That conclusion quickly led to a preliminary design of a cavity around the vacuum chamber supporting the molecular beam. Figure 8 shows the second generation of such a cavity. An essential difference between a molecular-beam laser and a heat pipe relates to the relaxation process of population reaching the lower laser level (level 3 in Fig. 7). Such relaxation, removing continuously population from the lower laser level, is needed to allow continuous laser operation. In a heat pipe relaxation is dominated by collisions. In the molecular-beam, however, it is the directional flow that continuously transports new molecules in low-lying thermally populated levels (level 1 in Fig. 7) into the region of the laser cavity and at the same time removes molecules in level 3 from the cavity. These latter molecules do not experience further collisions. They remain in the highly vibrationally excited level and are available for collision experiments. Rotating the birefringent filter allows choosing which vibrational level is populated.

It was a very crucial moment when Uli Hefter and Pat Jones tried for the first time to get the molecular-beam laser going. It rarely happens that such experiments work at the first try. In this case it did happen on July 28, 1981. The cavity was aligned

**Fig. 8** The optical laser cavity built around the molecular beam in a vacuum chamber. IP is the pump laser and IMB the molecular-beam laser radiation. Only a small fraction of the pump laser is absorbed in the beam. Therefore, the pump laser exits again through the mirror M3. BF is the birefringent (a tunable optical) filter that controls to which vibrational level v- in the electronic ground state lasing is possible

and the molecular beam was operating. Upon turning on the pump laser lasing was immediately observed. The laser operated for about 15 min and then it went off. It took us 6 months (!!) to get it back into operation. This task was accomplished by Uli Gaubatz, who had joined the group shortly after the first operation of the laser was observed. Without these crucial 15 min, proving that the concept works, we would have probably given up after a few months and, most likely, the successful path towards realizing STIRAP would have been abandoned.

It turned out that one of the problems was the alkali deposit on the intracavity windows (see Figs. 8 and 9), separating the vacuum region from the outside. After several trial-and-error modifications, the problems were overcome. One of the measures was the heating of the windows to high temperatures what required the use of metal rings for vacuum-tight sealing. Furthermore, the installation of small pipes that directed a flow of Argon atoms away from the windows reduced the rate of deposit of alkali atoms and molecules on the windows. With these measures the molecular-beam laser could be routinely operated and in 1986 we set out to determine the properties of the transfer process, in particular its efficiency and selectivity [26].

The delay of a couple of years between the first demonstration of successful laser operation and the attempt to use such a laser for quantitative population transfer was in part due to the fact that no apparatus was available for such experiments. Students needed to first complete their ongoing experiments. The time was used to further explore the physics of the molecular-beam laser [27, 28]. Later we also built a molecular-beam laser with iodine molecules using a slit-nozzle expansion. For the latter system an optical pump-power threshold for starting the laser operation as low as 250 nW [29] was demonstrated.

**Fig. 9** View through a window into the gain region of the molecular beam laser. The glowing red part is the molecular beam source with a cooler (darker) thermocouple attached. The thin yellow trace marks the pump laser beam which excites also background molecules. The thicker yellow region is seen because of radiation diffusion between particles along streamlines of the molecular beam. The gain region is the crossing between the two yellow traces

#### **5 The Breakthrough**

Figure 10 explains how the molecular-beam laser induced transfer efficiency was

**Fig. 10** Schematic of the set up (left part) for the calibration of the transfer efficiency induced by the molecular beam laser from level 1 (v-- = 0) to level 3 (v-- = 31). The element B is used to block the cavity. The related level scheme is shown in the right. Levels other than 1 and 3 that can also be reached from level 2 by spontaneous emission are summarized as level 4. The transfer into level 3 is probed by laser-induced fluorescence from level 5

**Fig. 11** The right part shows the P- and S- laser with coincident axes, as realized in the molecularbeam laser arrangement of Fig. 8. The left part shows the S-before-P arrangement of the lasers as also seen in the lower part of Fig. 1. The position zA marks the location near which the pump laser would start pumping molecules out of level 1

determined. The pump laser excited molecules from the level v-- = 0 to v- = 17 in the B-state of Na2, followed by spontaneous emission. The population in the target level of interest, e.g. v-- = 31, was probed by laser-induced fluorescence further downstream with the cavity blocked. Because the optical transition probabilities are known and after confirming that the population in level 1 was entirely depleted, the transfer efficiency of the population reaching level v-- = 31 by spontaneous emission could be determined. Unblocking the cavity allowed the molecular-beam laser to operate and the population in the target level increased. The increase of the population in relation to the known transfer efficiency by spontaneous emission yielded the beam-laser induced transfer efficiency. It was found [26] that the transfer efficiency was as large as 75% (larger than any other scheme would allow) but it was still far from the goal of ≈100%. The question thus was: what limits the transfer efficiency to about 75%?

The solution, which paved the final segment on the path to STIRAP, was surprisingly simple, as shown in Fig. 11. Results from earlier work on the consequences of optical pumping in two-step photoionization [30] led the way. The right part of Fig. 11 shows the profiles of the pump laser and the molecular-beam laser which appears after the pump laser is switched on. The axes of the two laser fields coincide. The molecules travel from left to right. As soon as the molecules reach the wings of the pump laser profile (P) they are efficiently pumped to the upper laser level. At the location zA, however, the local molecular-beam laser intensity, which is supported by molecules that had already crossed the cavity, is still weak. Therefore, stimulated emission induced by the radiation field S that is supposed to populate level 3 (the target level) cannot yet compete with spontaneous emission. In fact, the transit time of the molecules across the cavity is about one order of magnitude longer than the radiative lifetime in level 2, the upper laser level. Therefore, only a fraction of the relevant molecules reaches the axis of the cavity where the beam laser is sufficiently strong to allow stimulated emission to compete successfully with spontaneous processes.

The final conclusion was again straightforward. Despite the significant effort invested to realize it, the molecular-beam laser approach had to be abandoned. In addition to the pump laser, an external laser for driving the stimulated emission process was needed, with its axis placed upstream of the axis of the pump laser, as shown in the left part of Fig. 11. As soon as the molecules enter the wings of the pump laser profile, they should be exposed to the maximum possible intensity of the S-Laser to optimize the chance for successful competition of stimulated emission with spontaneous processes. The informed reader realizes that the above argument, with optical pumping processes in mind, doesn't yet properly catch an essential part of STIRAP physics. It provided, however, the rationale for placing the axis of the S-beam upstream of the one for the P-beam, with a suitable overlap between the two.

It was very fortunate that Piotr Rudecki, a visiting scientist from Torun/Poland, arrived in the fall of 1987 a few days after I had come to the conclusion that a S-before-P arrangement was needed. The experiment which he wanted to join was not ready yet. However, he had some experience in modeling radiative processes. Therefore, I asked him to take our code for simulating the molecular-beam laser, which we had developed with the help of Wellegehausen, and modify it in accordance with the new geometry. The hope was to quantitatively understand the benefit of the S-before-P configuration from results of simulation studies. I certainly did expect a transfer efficiency of more than 75%.

About 10 days later, Rudecki presented his results: nearly 100% transfer. Expecting something near 90%, my reaction was: "hard to believe, please check for errors". A few days later, Piotr joined the group for the traditional after-lunch coffee-and-discussion meeting at a round table near the lab, presented the results of new calculations (see Fig. 12) and stated firmly "no errors—100% is correct". I

clearly recall my prompt reaction: "congratulation—this result will be a big bang in the community and will determine what we do in the lab for the next 10 years".

A few months later, in early 1988, we managed to demonstrate that a very high population transfer can indeed be realized in the S-before-P configuration. In a first short publication [31], we showed results, but the theoretical basis had not yet been clearly sorted out. This gap was closed by the May 1, 1990 publication in J. Chem. Phys. [1], with Uli Gaubatz being the leading graduate student who had noted the close connection of what we did with the work by Claude Cohen-Tannoudji [32]. Prior to submitting the manuscript of paper [1], a detailed discussion of the adiabatic condition was published [2]. The paper [1] remains a standard reference for STIRAP and indeed, all the basic features that make STIRAP unique are experimentally documented and theoretically properly analyzed in that work. In that paper, also the acronym STIRAP was introduced. Regarding the latter, I had learned earlier that it is important to give a name to a new technique, method or process before others will do it. After the first rough draft of the paper was ready, I told my students that instead of joining the after-lunch meeting, I will spend one hour or two in my home-office and come back with a suggestion for an acronym. At home, I wrote down all physical processes or phenomena that had some connection to the transfer process, looked at the initial letter or letters and wrote them down in different orders. The criteria were: the acronym should have no more than two syllables and pronunciation should be easy. I returned to the lab with the suggestion STIRAP, which was accepted by all involved.

We defined the publication date or ref [1], May 1st, 1990 the "birthday" of STIRAP and celebrated its 25th anniversary in September 2015 with a well-attended and well-received international conference in Kaiserslautern [33].

#### **6 Some STIRAP Highlights that Followed**

The most recent compilation of some highlights regarding STIRAP applications can be found under [7]. Here a few topics are listed, with only one or two references given:

The application of the STIRAP-approach to acoustic waves [48] is a particularly nice example for how far reaching the concept is. It also underlines that STIRAP is not a purely quantum mechanical process.

There are also a number of proposals with a detailed analysis of the feasibility of STIRAP applications, such as

The example (SAP, in earlier years also called CTAP—coherent transport by adiabatic passage) would be a particularly intriguing demonstration of the STIRAPconcept: Consider three traps A, B and C in a linear arrangement and close proximity. Each trap is able to hold a single particle. Assume that one particle is initially in trap A while B and C are empty. When the coupling between the traps is properly varied as required for STIRAP (e.g. by lowering the barriers between them while keeping the quantum states in the traps in resonance) the particle is removed from A and appears in C without establishing a significant transient population in B.

#### **7 Final Remarks**

Following the original publication in 1990, the STIRAP concept has been systematically developed, both experimentally and theoretically, in Kaiserslautern (with too many publications to be all listed), also for applications beyond the canonical threelevel system. That work benefitted greatly from the contributions of the visiting scientists Bruce W. Shore, Leonid P. Yatsenko, Razmik Unanyan, Matthew Fewell, and Nikolay V. Vitanov. Experimental progress was achieved through the dedicated work of many excellent students: Axel Kuhn, Stefan Schiemann, Jürgen Martin, Thomas Halfmann, Heiko Theuer, and Frank Vewinger to name at least some. The postdocs George Coulston, Horst-Günter Rubahn, and Stéphane Guérin also contributed significantly to the successful developments.

At the occasion of my first public presentation of the concept in the colloquium at JILA/Boulder on March 1st, 1990 (i.e. prior to the publication of [1]) I had the chance to discuss STIRAP with Peter Zoller. It was the follow-up theoretical and experimental work from the groups of Peter Zoller and Bill Phillips [59, 60], respectively, on matter-wave mirrors and beam splitters and the experimental work in the group of Steve Chu on atom interferometry [39], that made STIRAP quickly known in the AMO community. Nevertheless, it took more than 10 years after the original publication [1] before STIRAP was used in many laboratories and in different areas of research.

