Quantum mechanics has had enormous success in handling the physics and chemistry of microscopic world in the past century. Basically, the theory system of quantum mechanics involves two parts, the first is the operational methodology and the second is the conceptual basis of physics. The operational methodology of quantum mechanics includes the wave function (which represents quantum states) and its statistical interpretation, Schrödinger’s equation governing the evolution of wave functions and quantum states, and operators representing physical measurements against quantum states. Numerous experiences and practices have confirmed that this methodology works perfectly successfully. However, the conceptual basis of the physics of quantum mechanics has been a controversial issue since its foundation nearly over a century ago.

It is a general consensus that the old and well-known principle of wave-particle duality (WPD) (or more generally the principle of complementarity) stands at the central conceptual core of quantum physics. This principle states that all microscopic particles exhibit mutually exclusive behaviors of two intrinsic attributes of the wave nature and particle nature, namely: they behave either as wave or as particle, depending on how they are observed and measured, but never both.^[1–8] This principle was set up by Bohr and his Copenhagen school colleagues at the start of quantum mechanics, around 1927, right at the first Einstein–Bohr debate on the conceptual basis of the newborn quantum mechanics. Numerous theoretical and experimental studies, either realistic or gedanken, have since been made to test the principle of WPD based on various interferometers, such as Mach–Zehnder interferometer (MZI) and Young’s two-slit interferometer (YI), for either massive particles such as electrons, neutron, and atoms^[9–19] or massless particles such as photons.^[20–38] In some sense, it is a mystery why no one has ever been able to simultaneously observe the wave and particle behavior of microscopic particles, which strictly upholds the WPD principle—no matter how experimental technologies have developed and advanced beyond the era of the Einstein–Bohr debate. In particular, many single particle manipulation technologies, to which several Nobel prizes have been awarded,^[39–41] have been invented, and many advanced optical and atom interferometers have been developed.^[42–44] These new technologies should have stimulated new insights and ideas to shed a new light on this very old fundamental problem of how to explore and correctly understand the principle of WPD and other quantum physics conceptual issues. Nonetheless, little progress has been made in this area, despite these technological advancements.

The physics of why these conventional interferometers do not allow for simultaneous observation of the wave and particle nature of a microscopic particle has reached popular consensus after a long history of intensive and extensive theoretical and experimental discussions, debates, and investigation. Typical examples of investigation include the famous classical gedanken schemes such as Einstein’s recoiling slit^[1] or Feynman’s light microscope,^[2] where it is generally believed that Heisenberg’s position–momentum uncertainty relation works to uphold the principle of WPD. More recently, several advanced gedanken and realistic schemes have been proposed and examined,^{[14–18,20–24]} including Scully’s micromaser-cavity atom interferometer^[20] and the Rempe’s optical-lattice Bragg-grating atom interferometer.^[14] In these relatively new works, most researchers believed that Heisenberg’s position–momentum uncertainty relation does not play an active role but rather another mechanism called quantum entanglement works to uphold the principle of WPD. Despite this fashionable conceptual innovation, controversies have occurred as to what physical mechanism, uncertainty principle versus quantum entanglement, upholds the principle of WPD in these new atom interferometers.^[20–24] However, in spite of this discrepancy, there is one consensus: no one has challenged the principle of WPD or even thought of a new regime of microscopic world going beyond the principle of WPD, namely: a world where one can simultaneously observe the wave and particle nature of a microscopic particle. More recently, some conceptual innovations have been made in regard to the principle of WPD and other quantum physics conceptual issues.^[27–38] For instance, whereas the classical wisdom shows via the so-called Wheeler’s delayed-choice setup that a microscopic object such as a photon behaves either as wave or as particle,^{[3,9–20,25,26]} some years ago several theoretical and experimental studies demonstrated via the so-called quantum delayed-choice architecture that a photon can lie in an intriguing quantum superposition state showing simultaneously partial wave and partial particle behaviors.^[27–30,33] However, these architectures still do not allow for simultaneous observation of full wave and full particle behaviors, and thus still work to uphold the principle of WPD.

There are very rare, little, or essentially no true hard challenges to the principle of WPD in the long history of quantum physics. This situation continues today, even when so many advanced single-particle quantum control and measurement technologies have become a reality to provide powerful experimental means to thoroughly check this fundamental issue of quantum mechanics.^[39–44] There may be several reasons for this situation. First, the foundation of quantum mechanics as settled down by the Copenhagen school pioneers of quantum mechanics is so fine to most contemporary scholars that there is no need to think over this problem or make any change. Second, even if there is conceptual issue in the physics foundation of quantum mechanics to some scholars, the operational methodology of quantum mechanics works so well in the microscopic world that there is no stringent demand to touch this foundation basis. Third, this conceptual issue about the physics foundation of quantum mechanics is so complicated and subtle, and the history is so long and obscure that it has become an extremely difficult task to resolve this long-standing puzzle. As a result, even those people who think that we need to rethink this conceptual issue are reluctant to pour their wisdom and enthusiasm into this old problem, seemingly in vain. Finally, for those who really think hard about this conceptual problem, the tradition of study and investigation is largely via conceptual or in some sense philosophical arguments rather than practical mathematical derivation and solution to specific gedanken or practical interferometer experiments based on strict quantum mechanical formulation. Yet, obviously this tradition has proven to be unable to find a promising route towards a new, deeper, and more delicate understanding of this complicated problem.

