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Atomic “bomb testing”: the Elitzur-Vaidman experiment violates the Leggett-Garg inequality

Carsten Robens,1 Wolfgang Alt,1 Clive Emary,2 Dieter Meschede,1 and Andrea Alberti1, ∗ 1Institut für Angewandte Physik, Universität Bonn, Wegelerstr. 8, D-53115 Bonn, Germany 2Joint Centre Durham-Newcastle, Newcastle University, Newcastle upon Tyne NE1 7RU, UK (Dated: September 21, 2016) Abstract. Elitzur and Vaidman have proposed a measurement scheme that, based on the quantum super- position principle, allows one to detect the presence of an object in a dramatic scenario, a bomb without interacting with it. It was pointed out by Ghirardi that this interaction-free measurement scheme can be put in direct relation with falsification tests of the macro-realistic worldview. Here we have implemented the “bomb test” with a single trapped in a -dependent optical lattice to show explicitly a violation of the Leggett-Garg inequality a quantitative criterion fulfilled by macro-realistic physical theories. To perform interaction-free measurements, we have implemented a novel measurement method that correlates spin and position of the atom. This method, which quantum mechanically entangles spin and position, finds general application for spin measurements, thereby avoiding the shortcomings inherent in the widely used push-out technique. Allowing decoherence to dominate the evolution of our system causes a transition from quantum to classical behavior in fulfillment of the Leggett-Garg inequality.