Several proposals did appear in the literature discussing the prospects for applying STIRAP to poly-atomic molecules. However, nearly all of them are based on model systems that did not adequately include relevant properties, such as the realistically modelled (detrimental) high level density. The consequences of the inclusion of these properties are carefully analyzed in an extensive simulation study [61] involving the HCN molecule. To the best of my knowledge, SO2 is still the largest, or most complex, molecule to which STIRAP has been successfully applied [62] in an experiment.

As explained in Sect. 3, the STIRAP-concept was developed with reaction dynamics experiments involving vibrationally excited molecules in mind. One early experiment of that kind has been completed in Kaiserslautern (Na2(v--) + Cl → NaCl + Na\* [63]). Using STIRAP was also essential in the recent observation of bimolecular reactions at ultracold temperatures [64]. However, the initial motivation had reactions of relevance to atmospheric chemistry in mind. Such an application still awaits its realization. Because of recent developments of coherent radiation sources for the region λ < 200 nm, this situation may change soon. The related requirements for the molecules H2, N2, O2, and OH are discussed in the appendix of [5].

#### **References**

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#### **Chapter 20 Manipulation and Control of Molecular Beams: The Development of the Stark-Decelerator**

#### **Gerard Meijer**

**Abstract** State-selective manipulation of beams of atoms and molecules with electric and magnetic fields has been crucial for the success of the field of molecular beams. Originally, this manipulation only involved the *transverse* motion. In this Chapter, the development of the Stark-decelerator, that allows to also manipulate and control the *longitudinal* motion of molecules in a beam, is presented.

#### **1 Introduction**

"*Born in leaks, the original sin of vacuum technology, molecular beams are collimated wisps of molecules traversing the chambered void that is their theatre [...]. On stage for only milliseconds between their entrances and exits, they have captivated an ever growing audience by the variety and range of their repertoire*". This is how John B. Fenn affectionately phrased it over 30 years ago, when he reflected on the long and rich history of molecular beams in his foreword to one of the classic books on this subject [1]. He could not have foreseen the spectacular leap forward that the level of control over molecular beams would take. In particular, methods that have been developed since then to slow down and store molecular beams – thereby stretching the duration of their performance on stage by orders of magnitude – have made whole new classes of experiments possible.

The motion of neutral molecules in a beam can be manipulated and controlled with inhomogeneous electric and magnetic fields. Static fields can be used to deflect or focus molecules, whereas time-varying fields can be used to decelerate or accelerate beams of molecules to any desired velocity. In this paper we present an historical overview, emphasizing the important role of molecular beam deflection and focusing experiments in the development and testing of quantum mechanics. We describe the original attempts and the successful implementation of schemes to decelerate and accelerate molecular beams with electric fields, that is, the development of the Starkdecelerator. The various elements, using electric as well as magnetic fields, that have

G. Meijer (B)

Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2021

B. Friedrich and H. Schmidt-Böcking (eds.), *Molecular Beams in Physics and Chemistry*, https://doi.org/10.1007/978-3-030-63963-1\_20

been developed for the manipulation and control of molecular beams since the first successful demonstration of the Stark-decelerator in 1998, have resulted in setups in which the molecules can be stored in stationary traps or injected in a molecular storage ring or synchrotron, for instance. Novel crossed-beam scattering studies at low collision energies, high-resolution spectroscopy studies on trapped or slow molecules and lifetime measurements of trapped molecules have become possible [2].

#### **2 Deflection and Focusing of Molecular Beams**

Atomic and molecular beams have played central roles in many experiments in physics and chemistry – from seminal tests of fundamental aspects of quantum mechanics to molecular reaction dynamics – and have found a wide range of applications [1]. Nowadays, sophisticated laser-based methods exist to perform sensitive and quantum state selective detection of the atoms and molecules in the beams. In the early days, such detection methods were lacking and the particles in the beam were detected, for instance, by a "hot wire" (Langmuir-Taylor) detector, by electronimpact ionization or by deposition and *ex-situ* investigation of the particles on a substrate at the end of the beam-machine. To achieve quantum state selectivity in the overall detection process, these methods were combined with inhomogeneous electric and magnetic field sections to influence the trajectories of the particles on their way to the detector.

The first paper discussing the degree of deflection for a beam of polar molecules passing through an inhomogeneous electric field was submitted almost a century ago, at the end of July 1921, to the *Zeitschrift für Physik* [3]. The paper was written by Hartmut Kallmann and Fritz Reiche, coworkers of Fritz Haber at the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry in Berlin, the present Fritz-Haber-Institut der Max-Planck-Gesellschaft. Kallmann and Reiche write in their paper that they performed their analysis in support of experiments that "are ongoing at the institute" to determine whether the dipole moment is a property of an individual molecule or whether this is only induced in the molecule when it is in close proximity to other molecules, an issue that was intensively debated at that time. By passing a molecular beam, that is, a dilute but highly collimated sample of molecules, through an inhomogeneous electric field, they argued, it should be possible to monitor the deflection of the molecules and to thus determine the value of their dipole moment – provided they have one. In the introduction of their paper, they discuss in general terms the forces on a moving dipolar molecule due to inhomogeneous electric fields as well as due to inhomogeneous magnetic fields. In their analysis, however, they restrict themselves to the electric field case, and assume that the dipole moment of the diatomic molecule is perpendicular to the angular momentum vector, i.e., that the component of angular momentum along the internuclear axis is zero.

Shortly after its submission, the paper by Kallmann and Reiche came to the eyes of Otto Stern, pressing him to write up the theory behind an experiment that – as he wrote – "he had been involved in for some time with his colleague Walther Gerlach". Stern submitted his paper only about one month later, at the end of August 1921, to the same journal. In a footnote to his paper, he explicitly acknowledges that the reason for publishing his paper is the upcoming paper of Kallmann and Reiche, and concludes that both papers are nicely complementary as his paper "discusses the case in which the dipole moment is parallel to the angular momentum vector, as is generally the case for magnetic atoms" [4]. In his paper, Stern describes a method to experimentally test space quantization via measuring the deflection of a beam of atoms with a magnetic moment when moving through an inhomogeneous magnetic field [4]. No further account on the early electric deflection experiment in Berlin is to be found in the literature or in the archives, while Stern and Gerlach performed their famous experiment within one year [5], in February of 1922. Electric deflection of a beam of polar molecules was first demonstrated by Erwin Wrede, a graduate student of Stern, several years later in Hamburg [6]. It is interesting to note that up to our re-discovery of the article by Kallmann and Reiche in 2009, this article went largely unnoticed, being cited only seven times; the important mentioning of the article by Otto Stern in a footnote in his paper could not include the final reference yet. Since then, it has been cited more than once per year.

All the original experimental geometries were devised to create strong magnetic or electric field gradients to efficiently deflect particles from the beam axis. In 1939, Isidor Rabi introduced the molecular beam magnetic resonance method by using two magnets in succession to produce inhomogeneous magnetic fields of oppositely directed gradients. In this set-up the deflection of particles caused by the first magnet is compensated by the second magnet, such that the particles are directed on a sigmoid path to the detector. A transition to "other states of space quantization" induced in between the magnet sections can be detected via the resulting reduction of the detector signal. This provided a new method to accurately measure nuclear or other magnetic moments [7]. Later, both magnetic [8, 9] and electric [10] field geometries were designed to focus particles in selected quantum states onto the detector. An electrostatic quadrupole focuser, i.e., an arrangement of four cylindrical electrodes with alternating positive and negative voltages, was used to couple a beam of ammonia molecules into a microwave cavity. Such an electrostatic quadrupole lens focuses ammonia molecules that are in the so-called low-field seeking, upper level of the inversion doublet while it simultaneously defocuses those that are in the lower, high-field seeking, level. The inverted population distribution of the ammonia molecules that is thus produced in the microwave cavity led to the invention of the maser by Gordon, Zeiger and Townes in 1954–1955 [11, 12]. Apart from the spectacular observation of the amplification of microwaves by stimulated emission, these focusing elements more generally enabled the recording, with high resolution and good sensitivity, of microwave spectra in a molecular beam. By using several multipole focusers in succession, with interaction regions with electromagnetic radiation in between, versatile set-ups to unravel the quantum structure of atoms and molecules were developed. In scattering experiments, multipole focusers were exploited to study steric effects, that is, to study how the orientation of an attacking molecule affects its reactivity [13]. Variants of the molecular beam resonance methods as well as scattering machines that employed state-selectors were implemented in many laboratories, and have yielded a wealth of detailed information on stable molecules, radicals and molecular complexes, thereby contributing enormously to our present understanding of intra- and inter-molecular forces.

#### **3 Early Attempts to Decelerate or Accelerate Molecular Beams**

The state-selective manipulation of beams of atoms and molecules with electric and magnetic fields is thus about as old as the field of atomic and molecular beams itself, and it actually has been crucial for the success of the latter field. In his autobiography, Norman Ramsey, who himself later invented the separated oscillatory fields method and wrote a very influential book on molecular beam methods [14], recalls that Rabi was rather discouraged about the future of molecular beam research when he arrived in Rabi's lab in 1937, and that this discouragement only vanished when Rabi invented the molecular beam magnetic resonance method [15]. However, even though the manipulation of beams of molecules with external fields has been used extensively and with great success in the past, this manipulation exclusively involved the *transverse* motion of the molecules.

When the velocity distribution in a molecular beam is rather broad, the stateselective deflection fields can be used to provide some velocity selection. Offset or angled molecular beam geometries have been used, in which deflection fields cause only the slow (fast) atoms or molecules to obtain sufficiently large (small) deflections to pass through the apparatus and to reach the detector, for instance. This approach has been attempted to selectively load slow ammonia molecules in a microwave cavity to produce a maser with an ultra-narrow linewidth [16]. At Bell telephone laboratories an electrostatic parabolic reflector was designed to selectively couple slow ammonia molecules in the microwave cavity, that is, after deflecting them by 180 degrees [17]. These approaches suffered from the deficiency that, as stated then, "it is generally known that the velocity distribution of molecules emanating from a hole in a box is not Maxwellian, but departs from it by having fewer low velocity molecules", possibly caused by collisions with fast molecules "from behind" [17].