Recently, we revisited the working principle of MZI for photons in the framework of Wheeler’s delayed-choice architecture using a different paradigm of study.^[34–36] We investigated the wave function of photon travelling and evolving in the MZI and calculated the interference pattern (quantified by the value of fringe visibility V) and the which-path information (quantified by the value of path distinguishability D) by doing some simple homework of quantum mechanics. The calculation directly discloses fruitful information and allows us to judge whether it is possible, or to what extent one can make simultaneous observation of the wave and particle nature of photons in these schemes of MZI.^[35] Based on the quantitative analysis, a completely innovative scheme of interferometer called the weak-measurement (WM) MZI was proposed and the wave function solution was obtained by doing another quantum mechanics homework.^[36] The result indicates that the power to simultaneously observe the wave and particle nature of photons can be significantly advanced (to a nearly perfect status) by using this new WM-MZI as compared with the conventional delayed-choice scheme of MZI, which has been intensively discussed by Wheeler and others.^[36] In the current work, we aim to follow such a new paradigm of study and revisit the working principle of interferometers for massive particles, particularly atom interferometers, by doing the homework of quantum mechanics, which is now a much more complicated and troublesome task as compared with photon MZI but is still not an intractable task, where care and caution are the most that one needs to yield a correct and reasonable solution. We will propose a weak-measurement atom interferometer called WM-AI and make a standard quantum mechanical analysis on the wave function evolution in this WM-AI, as well as classical MZI and YI based on a full and strict solution to Schrödinger’s equation. This study aims to shed a new light on the principle of WPD for massive particles as atoms in a more solid ground of quantum mechanics operation methodology.

The rest of this paper is arranged as follows. In Section 2, we make a quantum mechanical analysis of several classical AIs working in both the scheme of MZI and YI, such as Einstein’s recoiling slit^[1] or Feynman’s light microscope.^[2] In these AIs, the which-path detector is in direct interaction with the atom center-of-mass to identify its path unanimously and as a result disturb its wave function strongly and uncontrollably. We solve the evolution of the atom wave function input, within, and output from the interferometer, calculate the quantity V and D, discuss the perspective of simultaneous observation of the wave and particle nature of the atom using these interferometers, and analyze how the Heisenberg’s position–momentum uncertainty relation works to uphold the principle of WPD. In Section 3, we systematically investigate the scheme of WM-AI as first proposed by Scully and coworkers, where the which-path detector is a micromaser cavity in direct strong and deterministic interaction with the outer Rydberg quantum state of the atom while in very weak indirect interaction with the atom’s center-of-mass to disturb its wave function nearly negligibly. We make a quantum mechanical analysis over the wave function evolution, we discuss the consequent perspective of this WM-AI to observe the wave and particle nature of the atom, and compare with previous analyses made by Scully and coworkers. We then present an alternative scheme of MZI-type WM-AI based on a new architecture of interference pattern measurement and evaluation. In Section 4, we discuss a more practical scheme of WM-AI as first proposed by Rempe and coworkers, where the which-path detector is a standing-wave light Bragg grating to directly interact with the internal atomic quantum state of electrons and impose an optical force to cause splitting of the atom beam in different directions depending on the internal electronic state to identify the which-path information. Such an indirect interaction with the atom center-of-mass does not disturb its spatial wave function profile uncontrollably and maintain the coherence of the atomic beam travelling in different directions. We then make a quantum mechanical analysis over the atom center-of-mass wave function evolution and discuss the perspective of this WM-AI to observe the wave and particle nature of the atom. We further solve the wave function evolution of the internal electronic quantum states of the atom in both the real space and inner-state space when the atom travels through the interferometer and discuss its role in the atom which-path probe and interference pattern observation. We will point out that the internal quantum state of the atom also exhibit certain space distribution and interference patterns. When one attributes this pattern to the true atom’s center-of-mass interference pattern, ambiguity will take place to drastically complicate and even mislead the interpretation of experimental observation in regard to the test of WPD. In Section 5, we further propose a general scheme of WM-AI that involves a crucial device called polarization beam splitter that allows us to encode the which-path information within the internal quantum state of the atom and other massive particles for evaluating the principle of WPD. Finally, in Section 6, we summarize this paper, echo several crucial discoveries made in this work, compare with previous schemes and analyses of the atom interferometers, and present a perspective of future studies in the research of WPD and other quantum physics conceptual issues.

