. This isdemonstrated, for example, by the relatively long-termmaintenance of spatial coherence that can be achieveddespite a temperature gradient from 300 K to 100 nK overa distance of just 5 µ m [55]. Coherence of internal degreesof freedom close to the surface has also been measured tobe very high [56].This review is organized into the following sections:II. Particle Sources: a brief discussion of how theatom chip complements and extends the century-longuse of atomic and molecular beams in Stern-Gerlachexperiments;III. The Atom Chip Stern-Gerlach Beam Splitter:detailing relevant aspects of the atom chip and its basicoperating characteristics as a platform for SGI;IV. Half-Loop Stern-Gerlach Interferometer: firstrealization of SGI with spatial fringe patterns;V. Full-Loop Stern-Gerlach Interferometer: first realiza-tion of the four-field complete SGI with spin populationfringes;VI. Applications: clock interferometry and complementar-ity, the matter-wave geodesic rule and geometric phase,and a T interferometer realizing the Kennard phase;VII. Outlook: extending the atom-chip based SGI experi-ments to ion beams and to massive particles.Finally, we note that the SG effect, in conjunction withthe atom chip, may also be used for novel applicationswithout the use of interferometry. For example, we haveused the SG spin-momentum entanglement to realizea novel quantum work meter. In this work, done inconjunction with the group of Juan Pablo Paz, we wereable to test non-equilibrium fluctuation theorems [57].As we hope to show in this review, we believe that theatom chip provides a novel and powerful tool for SGinterferometry, with much yet to learn as SG studies entertheir second century. May we continue to find surprises,fundamental insights, and exciting applications. type source species temperature σ z σ v z k = σ p z / ¯ h σ z σ p z / ¯ h Ref.(K) ( µ m) (mm/s) ( µ m − )diffraction beam He 10 −
20 14 0.9 18 [58]diffraction beam He not given 50 43 2.6 130 [59]T-L interference beam macromolecules not given 0.266 0.04 16 4.3 [60]interference BEC Rb 40x10 − ion trap Ca + – 0.006 900 5x10 [61, 62]first realization beam Ag 1300 30 230 400 1x10 [1]TABLE I. Parameters relating to diffraction experiments using He atomic beams [58, 59], Talbot-Lau interference experimentswith macromolecules [60], and interference experiments using BEC’s [17] as described in this review. The temperature shown forthe beam experiments corresponds to the velocity spread superimposed on the moving frame of the longitudinal most-probablevelocity. The position spread σ z for the He beams is the beam collimator width, while the velocity spread σ v z is calculatedfrom the beam angular divergence and its most-probable longitudinal velocity [63] ( v x = v mp ≈
288 m / s for a He beam sourcetemperature of 8 K). For the macromolecular beam, the parameters are taken from the grating period, interferometer length, andthe stated longitudinal deBroglie wavelength. Corresponding parameters for the BEC are calculated using the Thomas-Fermiapproximation and a temperature at which the BEC is about 90% pure. Parameters for the original Stern-Gerlach experimentare shown for comparison in the last line. All species are in their ground electronic state. The x - and z - co-ordinates refer tothe horizontal and vertical directions respectively, where the beam experiments are horizontal (so z is the transverse direction)while the BEC experiments are vertical (so z is the longitudinal direction). We do not give parameters for the “beaded atom”experiments [36] since we believe that spatial interference fringes were not observed, as explained in [64]. Talbot-Lau interference, as applied to matter-wave interference studies, is described in detail in [65]. The particle species inthe quoted study are functionalized oligoporphyrin macromolecules with up to 2000 atoms and masses > These parameters are for a proposal for SGI using ion beams that will be discussed in Sec. VII A.
II. PARTICLE SOURCES
Molecular beam experiments exhibiting quantum interfer-ence, diffraction, and reflection have been brought veryskillfully into the modern era in presentations at this Con-ference by Markus Arndt, Maksim Kunitski, and WielandSch¨ollkopf, and as outlined in the keynote address byPeter Toennies. In particular, Stern’s vision – and real-ization – of diffraction of atomic and molecular beams(see, for example [4]) have found their modern expressionin the work of all these experts, and many others. Herewe will concentrate on a complementary approach to pre-cisely specify internal and external quantum states andhow they can be used to study interference phenomenain particular.Let us begin by comparing experimental parameters usedin the ultra-cold atomic environment in our laboratory,typically achieved with BECs of Rb, with correspond-ing state-of-the-art parameters for atomic beams. Table Isummarizes parameters that are most relevant for theseexperiments. Note that the beam experiments are con-ducted in a horizontal plane, transverse to the beampropagation direction, while our BEC interference ex-periments are conducted in an exclusively longitudinaldirection with the atoms falling vertically due to gravity(and with all applied forces also acting in the longitudinaldirection).We see that ultra-cold atom localization and velocityspreads are on the same order as transverse localizationfrom the exemplary atomic and molecular beam experi-ments quoted here but, of course, ultra-cold atoms are also localized in all three dimensions, whereas the beamtechniques do not achieve localization along the beampropagation axis.
III. THE ATOM CHIP STERN-GERLACHBEAM SPLITTER
In order to apply Stern-Gerlach splitting, our ultra-cold atomic sample needs to have at least two spinstates. However, our initial atomic sample is purely inthe | F, m F (cid:105) = | , (cid:105) state of Rb. After preparing a BECon the atom chip, our SG implementation therefore beginsby first releasing the magnetic trap, and then applying aradio-frequency (RF) π/ √ ( | (cid:105) + | (cid:105) ),where | (cid:105) and | (cid:105) represent the m F = 1 and m F = 2Zeeman sub-levels of the F = 2 manifold in the groundelectronic state [66]. Transitions to other m F levels areavoided by retaining a modest homogeneous magneticfield even after trap release. A field of about 30 G is suffi-cient to create an effective two-level system by pushingthe m F = 0 sub-level about 200 kHz out of resonancewith the | (cid:105) → | (cid:105) RF transition due to the non-linearZeeman effect. The intensity of the RF Rabi pulses is cal-ibrated such that a pulse duration of 20 µ s corresponds toa complete population inversion between the two states, i.e., a π -pulse. This corresponds to a Rabi frequencyof Ω RF = 2 π ·
25 kHz.We now consider the second factor crucial to the successof our SGI experiments: fast and precise magnetic fields,in both magnitude and direction, may be delivered bypulsed currents passed through micro-fabricated wireson the atom chip. Simple Biot-Savart considerationsfor atom chip wires, as used in our experiments, yieldmagnetic field gradients of about 200 G / mm at ∼ µ mfrom the chip, which is the starting distance for most ofour experiments. Accurate control of this initial position,which is also crucial for the success of the experiments, isensured by accurate control of chip wire currents and thehomogeneous magnetic field referred to above. In addition,the straight atom chip wires have very low inductance,thereby enabling the generation of well-defined magneticforce pulses with currents that are typically tens of µ s long.Such pulses are, in principle, able to induce momentumchanges of hundreds of ¯ hk . Our earliest implementationsof these experimental characteristics [67] were improvedin subsequent apparatus upgrades [64].Since the experiments proceed after turning off the mag-netic trap, the observation time is limited by the time-of-flight (TOF) of the falling atoms and the field-of-view ofour absorption imaging detection system. The latter islimited to about 4 mm, corresponding to a maximum TOFof about 28 ms. The optical detection system has a spatialresolution of about 5 µ m, an important consideration formeasuring spatial interference patterns (Sec. IV). Furtherexperimental details may be found in several recent Ph.D.theses from our laboratory [64, 67, 68].The Stern-Gerlach beam splitter (SGBS), first imple-mented in [16], begins with an equal superposition of | (cid:105) and | (cid:105) as described above and depicted