I. INTRODUCTION scopically distinct states. In a macro-realistic world- view, it is plausible to assume that a negative out- Measuring physical properties of an object whether come of a measurement cannot affect the evolution of macroscopic or microscopic is in most cases associated a macroscopic system, meaning that negative measure- with an interaction. For example, scattering off ments are non-invasive [8]. In order to rigorously test an object allows one to detect its presence in a given the macro-realistic point of view, Leggett and Garg have region of space. However, this also produces a small per- derived an inequality from the assumptions of macro- turbation of its state by direct momentum transfer. It is realism and non-invasive measurability, which can be vi- well known from numerous discussions on the physics of olated by quantum-mechanical superposition states [9]. the quantum measurement process (see, e.g., Refs. [1,2]) Relying on the implementation of an ideal negative- that a measurement in general modifies the quantum evo- measurement protocol a prerequisite for any genuine lution unless the object is already in an eigenstate of the test of the Leggett-Garg inequality experimental vio- measurement apparatus [3]. This is even the case when lations of the macro-realistic worldview have been ex- the measurement yields a negative outcome, that is, when perimentally demonstrated with phosphor impurities in we did not find the on a certain trajectory that silicon by Knee et al. [10] and with trapped by had originally a non-vanishing to Robens et al. [11]. be occupied. For example, in a double-slit experiment, The definition of the degree of “macroscopic distinct- quantum interference is suppressed as soon as a mea- ness” has been a matter of discussion in the literature surement detects the which-way information, regardless ever since [12], and is likely to remain as such till an of whether the information is acquired by direct interac- experiment [13] will shed new light, for example, discov- tion or indirect negative inference. Quantum ering a physical “classicalization” mechanism that causes formalizes the loss of interference in terms of the quan- an objective reduction of packets. Recently, Nimm- tum measurement process, showing that measurements richter and Hornberger proposed a quantitative criterion are generally invasive as they entail a modification of the based on a minimal macro-realistic extension of quan- subsequent quantum evolution. While the quantum mea- tum mechanics to quantify the macroscopicity of an ob- surement process is still intensely debated in the litera- ject [14]. Their objective criterion of macroscopicity al- ture [4], we adopt here the pragmatic view that a mea- lows us to experimentally test the behavior of a single surement applied to a superposition state causes a sudden trapped atom however microscopic it is, according to reduction of the to a smaller . our intuition under the hypothesis of macro-realism, as we can put its macroscopicity directly in relation to that arXiv:1609.06218v1 [quant-ph] 20 Sep 2016 Ideal negative measurements, that is, measurements without direct interaction, play an important role in a of other, ideally more massive physical objects. physical scenario known as macro-realism [5–7]. This It was pointed out by Ghirardi [15] that a Leggett-Garg scenario advocates a classical worldview describing the test of macro-realism is naturally related to the notion of state of macroscopic objects, according to which macro- interaction-free measurements introduced by Elitzur and scopic objects are always in one of several possible macro- Vaidman [16]. In a rather dramatic scenario, Elitzur and Vaidman proposed a quantum device able to single out live “bombs” from a collection containing also duds with- out triggering them nor interacting with them. While the ∗ [email protected] first realizations of the Elitzur-Vaidman experiment em- 2 ployed “flying” photons [17] and “flying” neutrons [18], we without touching it, therefore leaving the “bomb” intact. here implement a variation of this experiment with neu- Following Elitzur and Vaidman, we call this measurement tral atoms trapped in a one-dimensional optical lattice. scheme interaction-free [20]. A convenient scheme for interaction-free measurements We obtain further insight in the Elitzur-Vaidman with neutral atoms has been demonstrated by Robens “bomb test” by analyzing its outcomes from the perspec- et al. [11] exploiting state-dependent optical potentials. tive of statistical hypothesis testing, where the Elitzur- Following the idea of Ghirardi, we tested the hypoth- Vaidman quantum device is employed to test whether the esis of macro-realism with our atomic implementation “bomb” is live (positive test result). The typical figure of of the Elitzur-Vaidman “bomb testing” Gedankenexperi- merit in hypothesis testing is the statistical power 1 − β, ment. Our experiment shows explicitly that the Leggett- that is, the fraction of “live bombs” which are correctly Garg inequality is violated by 21 σ. In , trapped identified (without being triggered) and rescued. In their atoms can be held for long . By controlling the original work [16], Elitzur and Vaidman showed that this duration of a suitably chosen wait interval, it is straight- fraction amounts to 25 %(true positives). It is worth forward to study the influence of decoherence and exper- extending the analysis of “bomb testing” to quantum de- imentally observe the gradual transition from quantum vices that, under realistic conditions, are impaired by to classical behavior. decoherence. We assume that decoherence reduces the It is our pleasure to honor with these recent experi- fringe amplitude of the MZ interferometer in an unbi- mental results Theodor W. Hänsch. For many decades ased way, resulting in a contrast C less than unity. In he has laid the foundations in physics and technol- the presence of decoherence, the statistical power of the ogy, without which present-day laser control of quantum test remains unchanged, since this quantity describes the is hardly conceivable. In this sense, the objec- situation of a “live bomb”, where the of the tive of this article