A schematic configuration of the well-established traditional AI in the scheme of MZI is illustrated in Fig. 1(a). It is similar to the classical MZI for photons, except that all the devices such as beam splitters, mirrors, path delayer, and interference pattern detector now must be replaced by atom optics devices.^[42] This atom interferometer is thus called as MZI-AI. Two path detectors are inserted into the two arms of MZI-AI to probe the which-path information of the atoms. It is well-known that this classical MZI-AI does not allow us to simultaneously observe the wave and particle nature of a microscopic massless and massive particle, no matter how smart it is designed, like in the scheme of Wheeler’s delayed-choice experiment.^[35,36] Another popular traditional scheme to test the principle of WPD is the atomic two-slit YI (hereafter called YI-AI) as illustrated schematically in Fig. 1(b). As a key composite, two path detectors (PDs) are placed around (either right before, or right after, or surrounding) the two slits of YI-AI to probe the which-path information of the atoms. Perhaps due to its simplicity in physics and geometry, this type of YI-AI dominantly becomes the historical gedanken experimental setups made by Einstein and Feynman to discuss the principle of WPD. However, numerous studies have clearly shown that this classical YI-AI also does not allow us to simultaneously observe the wave and particle nature of a microscopic massless and massive particle, no matter how smart it is designed.

Fig. 1.

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Fig. 1. Schematic illustration of classical strong-measurement atom interferometer used to test the WPD for atoms in the scheme of (a) Mach–Zehnder two-arm interferometer and (b) Young’s two-slit interferometer involving the crucial composite device of (c) the strong-measurement path detector. The SM-AI includes historical schemes of AI such as Einstein’s recoiling slit and Feynman’s light microscope. Each SM-PD is designed to probe unanimously the passage of the atom via directly and strongly interacting with the atom center-of-mass, thus inducing an inevitable perturbation to the atom wave function as characterized by a random phase shift δφ of large strength in each channel of AI. The action of SM-PD thus smears out the interference fringe pattern in the output observation screen, upholding the principle of WPD via the mechanism of the Heisenberg’s position–momentum uncertainty relation.

The logic of reasoning and argument on how to uphold the principle of WPD in these historical MZI-AI and YI-AI setups is simple. If no path detector is used, either removing the path detector or turning them off in Figs. 1(a) and 1(b), then one can observe perfect interference pattern of the atom de Broglie wave. In contrast, once the path detector is in position and works to probe and identify unanimously the which-path information of the atom, the interference pattern disappears completely and immediately. The logic is indeed very simple, yet it works based on the fundamental assumption that the path detector, when it works, will inevitably cause an uncontrollable strong perturbation to the motion (more strictly, the wave function) of the atom. Then, if one wishes to shed a new light on deeper understanding the conceptual basis of quantum physics, it is worthwhile to revisit this assumption due to two reasons. The first is to have a more clarified picture of why and how the which-path probe perturb the atomic wave function, thus upholding the principle of WPD. The second and more important one is to see whether one can find a smarter AI setup via deliberate design to go beyond this principle of WPD. The lesson from our recent works on the design and analysis of photonic WM-MZI in Refs. [35] and [36] shows that it is indeed worthwhile to make a thorough analysis on this problem and nothing is impossible when new insights are poured into an issue. Indeed, the design of photonic WM-MZI is a direct inspiring point to simulate the study presented in the current work for atom interferometers: is it also possible to design a workable WM-AI?

The key to answer this question relies on the thorough analysis of the path-detector, as illustrated in Fig. 1(c), and its role in perturbing the motion and wave function of the atom during their mutual interaction for the which-path information to be recorded unanimously. This can only be done by deliberate mathematical and physical analyses, and, more importantly, made strictly in the framework of standard quantum mechanics methodology, the only working physical theory of the microscopic world. Nonetheless, before we do regular quantum mechanics homework of solving Schrödinger’s equation and wave function evolution in these AI setups, we should first clarify several important conceptual issues so that it is relatively easier to reach a consensus after mathematical solution is found. First, we emphasize that the subject concerning us is the WPD of the atom itself, namely, about the which-path information and interference pattern of the atom, more precisely the atomic center-of-mass, rather than anything else, like the internal electrons and nucleus of the atom. This is the conceptual core of all discussions, either theoretical or experimental. Second, in principle, all wave and particle information has and should have already been involved within and can be extracted from the atom wave function following the well-established procedures of quantum mechanical operation to find the calculation and experiment outcome of physical quantities and operators (e.g., the which-path distinguishability D and interference fringe visibility V). Third, the role of the internal electronic quantum states or the which-path detector against the atom should be taken into full account, but their contributions can be and should be handled reasonably and accurately without either underestimate or overestimate, by solving their evolution of quantum state wave function and influence to the atom de Broglie wave again in the framework of quantum mechanics. But great caution should be taken to insure that their wave function evolution is not meddled with the atom wave function either in the theoretical calculation or in the interpretation to experimental data about the wave and particle information, and that the physical and mathematical connection is reasonably interpreted. We will see that many of the previous realistic and gedanken experiments have problems in these issues.