At the end of the nineteen-fifties, alternative approaches, with electric fields designed such as to create "multiple retardation barriers" have been proposed to actively manipulate the *longitudinal* motion of molecules in a beam, that is, to slow down ammonia molecules [18]. A rather compact experimental setup consisting out of a source chamber, a deceleration chamber and a slow-molecule deflection and detection chamber was constructed for exactly this purpose in the physics department of MIT, under the supervision of John G. King [19]. The approximately 20 cm long decelerator consisted of a linear array of ten parallel plate capacitors, capable of maintaining a voltage difference of 30 kV across 1 mm plate separation. Ammonia molecules in the low-field seeking, upper level of the *J* = *K* = 1 inversion doublet lose kinetic energy when entering the high electric field region of the capacitor. When the electric field is slowly turned to zero while the molecules are in the homogeneous electric field inside the capacitor, the molecules do not regain the lost kinetic energy when exiting the capacitor and the process can be repeated. The experiments were performed using continuous beams and the same high voltage was applied to the whole array of electrodes in the form of a sine wave with a fixed frequency of 6 kHz. Therefore, the distances between the adjacent parallel plate capacitors as well as their lengths needed to be made such that it takes the ever slower molecules always exactly the same amount of time to reach the next stage, i.e., these distances and lengths needed to gradually decrease along the molecular beam. The machine was designed to slow down ammonia molecules from an initial speed of 200 m/s to a final speed of 35 m/s, selectively detecting the slowed molecules at the exit. The project suffered from the same deficiency as mentioned earlier, namely that there were not enough ammonia molecules at the initial speed of 200 m/s to yield a detectable signal. The work is described in detail in the Ph.D. thesis of Robert Golub [20], but no further publication has resulted from this work.

In the physical chemistry community the experimental efforts of Lennard Wharton, to demonstrate electric field *acceleration* of a molecular beam, are much better known. In the mid nineteen-sixties, at the University of Chicago, he constructed a molecular beam machine, containing a thirty-three foot long accelerator. The accelerator consisted of an array of 600 acceleration stages intended to increase the kinetic energy of LiF molecules in high-field seeking states from 0.2 eV to 2.0 eV, that is, speeding the LiF molecules up from 1200 m/s to 3800 m/s, with the aim to use these high energy beams for reactive scattering studies. Each acceleration stage consisted of two hemispherically ended rods with a diameter of 0.5 mm, spaced 0.5 mm apart. The beam was transversely kept together using additional alternate-gradient lenses. A popular scientific account of this work, together with a schematic drawing of the acceleration principle and a photograph of the "Chemical Accelerator", appeared in *Scientific American* in 1968 [21]. The photograph of the about eleven meter long machine is reproduced in Fig. 1. Also in this case, continuous molecular beams were used and the high voltage was applied as a sine wave with a fixed frequency to the array of electrodes, implying that the adjacent electric field stages needed to be put ever further apart to compensate for the molecules being ever faster, explaining the length of the molecular beam machine. An excellent paper, in which the focusing of beams of polar molecules in high-field seeking states was theoretically analyzed, resulted from this work [22]. The acceleration experiment was not successful, however, not only because the alignment of the array of electrodes is very critical in an alternate-gradient setup but also due to an overly optimistic view on the magnitude of the electric fields that could be stably obtained when designing the accelerator module. This is what the Ph.D. student who was working on this project, Edward A. Bromberg, concluded in his Ph.D. thesis in which he summarised that "it has not been possible to show either that particles have been accelerated, or that neutral particles cannot be accelerated" [23]. Both, the deceleration experiments of John King and the acceleration experiments of Lennard Wharton were not continued after the Ph.D. students completed their theses. Whereas interest in slow molecules as a maser

**Fig. 1** Photograph of the "Chemical Accelerator" built at the University of Chicago by Lennard Wharton, to produce molecular beams with an energy of about two electron volts. The photographs show the long dipole accelerator from both ends. Molecules produced by heating in an oven (left) are accelerated down the long tube by electric fields to the reaction chamber (right). Reproduced from [21]

medium declined owing to the invention of the laser, the molecular beam accelerator was made obsolete by the seminal demonstration of John Fenn and co-workers of gas dynamic acceleration of heavy species in seeded supersonic He and H2 beams [24]. Because of these unsuccessful early attempts, researchers who wanted to pursue molecular beam deceleration with electric or magnetic fields for high-resolution spectroscopy and metrology in the following decades, had difficulties getting this financed [25].

#### **4 Deceleration of CO (a3***-***) Molecules with Electric Fields**

I studied physics at the Katholieke Universiteit Nijmegen in Nijmegen, The Netherlands – since 2004 renamed into Radboud University – and performed my undergraduate and Ph.D. research in the Atomic and Molecular Physics department, headed by Antoni Dymanus and Jörg Reuss. In this department, I was exposed from day one, that is, from February 1984 on, to a wide variety of molecular beam machines; electrostatic and magnetic state-selectors and focusing elements were used throughout. The magnetic properties and the molecular quadrupole tensor of the water molecule [26] as well as the electric and magnetic properties of carbon monoxide [27], for instance, were measured with those machines by Antoni Dymanus and his students already in the nineteen-seventies and are still the standards in the field. During my time as Ph.D. student, Jörg Reuss wrote the chapter entitled "State-selection by non-optical methods" as contribution from Nijmegen to the earlier mentioned classic book on "Atomic and Molecular beam methods" [1].

After post-doctoral periods abroad, I was offered the opportunity to start up my own research program in Nijmegen, in what was then just renamed the Molecular and Laser Physics department, and I became the successor of Jörg Reuss per January 1995. With my Ph.D. student Rienk Jongma, we performed two-dimensional imaging of metastable CO (a3-) molecules, to study with modern tools the phenomenon of mass-focusing in a seeded molecular beam as well as the performance of an electrostatic hexapole focuser [28]. We showed that metastable CO molecules, prepared with a pulsed laser at a well-defined time, at a well-defined position and in a single quantum state, are ideally suited to study velocity distributions and spatial distributions in molecular beams in general, and to study electric and magnetic field manipulation of molecular beams in particular. The metastable CO molecules live several milliseconds [29], long enough for molecular beam experiments, and with about 6 eV internal energy they can be efficiently detected – temporally and spatially resolved – via the Auger electrons that are released when they impinge on a surface. Whereas the Stark shift of the rotational levels of CO in its electronic ground state is very small, the shift of the *J* = 1 rotational level in the metastable a3-<sup>1</sup> state is on the order of 1 cm−<sup>1</sup> in electric fields of about 100 kV/cm. When the CO molecules are state-selectively prepared in the metastable state with a pulsed laser *inside* a strong electric field, the interaction with the electric field is suddenly "switched on". This led us to propose a scheme for confining metastable CO molecules in stable "planetary" orbits in an appropriately shaped electrostatic trap [30]. As this confinement scheme would only work for CO molecules with speeds below about 22 m/s, we discussed how such slow molecules could be obtained, and in the final section of our paper we mentioned the possibility to slow down polar molecules with electric fields [30]. The latter was critically commented upon by the (anonymous) referee of our paper, which motivated us to try to experimentally demonstrate that when CO (a3-) molecules are prepared in a high-field seeking state *inside* a large electric field, the molecules will indeed slow down when leaving the high-field region, loosing an amount of kinetic energy that is identical to the Stark-shift of the levels. Even when seeded in Xe, however, the kinetic energy of the CO molecules in our beam was too large, to be able to unambiguously detect the deceleration effect of a single parallel-plate electric field stage of 140 kV/cm. It was clear that we would need either a considerably higher electric field or more electric field stages. When I informed Jörg Reuss in the spring of 1997 about these experiments, he told me that he vaguely remembered an experiment along similar lines by "Wharton in Chicago", but he did not know any further details. This was the first time I learned about Lennard Wharton's earlier experiments, which was comforting as I had been wondering for quite a while already why acceleration or deceleration of molecular beams with electric fields had never been demonstrated – at least now I knew it had been tried.

In December 1997, Rick Bethlem started in my department as Ph.D. student. He was hired on a project to study the physical properties of endohedral fullerenes, which would include the synthesis of small molecules like CO encapsulated in a fullerene cage, and the subsequent spectroscopic characterization of the motion of such a "dumbbell in a box". In one of our first meetings, prior to his start as Ph.D. student, I informed him about our proposed scheme to confine slow CO molecules in stable orbits and about our attempt to demonstrate molecular beam deceleration with a single electric field stage. Rick was fascinated by this topic, that actually fitted better to his background and interest, and we decided that he would switch research projects and start on the deceleration of metastable CO molecules with electric fields. At that time, I was also actively involved with other members in my department in experiments with an infrared free electron laser [31] and over lunch we had discussions about the operation principle of a linear accelerator (LINAC) for electrons, and whether and how the equivalent device for neutral, polar molecules could be constructed. We discussed in particular whether, as proposed by Basov [18], the "multiple retardation barriers" would need to be combined with electromagnetic radiation fields to be able to repeat the deceleration process, or that one could also rapidly switch the fields on and off; we concluded that both should be possible and decided to go for the approach with the time-varying electric fields. Searching for the term "slow molecules" on the internet gave us then – quite unexpectedly – as one of the first hits the one-page conference abstract by John King from 1959 in which he briefly described the ongoing ammonia deceleration experiments in his laboratory [19] (he there mentioned a 1 m long, 25-stage array of parallel-plate capacitors, using electric fields of 100 kV/cm, i.e., longer but with lower electric fields than what Robert Golub reported upon later [20]), to which we had found no reference in the later work from Lennard Wharton. Different from the earlier attempts by John King and Lennard Wharton, we used seeded pulsed beams and we did not have to rely on time-varying high voltages at a certain fixed frequency. Instead, we could make use of commercially available high voltage switches that can be rapidly (sub-μs) switched on and off in any pre-programmed time-sequence. This made it possible to design an array of – in our first experiment – 63 equidistant electric field stages with a center-to-center distance of 5.5 mm and to have complete flexibility in the input and output velocity that we would like to use. The electric field in each stage is formed by applying a high voltage to two parallel 3-mm-diameter cylindrical rods, centered 4.6 mm apart, leaving 1.6 mm opening for the molecular beam. The two opposing rods are simultaneously switched by two independent highvoltage switches from ground potential to voltages of +10 kV and −10 kV, yielding maximum electric fields of 125 kV/cm in a geometry that also provides transverse focusing. To obtain transverse focusing in two dimensions, adjacent electrode pairs are alternately positioned horizontally and vertically. All horizontal and all vertical electrode pairs are electrically connected and alternately switched, requiring a total of four independent high-voltage switches. A photograph of the prototype, 35 cm long, so-called "Stark-decelerator" as well as a scheme of the experimental set-up is shown in Fig. 2. The two electric field configurations between which switching takes place are schematically shown on the right hand side of Fig. 2.