is to experimentally demonstrate yet plays no role, see Fig.1(b). We therefore intro- another example of exploring the world with and duce a second figure of merit accounting for decoherence, laser-controlled atoms at the quantum-classical bound- which is given by the statistical error of type I, that is, the ary. probability α that we erroneously rescue the bomb even though the bomb is a dud (false positive). One can show that the probability of type I errors is α = (1 − C)/2. II. BRIEF REVIEW: THE ELITZUR-VAIDMAN This measure vanishes only for a decoherence-free “bomb INTERACTION-FREE “BOMB TEST” tester” and becomes α = 1/2 in the completely inco- herent limit (equivalent to random selection). It is also Let us illustrate the concept of the interaction-free “bomb test” as presented in the original work by Elitzur and Vaidman [16], based on a single photon travelling (a) A on a of trajectories along the two paths of a Mach-Zehnder (MZ) interferometer. D1 BS BS The two branches A and B of a MZ interferometer can 100% be balanced in such a way that one of the two output ports, say D2, is always bright while the other one, D1, B D2 is always dark, see Fig.1(a). If any object (e.g, a “live counts bomb” triggered by a single photon) intercepts the tra- (b) jectory of the photon in the lower branch B, the photon A field is removed from that branch and the balance of the D1 MZ interferometer is disturbed. As shown in Fig.1(b), BS BS % the photon has a 50 % probability to be blocked without 2 5 ever reaching the detectors. This event is of course highly B invasive: the “bomb” is triggered to explode by absorbing D2 the travelling photon. counts In all other events, the photon must have followed branch A, thus avoiding interacting with the object, and is subsequently routed to the detectors D1 and D2 with Figure 1. Bomb testing with a Mach-Zehnder interferometer operated with a single photon at a . The beam splitters equal probability. In case detector D2 clicks, insufficient have split ratio 50:50. (a) The phase difference between the knowledge is gained to conclude on whether the object is two arms is adjusted such that all photons are directed to present, as this outcome also occurs for no object present detector D2. The object situated close to the lower branch (“dud bomb”). If the dark detector D1 lights up, however, B (e.g., a dud bomb not equipped with the trigger) does not the presence of an object in one of the two arms is sig- intercept the photons. (b) The object (e.g., a “live bomb” naled with certainty. Since the photon could not reach equipped with an avalanche photodiode which is triggered the detector if it had touched the object, finding a pho- by a single photon [19]) intercepts the photons in the lower ton with detector D1 detects the presence of the object branch B. 3 worth mentioning that, by allowing for repeated mea- (A1) and (A2) must not hold. In conducting a Leggett- surements (this is always possible until the “bomb” has Garg test, it is therefore crucial that assumption (A2) of not exploded), the statistical power 1−β can be straight- non-invasive measurability cannot be simply dismissed by forwardly increased to 1/3 for 50:50 beam splitters and a macro-realist claiming that a measurement operation, to 1/2 for beam splitters with different split ratios. Fur- due experimental inadvertence, has influenced the sub- thermore, a more complex variant of this scheme based on sequent evolution of the particle; or else, even in case of the quantum Zeno effect has been suggested [21] and ex- an observed violation, no claim can be made concerning perimentally demonstrated using traveling photons [22], assumption (A1) of macro-realism. In the literature, this allowing one to approach unity efficiency (β = 0). type of objection is addressed as the clumsiness loophole Let us now reconsider the situation illustrated in [24]. To circumvent it, Leggett and Garg suggested using Fig.1(b) from the perspective of a macro-realist, who ideal negative measurements, which, to say it à la Vaid- conducts the same experiment but with a massive, ideally man and Elitzur, are interaction free. It should also be macroscopic object traveling along the two branches of noted that in standard both assumptions the MZ interferometer. For a macro-realist, the massive do not hold. In fact, standard quantum theory postulates particle travels either along branch A or along branch B, (1) no limit on the and split distance of a super- but not in a superposition state of the trajectories A and position state and that (2) even a negative measurement B. If one discards by post-selection all events where the can cause the ’s reduction. The latter ’s presence on one of the two trajectories has been is indeed central to the Elitzur-Vaidman interaction-free directly detected through the absorption by the object, experiment. then only interaction-free measurements (i.e. ideal nega- In the Elitzur-Vaidman experiment, the object that tive measurements) of the particle’s position are consid- is put to the test of the Leggett-Garg inequality is the ered. By intercepting the particle at one time in branch single photon travelling along the two paths of the MZ A and at another time in branch B, the macro-realist interferometer. In the following we define the three mea- learns about the particle’s position in the MZ interfer- surement operations performed on the photon and their ometer avoiding any interaction with it. Therefore, to a assigned values Q(ti), which are employed to perform the macro-realist, ideal negative measurements must appear Leggett-Garg test in Eq. (1): non-invasive, since the subsequent evolution of the par- Q(t1): We identify the first measurement at t1 with the ticle could have not been influenced by the presence of preparation of the initial state a photon in the not an object where the particle was . Non-invasive mea- input of the MZ interferometer. This measure- surements constitute an important prerequisite for any ment is by definition non-invasive, as it leaves the rigorous test of macro-realism through a violation of the particle in the initial state. We assign to this mea- Leggett-Garg inequality [23]. surement the value Q(t1)=+1.