Now we can begin our homework of quantum mechanical analysis. The quantum mechanics for the AI setup are illustrated in Fig. 1, which satisfies the following Schrödinger equation:

For the AI illustrated in Fig. 1, the Hamiltonian can be written as

Nonetheless, despite the difference in the configuration of MZI-AI and YI-AI setups, the most important issue that concerns us the most is the same, namely: how the originally incoming free-propagation atom interacts with the which-path detectors and how this interaction changes or disturbs the wave function of the atom. At this crucial stage of free atom beam passing through the which-path detector, the unperturbed Hamiltonian only contains the momentum energy term and reads , where R is the space coordinate of the atom, or more precisely, the atom center-of-mass. The interaction Hamiltonian H_int (R) describes how the atom interacts with external detectors, either through direct interaction (called direct scheme) with the atom center-of-mass (abbreviated as “acm”, and with coordinate R), called as , or through indirect interaction (called the indirect scheme) with the internal electronic quantum state (abbreviated as “iqs”, and with coordinate r) of the atom, called .

To quantitatively evaluate the principle of WPD in these AIs, we need to examine two relevant quantities. One quantity is the profile of the atom de Broglie wave interference pattern, which can be described quantitatively by the interference fringe visibility V, and the other quantity is the which-path information recording of the atom, which can be described by the path distinguishability D.^[35,36] The estimate and calculation of V and D must be placed in the framework of quantum mechanics via solution of Eqs. (1) and (2) in various geometric and physical configurations of AI for the wave function Ψ(R,t). The key is how to correctly solve Ψ(R,t) and how to rationally interpret its physical meaning in connection with the interference pattern and which-path information. At first glance, this seems to be easy; however, recall of the history proves that this is a very subtle issue.

A close and careful analysis indicates that several famous historical AI setups such as Einstein’s recoiling slit^[1] or Feynman’s light microscope^[2] adopt the direct-scheme which-path detector to probe the path of the atom or other massive particles, similar to the AI setups that are schematically depicted in Fig. 1. In the direct scheme, the Hamiltonian is

In the direct scheme, the which-path detector directly interacts with the atom center-of-mass to probe its path. To unanimously unveil the which-path information, the detector must be able to localize the atom within a sufficiently small size.^[1,2] In Feynman’s light microscope, this size is on the order of light wavelength, so that . According to the Heisenberg’s position–momentum uncertainty principle, the atom position localization will produce a momentum uncertainty on the order of . This momentum uncertainty arises from the momentum kick transferred by the scattered photon, and it is so large that the interference pattern of the atom de Broglie wave will be smeared out completely. Such a well-known qualitative analysis was used by Bohr and his Copenhagen colleagues to safeguard the principle of WPD against Einstein.^[1,2]

These qualitative physics analysis can be placed into a more formal framework of quantum mechanics by tracing back to the solution of wave function. Although the exact form of transmission wave function is hard to know without the knowledge of the exact form of , a quantitative estimate can still be made. Suppose that the incident wave function will become after the atom passes through the which-path detector. By taking another approximation that the which-path detector only induces a loss of the atom beam intensity (via scattering or reflection) described by the coefficient and a phase shift described by the factor of , where is the wave vector change due to the momentum transfer during the atom–detector interaction, and d is the interaction path length. The energy transfer due to the interaction Hamiltonian is on the order of .

In the MZI-AI and YI-AI setups as illustrated in Fig. 1, for the situations where the which-path detectors are removed, the relevant wave function probed by the interference pattern detector is given by

The wave function can also be estimated and calculated when adding the which-path detector. In early schemes like Einstein’s recoiling slit or Feynman’s light microscope, the atom–detector interaction will induce at least a random phase shift and in both paths of interferometer, and the relevant wave function up for detection by the interference-pattern detector is

This direct scheme is more or less a strong-measurement (SM) scheme, where the atom which-path information is acquired by directly localizing the position of the atom center-of-mass to a high precision in space, thus unavoidably causing strong perturbation to the wave function and smearing out the interference of the atom de Broglie wave. However, is this awkward situation of SM-AI setup avoidable? The answer relies on whether one is willing to take an open vision and attitude, look into the warehouse of modern advanced atomic and optical fine measurement technologies and search for better ways to probe the path of the atom in MZI-AI and YI-AI that go beyond the constraint of the conventional wisdom rooted in the Einstein–Bohr debate. Anyhow, 90 years have passed and technologies have evolved that are far beyond the imagination of either Einstein or Bohr, as well as their contemporary pioneering colleagues of quantum physics.