In Fig. 3, the original measurements are shown that gave us the first hint of signal due to decelerated molecules, indicated by the blue dashed line as a "guide to the eye" – which is needed here. Rick Bethlem had been scheduled to give an oral presentation on the deceleration experiments at the annual meeting of the Dutch AMO community in Lunteren, The Netherlands, in November 1998, and there he presented these results, that had been obtained just a few days before that. In the weeks following this meeting, we managed to significantly improve upon these first results and we demonstrated deceleration of metastable CO (*a*<sup>3</sup>-<sup>1</sup>, *J* = 1) molecules in low-

**Fig. 2** Rick Bethlem, holding the prototype Stark-decelerator in his hands. The hexapole that is used to optimize the coupling of the beam into the Stark-decelerator can be seen sticking out on the left. A scheme of the experimental setup, omitting the hexapole, is shown underneath the photograph. A pulsed molecular beam is produced by expanding a mixture of 5% of CO seeded in Xe through a pulsed valve into vacuum. After passing through a skimmer, the CO molecules are excited with a pulsed laser to the low-field seeking, upper -doublet component of the *J* = 1 level in the a3-<sup>1</sup> state. The metastable molecules then pass through the Stark-decelerator and are detected by monitoring the electrons that are emitted when they impinge on a clean gold surface. On the right, the two electric field configurations that are alternately used to slow down the metastable CO molecules are schematically shown.

field seeking states from 225 m/s to 98 m/s reducing their kinetic energy by almost 0.8 cm−<sup>1</sup> per electric field stage [32]. Two years after this, we demonstrated phasestable acceleration (and deceleration) of – again – CO (*a*<sup>3</sup>-<sup>1</sup>, *J* = 1) molecules, but this time in high-field seeking states, using an array of twelve dipole lenses in alternate-gradient configuration [33]. These were the successful demonstrations of the experiments that John King and Lennard Wharton set out to perform almost forty years earlier, made possible by the choice of our system, that is, by the advantages that laser-prepared, metastable CO molecules offered, and by the advances in highvoltage switching technology.

When our first manuscript on the Stark-decelerator was still under review, I presented the main results and our future plans during a workshop at ITAMP, in July 1999. There, Daniel Kleppner informed me that although none of the attempted deceleration work of John King had been published, there should exist a Ph.D. thesis from one of his students, and we subsequently traced down the Ph.D. thesis of Robert Golub. Interestingly enough, Robert Golub mentions in his Ph.D. thesis from 1967 that "there has recently been a proposal to use a similar scheme to accelerate molecules by L. Wharton" [20]; it remains unclear in how far Lennard Wharton was

**Fig. 3** Observed arrival time distribution of metastable CO molecules when an increasing number of deceleration stages is being used, from none at all (lowest curve) to all of them (uppermost curve). With no fields applied, the arrival time is centered around 2.5 ms after firing of the pulsed laser, i.e., 2.5 ms after preparation of the metastable CO molecules, with a more-or-less Gaussian distribution. The spikes in the signal prior to the arrival of the molecules on the detector are due to electrical noise from the high-voltage switches, and indicate how often the fields have been switched. When electric fields are applied, transverse focusing takes place, and more molecules are seen to arrive on the detector, with a highly structured temporal distribution. The blue, dashed line indicates the arrival time of the synchronous molecules, i.e., those CO molecules that experience the aimed-for, constant amount of deceleration per stage, and these are seen to arrive later when more deceleration stages are being used. In the inset, the measured (solid dots and squares) and calculated Stark shift of the components of the *<sup>J</sup>* <sup>=</sup> 1 level in the a3-<sup>1</sup> state of CO are shown in electric fields up to 150 kV/cm.

aware of John King's experiments. During the same workshop, Hossein Sadeghpour informed me on ongoing experiments in the group of Harvey Gould in Berkeley, aimed at decelerating molecules with electric fields. Shortly after our work was published, they published an article in which they presented data on the deceleration of Cs atoms with time-varying electric fields [34].

#### **5 Concluding Remarks**

The Stark-decelerator has made it possible to produce packets of state-selected molecules, oriented in space, with computer-controlled six-dimensional phase-space distributions. This level of control of molecular beams has first been demonstrated with electric fields, as outlined in this Chapter, but has also been obtained using timevarying magnetic fields and – to a lesser extent – using electro-magnetic radiation fields by now. Together, these methods have made a whole variety of new experiments possible. It would go too far, to (try to) list these here, and the interested reader is referred to the earlier mentioned Review from 2012 [2], and the references therein. As selected highlights that have appeared since then, I would like to mention the experimental realization of a molecular fountain by Rick Bethlem and co-workers [35], the demonstration of a cryogenic molecular centrifuge in the group of Gerhard Rempe [36], the magnetic deceleration and trapping of molecular oxygen in the group of Edvardas Narevicius [37] and the high-resolution collision experiments in the group of Bas van de Meerakker [38].

"If one extends the rules of two-dimensional focusing to three dimensions, one possesses all ingredients for particle trapping." This is how Wolfgang Paul stated it in his Nobel lecture [39], and as far as the underlying physics principles of particle traps are concerned, it is indeed as simple as that. To experimentally realize the trapping of neutral particles, however, the main challenge is to produce particles that are sufficiently slow that they can be trapped in the relatively shallow traps that can be made. When the particles are confined along a line, rather than around a point, the requirements on the kinetic energy of the particles are more relaxed, and storage of neutrons in a one meter diameter magnetic hexapole torus could thus be demonstrated first [40]. Trapping of atoms in a 3D trap only became feasible when Na atoms were laser cooled to sufficiently low temperatures that they could be confined in a quadrupole magnetic trap [41]. The Stark-decelerator enabled the first demonstration of 3D trapping of neutral ammonia molecules in a quadrupole electrostatic trap [42] even before it was used in the demonstration of an electrostatic storage ring for neutral molecules [43].

There obviously are large similarities between the manipulation of polar molecules and the manipulation of charged particles, and concepts used in the field of charged particle physics can and have been applied to neutral polar molecules, and *vice versa*. Both Hartmut Kallmann and Wolfgang Paul worked on the deflection and focusing of beams of neutral molecules before they turned their attention to controlling the motion of charged particles; it is interesting to realize that multipole fields were actually used

**Fig. 4** Photograph of the 0.5m diameter molecular synchrotron, consisting out of 40 straight hexapoles arranged in a circle. Deuterated ammonia molecules with a velocity of 125m/s are injected in the synchrotron and stay confined to a 2.5mm long packet, also after having made more than thousand round trips, i.e., after having travelled a distance of over one mile. The inset shows that the amount of trapped molecules decays with a 1/*e* time of 3.2 s, caused in equal parts by optical pumping due to black-body radiation and collisions with background gas.

in molecular beam physics first. Inspiration from charged particle physics has been instrumental for the development of the Stark-decelerator, the LINAC for neutral, polar molecules. It has also inspired the realization of a molecular synchrotron, shown in Fig. 4, in which state-selected, neutral molecules are kept together in a compact packet for a distance of over one mile, extending their duration on stage to many seconds [44].

#### **References**

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#### **Chapter 21 Quantum Effects in Cold and Controlled Molecular Dynamics**

**Christiane P. Koch**

**Abstract** This chapter discusses three examples of quantum effects that can be observed in state-of-the-art experiments with molecular beams—scattering resonances as a probe of interparticle interactions in cold collisions, the protection of Fano-Feshbach resonances against decay despite resonant coupling to a scattering continuum, and a circular dichroism in photoelectron angular distributions arising in the photoionization of randomly oriented chiral molecules. The molecular beam setup provides molecules in well-defined quantum states. This, together with a theoretical description based on first principles, allows for excellent agreement between theoretical prediction and experimental observation and thus a rigorous understanding of the observed quantum effects.

#### **1 Introduction**

When you ask young students entering a university physics course today for the term they associate most with quantum mechanics, many of them will respond with "entanglement". This reflects the rise of quantum information science out of an often ridiculed ivory tower to the decision-making levels of the big tech companies and to the headlines of well-respected media outlets. In the waves of excitement created by the "second quantum revolution" [1, 2], it may be overlooked that features more traditionally associated with quantum mechanics continue to fascinate and challenge our classically trained intuition.

This chapter reviews three examples of such quantum effects beyond entanglement from recent work of my group—tunneling resonances that emerge in cold collisions and that can be used to probe interparticle interactions [3], Fano-Feshbach resonances that can be protected against decay by a suitable phase condition [4], and quantum pathway interference in the circular dichroism of photoelectrons that is observed after the photoionization of chiral molecules [5, 6]. All three examples share a rigorous theoretical description based on first principles. More importantly

C. P. Koch (B)

Arnimallee 14, 14195 Berlin, Germany e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2021

B. Friedrich and H. Schmidt-Böcking (eds.), *Molecular Beams in Physics and Chemistry*, https://doi.org/10.1007/978-3-030-63963-1\_21

for the present contribution, the quantum effects discussed here have been observed in experiments with molecular beams [3, 4, 7]. They thus testify to the topicality and continuing significance of the molecular beam technique developed by Otto Stern and colleagues.

#### **2 Quantum Scattering Resonances in Cold Collisions**

The wave nature of colliding particles emerges most prominently at low scattering energies [8, 9], where quantum resonances dominate the scattering cross section before threshold laws take over [10]. While quantum resonances are also present at higher scattering energies, this presence is hidden in the ensemble average over all quantum states that are populated at a given energy. In other words, collisions are "cold" when only a few partial waves contribute to the scattering cross section [8]. At the corresponding collision energies, the dynamics are often dominated by the long-range behavior of the interparticle interactions [10], easing the interpretation of the collision studies.

In order to observe quantum scattering resonances experimentally, one needs to ensure a sufficiently narrow velocity distribution—narrower than the resonance width—in addition to the capability to finely tune the relative kinetic energy of the colliding particles to very low values. Both requirements can be met with merged neutral beams [11, 12], the use of which has allowed for the observation of tunneling resonances in Penning ionization reactions [13]. Since then, scattering resonances have also been observed in inelastic low-energy collisions, see Ref. [14] and references therein.

Penning ionization reactions occur when the excitation energy of a particle prepared in a metastable quantum state is sufficient to ionize its collision partner [15]. Metastable nobel gas atoms, for example, feature excitation energies of more than 10 eV which is above the ionization potential of most molecules. The scattering resonances in this case are tunneling, or shape, resonances that occcur when the kinetic energy of the colliding particles matches the energy of a quasi-bound state that is trapped behind the rotational barrier of an otherwise barrier-less potential, cf. Fig. 1. Peaks in the ionization cross section indicate not only the presence of a tunneling resonance but also highlight the corresponding quantization of the intermolecular motion. In a more pictorial description, when the colliding particles hit a resonance upon tunneling through the rotational barrier, their amplitude gets trapped at short interparticle distances. This results in peaks in the cross section since the ionization probability grows inverse-exponentially with interparticle distance [15].