Q(t2): This measurement is performed at time t2 when III. RELATION TO THE LEGGETT-GARG the photon is either in branch A or B of the INEQUALITY MZ interferometer. It detects in which branch the photon travels by removing the photon at one time from branch A, at another time from Based on two assumptions, (A1) macro-realism and branch B. The events in which the photon was (A2) non-invasive measurements, the Leggett-Garg in- directly intercepted by the object (i.e., when the equality bounds a linear combinations of two-time corre- “bomb” exploded) are discarded by post-selection lation measurements, in order to ensure that only ideal negative mea- surements are performed; this measurement must K = hQ(t2)Q(t1)i+hQ(t3)Q(t2)i−hQ(t3)Q(t1)i ≤ 1, (1) thus appear to a macro-realist as non-invasive as it avoids any direct interaction with the photon it- where Q(ti) denote the outcome of measurements carried self. We assign to this measurement the constant out on the object at three subsequent times (i = 1, 2, 3). value Q(t2)=+1 regardless of which trajectory The values assigned to the individual measurement out- the photon has taken. It should be noted that comes have to fulfill the condition |Q(ti)| ≤ 1 but can this measurement is not performed when evalu- otherwise be freely chosen. Standard quantum mechan- ating the correlation function hQ(t3)Q(t1)i of the ics and macro-realism are distinguished from each other Leggett-Garg test. In fact, from the perspective by violating and fulfilling this inequality, respectively. of a macro-realistic who advocates (A1) and (A2), Observing a violation of the inequality (1) allows one an ideal negative measurement could not have not falsify to refute (i.e., ) the assumptions underlying the influenced the evolution of the particle. Leggett-Garg test in the range of parameters investigated by the experiment, which in general are represented by Q(t3): The final measurement is performed at time t3 the mass of the superposition states and their spatial split when the photon has reached the output of the distance [14]. For simple logical reasons, a violation of MZ interferometer. Depending on whether de- Eq. (1) implies that at least one of the two assumptions tector D1 or D2 has clicked, we assign to this 4