The mixture-type WM-AI illustrated in Fig. 4 has involved two major elements of technology to implement the fundamental purpose of testing the principle of WPD for atoms. The first is the use of microwave pulses to prepare the internal electronic state of the atom into a predesignated quantum state that has definite and unique correlation with the path of motion that the atom takes so that the which-path information is encoded within the internal electronic state. The second is the use of two 1D optical-lattice Bragg gratings to serve as the atomic beam splitter that can selectively separate the incoming atom beam into different directions dependent on the internal electronic state of the atom while maintain the good coherence of the atomic beam by imposing as small as possible a perturbation to the atomic beam wave function. The success of this WM-AI for testing the principle of WPD completely originates from these two technological innovations. After almost 20 years, these detection technologies should have evolved to an even much higher level so that we are now much better at performing again similar experiments to test the principle of WPD.

When we are only concerned with one specific channel of interference pattern observation, such as beams D and E or beams F and G, then the functionality of the mixture-type WM-AI can be modeled and summarized into a simplified version of the MZI-type WM-AI, as schematically illustrated in Fig. 5. This simple simplification and extension would result in a new scheme of WM-AI that can describe a much broader range of AI scheme than the WM-AI, as illustrated in Fig. 4. The crucial composite device in the new scheme of WM-AI is designated as the polarization beam splitter (PBS), which would split the incoming atomic beam into the direct transmission path (path Y) for atoms with the internal electronic quantum state and into the 90° reflection path (path X) for atoms with the internal electronic quantum state . If the incident atom beam is in the electronic state of , after passing through the PBS, it would be split into two beams by 50:50, one along the path Y and in the electronic state , while one along the path X and in the electronic state . Obviously, the PBS in the current MZI-AI is similar in functionality to the well-known true PBS, which is a popular optical device in doing quantum optics and quantum information experiments for photons. In fact, we directly borrow the popular nomenclature in optics and photonics and apply it to atomic physics in the current work. Of course, the concept of PBS can apply to any particles other than atoms with two internal quantum states.

Fig. 5.

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Fig. 5. A general scheme of MZI-type WM-AI involving an internal quantum state correlated beam splitter as the WM-PD. The wave nature of the atoms, the interference pattern, is probed via observation of the transverse interference pattern while the particle nature of the atoms, the which-path information, is probed automatically via the PBS. The incident beam is described by a composite wave function , which involves the usual coherent spatial wave function but dressed with a coherent superposition of internal quantum state and . The PBS selectively transmits the beam component with state along the path Y, while it reflects the beam component with state along the path X, thus automatically offering the which-path information. The transverse interference pattern arising from the two atom beam cross over is recorded by a deliberately designed interference screen that is only sensitive to the atom center-of-mass spatial state , but is insensitive to the internal electronic state and . This simple scheme of PBS WM-AI can also offer a superior power to simultaneously observe the wave and particle nature of the atoms, and can be applicable to other massive particles.

Obviously, the major difference of the WM-AI scheme in Fig. 5 from the WM-AI scheme illustrated in Figs. 2(a) and 3 is that the PBS is used to replace the beam splitter. For an incoming atom beam that is described by the atom–electron composite wave function as , the beam transporting along the path X carries the atom–electron wave function as , while another beam transporting along the path Y carries the atom–electron wave function as . Note that an appropriate design of this PBS should ensure that the internal state selective beam splitting operation of the PBS does not cause any loss to the coherence of the atom beam, much like its optical counterpart does to photons.

According to the standard interpretation of the principle of WPD, with Ref. [14] serving as a representative example, because the internal quantum state ( or ) has provided a natural way to identify the which-path information of each atom, in principle there should be no way to observe the interference pattern of the atom at the IS. In this situation, the principle of WPD is upheld by the entanglement or correlation between the internal quantum state encoded which-path information and the atom motion. Once this which-path information is erased, the interference pattern is recovered.^[14,15,20] These physical arguments can be placed in more mathematical terms. According to Refs. [14], [15], and [20] the interference pattern is described by