schematicallyin Fig. 1. We then apply a magnetic field gradient ∇| B | for duration T , which creates a state-dependent force F m F = m F g F µ B ∇| B | on the atomic ensemble, where µ B , g F , and m F denote the Bohr magneton, the Land´e fac-tor, and the projection of the angular momentum on thequantization axis, respectively.The magnetic potential created by the atom chip canbe approximated as a sum of a linear part with char-acteristic force F and a quadratic part with character-istic frequency ω . After this magnetic gradient split-ting pulse, the new state of the atoms is given by ψ f = √ ( | , p (cid:105) + | , p (cid:105) ), where p i = F i T ( i = 1 , We express the momentum transfer in units of ¯ hk , a referencemomentum of a photon with 1 µ m wavelength, in order to comparewith atom interferometry based on optical beam splitters. FIG. 1. The Stern-Gerlach beam-splitter (SGBS) atwork [16, 67]. SGBS (a) input and (b,c) output images, andthe corresponding schematic descriptions. The top row depictsour atom chip, with a pulsed current I being used to generatethe magnetic gradient ∂B/∂z (we currently use three parallelwires with equal currents but opposing polarities). The chipfaces downwards so that atoms can separate vertically duringtheir free fall. (a) A magnetically trapped BEC in state | (cid:105) before release. (b) After a weak splitting of less than ¯ hk usinga 5 µ s magnetic gradient pulse and allowing a TOF of 14 ms.(c) After a strong splitting of more than 40 ¯ hk using a 1 msmagnetic gradient pulse and allowing a TOF of 2 ms. Inter-ferometric signals are formed either as spatial interferencefringes by passively allowing overlap of the wavepackets (the“half-loop” SGI), or as spin-state population oscillations uponactively recombining them (the “full-loop” SGI), as describedin Secs. IV and V respectively. Adapted from [16]. ference patterns analogous to a double-slit experiment,and a “full-loop” configuration in which the wavepacketsare actively re-combined, analogous to a Mach-Zehnderinterferometer.By applying additional pulses with different timing, thesemethods have been used to demonstrate, to the best of ourknowledge, the first Stern-Gerlach spatial fringe interfer-ometer (Sec. IV, [16, 17]), the first full-loop Stern-Gerlachinterferometer (Sec. V, [18, 19]), and several applicationsthat we will describe in Sec. VI, including experimentsto simulate the effect of proper time on quantum clockinterference [20, 21]. IV. HALF-LOOP STERN-GERLACHINTERFEROMETER
The two separated wavepackets generated by the SGBSinitiate the pulse sequence shown in Fig. 2. Just afterthe SG splitting pulse, another RF π/ µ s dura- FIG. 2. Schematic depiction of the longitudinal half-loop SGIgiving rise to spatial interference fringes (vertical position z inthe center-of-mass frame vs. time). The initial wavepacket | (cid:105) (extreme left) is subjected to a π/ | (cid:105) + | (cid:105) . Thefirst magnetic gradient pulse of duration T (purple column)induces a Stern-Gerlach splitting into | (cid:105) (green curve) and | (cid:105) (purple curve) having momenta p and p , respectively. Wethen immediately apply a second π/ | (cid:105) and | (cid:105) states into equal superpositions | (cid:105) ∓ | (cid:105) as shown. The delay time T d allows these wavepackets tospatially separate (in the z direction). The duration T of asecond gradient pulse is tuned to bring the momentum dif-ference between the | (cid:105) components close to zero (see text),allowing their space-time trajectories to become parallel (solidpurple curves) while expelling the | (cid:105) components (dottedgreen trajectories). The atoms then fall freely under gravity.Given sufficient time-of-flight, the two | (cid:105) wavepackets expand(dotted purple lines) and eventually overlap to generate spatialinterference fringes, which are measured by taking an absorp-tion image of the atoms. We note that due to the curvatureof the magnetic field forming the magnetic gradient pulse, thelong T pulse also focuses the wavepackets, as depicted in thefigure. In fact, this focusing accelerates the process of finalexpansion, thereby creating the two-wavepacket overlap ina shorter time. Adapted from [17] with permission © IOPPublishing & Deutsche Physikalische Gesellschaft. CC BY 3.0 tion) is applied, creating a wavefunction consisting of fourwavepackets [67], of which we are concerned only with thetwo | (cid:105) wavepackets having momenta p and p [the | (cid:105) components can be disregarded since they appear at dif-ferent final positions on completing the pulse sequenceand a time-of-flight (TOF) period].The time interval between the two RF pulses (in whichthere are only two wavepackets, each having a differentspin) is reduced to a minimum ( ∼ µ s) to suppress thehindering effects of a noisy and uncontrolled magneticenvironment, thereby removing the need for magneticshielding. After a magnetic gradient pulse of duration T designed to stop the relative motion of the two wavepack-ets, the atoms fall under gravity for a relatively long TOF,expanding freely until they overlap to create spatial in-terference fringes as shown schematically in Fig. 2 andexperimentally in Fig. 3. The period of the interference fringes must be large enoughto be observable with the spatial resolution of our imagingsystem (about 5 µ m). This is accomplished if two con-ditions are fulfilled. First, the distance between the twowavepackets, d , should not be too large, since in principlethe fringe periodicity varies as ht/md when the relativemomentum is zero, where h , t , and m are the Planckconstant, TOF duration, and the atomic mass, respec-tively. Second, the momentum difference between thetwo wavepackets should be smaller than their momentumwidth to avoid orthogonality. This is accomplished bytuning the duration T of the second gradient pulse, whichcan stop the relative motion of the two | (cid:105) wavepackets;despite being in the same spin state, the slower wavepacketexperiences a stronger impulse than the faster one sinceit is considerably closer to the atom chip after the rela-tively long delay time T d . We have found that zeroingthe momentum difference between the two wavepacketsis very robust [67].Given that the final momentum difference between the twointerfering wavepackets is smaller than their momentumspread, they overlap after a sufficiently long TOF and aninterference pattern appears with the approximate form: n ( z, t ) = A exp (cid:20) − ( z − z CM ) σ z ( t ) (cid:21) × (cid:20) V cos (cid:18) πλ ( z − z ref ) + φ (cid:19)(cid:21) , (1)where A is the amplitude, z CM is the center-of-mass (CM)position of the combined wavepacket at the time of imag-ing, σ z ( t ) ≈ ¯ ht/ mσ is the final Gaussian width, λ ≈ π ¯ ht/md is the fringe periodicity ( d = | z − z | is thedistance between the wavepacket centers), V is the in-terference fringe visibility, and φ = φ − φ is the globalphase difference. The vertical position z is relative toa fixed reference point z ref . The phases φ and φ aredetermined by an integral over the trajectories of the twowavepacket centers. We emphasize that Eq. (1) is not aphenomenological equation, but rather an outcome of ouranalytical model [16].In order to characterize the stability of the phase, whichis the main figure of merit in interferometry, we averagemultiple experimental images with no post-selection oralignment (each single-shot image is a result of one ex-perimental cycle). Large fluctuations in the phase and/orfringe periodicity in a set of single-shot images would re-sult in a low multi-shot visibility, while small fluctuationscorrespond to high multi-shot visibility. The multi-shotvisibility is therefore a measure of the stability of thephase and periodicity. Single-shot and multi-shot visibili-ties are all extracted by fitting to Eq. (1) after averagingthe experimental images along the x direction (see Fig. 3)to reduce noise. We note that these procedures have beenused over several years of half-loop SGI studies [16, 17],while the experimental results were simultaneously be- FIG. 3. Spatial interference patterns from the Stern-Gerlachinterferometer. (a) A single-shot interference pattern of a ther-mal cloud with a negligible BEC fraction, fitted to Eq. (1) witha visibility of V = 0 .
65 (only slightly lower than single-shot vis-ibilities typically measured for a BEC). (b) A multi-shot imagemade by averaging 40 consecutive interference images usinga BEC (no correction or post-selection) with a normalized vis-ibility of V N = 0 .