measurement the value Q(t3)=+1 or Q(t3)=−1, the which-way information acquired through interaction- respectively. Because we are not interested in free measurements yields a vanishing contrast, that is, the system’s evolution after t3, non-invasiveness hQ(t3)iwith Q2 = 0. of this measurement operation is not required.

Previous Leggett-Garg experimental tests prior to IV. INTERACTION-FREE MEASUREMENT Robens et al. [11] only considered dichotomic designa- WITH TRAPPED ATOMS tions of Q(t2), as opposed to the constant choice of Q(t2)=+1 here. By deliberately disregarding the unnec- Our atomic realization of the Elitzur-Vaidman experi- essary, dichotomic constraint, we allow the Leggett-Garg ment employs single neutral atoms trapped in an optical correlation function of the Elitzur-Vaidman experiment potential instead of flying photons. Instead of delocaliz- to reach the maximum value, K = 2, as permitted by ing the particle on two distinct trajectories as in the MZ quantum theory for a two level system (i.e., the photon interferometer shown in Fig.1, we let the particle evolve in a superposition state of the trajectories A and B). It in a superposition of two long-lived internal states, which can be shown that a violation of the inequality in Eq. (1) we denote by |↑i and |↓i hereafter. In our experiment, is also produced in the Elitzur-Vaidman experiment for a an atomic Ramsey interferometer plays the role of the dichotomic choice of Q(t2) (see AppendixC). However, in optical MZ interferometer. this case the maximum value of K predicted by quantum theory is only 3/2 [25, 26] instead of 2. Q(t ) Taking into account our specific designation of i , A. Experimental apparatus we can recast Eq. (1) into a simpler form. Since Q(t2)=1, the correlation function hQ(t3)Q(t2)i equals State-dependent optical conveyor belts. hQ(t3)iwith Q2, that is, the average value of Q(t3) At the core of conditioned on a negative result of the measurement our realization of the Elitzur-Vaidman experiment with Q(t2). Likewise, since Q(t1)=1, the correlation function trapped atoms are polarization-synthesized (PS) optical hQ(t3)Q(t1)i simplifies to hQ(t3)iwithout Q2, that is, the lattices, which were recently introduced by Robens et average value of Q(t3) without having measured the po- al. [29]: two one-dimensional, periodic optical potentials sition of the photon at t2. Our experiment can thus be can be independently shifted along their common longi- analyzed with a simplified version of Eq. (1), tudinal direction to selectively transport atoms in either one of two internal states, |↑i and |↓i. In essence, two co- K = 1 + hQ(t3)iwith Q2 − hQ(t3)iwithout Q2 ≤ 1 . (2) propagating laser beams of opposite circular polarization interfere with a third, counterpropagating, linearly polar- It is interesting to go one step further to provide an in- ized beam. Their interference gives rise to two standing terpretation of the Leggett-Garg correlation function K of left- and right-handed polarization, whose posi- from the point of view of quantum theory. We know that tions are actively controlled by means of two independent the values of Q(t3)= ± 1 are evenly distributed when the optical phase-locked loops. We obtain a residual jitter Q(t2) measurement is performed, since the latter reveals of their relative position on the order of 1 Å, which is the which-way information; hence hQ(t3)iwith Q2 = 0. much smaller than the longitudinal extent of the atom’s Moreover, one can prove that hQ(t3)iwithout Q2 is iden- wave function of ≈ 20 nm. At the so-called magic wave- tical to the contrast C of the MZ interferometer (see Ap- length λL = 866 nm of cesium atoms, the internal state + pendixD). Thus, we find that the correlation function K |↑i = |F = 4, mF = 4i interacts exclusively with the σ - takes the suggestive form polarized component, while |↓i = |F = 3, mF = 3i pre- dominantly interacts with the σ−-polarized component K = 1 + C. (3) [30]. Moreover, we choose a relatively deep lattice with a depth of U/kB ≈ 80 µK to prevent tunneling between dif- Intuitively, the function K provides a quantitative in- ferent sites. Hence, atoms in the two internal states are dication, say a witness, of the amount of superposition bound to two spatially superimposed, but orthogonally involved in the evolution of the quantum particle. Note polarized lattices, which can be individually shifted much that K can be put in relation to the first quantum wit- like two independent optical conveyor belts: atoms in the ness W of superposition states introduced in Ref. 27 (also |↑i and |↓i states follow, nearly rigidly, the σ+-polarized described as no-signaling in time in Ref. 28). In fact, and σ−-polarized standing waves, respectively. W = |K − 1| = C as shown in Ref. 11. This demon- State-dependent optical lattices have been pioneered strates that the figure of merit α of a partially decohered first in the MPQ laboratories [31, 32], demonstrating “bomb tester” (see Sec.II) is directly related to the quan- another example of Theodor W. Hänsch’s legacy as an tum witness W since α = (1 − W )/2. Furthermore, our inspiration for future generations of experiments. Com- results shows that any quantum particle exhibiting a non- pared to former realizations of state-dependent optical vanishing interference contrast should allow, according to lattices, PS optical lattices have replaced the polariza- quantum theory, for a violation of Eq. (1), provided that tion control formerly based on an electrooptic modulator one can additionally show through an experiment that by a direct synthesis of light polarization, which enable 5