The experiments leading to the first observation of tunneling resonances in Penning ionization reactions were carried out with metastable helium colliding with argon atoms and dihydrogen molecules [13]. At the time, no qualitative difference was observed for the Penning ionization of an atom compared to that of a molecule. In general, however, one would expect the molecular degrees of freedom—rotations or vibrations—as well as the anisotropy of the interparticle interaction potential (the

difference between the molecule hitting the metastable atom head-on or in a *T* shaped geometry) to come into play and modify the cross section, respectively the reaction rate. For low-energy collisions involving dihydrogen molecules, a change of the H2 vibrational state is energetically not accessible. In order to assess the role of the anisotropy of the interparticle interaction, it is useful to expand the angular dependence of the interaction potential into Legendre polynomials,

$$V(R,\theta) = V\_0(R) + V\_2(R)\cos^2\theta + \dotsb \dots$$

Here, *R* and θ denote the interparticle separation and the angle between intermolecular axis and collision axis, respectively. A comparison of the magnitude of *V*2(*R*) to the rotational constant of the molecule yields, for H2, energy equivalents of 1 K versus roughly 200 K. The collision is thus highly unlikely to induce changes of the rotational state. Nevertheless, the anisotropy of the interaction does affect the reaction rate! It determines the occurrence of quantum scattering resonances in slow barrier-less reactions [3] and dictates the scaling of the reaction rate coefficient with collision energy in fast barrier-less reactions [16].

The role of the anisotropy for the occurence of quantum scattering resonances is most readily understood using an adiabatic separation of interparticle motion and internal molecular rotation [17]. Denoting the eigenstates of the internal molecular rotation by | *j*, *m <sup>j</sup>*-, the potential energy operator is effectively diagonal in this basis (reflecting the absence of rotational state changing collisions), but the diagonal matrix elements differ for *j* = 0 and *j* = 1: While the matrix element for *j* = 0 contains only the isotropic part of the interaction, *V*0(*R*), those for *j* = 1 also depend on the anisotropy, *V*2(*R*). This allows one to directly probe the role of the anisotropy

**Fig. 2** Rate coefficient for Penning ionization of dihydrogen colliding with metastable helium: The tunneling resonance at a collision energy of k*<sup>B</sup>* × 270 mK appears only if the molecule is rotationally excited. This is due to the anisotropy of the interparticle interaction (equal to *V*<sup>2</sup> in leading order). Adapted from Ref. [3]

in atom-molecule collisions via quantum scattering resonances [3], provided the molecule can be selectively prepared in *j* = 0 or *j* = 1.

Rotational state selection in H2 molecules is made possible by the large rotational splitting together with the nuclear spin symmetry. The latter implies that para-H2 must have an even rotational wavefunction, thus *j* = 0, whereas the rotational wavefunctions for ortho-H2 are odd, i.e., *j* = 1 for ortho-dihydrogen. Higher rotational levels are not populated in molecular beam experiments due to the large rotational splitting. Samples of almost pure para-H2 are readily obtained by catalytic ortho-para conversion. It is thus comparatively straightforward to carry out molecular beam studies with rotational state selection [3, 16].

Repeating the Penning ionization experiments of Ref. [13] with dihydrogen molecules in a well defined rotational state led to the surprising observation, cf. Fig. 2, of the low-energy resonance (at a collision energy of k*<sup>B</sup>* × 270 mK) occuring only for ortho-H2, i.e., only for rotationally excited molecules. A theoretical model capturing this phenomenon requires spectroscopic accuracy. Systematic corrections to high-level state-of-the-art quantum chemistry combined with coupled channels calculations allow for reaching this accuracy. This is illustrated by the green, blue and red curves in Fig. 2, depicting the results obtained with a potential energy surface derived by coupled cluster theory with single, double and non-iterative triple excitations (green), including the full CI correction (blue) and further improved by uniformly scaling the correlation energy by 0.4% (red) [3].

The excellent agreement between the theoretical results and the experimental data observed in Fig. 2 allows for a more indepth examination of the role of the anisotropy. In the theoretical model, we can easily modify the relative weight of the anisotropic part of the interaction potential, denoted by α in Fig. 2. Increasing α up to 50% does not introduce a noticeable change on the results obtained with para-H2, i.e., molecules in their rotational ground state [3]. On the other hand, the low-energy resonance observed only for rotationally excited H2 in Fig. 2 depends very sensitively on the scaling factor of the anisotropic potential, shifting to lower energies with decreasing anisotropy [3].

The reason underlying this behavior can be unveiled using the adiabatic approximation mentioned above [17]. As is often the case with perturbation theory, it does not yield a quantitative description but qualitatively provides the correct picture. When examining the adiabatically separated scattering channels, a bound state just below the dissociation threshold occurs for rotational ground state para-dihydrogen (with = 3, *j* = 0, and *J* = 3). For rotationally excited ortho-dihydrogen, the anisotropic part of the interparticle interaction potential introduces an energy shift that is added to the effective potential. This pushes, what is a weakly bound state for para-dihydrogen, above the dissociation threshold for ortho-dihydrogen, turning the bound state into a shape resonance (with = 3, *j* = 1, and *J* = 3) [3], thus solving the riddle of resonance (dis)appearance.

To conclude this section, quantum scattering resonances testify to the emergence of the wave nature of matter in "cold" collisions. Using dihydrogen molecules in merged beam studies allows for simple rotational state selection which in turn can be used to probe the anisotropic part of the interparticle interaction governing Penning ionization reactions [3, 16]. While an excellent agreement between theory and experiment can be reached when including appropriate corrections for electron correlations, the calculation of quantum resonances involving metastable states continues to present a significant challenge even for the highest available levels of first principles based theory.

#### **3 Phase-Protection in Fano-Feshbach Resonances**

After understanding the dramatic effects that very small shifts in energy may have on a shape resonance, in the present section, another type of quantum resonance will highlight the sensitivity of resonances to small changes in phase. A Fano-Feshbach resonance describes the decay of a bound quantum state due to coupling with a continuum of scattering states. In contrast to shape resonances, it involves two distinct scattering channels characterized by different quantum numbers, as sketched in Fig. 1. In the original theory, the coupling between bound and continuum states described an effective nucleon-nucleon interaction [18], resp. configuration interaction in autoionization [19]. In recent years, Fano-Feshbach resonances due to the hyperfine interaction between different nuclear spin states have attracted attention as key tool for control in ultracold gases [20]. The present example considers rovibrational predissociation resonances due to the spin-orbit interaction [21].

Predissociation resonances may be populated by associative ionization which accompanies a Penning ionization reaction, provided the dependence of the Penning ionization rate on interparticle separation matches the ionic potential energy curve [15]. The ionization then populates bound levels of noble gas diatomic molecules such as HeAr+, NeAr+, or HeKr<sup>+</sup> which may or may not be spin-excited. These seemingly simple molecules with very similar electronic structure possess (spin-excited) predissociation resonances with surprisingly different lifetimes.

In order to estimate lifetimes, one typically inspects the coupling responsible for the decay. When comparing predissociation in HeAr<sup>+</sup> and NeAr+, the spin-orbit splitting in the two cases is identical. The term dominating the coupling between spin-excited and ground states is radial coupling [4] which scales as /μ, where μ is the reduced mass. The lifetime of the resonances then scales with μ2. Since, for the two diatoms, the reduced masses differ by a factor of approximately four, the lifetime of NeAr<sup>+</sup> would be expected to be larger than that of HeAr<sup>+</sup> by about an order of magnitude. This expectation derived from scaling arguments ignores, however, a phase dependence of the lifetimes which may entirely alter the picture, as shown next.

A qualitative understanding of the resonance lifetimes can be obtained in first order perturbation theory, using Fermi's golden rule:

$$
\sigma\_{\varphi}^{-1} \sim \left| \langle k\_{\rm res} | \mathbf{W}\_{\rm coul} | \varphi \rangle \right|^2,\tag{1}
$$

where |ϕ denotes the quasi-bound state, **W**coupl the coupling operator, and |*k*res the continuum state with scattering momentum *k*res, in resonance with the quasi-bound state, cf. Fig. 1. Since the continuum state describes the scattering off a potential, a phase shift δ is associated with the position of the repulsive wall. It is this phase shift that determines the value of the complex overlap in Eq. (1). In particular, there exist combinations of *k* and δ for which the overlap vanishes such that τϕ → ∞ despite non-zero coupling! We have termed this phenomenon "phase protection" [4]. In fact, it is straightforward to show that vanishing overlap is equivalent to

$$\arg \left( \tilde{\varphi}(k) \right) + \delta = m\pi \quad \text{with } m \in \mathbb{Z} \tag{2}$$

with ϕ(˜ *k*) the Fourier transform of the quasi-bound state, when neglecting the energy dependence of the phase shift δ and assuming *s*-wave scattering.

In a real molecule, the resonance lifetime will never strictly go to infinity since eventually other decay mechanisms will become relevant. However, the lifetime can indeed become very large. This is shown in Fig. 3 displaying the potential energy curves, derived from spectroscopic data, respectively coupled-cluster calculations,

**Fig. 3** Potential energy curves (left), lifetimes (center), and resonance positions (right) of predissociation resonances in spin-excited noble gas dimers. Adapted from Ref. [4]

for HeAr<sup>+</sup> and NeAr+, together with the lifetimes obtained with a complex absorbing potential and including relativistic and angular couplings [4]. The potential energy curves shown in the left-hand part of Fig. 3 highlight the similarity of the potential energy curves and spin-orbit coupling for the two molecules. In contrast, a striking difference is observed in the middle panel of Fig. 3, where the predissociation lifetimes of spin-excited NeAr<sup>+</sup> span more than four orders of magnitude whereas those of HeAr<sup>+</sup> differ by only a factor of 10. This difference is rationalized in terms of the condition for phase protection, cf. Eq. (2), visualized in the right part of Fig. 3. The lines indicate the combinations of *k* (respectively scattering energy) and δ for which the overlap in Eq. (1) vanishes. For NeAr+, several resonance positions are located very close to those lines. The closer the resonance position to its corresponding phase protection line, the larger becomes the corresponding lifetime. In contrast, for HeAr+, none of the resonances fulfills even approximately the condition for phase protection which is why the lifetimes span a much smaller range.

Experimental evidence for the lifetimes of spin-excited NeAr<sup>+</sup> spanning several orders of magnitude is provided by two experiments [4], using molecular beams. First, velocity-map imaging (VMI) of the reaction products of Penning and associative ionization of argon by metastable helium revealed predissociation for HeAr<sup>+</sup> to occur on a timescale smaller than the time of flight of the VMI setup, of the order of 10 μs. In contrast, the predissociation feature was missing in the VMI for NeAr+, suggesting lifetimes significantly larger than the time of flight [4]. A second experiment, designed to assess the range of NeAr<sup>+</sup> lifetimes more directly, injected the molecular ions generated by associative ionization into an electrostatic ion beam trap [22] and recorded oscillations of the molecules between two electrostatic mirrors for several hundreds of milliseconds via trap loss of neutral particles. The experimental data was found to decay non-exponentially, with decay times ranging from below 50 µs up to 100 ms [4], in excellent agreement with the predictions of Fig. 3.