99. (c) Polar plot of phase 0 ◦ ≤ φ ≤ ◦ vs. visibility 0 ≤ V ≤ T , T d , T ) = (4 , , µ s. Adapted from [64]. ing greatly improved by significant modifications to theoriginal apparatus [64, 67].For a pure superposition state, as in our model, perfectfringe visibility V would be 1. A quantitative analysisof effects reducing V appears in [17, 64]. Some of theseeffects are purely technical, e.g., imperfect BEC purityand wavepacket overlap in 3D, as well as various imaginglimitations etc. Such technical effects are irrelevant to thephase and periodicity stability shown by the multi-shotvisibility, so we normalize the latter to the mean of thesingle-shot visibilities taken from the same sample: V N ≡ V avg / (cid:104) V s (cid:105) , where V avg is the (un-normalized) visibility ofthe multi-shot average extracted from the fit, and (cid:104) V s (cid:105) is the mean visibility of the single-shot images whichcompose that multi-shot image. The normalized multi-shot visibility thus reflects shot-to-shot fluctuations of theglobal phase φ and the fringe periodicity λ . We note thatsome BEC intrinsic effects, such as phase diffusion, wouldnot lead to a reduction of the single-shot visibility, butmay cause the randomization of the shot-to-shot phase.However, such effects are expected to be quite weak,since atom-atom interactions rapidly become negligibleas the BEC expands in free-fall, and the experiment maybe described by single-atom physics.Representative results from the above analysis are shownin Fig. 3. The very high (normalized) visibility shownin (b) demonstrates that the phase and periodicity arehighly reproducible for each experimental cycle, the for-mer being particularly emphasized in plot (c). High-visibility fringes ( V > .
90) were observed over a wide va-riety of experimental parameters, covering a range of max-imum separations and velocities between the wavepackets.In particular, we conducted experiments at the apparatus-limited maximum value of T d = 600 µ s (which also re-quired a long TOF=21 .
45 ms) in order to maximize the spatial separation of the wavepackets during their timein the interferometer. These measurements achieved aseparation d = 3 . µ m, a factor of 20 larger than theatomic wavepacket size (after focusing, see Fig. 2), whilemaintaining a normalized visibility of V N = 0 .
90 [17].Given that our observed stable interference fringes arisefrom such well-separated paths, these experiments demon-strate what is, to the best of our knowledge, the first im-plementation of spatial SG interferometry. This achieve-ment is due to three main differences compared withprevious SG schemes. Firstly, we have used minimal-uncertainty wavepackets (a BEC) rather than thermalbeams. Secondly, while the splitting is based on two spinstates, the wavepackets in the two interferometer armsare in the same spin state for most of the interferometriccycle, thus reducing their sensitivity to disruptive externalmagnetic fields. Finally, chip-scale temporal and spatialcontrol allows the cancellation of path difference fluctu-ations. It should also be noted that a longitudinal SGI,based on a particle beam source, cannot take imagesof spatial fringes due to the high velocity of the fringepattern in the lab frame.This, however, is not yet the four-field SGI originallyenvisioned shortly after the original Stern-Gerlach ex-periments (as recounted in [26]), since the separatedwavepackets are not actively recombined in both positionand momentum. The two remaining magnetic gradientsrequired to complete such a “closed” SGI are discussedin the following section.
V. FULL-LOOP STERN-GERLACHINTERFEROMETER
Clearly, if a wavepacket can be coherently reconstructedafter SG splitting and recombination in a four-field con-figuration [26], it should be possible to observe an inter-ference pattern at the output of such an SGI. To the bestof our knowledge however, no such interference patternhas heretofore been measured experimentally, and this isthe task that we now describe, many details of which aretaken from [64] and references therein.The device envisioned consists of four successive regionsof magnetic gradients giving rise to the operations ofsplitting, stopping, reversing and, finally, stopping thetwo wavepackets, as shown schematically in Fig. 4(a). Ifexecuted perfectly, the two wavepackets would arrive atthe output of such an interferometer with a minimal rela-tive spatial displacement and momentum difference, sothat an arbitrary initial spin state should be recoverable,using the spin state of the recombined wavepacket as theinterference signal. However, the operation of such aninterferometer was considered to be technically impracti-cal, since coherent recombination of the two beam pathswould require extremely precise control of the magneticfields [24].
FIG. 4. The longitudinal full-loop SGI giving rise to spin population oscillations, plotted in the center-of-mass frame asin Fig. 2. (a) The sequence consists of RF pulses (blue) to manipulate the inner (spin) degrees of freedom and magneticgradients (purple) to control the momentum and position of the wavepackets. The interferometer is prepared from the initialwavepacket | (cid:105) (extreme left) by applying a π/ | (cid:105) + | (cid:105) [Blochsphere shown in (b)]. The first magnetic gradient pulse at t = 0 induces a Stern-Gerlach splitting into | (cid:105) (green curve) and | (cid:105) (purple curve). Three additional magnetic gradient pulses are used to stop the relative motion of the wavepackets (at theirmaximum separation ∆ z max ), reverse their momenta, and finally stop them at the same position along z . The re-combinedwavepacket at t = 2 T is therefore written as ψ ( z, T ) | (cid:105) + ψ ( z, T ) | (cid:105) , shown in (c) for an arbitrary interferometer phase δ Φ.After recombination, the population in | (cid:105) is measured by applying a second π/ ϕ RF , followed by amagnetic gradient to separate the populations and a subsequent pulse of the imaging laser. We expect to observe spin populationfringes, i.e., oscillations in the m F = 1 population, as we scan ϕ RF , as indeed shown by the experimental results in (d), forwhich the measured visibility is 95%. The Bloch spheres in (d) show the particular case in which the initial vector (dashedblack arrow) acquires an interferometer phase δ Φ = π/ x ( ϕ RF = π/
2) or − x ( ϕ RF = 3 π/
2) axes respectively (red arrows). The states | F, m F (cid:105) = | , (cid:105) ≡ | (cid:105) and | , (cid:105) ≡ | (cid:105) are defined along the z axis inthe Bloch spheres. Adapted from [64]. Our experiments begin, as before, with a π/ | (cid:105) and | (cid:105) of Rb that is subsequently split into two momentumcomponents by a magnetic gradient pulse (along the ver-tical axis z ) as described in Secs. III and IV. Additionalmagnetic gradient pulses are needed to “close” the loopof such an interferometer, i.e., to overlap the wavepack-ets spatially and with zero relative momentum. To stopthe relative motion of the two wavepackets after the firstpulse, and to accelerate them backwards, we reverse thecurrent on the atom chip, causing the force applied bythe magnetic field gradient to be in the opposite direction.Alternatively, we can apply a spin inversion procedureby using a π Rabi pulse that inverts the population be-tween the two internal states, following which a magneticgradient pulse will then apply the opposite differentialmomentum to the two wavepackets. We obtain the signalwith the help of a second π/ ϕ RF of this π/ V of the observed spin fringes,is expressed as [29] V = exp (cid:40) − (cid:34)(cid:18) ∆ z (2 T ) σ z (cid:19) + (cid:18) ∆ p z (2 T ) σ p (cid:19) (cid:35)(cid:41) , (2)where ∆ z (2 T ) and ∆ p z (2 T ) denote the mismatch be-tween the wavepackets in their final position and momen-tum respectively, after the interferometer duration 2 T [Fig. 4(a)], and σ z and σ p are the corresponding initialwavepacket widths. Equation (2) summarizes the mainresult of the HD papers in relation to our experimentalobservable. We emphasize that this reduction in visi-bility has nothing to do with effects of decoherence dueto some coupling with the environment. We also notethat the above HD calculation is done for a minimal-uncertainty wavepacket. For the general case, one canidentify l z = ¯ h/σ p and l p = ¯ h/σ z as the relevant scalesfor coherence [26, 29], where l z and l p are the spatialcoherence length and the momentum coherence width,respectively.Let us discuss the meaning of this equation. The quanti-ties σ z and σ p characterize the initial atomic wavefunction,and are thus microscopic quantities. The quantities ∆ z and ∆ p z describe the experimental imprecision in thefinal recombination. In a “good” SG experiment ( i.e., onewhich allows “unmistakable” splitting [29]) the maximumvalues of splitting in position and momentum should bemuch larger than their respective initial widths, mean-ing they should be macroscopic. On the other hand,according to Eq. (2), a nearly perfect maintenance of spincoherence ( V (cid:39)