t1 State preparation Q1

π/2 pulse

Without Q2 With Q2

t2

long state-dependent shift (removal)

π/2 pulse π/2 pulse

or

Spin position mapping Q3

or t3

time D1 D1 D2 D2

Figure 2. Illustration of the Elitzur-Vaidman experiment using single atoms. Atoms are trapped in state-dependent optical lattices, which consist of two independently movable, periodic optical potentials for atoms in the internal states |↑i and |↓i. The two atomic states form a spin-1/2 system, which is represented on the at different moments of the time evolution; short microwave pulses allow us to rotate the spin. On the left-hand side, protocol of a Ramsey interferometer, whose pulses are configured to produce the state |↓i; this situation is equivalent to that in Fig.1(a). The spin information is eventually mapped onto two different positions on the lattice, D1 and D2, which are efficiently detected by fluorescence imaging. On the right-hand side, protocol of a Ramsey interferometer where an interaction-free measurement (i.e., an ideal negative measurement) of the spin state is performed at the intermediate time t2. This measurement intercepts only atoms in one spin state by transporting them far apart (grayed lattice regions); this situation is equivalent to that in Fig.1(b). arbitrary, state-dependent displacements of atoms. Po- Non-destructive spin-state measurement. Exploiting larization synthesis is realized through rf-control of the PS optical lattices, we devised a novel measurement optical phases (0.1◦ RMS phase jitter) of two overlapped method to detect the internal state of the atom in beams with opposite circular polarization [29]. the most gentle way possible. In quantum mechan- Microwave control. We employ microwave radiation at ics, least perturbative measurements are called non- the cesium clock frequency of 9.2 GHz to induce coher- destructive (or, equivalently, quantum non-demolition ent oscillations between the two atomic hyperfine states, measurements): the state of the object is preserved af- |↑i and |↓i with a Rabi frequency of 55 kHz. Therefore, ter the measurement in the eigenstate of the measured the application of the microwave radiation field for 4.5 µs quantity corresponding to the observed outcome [33]; a realizes a so-called π/2 pulse, which transforms a pure in- repeated measurement of the same quantity would there- ternal state into an equal superposition of |↑i and |↓i. In fore leave the state unchanged. Conversely, the widely our realization of the Elitzur-Vaidman experiment, mi- employed push-out method, which expels atoms in one crowave π/2 pulses represent the atomic analogue of the particular internal state from the trap by applying state- beam splitters for photons, which are illustrated in Fig.1. selective radiation pressure [34], represents a destructive 6 measurement. a trapped atom. Note that this situation is fully equiva- Our method is closely related to an optical Stern- lent to the unobstructed MZ interferometer of Fig.1(a), Gerlach experiment [35, 36], where spin-position entan- where the atomic internal states here take the place of glement is created by state-dependent light fields, and is the distinct trajectories of the photon. Here we adjust reminiscent of the non-destructive Stern-Gerlach exper- the microwave phase φ beforehand in order to ensure the iment by Dehmelt [37, 38]. We realize non-destructive highest probability to detect |↓i at time t3, much like in measurements by displacing atoms by a discrete number the MZ interferometer one must balance the two branches of lattice sites conditioned on the internal state, thereby to route all photons to detector D2. transferring spin states to well-separated positions. The Let us turn to the application of the atomic “bomb position can be detected efficiently at a later time by test” illustrated on the right-hand side of Fig.2. Af- fluorescence imaging under molasses illumination with- ter the initial microwave π/2 pulse, which puts the atom out any atom loss [39]; we identify the correct position, in a coherent superposition state (for a macro-realist, a and therefore the spin state, with > 99 % reliability. The stochastic mixture of both states), we spatially remove translational invariance of the optical lattice ensures that atoms, in different experiments, at one time in state |↑i this measurement protocol constitutes a non-destructive and at another time in state |↓i by transporting them measurement of the internal state. Note also that it is apart by seven lattice sites in about 200 µs. The number not necessary that the position read-out immediately fol- of sites is chosen sufficiently large to avoid any error in lows the state-dependent displacement. The possibility the final position measurement, and therefore in the spin to postpone the “destructive” fluorescence image of the reconstruction. For the Leggett-Garg test, we post-select atoms till the end of the evolution allows us to leave the events where an atom is indeed found at position D1 the evolution of the system minimally perturbed as re- or D2, meaning that it has not been removed at time t2 quired by a non-destructive spin measurement. We use and its spin has thus been measured by an ideal negative this technique for the ideal negative measurement Q(t2), measurement Q(t2), as was argued in Sec.III. The ex- in which case only one spin component at a time is dis- cluded events instead correspond to the atoms removed placed, leaving the other unperturbed (see Sec.IVB). from the Ramsey interferometer, and are tantamount to It should also be noted that non-destructive measure- having triggered the “bomb” in one of the two branches ments are not strictly needed in order to perform the ideal of the MZ interferometer of Fig.1(b). negative measurements that are instead required for the Elitzur-Vaidman experiment and the Leggett-Garg test [21]. This fact is illustrated in Fig.1 where the photon re- V. EXPERIMENTAL RESULTS moval represents indeed a destructive measurement, since the particle is destroyed if intercepted by the “bomb”. In panels Fig.3(a-c), we show the raw data correspond- ing to our atomic realization of the Elitzur-Vaidman ex- periment for three different situations: (a) without the B. Measurement protocols “bomb”, (b) with the “bomb” removing atoms in |↑i, and (c) again with the “bomb” but removing atoms in |↓i. We start each experimental sequence with, on average, From the dataset (a) we reconstruct the correlation func- 1.2 atoms loaded into the lattice. The loading procedure tion hQ(t3)iwithout Q2, while by merging the datasets (b) is stochastic and only atoms sitting at sufficiently sepa- and (c) we obtain the correlation function hQ(t3)iwith Q2. rated lattice sites are considered. Atoms are cooled to The combination of these two correlation functions pro- the longitudinal using first molasses cool- duce a violation of the Leggett-Garg inequality as de- ing and then microwave sideband cooling [40]. With the fined in Eq. (2). In Fig.3(d) we present the recorded axis chosen along the lattice direction, op- Leggett-Garg correlation function K for different dura- tical pumping by a σ+-polarized laser beam initializes tions of the waiting time, which separates the two mi- > 99 % of the atoms in the state |↑i. crowave π/2 pulses of the Ramsey interferometer. For We outline in Fig.2 the protocols employed to mea- a minimum waiting time of 5 µs, we record a value of sure the two terms forming in the Leggett-Garg inequal- K = 1.958 ± 0.033, which violates the Leggett-Garg in- ity in Eq. (2). On the left-hand side, we present the pro- equality by 21 σ. While this value of K lies very close cedure with no “bomb” present, which comprises Q(t1) to the decoherence-free prediction (K = 2), the recorded (preparation of the initial spin state preparation) and values of K decrease visibly for longer waiting times (i.e., Q(t3) (detection of the final spin state) measurements, increasing decoherence) till they reach the value of 1 but not Q(t2) (the “live bomb” in one of the interferom- for fully decohered spin dynamics, in fulfillment of the eter’s branch): the spin preparation is followed by a π/2 Leggett-Garg inequality. pulse, a variable waiting time, a second π/2 pulse with In the present experiment, we attribute the main adjustable microwave phase φ, and a non-destructive spin source of decoherence to scalar differential light shift measurement mapping the spin state onto different posi- [41], causing inhomogeneous spin of the atoms, tions as described in Sec.IVA. This sequence describes which in our case are thermally distributed over more a Ramsey experiment interrogating the spin coherence of than 100 vibrational levels in the directions transverse to 7