One may wonder whether there is a way to predict, for a given molecule, the chance of phase protection to occur. Assuming one can approximate the potential energy curve by a Morse potential, the lifetime of level v is determined by a phase and the Fourier transform of the corresponding eigenfunction, ψ˜ v(*k*),

$$
\pi\_v^{-1} \propto \sin\left(\delta + \phi\_v(k)\right)^2 \left|\tilde{\psi}\_v(k)\right|^2,
$$

where φv(*k*) = arg[ψ˜ v(*k*)]. On top of a smooth variation with *k* (or energy) due to - - ψ˜ v(*k*) - - - 2 , the lifetimes τv oscillate and vanish whenever the condition for phase protection, cf. Eq. (1), is fulfilled. In order to answer the question how often the latter will happen, one can exploit the fact that, for a Morse potential, ψ˜ v(*k*) is known analytically, in terms of complex -functions [23]. This allows for deriving the scaling of the lifetimes with the parameters of the Morse potential and the reduced mass: The number of times τv vanishes, and thus the chance for phase protection, increases with μ as well as the well depth and equilibrium distance and decreases with the potential width [4]. This agrees well with the observations for NeAr<sup>+</sup> and HeAr<sup>+</sup> in Fig. 3 since the reduced mass and the well depth are larger for NeAr<sup>+</sup> than for HeAr+, while the position of the minimum and the potential width are very similar. Moreover, this scaling argument predicts an isotope effect on predissociation lifetimes which has indeed been observed experimentally for N<sup>+</sup> <sup>2</sup> [24] and Ne<sup>+</sup> <sup>2</sup> [25].

To summarize this section, phase protection refers to the fact that the lifetime of a quasi-bound quantum state can become very, very large despite non-zero coupling with a continuum of scattering states. The protection from decay occurs whenever the relative phase in the overlap of bound and scattering state becomes a multiple of π. Experiments with spin-orbit excited noble gas dimers have confirmed our theoretical prediction of phase protection. The probability of phase protection increases with reduced mass, well depth and equilibrium distance of the potential supporting the quasi-bound state, providing a blueprint to identify quantum states that are intrinsically protected against undesired decay.

#### **4 Photoelectron Circular Dichroism and Its Coherent Control**

The third example showcases a quantum effect that is observed in experiments with molecular beams when circularly polarized light ionizes molecules which are chiral [7]. Remarkably, when a molecule is chiral, i.e., when its nuclear scaffold exhibits a handedness, ionization of randomly oriented molecules with right circularly polarized light does not yield the same photoelectron angular distribution as that with left circularly polarized light [26]. The difference between the photoelectron angular distributions obtained with left and right circularly polarized light is called photoelectron circular dichroism (PECD). Instead of exchanging the polarization direction of the light, one can also exchange the handedness of the molecule to observe PECD [26]. Unlike other dichroic effects, PECD does not involve a magnetic dipole moment and is obtained merely within the electric dipole approximation of the light-matter interaction [26].

First order perturbation theory for the photoionization cross section reveals the mechanism underlying the dichroic effect [26]: For a chiral molecule, the photoionization matrix elements with opposite spatial orientation do not cancel when averaging over the Euler angles. This gives rise to terms in the cross section that are odd under inversion of the polar angle and thus to dichroism. More intuitively, the photoionization cross section involves two rotations—of the polarization axis into the molecular frame and of the photoelectron momentum into the lab frame—which are sufficient to be sensitive to the handedness of the molecular scaffold. This geometric picture can be made more rigorous by noticing that the angle-resolved photoionization cross section is a vector observable which can be expressed as a triple product [27]. For non-coplanar vectors forming the triple product, the observable becomes enantiosensitive [27].

While theoretically predicted in the mid-1970s [26] and first observed with synchrotron radiation in the early 2000s [28], femtosecond laser pulses driving multiphoton ionization have made PECD accessible in table-top experiments [7, 29–32]. Since multi-photon ionization probes intermediate electronically excited states [7, 33], a theoretical description beyond first order in the perturbation theory for the light-matter interaction is called for. At the same time, the bicyclic ketones with which the experiments have been carried out, for example, fenchone, camphor, or limonene, are amenable to a high-level treatment of their electronically excited states. In contrast, modeling the photoionization continuum from first principles is rather challenging.

A way to address this challenge, applied to the specific example of resonantly enhanced (2+1) multi-photon ionization (REMPI) of fenchone and camphor [5], separates the non-resonant two-photon excitation from the one-photon ionization. The former can be described with coupled cluster theory whereas for the latter a single active electron approach using hydrogenic orbitals captures the essential physics of the photoelectron moving in a Coulombic potential [5]. Neglecting any coherent effects during the excitation, the ionization probes an anisotropic distribution of electronically excited molecules. This is due to the anisotropy of the two-photon absorption tensors of the molecules [5]. The properties of the two-photon absorption alone are, however, not sufficient to determine which intermediate state is probed by the REMPI process. Based on energetic arguments, there are five possible candidate states, only one of which is ruled out when using the information of the two-photon absorption tensors in the calculation of the photoionization cross section [5]. When including also the properties of the probed electronically excited states, only one out of the five candidates is in agreement with the experimental data [5]. Interestingly, the theoretically predicted intermediate state differs for fenchone compared to camphor [5]. This might explain the different photoelectron angular distributions and different PECD observed for these two chemically very similar molecules [7].

There are many intriguing questions that arise in the context of PECD, for example whether the effect is determined by the initial, the final, or the intermediate state of the process. In order to answer such questions, a more rigorous description of PECD and related effects in the ionization of chiral molecules is required. In particular, a better description of the photoionization continuum including electron correlation effects and the ability to treat coupled electronic and vibrational motion are called for. At the same time, PECD provides a very convenient experimental handle to probe the chiroptical response of molecules, and it is natural to ask whether this response can be enhanced by suitably shaping the ionizing field, in the spirit of coherent control of photoinduced dynamics [34, 35].

This question can be answered by combining the time-dependent configuration interaction singles (CIS) method for the electronic structure [36], second order perturbation theory for the light-matter interaction, and parameter optimization for the electric field using sequential parametrization updates [37]. Figure 4 illustrates the ionization pathways that are accounted for in this description. Pathways ending at the same final kinetic energy of the photoelectron (within the spectral bandwidth of the pulse) can interfere with each other, cf. the red and green arrows in Fig. 4. This can be thought of as a spectral realization of a double-slit experiment, with the additional advantage that the relative phase between the two pathways can be adjusted by properly tuning the ionizing electric field in its amplitude and phase [35].

A bichromatic pulse driving the red and green ionization pathways in Fig. 4 is thus a natural starting point for optimizing the electric field. When ionizing a chiral methan derivate with such a pulse, assuming a flat spectral phase, a PECD of about 4% (relative to the isotropic ionization yield) is obtained [6]. When optimizing the

**Fig. 4** One-photon and two-photon ionization pathways within the time-dependent CIS framework

#### 21 Quantum Effects in Cold and Controlled Molecular Dynamics 487

**Fig. 5** PECD after ionization of CHBrClF with a bichromatic guess pulse with flat spectral phase (left), optimized bichromatic (center) and freely optimized (right) electric fields. The percentage is taken with respect to the isotropic yield, and the increase is due to interference between one-photon and two-photon pathways (center), respectively between different two-photon pathways probing different intermediate states (right). Adapted from Ref. [6]

**Fig. 6** The freely optimized field drives various two-photon ionization pathways that, depending on the linear chirp of the low-frequency component of the ionizing field, respectively the timedelay between low-frequency and high-frequency pulse compontents, interfere constructively or destructively. Adapted from Ref. [6]

field, constraining the pulse to be bichromatic, the PECD is pushed up to about 20%. This is shown in the center panel of Fig. 5. Compared to the initial guess pulse, the photon energies are lowered and, more importantly, the spectral phases of the two pulse components are adjusted to ensure that one-photon and two-photon ionization pathways interfere constructively. In contrast, with a flat spectral phase, the interference is mainly destructive. Indeed, constructive ionization of one-photon and two-photon pathway makes up for 17% out of the calculated 20% of PECD [6].

An even larger increase in PECD, up to almost 70%, is obtained when freely optimizing the ionizing electric field [6], cf. right panel of Fig. 5. This increase is driven by interference between various two-photon ionization pathways probing different intermediate, electronically excited states, illustrated in Fig. 6 (right). The optimized field turns out to have two spectral components, with the lower one exhibiting a linear spectral phase [6]. Such a "chirp" provides a very convenient handle to analyze the interference pattern in more detail, since changing the chirp rate, i.e., the slope of the spectral phase, amounts to changing the relative time delay between the low-frequency and the high-frequency component of the optimized pulse. If the lowfrequency component precedes the high-frequency one, the PECD does not depend on the specific time delay and amounts to about 10%, cf. red-shaded parts in Fig. 6. If, in contrast, the high-frequency component arrives first, an electronic wavepacket is created which is then ionized by the low-frequency component of the pulse, cf. blue-shaded parts in Fig. 6. In this case, PECD sensitively depends on the time-delay, reflecting the time evolution of the electronic wave packet, oscillating between 6% and 62%. The largest PECD is obtained when the two spectral components of the optimized pulse overlap in time [6]. This allows for maximum interference between all the two-photon ionization pathways, highlighted by the yellow-shaded parts in Fig. 6.

In summary, photoelectron circular dichroism is a sensitive chiroptical probe of electron dynamics in a chiral potential, measured in experiments with molecular beams. Time-independent perturbation theory of resonantly enhanced (2+1) photoionization of camphor and fenchone molecules, combined with an *ab initio* description of the bound electronic spectrum based on coupled cluster theory, yields semi-quantitative agreement with the experimental data [5]. It emphasizes the role of orientation-selective excitation in multi-photon ionization and allows to identify the intermediate state that is probed by the REMPI process [5]. The magnitude of the chiroptical response can be enhanced by suitably shaping the ionizing field [6]. Time-dependent perturbation theory allows for directly identifying the quantum pathway interference responsible for the enhancement [6]. Whether there exists an upper bound for a chiroptical response such as photoelectron circular dichroism is one of the many open questions in the quantum control of molecular chirality.

#### **5 Conclusions**

This chapter reviewed three examples of quantum effects beyond entanglement the (dis)appearance of a tunneling resonance in cold collisions, the protection of predissociation resonances by a phase, and the circular dichroism observed in the photoelectron spectrum of chiral molecules. In each case, a theoretical description based on first principles was key to elucidating the rather surprising observations made in experiments with molecular beams. The examples thus testify to the intriguing nature of quantum mechanics as well as to the topicality of Otto Stern's legacy.