1) requires both ∆ z (cid:28) σ z and ∆ p z (cid:28) σ p .Consequently, Eq. (2) tells us that we need to recom-bine macroscopic quantities with a microscopic level ofprecision. This is the challenge facing SG interferometerexperiments.It is interesting to note that in the half-loop experiments,we found that ∆ p z can be quite large (rendering thetrajectories during the TOF period in Fig. 2 slightlynon-parallel) without significantly reducing the measuredspatial interference fringe visibility, so the stability of thehalf-loop experiments cannot be used to examine the HDequation. This robustness of the half-loop may also beunderstood by considering the fact that the expansionof the wavepackets creates an enhanced local coherencelength, since for every region of space the k vector variancebecomes smaller as TOF increases (see also [69, 70]).A practical full-loop SG experiment must consider andaddress two effects. First, as noted above, the HD ef-fect requires accurate recombination, namely, small ∆ z and ∆ p z . These small values must be maintained formany experimental cycles, and thus a high level of stabil-ity in these values is also important. Achieving accuraterecombination means that the overlap integral, calculatedin Eq. (2), will have a significant non-zero value. Second,one must maintain a stable interferometer phase δ Φ, sothat it has the same value shot-to-shot. This requiresthat the coupling to external magnetic noise is kept to aminimum, either by shielding the experiment and stabiliz-ing the electronics ( e.g., responsible for the homogeneousmagnetic fields), or by conducting the experiment ex- tremely quickly so that such environmental fluctuationsdo not have time to introduce significant phase noise.Our full-loop SGI yields a visibility up to 95% [Fig. 4(d)],proving that we are able to use the SG effect to build afull-loop interferometer as originally envisioned almost acentury ago. We note three differences between our realiza-tion and the scheme considered in the HD papers: 1. Weuse a BEC, which is a minimum-uncertainty wavepacket,whereas the HD papers considered atomic beam experi-ments with large uncertainties on the order of σ z σ p (cid:39) ;2. We implement fast magnetic gradient pulses generatedby running currents on the atom chip, in contrast to usingconstant gradients from permanent magnets that wereconsidered in the original proposals; 3. Our interferometeris a 1D longitudinal interferometer, while the originallyenvisioned SGI was 2D, i.e., it enclosed an area.The full-loop experiments include a wide variety of opti-mizations and checks (see [64] for additional details). Tomake sure the spin superposition is not dephased due tosome slowly varying gradients in our bias fields, we add π pulses giving rise to an echo sequence. To access a largerregion of parameter space and to ensure the robustness ofour results, we use several different configurations by, forexample, implementing the reversing pulse ( T ) by invert-ing the sign of the atom chip currents vs. inverting thespins with the help of π pulses. We also utilize a variety ofmagnetic gradient magnitudes, and scan both the splittinggradient pulse duration T and the delay time betweenthe pulses T d . All results are qualitatively the same. Forweak splitting we observe high visibility ( ∼ hk the visibilityis still high ( ∼ i.e., there was no recombination) and wavepackets emergedfrom the interferometer with the same separation as themaximal separation achieved within (see Fig. 2 of [40] andfootnote [10] of [43]). We have not found anywhere in themany papers published by this group (only some of whichare referenced here) evidence of four operations beingapplied as required for a full-loop configuration, whetherthe experiment was with longitudinal or transverse gra-dients. In addition, no spatial interference fringes wereobserved, as the spatial modulation they observed was asignature of multiple parallel longitudinal interferometers,each having its own individual relative phase between itstwo wavepackets.To conclude, we have shown that a full-loop may berealized. In addition, as previously shown in Heisenberg’sargument, the momentum splitting is the figure of meritin determining the phase dispersion. In our experiment,coherence is observed up to a momentum splitting ashigh as ∆ p z ( T ) /σ p = 60. However, in contrast, thevisibility is more sensitive to spatial splitting and weachieve ∆ z ( T ) /σ z = 4, much lower than for the half-loop, where we achieved ∆ z/σ z = 18. The splitting iscoherent but its limits in terms of the HD effect are yetto be explored quantitatively. Many mysteries remainto be solved, such as why is the observed reduction notsymmetric in momentum and spatial splitting, in contrastto Eq. 2. A simple answer, which is yet to be examined indetail, is the existence of some sort of spatial decoherencemechanism due to the environment.Having now described the SG beam-splitter, the SG half-loop, and the SG full-loop, we show in the next section howthese techniques may be used for different applications. VI. APPLICATIONS
The pulse sequence in the half-loop experiments createstwo spatially separated wavepackets in the state | (cid:105) withzero relative momentum [left-most frame of Fig. 5(a-c)].We now take advantage of the long free-fall period inthe experiment (labelled TOF in Fig. 2 i.e., after the“stopping pulse”) to further manipulate these wavepacketswhile they are allowed to expand and ultimately to overlap.The experiments are based on imposing a differentialtime evolution between the two wavepackets, which wemeasure as the interference patterns generated upon theirrecombination.In particular, we create a “clock” state for each of the twowavepackets by first applying an RF pulse that preparesthe atoms in a superposition of two Zeeman sublevels | (cid:105) and | (cid:105) whose coefficients depend on the Bloch sphereangles θ and φ . This superposition state is a two-levelsystem evolving with a known period, as in the regularnotion of an atomic clock. The RF pulse (duration T R )controls the value of C = sin θ , while a subsequent mag-netic gradient pulse (duration T G ) controls the valueof D I = sin( φ/
2) by changing the relative “tick” rate ∆ ω of the two clock wavepackets, as illustrated in Fig. 5(a-c).The quantities C and D I describe the clock preparationquality and the ideal distinguishability between the twoclock interferometer arms respectively, which we will findquantitatively useful in our discussion of clock complemen-tarity [see Eqs. (4) and (5) below]. We note that, althoughthe magnetic gradient pulse applies a different SG forceto each of the states within the clock, we have evaluatedthis effect for our experimental parameters and find thatit is smaller than our experimental error bars ( ≤ FIG. 5. Clock interferometry. (a) Timing sequence (not toscale): Following a coherent spatial splitting by the SGBS anda stopping pulse, the system consists of two wavepackets inthe | (cid:105) state (separated along the z axis) with zero relativevelocity, as in Sec. IV. The clock is then initialized with an RFpulse of duration T R (usually a π/ T R = 10 µ s) afterwhich the relative “tick” rate ∆ ω of the two clock wavepacketsmay be changed by applying a magnetic field gradient ∂B/∂z of duration T G . Clock initialization occurs 1 . . φ BS . Whenthe clock reading ( i.e., the position of the clock hand) in thetwo clock wavepackets is the same ( φ BS = φ + ∆ ωT G =0 , π ), fringe visibility is high. (c) When the clock readingis opposite (orthogonal, φ BS = φ + ∆ ωT G = π ), there is nointerference pattern. (d)-(f) Corresponding interference dataof the two wavepackets i.e., of the clock interfering with itself.All data samples are from consecutive measurements withoutany post-selection or post-correction. Single-shot patternsfor φ BS = φ + ∆ ωT G = π also show very low fringe visibility(see Fig. 2(c) of [20]). Adapted from [20] and reprinted withpermission from AAAS; (e) is adapted from [64]. A. Clock Interferometery