Our systematic analyses and discussions in Section 4 show that the physics represented in Eq. (37) is indeed the interference for the internal state rather than the atom de Broglie wave. This equation can be regarded as the convolution of the interference pattern formed by the atom center-of-mass de Broglie wave in the laboratory space and the interference pattern formed by the internal electronic state within the sub-atomic space, with the latter being the modulation factor to the former. Because of the negative modulation factor, the interference fringe pattern of the atom center-of-mass de Broglie wave seemingly disappears. However, this result does not represent the true disappearance of interference fringe for the atom center-of-mass de Broglie wave, which should be described by the following formula:

The working principle of the PBS-type WM-AI as depicted in Fig. 5, the optical-lattice Bragg-grating WM-AI as illustrated in Fig. 4, and the microcavity-type WM-AI as presented in Fig. 2 can be applied not only to atoms but also applicable to other types of massive particles (such as electrons, neutrons, molecules, etc.). These massive particles should have certain internal quantum states (e.g., spins and magnetic moments for electrons and neutrons) that can be probed with high sensitivity by some specific deliberately designed external advanced detectors. In addition, all of the detection operation energy with the internal quantum states should be set to be arbitrarily weak compared with the momentum energy of massive particles under the test of the WPD principle. Simple analyses, similar to those presented in the previous two sections, will automatically point to a simple conclusion that these WM-particle interferometers can arm people with an unprecedented power to simultaneously observe the wave and particle nature of all these massive particles. This would far exceed many historical interferometers governed either by the principle of Heisenberg’s uncertainty relation or by the principle of quantum entanglement. However, whether or not these interferometers work with massless particles such as photons remains an unresolved issue that needs additional systematic studies.

This discussion raises another issue about the relation between the concept of identical particles and the check of WPD. In many classical two-slit or two-arm interferometer experiments of particles such as electrons and neutrons, it is generally agreed that if the particles in two channels are identical, namely, if there is no way to identify their path information, then a perfect interference pattern is observed. However, this condition of an identical particle is not necessary when considering their internal freedoms, such as spins or others. In practical electrons and neutrons interference experiments, the incoming electron or neutron beam upon two-slit or two-arm interferometers does not have identical spins or other internal freedoms because no special means and measures are adopted to guarantee this condition. Strictly speaking, the incident electron and neutron beams are not identical particles in two channels, but all experiments have shown that this non-identical particle situation does not influence the observation of perfect interference pattern. The reason for this is simple: the interference pattern detectors in these interferometers are insensitive to these internal freedoms. In other words, it is possible to only observe the interference pattern of electron and neutron center-of-mass while completely neglecting their internal freedoms. One can either simply neglect these internal freedoms and miss the which-path information (like in classical interferometers ) or deliberately utilize them to serve as powerful which-path detectors and tell which-path the particle goes (like in our current work), although neither these operations nor measures influence the essence of wave nature of the particles.

In summary, we have presented a systematic theoretical analysis in the framework of quantum mechanics upon several historical gedanken or realistic atomic interferometers to test the principle of WPD and complementarity, including Einstein’s recoiling slit, Feynman’s light microscope, Scully’s micromaser-cavity interferometer, and Rempe’s optical-lattice Bragg-grating interferometer. These interferometers are in various operation schemes such as MZI-AI, YI-AI, or MZI-YI mixture-type AI. No matter how smart and delicate they were, they were long believed to strictly uphold the principle of WPD—either by the mechanism of the Heisenberg’s position–momentum uncertainty relation or quantum entanglement. All practical experiments have so far confirmed the orthodox interpretation of quantum physics that WPD is the intrinsic property of all massless and massive microscopic particles and no one is able to simultaneously observe the wave and particle nature of these microscopic objects.

To examine and evaluate these schemes of AI from a new point of view and with a hope to shed some new light on these old and somewhat unresolved fundamental quantum physics problems, we commit to present a reasonable and trustful quantum theoretical analysis strictly in the framework of orthodox quantum mechanics methodology by using Schrödinger’s equation to evaluate the time–space evolution of wave function under a specific model of Hamiltonian. Because a practical AI involves microscopic entities such as the atom center-of-mass and internal electrons bound to the nucleus, and their interaction with various composite devices to implement measurement of the wave and particle behavior of the atom, it is important to take some necessary steps to reasonably solve this fundamental problem. The first step is to construct a reasonable model of Hamiltonian and wave function to correctly describe relevant interaction and measurement processes. The second step is to solve the time–space evolution of various wave functions and identify their contribution to the observation of wave (interference pattern) and particle (which-way information). The third step is to correctly calculate the quantitative index of the interference fringe visibility V and the which-path distinguishability D for atom (more accurately, the atom center-of-mass de Broglie wave) and take caution not to mix with other microscopic entities, such as the internal bound electron quantum states and the which-path detector quantum states. The fourth and final step is to draw a complete and balanced picture about the ultimate power of a specific AI to observe the wave and particle nature of the atoms, and then tell whether it upholds or breaks down the principle of WPD.