**Acknowledgements** I would like to thank my students and postdocs, in particular Alexander Blech, Daniel Reich, Esteban Goetz and Wojtek Skomorowski, for their dedication and determination in carrying out the work that I have reviewed here. I am indebted to my colleagues Edvardas Narevicius and Robert Berger for many years of inspiring and fruitful collaboration. I am grateful for a Rosi and Max Varon Visiting Professorship to the Weizmann Institute of Science and for financial support from the German-Israeli Foundation (grant no. 1254), the State Hessen Initiative for the Development of Scientific and Economic Excellence (LOEWE) within the focus project Electron Dynamic of Chiral Systems (ELCH), and the Deutsche Forschungsgemeinschaft (Projektnummer 328961117—SFB 1319). This work was carried out while I was at the University of Kassel; and I would like to thank my Kassel colleagues for creating a friendly and stimulating research atmosphere.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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#### **Chapter 22 From Hot Beams to Trapped Ultracold Molecules: Motivations, Methods and Future Directions**

#### **N. J. Fitch and M. R. Tarbutt**

**Abstract** Over the past century, the molecular beam methods pioneered by Otto Stern have advanced our knowledge and understanding of the world enormously. Stern and his colleagues used these new techniques to measure the magnetic dipole moments of fundamental particles with results that challenged the prevailing ideas in fundamental physics at that time. Similarly, recent measurements of fundamental electric dipole moments challenge our present day theories of what lies beyond the Standard Model of particle physics. Measurements of the electron's electric dipole moment (eEDM) rely on the techniques invented by Stern and later developed by Rabi and Ramsey.  These experiments, and many others, are likely to benefit from ultracold molecules produced by laser cooling. We explain how laser cooling can be applied to molecules, review recent progress in this field, and outline some eagerly anticipated applications.

#### **1 Introduction**

It has been nearly a hundred years since Otto Stern and Walther Gerlach used an atomic beam to reveal one of the most striking aspects of the then-burgeoning quantum theory, space quantization [1]. Their work introduced new techniques that would later be used in countless experiments in physics and chemistry. Stern saw clearly the great promise of his new method, stating [2]

The molecular beam method must be made so sensitive that in many instances it will become possible to measure effects and tackle new problems which presently are not accessible with known experimental methods.

He was right. The molecular beam method has been at the heart of atomic and molecular physics ever since and remains the method of choice for a huge number of experiments.

491

N. J. Fitch · M. R. Tarbutt (B)

Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2021

B. Friedrich and H. Schmidt-Böcking (eds.), *Molecular Beams in Physics and Chemistry*, https://doi.org/10.1007/978-3-030-63963-1\_22

A more recent development—laser cooling—can also be traced back to Stern, via Frisch who used the molecular beam method to measure the photon recoil momentum [3]. By controlling this recoil, modern atomic physics experiments routinely cool atoms and ions to µK temperatures. Until recently, experiments with molecules lagged behind, usually because of the difficulty of cooling them. Nevertheless, molecules have many useful properties that are increasingly being exploited for a variety of applications including tests of fundamental physics. An important example is the measurement of the electron's electric dipole moment (eEDM) where the precision of molecular experiments exceeds that achieved using atoms. The desire to improve these measurements provides strong motivation to extend cooling and trapping methods to molecules, and this has been achieved in the last decade. Laser cooling has been used to collimate and decelerate molecular beams, capture molecules in magneto-optical traps, and then cool them to ultracold temperature. These ultracold molecules can be used to address a wide variety of important problems - exploring what new forces lie beyond the Standard Model of particle physics [4, 5], studying collisions and reactions at the quantum level [6, 7], simulating the behaviour of many-body quantum systems [8–10], and processing quantum information [11, 12].

In this article, we review some of these past developments and future prospects. We begin in Sect. 2 with a brief review of molecular beam sources. In Sect. 3 we consider how the development of molecular beam methods for measuring magnetic dipole moments eventually enabled measurements of the electric dipole moments of fundamental particles. We briefly review the current status of these experiments in Sect. 4 and explain the importance of laser cooling to the future of this endeavour. In Sects. 5 and 6 we explain how laser cooling works for molecules and present recent achievements in this field. Finally, in Sect. 7, we give a brief overview of some applications of ultracold molecules and how they might be realized.

#### **2 Molecular Beam Sources**

The molecular beam method developed by Stern [13, 14] is the foundation for innumerable experiments in atomic and molecular physics. Here, we give a brief review of the three main types of atomic and molecular beam sources in use today: effusive beams, supersonic beams, and cryogenic buffer gas beams. Their velocity distributions and flux are compared in Fig. 1 for the case of YbF molecules, one of the few species for which all three types of beam source have been realized.

Effusive sources typically use heated ovens to generate a sufficient vapour pressure of the atoms or molecules of interest. They operate at low pressure so that there are no collisions in the vicinity of the exit aperture. As first shown experimentally by Stern [13, 14], these sources produce beams with a broad velocity distribution, whose mean and width both scale as <sup>√</sup>*T*/*m*s, where *<sup>T</sup>* is the temperature of the source and *m*<sup>s</sup> is the mass of the species. As a consequence of the high oven temperature, effusive beams are characterized by a wide velocity distribution and low flux in any single quantum state.

**Fig. 1** Velocity distributions of effusive, supersonic, and buffer-gas beams. The vertical axis indicates the number of molecules per unit solid angle per unit time and per unit interval of velocity. The plots are for YbF molecules in their absolute ground state, chosen because, for this molecule, all three sources have been developed and characterised. Realistic operating conditions have been taken. For the effusive case, molecules are generated from an oven at 1500 K [15]. For the supersonic case, a carrier gas of argon with a reservoir temperature of 300 K is assumed, with an internal beam temperature of 4 K and a repetition rate of 25 Hz [16]. For the cryogenic buffer-gas case, a carrier gas of helium at 4 K creates a beam with moderate hydrodynamic boosting, operating at 10 Hz [17]

In supersonic sources [18, 19], a gas held at high pressure expands through a nozzle into a vacuum chamber. There are a large number of collisions in the vicinity of the nozzle. The slower particles are bumped from behind, while the faster ones bump into those ahead, so that all the particles end up travelling at nearly the same speed. In this way, the random thermal motion is converted into forward kinetic energy, producing a cold, fast beam—the mean velocity is high, but the velocity distribution is narrow, as illustrated in Fig. 1. The collisions also transfer the rotational and vibrational energy of the molecules to forward kinetic energy, resulting in a beam that is cold in all degrees of freedom. The first supersonic beams were continuous, but the method was soon extended by using pulsed valves with short opening times. In this way, intense pulses can be produced without an excessive gas load. A wide variety of methods have been developed to introduce atoms and molecules of interest into the supersonic expansion, including laser ablation, electric discharge, and photodissociation. Translational temperatures of 1 K are typical, and beam speeds are 1800 m/s when the carrier gas is room temperature helium, and 400 m/s for room temperature krypton.

The third type of molecular beam source is the cryogenic buffer gas source [20, 21]. Here, the molecules of interest are formed inside a cryogenically-cooled cell containing a cold buffer gas, often helium at 4 K. The molecules are commonly formed by laser ablation or introduced into the cell through a capillary. The internal and motional degrees of freedom of the molecules thermalize through collisions with the buffer gas, and then the molecules exit through a hole in the cell to form a beam. The density of buffer gas in the cell is determined by the gas flow rate and the size of the exit hole. When the density is low, there are few collisions near the aperture, so the beam tends towards the effusive regime. These beams are slow, especially for heavy species, for then the mass *m*<sup>s</sup> is large whereas the temperature *T* is small, typically two orders of magnitude lower than a standard effusive source. Beam speeds as low as 40 m/s have been achieved this way [22]. However, the molecular flux tends to be low in this regime because most molecules diffuse to the cell walls, where they freeze, instead of passing through the exit aperture. As the buffer gas flow is increased it sweeps more molecules out of the cell, increasing the beam flux. However, collisions near the aperture boost the beam speed. In the limit of high density the speed of the molecules reaches the supersonic speed of the buffer gas which scales as <sup>√</sup>*T*/*m*b, where *m*<sup>b</sup> is the mass of the buffer gas atoms. Very often, cryogenic buffer gas sources are operated in an intermediate flow regime where the flow is high enough to extract a substantial fraction of the molecules from the cell, but low enough for a moderate beam speed. Speeds in the range 100–200 m/s are typical, as illustrated in Fig. 1. Due to the high flux and low relative beam speeds, these sources are becoming increasingly popular, especially for experiments on laser cooling of molecules and tests of fundamental physics.

#### **3 Particle Dipole Moments**

Stern's pioneering experiments established the reality of space quantization and determined the magnetic dipole moments of the electron and proton [1, 23–25]. It is interesting to consider whether elementary particles might also have *electric* dipole moments. Just like the magnetic moment, such an electric dipole would have to be oriented along the particle's spin. Furthermore, this orientation must be fixed, since the particle would otherwise have an additional degree of freedom that would, for example, change the filling of electron orbitals in the periodic table. A spin defines a direction of circulation, as does a magnetic moment, so it seems natural for the two to be associated. Far less natural is the association of an electric dipole – a charge separation – with this direction of circulation. Indeed, such an electric dipole moment (EDM) implies a difference between left- and right-handed coordinate systems, and implies a fundamental arrow of time. To see this, consider the Hamiltonian for a particle with magnetic moment *µ* and EDM *d*, both fixed relative to the spin, interacting with magnetic and electric fields *B* and *E*:

$$
\mathcal{H} = -\boldsymbol{\mu} \cdot \mathbf{B} - \mathbf{d} \cdot \mathbf{E}.\tag{1}
$$

Reflection in a mirror, equivalent to the parity operation, reverses *E* but does not reverse *B*, *µ* or *d*. Conversely, reversing the direction of time reverses *B*, *µ* and *d*, but not *E*. We see that while the first term in *H* is even under both the parity and time-reversal operations, the second term is odd and so the existence of an EDM violates both symmetries.

Prior to the 1950s, it was generally considered that nature did not distinguish between left and right, or between forwards and backwards in time, and this seemed to be a powerful argument against the existence of fundamental electric dipoles, implying that |*d*| = 0. This idea was challenged by Purcell and Ramsey who insisted that it was "*a purely experimental matter*", noted that existing evidence was weak, and declared their intention to measure the EDM of the neutron [26]. Regarding the need for experimental evidence to determine whether parity (P), time-reversal (T) and charge conjugation (C) are symmetries of nature, Hermann Weyl was similarly emphatic, writing that [27]

a priori evidence is not sufficient to settle the question; the empirical facts have to be consulted.

Along this line of thought, in 1956 Lee and Yang [28] noted that, for the weak interaction,

parity conservation is so far only an extrapolated hypothesis unsupported by experimental evidence.

Within a few months it was discovered that the weak interaction violates P symmetry [29–31]. In 1964 it was found that the weak interaction also violates CP symmetry, the combined symmetry of charge conjugation and parity [32]. CP violation is equivalent to T-violation in most theories, and so the last theoretical objection to the existence of fundamental EDMs was removed. Suddenly, and ever since, the question was not whether particles could have electric dipoles, but why those dipoles are so small.