Let us first discuss the motivation for clock interferome-try [20]. Time in standard quantum mechanics (QM) isa global parameter, which cannot differ between paths.Hence, in standard interferometry [71], a height differencein a gravitational field between two paths would merelyaffect the relative phase of the clocks, shifting the interfer-ence pattern without degrading its visibility. In contrast,general relativity (GR) predicts that a clock must “tick”slower along the lower path; thus if the paths of a clockpassing through an interferometer have different heights,a time differential between the paths will yield “whichpath” information and degrade the visibility of the interfer-ence pattern according to the quantum complementarityrelation between the interferometric visibility and thedistinguishability of the wavepackets [72]. Consequently,whereas standard interferometry may probe GR [73–75],clock interferometry probes the interplay of GR and QM.For example, loss of visibility because of a proper time lagwould be evidence that gravitational effects contribute todecoherence and the emergence of a classical world [76].Here we describe the use of this new tool – the clockinterferometer – for its potential to investigate the role oftime at the interface of QM and GR. Since the genuine GRproper time difference is too small to be measured withexisting experimental technology, our experiments insteadsimulate the proper time difference between the clockwavepackets using magnetic gradients, thereby causingthe clock wavepackets to “tick” at different rates. Ourresults in this proof-of-principle experiment show thatthe visibility does indeed oscillate as a function of thesimulated proper time lag.In the ultimate experiment, each part of the spatial su-perposition of a clock, located at different heights aboveEarth, would “tick” at different rates due to gravitationaltime dilation (so-called “red-shift”). We can easily cal-culate the proper time difference between two arms ofthe clock interferometer as a figure-of-merit for this effect.Using a first-order approximation of gravitational timedilation, and assuming a large separation between thearms of ∆ h = 1 m, an interferometer duration of T = 1 syields a proper time difference between the arms ofonly ∆ τ (cid:39) T g ∆ h/c (cid:39) − s. Such a small time differ-ence means that a very accurate and fast-ticking clockmust be sent through an interferometer with a large space-time area in order to observe the actual GR effect. Bothrequirements are beyond our current experimental capa-bilities. Our “synthetic” red-shift is created by applyingan additional magnetic gradient (of duration T G ) thatcauses the clock wavepackets to “tick” at different rates.We denote the “tick” rate difference by ∆ ω .Our results, some of which are presented in Fig. 5(d-f), with more details in [20, 64], show that the rela-tive rotation between the two clock wavepackets affectsthe interferometric visibility. In the most extreme case, when the two clock states are orthogonal, e.g., one in thestate √ ( | (cid:105) + | (cid:105) ) and the other in the state √ ( | (cid:105) − | (cid:105) ),the visibility of the clock self-interference drops to nearzero [Fig. 5(e)]. By varying the duration of the magneticgradient T G and thereby scanning the differential rotationangle φ BS between the two clock wavepackets, we showquantitatively that the visibility oscillates as a functionof our “synthetic” red-shift with a period of ∆ ωT G = 2 π [Fig. 5(d,f)]. As an additional test of the clock inter-ferometer, we modulate its preparation by changing theduration of the clock initialization pulse T R , which influ-ences the relative populations of the two states composingthe clock. This changes the state of the system from ano-clock state to a full-clock state in a continuous manner.The results show that the visibility behaves as expectedin each case, further validating that it is the clock readingwhich is responsible for the oscillations in visibility thatwe observe as a function of T R [20]. B. Clock Complementarity
These measurements of visibility may naturally be ex-tended to study quantum complementarity for our self-interfering atomic clocks, which we again remark is at theinterface of QM and GR. Our central consideration hereis the inequality [77] V + D ≤ , (3)where V is the “visibility” of an interference patternsuch as discussed throughout this review, and D is the“distinguishability” of the two paths of the interferingparticle. The latter quantity can also be measured directlyin the clock experiments by controlling the angle φ BS ,where ( θ = π/ , φ BS = ∆ ωT G = π ) prepares two perfectlydistinguishable clocks such that D = 1 [Fig. 5(e)]. A briefaccount of recent work theoretically and experimentallyverifying this fundamental inequality is given by [21] andreferences therein.It is important to investigate clock complementarity, par-ticularly in view of recent theoretical work showing thatspatial interferometers can be sensitive to a proper timelag between the paths [78] and speculation (see Table 1 in[72]) that the inequality of Eq. (3) may be broken suchthat V + D > et al. summarize the importance of this work asfollows: “. . . on the one hand, if the ‘ticking’ rate ofthe clock depends on its path, then clock time provideswhich-path information and Eq. (3), developed in theframework of non-relativistic QM, must apply. Yet, onthe other hand, gravitational time lags do not arise innon-relativistic QM, which is not covariant and thereforenot consistent with the equivalence principle [79]. Henceour treatment of the clock superposition is a semiclassicalextension of quantum mechanics to include gravitationalred-shifts.”The experiments we conducted in [21] set out to testEq. (3) quantitatively. Imperfect clock preparation ( i.e., FIG. 6. Clock complementarity: (a-e) V , D I , and C mea-sured independently and (f-g) combined in the complemen-tarity relations of Eqs. (3)-(6). (a) The visibility of an idealclock ( C = 1) interference pattern vs. T G , fitted to | cos( φ/ | ;(b-c) the distinguishability is calculated from Eq. (5) using thedifference in relative angles φ − φ , each measured separatelyand shown in (b); and (d-e) the clock preparation quality C is calculated from Eq. (4) using the data in (d). Finally, (f)shows the combination of all three parameters ( V N ) +( C · D I ) for four values of C when D I is scanned and (g) shows thesame combination for four values of D I when C is scanned.Only the data point in (f) for T G near 22 µ s differs from unity,due to a relatively large experimental error in measuring theinterferometric phase. These data therefore verify clock com-plementarity. Adapted from [21] with permission © IOPPublishing & Deutsche Physikalische Gesellschaft, all rightsreserved. with θ (cid:54) = π/
2) reduces the measurable distinguishability D from its ideal value D I as D = ( C · D I ) , where C ≡ sin θ = 2 (cid:112) P (1 − P ) (4)and D I = | sin(∆ φ/ | (5)with P and 1 − P denoting the populations (occupationprobabilities) of the two energy eigenstates of the clockand ∆ φ BS ≡ φ u BS − φ d BS , where u and d denote the upperand lower paths of the interferometer, respectively.The experiment now has the task of measuring the threequantities V , D I and C independently. We use the nor-malized visibility V N as discussed in Sec. IV. We evalu-ate D I independently by measuring the relative phases intwo single-state interferometers, one for each of the twoclock states, and we measure C , also independently, ina separate experiment by measuring P after the clock is initialized. Our results for these independently-measuredquantities are shown in Fig. 6(a), (c), and (e), where theresults in (c) and (e) are based on analyzing the datain (b) and (d) respectively. We then combine these threequantities in the complementarity expression( V N ) + ( C · D I ) ≤ , (6)whereupon we see from Fig. 6(f-g) that the complemen-tarity inequality [Eq. (3)] is indeed upheld for the clockwavepackets superposed on two paths through our SGinterferometer.While the relation in Eq. (6) is specific to clock comple-mentarity, it is unusual in linking non-relativistic quantummechanics with general relativity. A direct test of thiscomplementarity relation will come when D I reflects thegravitational red-shift between two paths which traversedifferent heights. C. Geometric Phase