500 2.0 400 (a) (d) over, the analysis of the experimental results allows us to 300 50% understand the exact relation between the Leggett-Garg 200 1.8 Atoms 100 correlation function K and the interference contrast C of 0 500 1.6 Violation region the corresponding Ramsey interferometer, thus providing (b) 400 intuition about the quantum-to-classical transition of the 300 25% 1.4

200 K Leggett-Garg test. Atoms 100 0 It is also worth noting that our experiment demon- 500 1.2 400 (c) strates a non-destructive measurement technique of the 300 25% 200 1.0 spin state of the atoms, where spin-position entangle- Atoms 100 Classical region ment is used to transfer information from spin to posi- 0 −3 −2 −1 0 1 2 0.8 tion space. This mapping technique allows us to directly D2 Site D1 0 250 500 750 1000 Waiting time (μs) read out both spin states in a Ramsey interferometer, thus avoiding the shortcomings of the widely used push- Figure 3. Experimental violation of the Leggett-Garg inequal- out technique, where atoms are lost after the measure- ity in the quantum-to-classical transition. From (a) to (c), dis- ment and, most importantly, the measurement outcomes tributions at time t3 of the detected atom at sites D1 and D2 must be corrected for atom losses. Further, with the non- for a waiting time of 100 µs, corresponding to the solid point destructive measurement technique demonstrated here in (d) for three different protocols. (a) Without the Q(t2) we could recycle atoms multiple times. We finally antici- measurement (left-hand-side protocol in Fig.2). (b) With pate that non-destructive spin measurements, preserving the Q(t2) measurement shifting atoms in |↑i away at time t2 spatial coherence of atoms delocalized over several lattice (right-hand-side protocol in Fig.2). (c) The same but with atoms in |↓i shifted away. (d) Values of the Leggett-Garg cor- sites, can find application in the realization of dissipative relation function K of Eq. (2) for increasing waiting times be- quantum-walk protocols [45]. tween the two π/2 pulses. Decoherence gradually suppresses the quantum behavior of the atom. The shaded band repre- sents the theoretical quantum-mechanical prediction for co- herence times between 75 µs and 200 µs caused by differential scalar light shift [41]. Percentage values are referred to the ACKNOWLEDGMENTS total number of interrogated atoms in each dataset. The ver- 1 σ tical error bars represent statistical uncertainty. We are very grateful to Jean-Michel Raimond for in- sightful discussions. We acknowledge financial support from NRW-Nachwuchsforschergruppe “Quantenkontrolle the lattice. Besides producing a rigorous violation of the auf der Nanoskala,” ERC grant DQSIM, and EU Inte- Leggett-Garg inequality for a pseudo-spin-1/2 particle, grated Project SIQS. In addition, C.R. acknowledges sup- K our results show that the correlation function can be port from the BCGS program and the Studienstiftung des interpreted, from the point of view quantum theory, as a deutschen Volkes. quantum witness of superposition states, and employed to study decoherence one of the most basic mechanisms affecting atoms trapped in optical potentials [42].