Following this logic of reasoning and examination, we have first chosen to handle the historical YI-type AI schemes, such as Einstein’s recoiling slit and Feynman’s light microscope. Because these gedanken schemes of AI use an external position detector that directly interacts with the atom center-of-mass to determine unanimously the which-path information of the atom, they can be categorized as the SM-AI involving an SM-PD. For these AIs, we have introduced a quantum mechanical model that involves a Hamiltonian composed of the atom center-of-mass momentum energy term H_free (R) and the which-path measurement interaction Hamiltonian . The solution of Schrödinger’s equation in a reasonable approximation has introduced a random phase shift δ φ of significant strength to the free-transport wave function of the atom beam for each channel (one slit in YI-AI or one arm in MZI-YI), and the superposition of these two randomly fluctuated wave functions in the observation screen will inevitably result in an interference pattern of null visibility, namely: V = 0 when D = 1. Meanwhile, when no which-path information is probed by external detectors so that disappears, the superposition of the two unperturbed free-transport wave functions in the observation screen will result in an interference pattern of perfect visibility, namely, V = 1 but D = 0. Thus, this type of SM-AI involving SM-PD perfectly echoes with the principle of WPD, which is upheld by the mechanism of the Heisenberg’s position–momentum uncertainty relation in these AIs.

Having noticed that the SM-AI is unable to arm one with the power to simultaneously observe the wave and particle nature of the atom, we turn to another category of AI scheme called WM-AI and see whether these AI setups can allow for a better power to go beyond the limitation of WPD. By introducing WM-PD into the AI setup, we can now probe the which-path information by making the which-path detector only interact with the internal electronic quantum state of the atom (described by Hamiltonian with a very weak energy scale (e.g., microwave photon energy) that is far smaller than the atom’s center-of-mass momentum energy. The detection operation of the which-path information will cause deterministic internal transition of electronic quantum states but will only have a negligible influence on the free-transport atom center-of-mass wave function and also their interference pattern. In principle, these WM-AIs are able to offer us with a much higher capability of simultaneously observing the wave and particle behavior of the atom.

Following such an intuitive insight of physics, we consider two WM-AI . The first, as first analyzed by Scully and coworkers in 1991, is designed to test the principle of WPD for a Rydberg atom with two high-orbital electronic states. The most crucial part of this setup is two high-Q micromaser cavities to serve as the WM-PD, which can probe the passage of an atom via sensing the spontaneously emitting microwave photon originated from down transition of bound electron from the excited state to the ground state. We have considered the influence from various factors from the composite atom–electron–cavity system and built a quantum mechanical model involving all these features. The solution of Schrödinger’s equation to this complicated quantum mechanical problem indicates that in a reasonable approximation, this problem could be separated and simplified into a two-level quantum physics problem because of the weak perturbation nature of the which-path detector against the atom center-of-mass motion via the emission and probe of microwave photon. The first-level physics handles the deterministic probe of microwave photon by the micromaser-cavity which-path detector and deals with the microwave photon interaction with the internal electronic quantum states. The second-level physics deals with the microwave photon emission and detection induced recoiling perturbation to the atom center-of-mass wave function. Our analysis for a practical energy scale of the atom momentum energy and microwave photon energy indicates that the weak perturbation induced random phase shift to the atom wave function in two channels is so small that it only has a very minute degraded influence on the interference pattern and the fringe visibility. As a result, while the WM-PD can unanimously monitor the passage of each atom so that the path distinguishability is D = 1, the interference fringe visibility still maintains a nearly perfect level of . Thus, we see that this WM-AI in both the MZI and YI configurations can achieve and thus offer a much higher power to simultaneously observe the wave and particle nature of the atoms than classical schemes of SM-AI as Einstein’s recoiling slit and Feynman’s light microscope.

The second scheme of WM-AI, which is an MZI-YI mixture-type AI and was first analyzed by Rempe and coworkers in 1998, uses an atom of two ground hyperfine electronic states and one upper excited state for testing the principle of WPD. The crucial parts of this setup are two microwave pulses to prepare the bound electron in a specific hyperfine state, and two 1D optical-lattice Bragg gratings to serve as the first and second atomic beam slitter. Importantly, the two atomic beam splitters can selectively split the incoming atomic beam into different directions dependent on the internal electronic quantum state, thus the which-path information of the atom is automatically encoded within the internal state. Equally importantly, the operations of the which-path information encoding and beam splitting do not cause any considerable perturbation and degradation to the atomic beam coherence. Thus, the combinative action of beam splitters and microwave pulses serves as the WM-PD for the incoming atom beam. We then consider the influence from various factors of the composite interacting system involving the atom, beam splitters, and microwave pulses and build a quantum mechanical model in combination with the solution of Schrödinger’s equation. In addition, under the reasonable two-level physics model, we are able to solve the time–space evolution of both the atom center-of-mass wave function and the internal electronic state bound to atom. The results of such a thorough and systematic quantum mechanical analysis show that the microwave pulse interaction with the internal electronic state only induces a very weak recoiling perturbation and thus a negligible random phase shift to the atom wave function in two channels. Consequently, this operation of which-path information encoding and detection only cause very minute degradation to the interference pattern and the fringe visibility of the atom’s de Broglie wave. This analysis indicates that while the WM-PD can unanimously monitor the passage of each atom so that the path distinguishability is D = 1, the interference fringe visibility still maintains a nearly perfect level of . Therefore, we see that this MZI-YI mixture-type WM-AI can also reach an index of and thus offer a superior power to simultaneously observe the wave and particle nature of the atoms.