The geometric phase due to the evolution of the Hamil-tonian is a central concept in quantum physics and maybecome advantageous for quantum technology. In non-cyclic evolutions, a proposition relates the geometric phaseto the area bounded by the phase-space trajectory andthe shortest geodesic connecting its end points [80–82].The experimental demonstration of this geodesic ruleproposition in different systems is of great interest, es-pecially due to its potential use in quantum technology.Here, we report a novel experimental confirmation of thegeodesic rule for a noncyclic geometric phase by meansof a spatial SU(2) matter-wave interferometer, demon-strating, with high precision, the predicted phase signchange and π jumps. We show the connection betweenour results and the Pancharatnam phase [83].In the clock complementarity application just described,we scanned the third RF pulse (duration T R ) to vary theclock preparation parameter C = sin θ . In our case, a π/ T R = 10 µ s, so T R < µ splaces the Bloch vector in the northern hemisphere of theBloch sphere with P < P , while 10 < T R < µ s placesthe Bloch vector in the southern hemisphere ( P > P ) i.e., the selected hemisphere is a periodic function of T R such that an unequal superposition of | (cid:105) and | (cid:105) is cre-ated for each of the wavepackets unless θ lies on theequator. After applying this RF pulse (with some chosenduration T R ), we adjust the phase difference between thetwo superpositions by applying the third magnetic gradi-ent pulse of duration T G . This rotates the Bloch vectorsalong the latitude that was selected by the RF pulse topoints A and B in the northern hemisphere (or A (cid:48) , B (cid:48) inthe southern hemisphere) as shown in Fig. 7(a), therebyaffecting the phase difference ∆ φ BS , which we simplycall ∆ φ hereafter.The two wavepackets are allowed to interfere as in ourhalf-loop experiments, enabling a direct measurement of2 FIG. 7. Geometric phase. (a) Bloch sphere for the twowavepackets (green and red arrows labeled A and B , respec-tively) prepared by an RF pulse (duration T R , rotation an-gle θ ) and a subsequent magnetic gradient pulse (duration T G )that induces a rotation angle difference of ∆ φ . The rota-tion A → B lies along a constant latitude (solid purple line),while the returning geodesic B → A lies along the “great circle”curve (dashed purple line). Bloch vectors for correspondingwavepackets prepared in the southern hemisphere are shownas A (cid:48) and B (cid:48) . (b-c) Interference fringes generated by thehalf-loop SGI, averaged over a total of 330 experimental shotswith varying 0 < T R < µ s, while keeping a fixed valueof T G = 17 µ s (this value of T G corresponds to ∆ φ = π , seetext). The dashed green lines show that the maxima in (b) lieexactly where the minima occur in (c), corresponding to Blochvectors prepared in the northern and southern hemispheres,respectively. Adding all these interference patterns togetherin (d) shows near-zero visibility, i.e., they are completely outof phase. The fact that exactly the same pattern is observedwhile in the same hemisphere, independent of θ (durationof T R ), is called “phase rigidity”. (e) Total phase extractedfrom the interference fringes measured as a function of the RFpulse duration (lower scale) and the corresponding latitude θ (upper scale). Phase rigidity is clearly visible. (f-g) Dynami-cal and geometric phases extracted from the data in (e) andindependently measured values of θ and ∆ φ (see text). Therange of T R in (e-g) ( T G is fixed at 17 µ s) corresponds to a fullcycle from the northern hemisphere (0 < T R < µ s) throughthe southern hemisphere (10 < T R < µ s), and back to thenorth pole at T R = 40 µ s. Adapted from [84] with permission © the authors, some rights reserved; exclusive licensee AAAS.CC BY 4.0 the geometric phase. As usual, we extract the “total” in-terference phase (labeled Φ) by fitting the fringe patternsusing Eq. (1). For general values of θ and ∆ φ ( i.e., afterthe application of both T R and T G ), we write the totalphase between the two wavepackets as [84]Φ = arctan (cid:26) sin ( θ/
2) sin(∆ φ )cos ( θ/
2) + sin ( θ/
2) cos(∆ φ ) (cid:27) . (7) Measurements of Φ, combined with values of θ deducedindependently from the relative populations of states | (cid:105) and | (cid:105) , then allow us to fit ∆ φ to high precision as afunction of T G . These measurements verified that ∆ φ depends linearly on T G , and we found that ∆ φ = π occursat T G = 17 µ s.Figure 7(b-c) shows interference fringe images for thisspecific value of T G , from which we extract the totalphase as shown in Fig. 7(e). We see immediately thatthis phase is independent of θ within each hemisphere,an observation we call “phase rigidity”. Moreover, the(constant) phase in each hemisphere differs by π , whichcan also be deduced from the vanishing visibility shownin Fig. 7(d) in which we have combined the data fromboth hemispheres. Evidently, there is a sharp jump in thephase of the interference pattern as θ crosses the equator,as suggested by the singularities in Eq. (7) that arisewhen θ = π ( n + 1 /
2) (integer n ) and ∆ φ = π .To understand the non-cyclic geometric phase, we need tofurther examine the Bloch sphere. We see that the pathfrom A → B along the latitude θ and returning along thegeodesic (or “great-circle route”) from B → A enclosesan area [blue shading in Fig. 7(a)] in a counter-clockwisedirection, whereas the corresponding path from A (cid:48) → B (cid:48) and back again in the southern hemisphere proceeds in aclockwise direction. One-half of this area is the “geometricphase” that we now wish to calculate.The total phase change Φ for closed paths like A → B → A and A (cid:48) → B (cid:48) → A (cid:48) is a sum of two contributions, thedynamical phase Φ D and the geometric phase Φ G . Thedynamical phase is given by [80]Φ D = ∆ φ (cid:0) − cos θ (cid:1) , (8)which can be determined by measuring θ and ∆ φ inde-pendently. For the particular value of ∆ φ = π chosenas a sub-set of our experimental data, we are then ableto present Φ D in Fig. 7(f). Finally, we subtract thephases Φ D , as plotted in (f), from the total phases Φplotted in (e) (which, as noted above, are extracted di-rectly from the observed interference pattern) to obtainthe phases Φ G . Namely, we perform Φ − Φ D and get Φ G ,which is presented in Fig. 7(g). Let us emphasize thatthe total phase Φ is also the Pancharatnam phase [83],and thus our experiment is also a direct measurement ofthis phase.Our plot of Φ G exactly confirms the prediction shownin Fig. 4(d) of [81], also reproduced as the dashed blueline in Fig. 7(g). The predicted sign change as the lati-tude crosses the equator is clearly visible. The evidentphase jump is due to the geodesic rule. When ∆ φ = π ,the geodesic must go through the Bloch sphere pole forany θ (cid:54) = π/
2. As the latitude approaches the equator( i.e., increasing θ ), the blue area in Fig. 7(a) (twice Φ G )grows continuously, reaching a maximum of π in the limit3as θ → π/
2. As the latitude crosses the equator, thegeodesic jumps from one pole to the other pole, resultingin an instantaneous change of sign of this large area anda phase jump of π .Finally, our approach for testing the geodesic rule isunique for the following reasons: 1. the use of a spa-tial interference pattern to determine the phase in a sin-gle experimental run (no need to scan any parameterto obtain the phase); 2. the use of a common phase ref-erence for both hemispheres while scanning θ , enablingverification of the π phase jump and the sign change;and 3. obtaining the relative phase by allowing the twocoherently-prepared wavepackets to expand in free flightand overlap, in contrast to previous atom interferometrystudies that required additional manipulation of θ and ∆ φ to obtain interference. D. T Stern-Gerlach Interferometer
Here we consider an application of the full-loop SGIwherein we minimize the delay times between succes-sive SG pulses as much as allowed by our electronics. Insuch an extreme scenario, it is expected that the phaseaccumulation will scale purely as T , thus representingthe first pure interferometric measurement of the Kennardphase [19] predicted in 1927 [85, 86] (see also [87–89]).The theory for this experiment was done by the group ofWolfgang Schleich.In order to describe the phase evolution of an atommoving in a time- and state-dependent linear poten-tial, it is sufficient [90] to know the two time-dependentforces F u ≡ F u ( t ) e z and F l ≡ F l ( t ) e z acting on the atomalong the upper and lower branches, respectively, of theinterferometer shown in Fig. 8, where z is the axis ofgravity, the axis of our longitudinal interferometer, andalso the axis of our magnetic gradients.In the present case, these forces comprise the gravitationalforce F g = mg and the state-dependent magnetic forces F i = − µ B ( g F ) i ( m F ) i ( ∂ | B | /∂z ) e z , ( i = 1 , F u,l ( t ) = F g + F , F ( t ) , (9)where µ B , g F , and m F are the Bohr magneton, the Land´efactor, and the projection of the angular momentum onthe quantization ( y -)axis, respectively. The function F ( t )provides the time-dependent modulation shown as theorange curve in Fig. 8(b): F ( t ) ≡ Θ( t ) − Θ( t − T ) − Θ( t − T − T d )+Θ( t − T − T d )+ Θ( t − T − T d ) − Θ( t − T − T d ) . (10)Here we are using the Heaviside step function Θ( t ) andwe are assuming that the duration of each gradient pulseis identical, i.e., T , , = T , as are the two delay times, T d ,d = T d . We are also careful to ensure experimentally FIG. 8. Pulse sequence of our longitudinal T -SGI (not toscale). (a) Trajectories of the atomic wavepackets with in-ternal states | (cid:105) (green curve) and | (cid:105) (purple curve). Herewe are using the freely-falling reference frame (gravity up-wards), distinct from the center-of-mass reference frame usedfor Figs. 2 and 4. Also shown are the RF (blue) and magneticgradient (red) pulses. The magnetic field gradients resultin a state-dependent force along the z -direction while thestrong bias magnetic field along the y -direction defines thequantization axis and ensures a two-level system. (b) Timedependence of the relative force F = F ( t ) [orange curve,Eq. (10)] and the