Appendix A: Statistical errors VI. CONCLUSIONS In this work, the confidence intervals of the correla- In this paper, we have shown that the “bomb-testing” tion measurements represent 1 σ statistical uncertainty, experiment not only provides dramatic evidence of the which has been computed by fitting a Gaussian profile to “weirdness” of , as originally intended the bootstrapped distribution (i.e., the distribution ob- by Elitzur und Vaidman, but also can be recast in a tained by resampling with replacement). Independently rigorous test of the macro-realistic worldview based on from bootstrapping, we also computed the statistical un- the violation of the Leggett-Garg inequality. With our certainties using Monte Carlo resampling, where the sta- realization of the Elitzur-Vaidman experiment, we can tistical errors of position distributions are estimated with refute the macro-realistic assumptions for individual ce- binomial statistics (Clopper-Pearson method). The two sium atoms with a 21 σ statistical confidence. While estimation methods lead to consistent results. While from the point of view of “macroscopicity”, the present Monte Carlo analysis requires invariant statistical prop- Leggett-Garg experiment does not improve on the results erties to be valid, bootstrapping analysis remains valid obtained with quantum walks by Robens et al. [11], our also in the presence of slow drifts of experimental pa- atomic implementation of the Elitzur-Vaidman experi- rameters. The close agreement between the two statisti- ment can be directly extended in the future to superpo- cal analyses indicates that each correlation measurement sition states involving splitting distances on the macro- of K (lasting about 120 min) is performed under constant scopic scale of a millimeter and beyond [43, 44]. More- experimental conditions. 8

Appendix B: Systematic errors both Q(t2) and Q(t3) as +1 for |↑i and as −1 for |↓i. The measured value K = 1.503 ± 0.051 is consistent with the Provided that the experiment is performed under con- quantum mechanical expectation of K = 3/2. stant experimental conditions, systematic errors do not invalidate the result of a Leggett-Garg test. In fact, if we consider the three main mechanisms that bring about Appendix D: Relation with Ramsey contrast systematic fluctuations: (1) Imperfect initialization pre- pares < 1 % of the atoms in the wrong internal state. A Ramsey interference fringe is represented by the However, to derive the Leggett-Garg inequality, a statis- probability p↑ of measuring |↑i as a function of the Ram- tical mixture defining the initial state is perfectly admis- sey phase. Assuming the fringe is centered around the sible. (2) Imperfect reconstruction of the atom’s position average value p↑ = 1/2 (this is true in case of, e.g., pure can be accounted for in terms of a noisy measurement spin dephasing), the contrast can be expressed as apparatus. (3) Spontaneous spin flips due to the finite T1 time can be accounted for in terms of an additional C = p↑,max − p↑,min = 1 − 2p↑,min, (D1) stochastic process, which also contributes to determine the system’s evolution. We estimate that each of these where p↑,max/min is the maximum/minimum value of three mechanisms actually affects the position distribu- the fringe. Because the two Ramsey π/2 pulses in the tion by < 1 %, that is less than the statistical uncertainty. Elitzur-Vaidman experiment are set to have the same phase (Sec. IV), the value of p↑ measured when evaluat- ing hQ(t3)Q(t1)i corresponds to p↑,min. Hence, we obtain Appendix C: Dichotomic choice that the correlation function reads

We verified that our system produces a violation also hQ(t3)Q(t1)i = p↑ − p↓ = p↑ − (1 − p↑) = −C, (D2) with a dichotomic definition of Q(t2). We performed a Ramsey sequence using two subsequent π/3 pulses. In which together with the definition of the LG inequality this case, we set Q(t1) = 1 by preparation and designated in Eq. (1) proves Eq. (3).

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