Our quantum mechanical analyses clearly show that the well-established mechanism of the Heisenberg’s position–momentum uncertainty relation governing the SM-AI as Einstein’s recoiling slit and Feynman’s light microscope does not play an active role in these two WM-AI schemes. This observation is in accordance with previous theoretical and experimental studies by the Scully group and the Rempe group. However, there exists a drastic difference between our current quantum mechanical analysis and the studies made by these two groups, who claimed that in their WM-AI schemes the principle of WPD is still strictly upheld via the mechanism of quantum entanglement. To clarify the role of quantum entanglement, we investigate the space–time evolution of the internal state of the atom as well as the combined atom-detector system, which allows us to identify the microscopic origin of quantum entanglement, whose role has been emphasized in numerous previous literature. Our quantum mechanical analysis shows that if one only focuses on the atom center-of-mass de Broglie wave and looks at its interference pattern formation following the formula as , as should be done for the aim to test the principle of WPD for the atom, then one will also find that the mechanism of quantum entanglement does not play an active role in these two WM-AI . Nonetheless, if one does not take caution and make reasonable constraint to limit the interpretation only to the atom center-of-mass as the single object to test the WPD but rather expands with arbitrariness their perspective to the internal electronic states or the detector quantum states as the object to test the WPD following the formula as , then one will naturally introduce the mechanism of quantum entanglement, either between the atom center-of-mass which-path information with the which-path micromaser-cavity detector in Scully’s scheme or between the atom center-of-mass which-path information with the internal electronic state which-path detector in Rempe’s scheme. By doing these operations and manipulations over calculation and experiment, the principle of WPD is still upheld. However, from the point of view of rigorous scholarship, this route of reasoning and argument is out of logic to a large extent and will inevitably lead to an unphysical artifact in understanding the fundamental physics problem of WPD.

The critical key towards the superior power of the proposed and analyzed WM-AI setups in simultaneous observation of the wave and particle nature of the atom largely lies in the design and introduction of the WM-PD. This instrument deliberately interacts with the atomic internal freedoms to unanimously probe the which-path information of the atom, while only inducing negligible perturbation of the atomic center-of-mass motion. Another instrument directly interacting with the atomic center-of-mass while insensitive to the internal freedoms is used to monitor the atomic center-of-mass interference pattern. These two sets of instruments work separately and collectively to nearly perfectly enable simultaneous observation of both the wave and the particle nature of the particles, while avoiding the complicated role of Heisenberg’s uncertainty relation and quantum entanglement. The operation of WM-PD does not break down the coherent superposition of the atom center-of-mass transport state and turns it into an incoherent mixture state, thus we can eventually maintain a good atom-of-mass interference pattern. However, the price is that the operation will destroy the coherent superposition of the WM-PD quantum state (in either the internal electron of the atom or in the path detector), but fortunately this will not influence the atom center-of-mass quantum state and its interference pattern. Note that the current WM-PD has a very different operation principle than classical nondemolition measurement scheme.^[42]

Based on these physics insights and quantum mechanical analyses that were found using this technique, we are able to further propose a fruitful variety of WM-AI schemes that can help to elucidate the puzzling physics of WPD for atoms. Because the key of this class of new concept WM-AI is to use two sets of detection instrument to probe the wave (interference pattern) and particle (which-path information) behaviors separately, simultaneously, and without mutual repulsion and disturbance in different space–time domains, the principle and infrastructure of WM-AI can be straightforwardly extended to design and construct other types of weak-measurement particle interferometers working on different energy scales, different particles (electrons, neutrons, protons, atoms, molecules, etc.), difference internal states (electron, nuclear spin, etc.), and different path detectors. We hope that the new physics insights and analytical methodologies presented in this work will help to shed a new light on the exploration, understanding, and elucidation of the principle of WPD for microscopic objects and will also place quantum physics on a more solid conceptual ground.