corresponding relative momentum δp ( t )[blue dashed curve, Eq. (12)] between the wavepackets movingalong the two interferometer paths. In the experiment, weachieved the maximal separation ∆ z max = 1 . µ m in positionand ∆ p max /m Rb = 17 mm / s in velocity. Reprinted from [19]with permission © (2019) by the American Physical Society. that the magnetic field is linear in the vicinity of theatoms and acts only along the vertical ( z -)axis. As in the full-loop SGI experiments of Sec. V, we measurethe spin population in state | (cid:105) which, in this configura-tion, is a periodic function of the interferometer phase [91]. P = 12 [1 − cos ( δ Φ + ϕ )] , where δ Φ = 1¯ h (cid:90) T dt ¯ F ( t ) δz ( t ) , (11)with the total time T ≡ T + 2 T d . Note that the inter-ferometer will be closed in both position and momentum Magnetic field linearity is ensured to a good approximation bythe three-wire chip design and by carefully positioning the atomsvery close to the center of the quadrupole field that they produce,as well as by the short distances that the atomic wavepacketstravel ( ∼ µ m) compared to their distance from the chip ( ∼ µ m). We also adjust the duration of T slightly, relative to T ,to better optimize the visibility and account for any residualnon-linearity. See [19, 68] for further details. δp ( t ) = (cid:90) t dτ δF ( τ )and δz ( t ) = 1 m (cid:90) t dτ δF ( τ )( t − τ ) (12)both vanish at t = T . Here ϕ is a constant phasetaking into account possible technical misalignment,while ¯ F ( t ) ≡ [ F u ( t ) + F l ( t )] / F g + ( F + F ) F ( t )and δF ( t ) ≡ F u ( t ) − F l ( t ) = ( F − F ) F ( t ) are the meanand relative forces respectively. From Eq. (11) we finallyobtain δ Φ = mga B ¯ h (cid:18) µ − µ µ B (cid:19) (cid:0) T + 3 T T d + T T d (cid:1) + ma B ¯ h (cid:18) µ − µ µ B (cid:19) (cid:18) T + T T d (cid:19) , (13)with a B ≡ µ B ∇ B/m being the magnetic acceleration.As sketched in Fig. 8, the experiment begins with anon-resonance RF π/ | (cid:105) to an equal superposi-tion, √ ( | (cid:105) + | (cid:105) ). This π/ µ s (thefirst “dark time”), we apply an RF π -pulse that flipsthe atomic state to √ ( | (cid:105) − | (cid:105) ). After a second darktime of another 400 µ s, a second π/ π -pulse inverts the populationbetween the two states of the system thereby allowing anytime-independent phase shift accumulated during the firstdark time to be canceled in the second dark time. Theexperiment is completed by applying a magnetic gradientto separate the spin populations and a subsequent pulseof the detection laser to image both states simultaneously.As with all our previous full-loop experiments, the fourmagnetic field gradient pulses are produced by current-carrying wires on the atom chip. This magnetic pulsesequence sends the spin states | (cid:105) and | (cid:105) along differ-ent trajectories in the SGI and ultimately closes the in-terferometer in both momentum and position. Carefulcalibration measurements verified that reversing the wirecurrents (the current flow is reversed during T and T relative to T and T ) provides magnetic accelerationsthat are equal in magnitude (but opposite in sign) towithin our experimental uncertainty of < < T < µ s. From Eq. (13), itis apparent that the T dependence will be most evidentif T d (cid:28) T , which is satisfied for most of the experimentalrange by using a fixed experimental value of T d = 2 . µ s(limited by the speed of our electronic circuits). Note FIG. 9. Measurement of the cubic phase with the T -SGIpresented in Fig. 8. The solid red line represents a fit basedon Eq. (13), as described in the text. The dashed blue line isa fit with T d = 0, showing that the interferometer phase scalespurely as T for T > ∼ µ s. The visibility drops from 68%to 32% over 70 µ s with a decay time of 75 µ s. This reductionresults from inaccuracies in recombining the two interferometerpaths. The dashed gray horizontal lines depict the maximaland minimal values of the population P measured indepen-dently without magnetic field gradients. Reprinted from [19]with permission © (2019) by the American Physical Society. that T < ∼ µ s is limited by the duration of the seconddark time.The experimental data (dots) agree very well with thetheory (solid red line) based on Eq. (13), where the fittingparameters are the magnetic acceleration a B as well asthe decay constant of the visibility and a constant phase ϕ . The dashed blue line is obtained by setting T d = 0,leading to a pure T scaling that is indistinguishable fromthe full theoretical fit for T > ∼ µ s: δ Φ ( T ) ∼ = ma B h (cid:18) µ − µ µ B (cid:19) (cid:18) g + µ + µ µ B a B (cid:19) T . (14)The maximum visibility displayed by the gray lines isfirst measured by performing only the RF spin-echo se-quence ( π/ − π − π/
2) without the magnetic field gradi-ents and changing the phase of the second π/ a fit B =246 . ± .
09 m / s . Separate measurements were usedto independently determine the magnetic field gradientusing time-of-flight (TOF) techniques, which gave a valueof a TOF B = 249 ± / s . While these measurementsagree with one another, the difference in measurementerrors clearly shows that our T -SGI provides a muchmore precise measurement of the magnetic field gradient. These values for a fit B and a TOF B are different from those presentedin [19] due to a different fitting procedure used there. A fullanalysis and fitting procedures are presented in the Appendicesof [68]. T (cid:28) T d , such thatduring T d the relative momentum δp ≡ ma B T ( µ − µ ) /µ B between the paths is kept constant, i.e., we takethe magnetic field gradient pulses to be delta functions.In this limit the interferometer phase from Eq. (13) be-comes δ Φ ( T ) ∼ = δp h gT , (15)scaling quadratically with the total time T ∼ = 2 T d , since wenow maintain a piecewise constant momentum differencebetween the two arms. This is similar to the T -SGI [18]or the Kasevich-Chu interferometer [90], although themomentum transfer δp is provided by the magnetic fieldgradient in the case of the T -SGI, rather than by thelaser light pulse.We conclude our discussion of this unique T interfer-ometer by comparing the scaling of the interferometerphases δ Φ ( T ) and δ Φ ( T ) with the total interferometertime T , as given by Eqs. (14) and (15) respectively. Thedata in Fig. 10 are taken from Fig. 9 and from our T -SGI(when experimentally realizing the condition T (cid:28) T d ),showing clearly that the T -SGI significantly outperformsthe T -SGI with respect to total phase accumulation,even though the latter can currently operate for totaltimes T up to three times larger than the former. Finally,let us briefly note that this T realization has alreadybeen coined a proof-of-principle experiment for testingthe quantum nature of gravity [49].Looking into the future, we may ask if one may extendthe T scaling to yet higher powers of time. In theRamsey-Bord´e interferometer [92], the phase shift thatscales linearly with the interferometer time T originatesfrom a constant position difference between two pathsduring most of this time. In the Kasevich-Chu interfer-ometer [93, 94], the quadratic scaling of the phase withtime is caused by a piecewise constant velocity difference,while a piecewise constant acceleration difference betweenthe two paths results in the cubic phase scaling δ Φ ∝ T ,as presented above.One can generalize this idea to achieve any arbitraryphase scaling by having a piecewise difference in the n th derivative of the position difference between the two paths.By designing an interferometer sequence consisting ofpulses with a higher-order time-dependence of the forces,combined with careful choices of the relative signs anddurations of the pulses, the total phase can be madeto scale with the interferometer time as T n +1 for anychosen n > VII. OUTLOOKA. SGI with Single Ions
The discovery of the Stern-Gerlach effect led to lively dis-cussions early in the quantum era regarding the possibility
FIG. 10. Scaling of the interferometer phases δ Φ ( T ) [squares,Eq.( 14)] and δ Φ ( T ) [circles, Eq. (15)], as functions of the totalinterferometer time T . The solid red line is fitted to our datafor the T -SGI and the dashed blue line is fitted to our T -SGIdata when experimentally realizing the condition T (cid:28) T d .In its current configuration with T max = 285 µ s, the phaseof the T -SGI is almost six times larger than the phase ofthe best T -SGI, even though the magnetic field gradientsand the maximal time T max = 924 µ s are larger than thoseof the T -SGI by factors of 2.3 and 3.2, respectively. Forreference, the green square and green dot represent data forwhich the observed visibility is ≈
30% for both the T -SGIand T -SGI respectively. Adapted from [19] with permission © (2019) by the American Physical Society. of measuring an analogous effect for the electron itself (see e.g., [95, 96]). The Lorentz force adds the complicatingfactor of a purely classical deflection of the electron beamthat would smear out any expected SG splitting. Here wesummarize a generalized semiclassical discussion for anycharged particle of mass m and charge e from [62] (thoughwith the co-ordinate system in Table I). Assuming a beammomentum p x and a transverse beam spatial width ∆ z ,we calculate the spread of the Lorentz force ∆ F L due toa transverse magnetic gradient B (cid:48) as∆ F L = em p x B (cid:48) ∆ z. (16)Since the beam would be well collimated, ∆ p z < p x , so∆ F L > em B (cid:48) ∆ p z ∆ z ≥ e ¯ h m B (cid:48) = m e m e ¯ h m e B (cid:48) = m e m F SG , (17)where the second inequality uses the uncertainty principleand we have introduced the electron mass m e to relate F L to the Stern-Gerlach force F SG .The spatially inhomogeneous Lorentz broadening is there-fore larger than the SG splitting for electrons, at least inthis semiclassical analysis [97], and this lively controversyhas continued for decades though, as far as we know,without any conclusive experimental tests for electrons orfor any other charged particles (see [98–100] for reviews ofthe early history of this issue and recent perspectives). In6contrast, Eq. (17) shows no such fundamental problem ifwe take ions such that m e /m < − , thereby motivatingour proposals, including chip-based designs, for measure-ments using very high-resolution single ion-on-demandsources that have recently been developed using ultra-coldion traps [61, 101]. As a practical matter, we note thata suitable ion chip could be fabricated and implementedeither based on an array of current-carrying wires as ana-lyzed in [62] or on a magnetized microstructure like thoseimplemented in [102, 103].Although we did not extend our analysis to include thecoherence of the spin-dependent splitting, the suggestedion-SG beam splitter may form a basic building blockof free-space interferometric devices for charged parti-cles. Here we quote from our collaborative work withHenkel, Schmidt-Kaler and co-workers [62]. In additionto measuring the coherence of spin splitting as in the“Humpty-Dumpty” effect (see Sec. I), we anticipate thatsuch a device could provide new insights concerning thefundamental question of whether and where in the SGdevice a spin measurement takes place. The ion interfer-ence would also be sensitive to Aharonov-Bohm phaseshifts arising from the electromagnetic gauge field. Theion source would be a truly single-particle device [61] andeliminate certain problems arising from particle interac-tions in high-density sources of neutral bosons [104].Such single-ion SG devices would open the door for a widespectrum of fundamental experiments, probing for exam-ple weak measurements and Bohmian trajectories. Thestrong electric interactions may also be used, for exam-ple, to entangle the single ion with a solid-state quantumdevice (an electron in a quantum dot or on a Coulombisland, or a qubit flux gate). This type of interferometermay lead to new sensing capabilities [105]: one of thetwo ion wavepackets is expected to pass tens to hundredsof nanometers above a surface (in the chip configurationof our proposal [62]) and may probe van der Waals andCasimir-Polder forces, as well as patch potentials. Thelatter are very important as they are believed to give riseto the anomalous heating observed in miniaturized iontraps [106]. Due to the short distances between the ionsand the surface, the device may also be able to sense thegravitational force on small scales [107]. Finally, such asingle-ion interferometer may enable searches for exoticphysics. These include spontaneous collapse models, thefifth force from a nearby surface, the self-charge interac-tion between the two ion wavepackets, and so on. Eventu-ally, one may be able to realize a double SG-splitter withdifferent orientations, as originally attempted by Stern,Segr`e and co-workers [108, 109], in order to test ideaslike the Bohm-Bub non-local hidden variable theory [110–112], or ideas on deterministic quantum mechanics (see, e.g., [113]). Since ions may form the basis of extremelyaccurate clocks, an ion-SG device would enable clock inter-ferometry at a level sensitive to the Earth’s gravitationalred-shift (see the proof-of-principle experiments with neu-tral atoms in [20, 21]). This has important implications for studying the interface between quantum mechanicsand general relativity. B. SGI with Massive Objects
The main focus of our future efforts will be to real-ize an SGI with massive objects. The idea of usingthe SG interferometer, with a macroscopic object asa probe for gravity, has been detailed in several stud-ies [46, 47, 114, 115] describing a wide range of experi-ments from the detection of gravitational waves to testsof the quantum nature of gravity. Here we envision usinga macroscopic body in the full-loop SGI. We anticipateutilizing spin population oscillations as our interferenceobservable rather than spatial fringes i.e., density modula-tions. This observable, as demonstrated in the atomic SGIdescribed above, is advantageous because there is no re-quirement for long evolution times in order to allow thespatial fringes to develop, nor is high-resolution imagingneeded to resolve the spatial fringes. Let us note thatthere are other proposals to realize a spatial superpositionof macroscopic objects [70, 116].As a specific example, let us consider a solid object com-prising 10 − atoms with a single spin embedded inthe solid lattice, e.g., a nano-diamond with a single NVcenter. Let us first emphasize that even prior to anyprobing of gravity, a successful SGI will already achieveat least 3 orders of magnitude more atoms than the stateof the art in macroscopic-object interferometry [60], thuscontributing novel insight to the foundations of quantummechanics. Another contribution to the foundations ofquantum mechanics would be the ability to test contin-uous spontaneous localization (CSL) models ( e.g., [117]and references therein).When probing gravity, the first contribution of such amassive-object SGI would simply be to measure little g .As the phase is accumulated linearly with the mass, amassive-object interferometer is expected to have muchmore sensitivity to g than atomic interferometers be-ing used currently (assuming of course that all otherfeatures are comparable). This is also a method toverify that a massive-object superposition can be cre-ated [114, 118, 119]. A second contribution would mea-sure gravity at short distances, since the massive objectmay be brought close to a surface while in one of the SGIpaths, thus enabling probes of the fifth force. Once the SGtechnology allows the use of large masses, a third con-tribution will be the testing of hypotheses concerninggravity self-interaction [48, 116], and once large-area in-terferometry is also enabled, a fourth contribution wouldbe to detect gravitational waves [46]. Finally, placingtwo such SGIs in parallel next to each other will enableprobes of the quantum nature of gravity [47, 120]. Letus emphasize that, although high accelerations may beobtained with multiple spins, we intend to focus on thecase of a macroscopic object with a single spin, sincethe observable of such a quantum-gravity experiment is7entanglement, and averaging over many spins may washout the signal.To avoid the hindering consequences of the HD effect,one must ensure that the experimental accuracy of therecombination, as discussed in Sec. V, will be betterthan the coherence length. Obviously it is very hard toachieve a large coherence length for a massive object,but recent experimental numbers and estimates seem toindicate that this is feasible. Another crucial problemis the coherence time. A massive object has a hugecross section for interacting with the environment ( e.g., background gas), but the extremely short interferometertimes, as discussed in this review, seem to serve as aprotective shield suppressing decoherence. We are nowpreparing a detailed account of these considerations [121]. Disclosure Statement
The authors declare that they have no competing financialinterests.
Acknowledgments
We wish to warmly thank all the members – past andpresent – of the Atom Chip Group at Ben-Gurion Uni-versity of the Negev, and the team of the BGU nano-fabrication facility for designing and fabricating innovativehigh-quality chips for our laboratory and for others aroundthe world. The work at BGU described in this review wasfunded in part by the Israel Science Foundation (1381/13and 1314/19), the EC “MatterWave” consortium (FP7-ICT-601180), and the German DFG through the DIPprogram (FO 703/2-1). We also acknowledge supportfrom the PBC program for outstanding postdoctoral re-searchers of the Israeli Council for Higher Education andfrom the Ministry of Immigrant Absorption (Israel). [1] W. Gerlach and O. Stern. Der experimentelle Nachweisder Richtungsquantelung im Magnetfeld.
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