In considering how the electric dipole moment of a particle might be measured, it is instructive to reflect on Stern's method for measuring magnetic moments, which is illustrated in Fig. 2a. The magnetic moment of an atom is proportional to its internal angular momentum, -*F*, so we often write *µ* = −*g*μ<sup>B</sup> *F*, where μ<sup>B</sup> is the Bohr magneton and *g* is a proportionality factor. Taking *B* in the *z*-direction, the interaction energy is *W*<sup>B</sup> = −*µ* · *B* = *g*μ<sup>B</sup> *BFz* = *g*μ<sup>B</sup> *B mF* , where *mF* is the projection of the angular momentum onto the *z*-axis. Space quantization is expressed by only discrete values being allowed for *mF* . In the inhomogeneous field of the Stern-Gerlach magnets there is a force on the atoms *F*<sup>B</sup> = −**∇***W*<sup>B</sup> = −*g*μB**∇***B mF* , leading to the deflection proportional to *mF* observed by Stern and Gerlach. Let us now consider an atom with an unpaired electron that has an eEDM, *de*. If an electric field *E* is applied in the *z*-direction, there is an interaction energy *W*<sup>E</sup> = −*RdeE mF* /|*F*|, analogous to the magnetic interaction. The proportionality factor *R* depends on the choice of atom and is often called the enhancement factor because, for heavy atoms, |*R*| can be considerably larger than 1 [33]. For example, for Cs, *R* = 120 [34]. Suppose we pass a beam of Cs atoms through a region of inhomogeneous electric field. Of course, the electric field produces an induced electric dipole in the atom, resulting in a force which is often used to deflect atoms and molecules. This force is the same for states of opposite *mF* and is not the one of interest here. The force

**Fig. 2** The evolution of molecular beam methods for measuring dipole moments. (a) Stern and Gerlach's experiment for measuring magnetic moments and demonstrating space quantization. (b) Rabi's improved method for measuring magnetic moments. Between the two deflectors there is a uniform magnetic field and an oscillating field which, when resonant, changes the spin state. (c) A further improvement by Ramsey separates the oscillating field into two short regions. (d) Method for determining the electron's electric dipole moment by measuring the spin precession in an applied electric field. In all its key aspects, the technique is the same as Ramsey's method

due to the permanent EDM is *F*<sup>E</sup> = −**∇***W*<sup>E</sup> = *Rde***∇***E mF* /|*F*|, and deflects states of opposite *mF* in opposite directions. Taking *de* = 10−<sup>29</sup> e cm, which is close to the current upper limit [35], a field gradient of 100 kV/cm2, and a speed of 200 m/s, the deflection after propagating 1 m is about 10−<sup>19</sup> m. Clearly, this is not a good way to measure *de*. Nevertheless, the subsequent development of molecular beam methods inspired by Stern's techniques became so sensitive that measurements of *de* soon became feasible.

In the late 1930s, Rabi introduced a new idea that greatly improved Stern's method of measuring magnetic moments [36]. Instead of using the magnetic deflector as a measuring device, he used a pair of them as state selectors, as illustrated in Fig. 2b. The deflectors are arranged such that molecules reach a detector provided they remain in the same state so that they have the same magnetic moment throughout. Between the two deflectors, Rabi produced a uniform magnetic field, *B*0, so that neighbouring *mF* states are separated by an energy ω<sup>0</sup> = *g*μ*B*0. In this region, an rf field of frequency ω ≈ ω<sup>0</sup> resonantly drives transitions from one *mF* state to another. Molecules that change *mF* are deflected by the second magnet and miss the detector, resulting in a dip in the detected signal at ω = ω0. This measurement of the resonant frequency, together with a measurement of *B*0, determines the magnetic moment. The precision of this measurement is proportional to the interaction time with the rf field, so it's desirable to make this as long as possible. In practice however, this time is limited by the difficulty of keeping the fields uniform enough.

By the 1950s, Ramsey had solved this problem by separating the rf region into two short sections driven by the same oscillator [37], as illustrated in Fig. 2c. The first deflecting magnet prepares the molecules in a chosen spin state, say spin up. The first rf region rotates the spin so that it is orthogonal to *B*0. The spin then precesses in the uniform magnetic field with angular frequency ω<sup>0</sup> for a time *T* . When ω = ω<sup>0</sup> the rf oscillation is in phase with the spin precession, so the second rf region rotates the spin in the same direction as the first, producing the spin-down state which will miss the detector. If there is a frequency difference ω − ω<sup>0</sup> = ±π/*T* , the extra half rotation means that the spin will be driven back to the spin-up state in the second rf region and will reach the detector. The signal at the detector oscillates as ω is scanned, allowing ω<sup>0</sup> to be determined with an uncertainty inversely proportional to the free precession time *T* . Ramsey's method has such high precision that it is suitable for measuring the tiny electric dipole moments of fundamental particles. All that is needed is to add to *B*<sup>0</sup> a uniform electric field *E*0, and then measure the change in the precession frequency when the direction of *E*<sup>0</sup> is reversed. This change is proportional to the EDM. This was the method used by Smith, Purcell and Ramsey in their first measurement of the neutron EDM [38], and the one used for all subsequent measurements of particle electric dipole moments.

Figure 2d illustrates a typical electron EDM measurement that uses Ramsey's molecular beam method. A molecular beam passes through a state selector, which could be a magnetic or electric deflector but in modern experiments is often a laser beam that optically pumps molecules to the desired state. Next, a first region of oscillating field aligns the spin along *x*. This can be done using an rf field, or a laser field that drives a Raman transition or optical pumping process. The spin now precesses around *z* due to the combination of μ interacting with *B*<sup>0</sup> and *de* interacting with *E*0. Finally the spin direction is measured, for example by mapping its direction to a pair of states that are easily distinguished spectroscopically. The change in the spin precession angle that correlates with the reversal of *E*<sup>0</sup> determines *de*.

#### **4 Current Status and Future Directions of eEDM Experiments**

In the Standard Model of particle physics, the eEDM is predicted to be *de* ≈ 10−<sup>38</sup> e cm [39]. Theories that extend the Standard Model often introduce new CPviolating interactions, which are needed to explain the observed asymmetry between matter and antimatter in the universe [40], and these new interactions lead to much larger eEDM values. Thus, eEDM measurements can be excellent probes of these theories. Early measurements used heavy atoms and yielded results consistent with zero, eventually assigning an upper limit of |*de*| < 1.6 × 10−<sup>27</sup> e cm [41, 42]. Although more difficult to produce and control, heavy polar molecules can be far more sensitive to the eEDM than atoms [43, 44]. The sensitivity is proportional to the degree of polarization and therefore to the electric-field-induced mixing of opposite parity states. In atoms, these are different electronic states whose spacing is typically ∼ 1 eV, but in molecules they are neighbouring rotational states whose energy spacing is about four orders of magnitude smaller, or the opposite parity states of an Ω-doublet where the spacing is even smaller still. Because the levels are closely spaced, only a modest electric field is needed to fully polarize the molecule. In this case, it is common to define an effective electric field *E*eff = *RE* which saturates to a maximum value *E*max eff . The effective field is enormous for some species, and its maximum value is often easy to reach. For example, *E*max eff ≈ 26 GV/cm for YbF and ≈ 78 GV/cm for ThO.

The enormous effective fields make eEDM experiments with molecules very attractive, and measurements have been made using beams of YbF [45, 46], beams of ThO [35, 47], a cell of PbO vapour [48], and trapped HfF<sup>+</sup> molecular ions [49]. The results of these measurements are all consistent with zero, and the best upper limit is currently |*de*| < 1.1 × 10−<sup>29</sup> e cm at the 90% confidence level [35]. Remarkably, this experiment and ones like it test theories that extend the Standard Model at an energy scale similar to, and even exceeding, the maximum collision energy of the Large Hadron Collider.

Given the great significance of eEDM experiments, it is natural to consider how to make the next leap in sensitivity. The uncertainty in measuring *de* scales as 1/(*T* <sup>√</sup>*N*) where *T* is the spin precession time and *N* is the number of molecules used in the measurement. In a molecular beam experiment of length *L*, where the spin precession region occupies most of the space and the beam has diverged sufficiently that it fills the detector, *T* is proportional to *L* but *N* falls as 1/*L*<sup>2</sup> because of the divergence of the beam. Consequently, there is no benefit in increasing *L*. This can be circumvented by cooling the molecules to much lower temperatures. A beam that is cooled in the transverse directions is collimated and can travel for long distances without spreading, allowing *T* to increase without reducing *N*. Going further, molecules that are cold enough can be launched into a fountain [50] or stored in a trap [51], giving access to even longer spin precession times. These ideas require molecules cooled to μK temperatures. Such low temperatures can be reached either by associating ultracold atoms into molecules [52], or by direct laser cooling [53]. EDM experiments using laser-cooled YbF, BaF, YbOH and TlF are all currently being developed [54– 57]. They will have unprecedented sensitivity and tremendously exciting potential for new discoveries in fundamental physics.

#### **5 Laser Cooling of Molecules: Principles**

#### *5.1 Laser Cooling Scheme*

Figure 3 illustrates the energy level structure of a typical diatomic molecule, showing the electronic, vibrational, rotational, and hyperfine structure and the notation used to label the levels. For molecules to be slowed, cooled and trapped by radiation pressure, they must scatter many photons from the light, typically 10<sup>4</sup> or more. This calls for a cooling scheme where an upper level decays to only a few lower levels, so that only a few transitions need to be addressed. The inset to Fig. 3 shows an example of such a scheme. The upper level is the lowest level of positive parity in the first electronically excited state, labelled here as A, v = 0, *R* = 0, +. Electric dipole transitions to the X state must change the parity and obey the selection rule Δ*R* = 0, ±1, which means that only the *R* = 1 rotational state is accessible.1 However, the molecule can decay to any vibrational state, since there is no selection rule dictating how v can change in an electronic transition. The branching ratio to each vibrational state is mainly given by the square of the overlap integral between the vibrational wavefunctions in the lower and upper electronic states, which is known as the Franck Condon factor. For molecules where the optically active electron is not involved in the bonding, the sets of vibrational wavefunctions for the two electronic states are very similar. In this case, the branching ratio is close to 1 for the Δv = 0 transition and diminishes rapidly with increasing Δv. These molecules are the ones best suited to laser cooling because only a few vibrational bands need to be addressed, requiring just a few lasers, as indicated in the figure. Hyperfine components of these transitions can usually be addressed by adding radio-frequency sidebands to each laser using acousto-optic or electro-optic modulators.

<sup>1</sup>Often, *R* is not a good quantum number because it is coupled strongly to other angular momenta in the molecule, such as the orbital angular momenta of the electrons. In this case, *R* may be replaced by the relevant coupled angular momentum and the same principles apply.

**Fig. 3** Energy level structure of a typical diatomic molecule, with indicative transition frequencies. The ground electronic state is labelled X, and the excited states are A, B, etc. Transitions between electronic states are usually in, or near, the visible part of the electromagnetic spectrum. Each electronic state has a set of vibrational states, labelled by v. Vibrational transitions are usually in the mid-infrared. Each vibrational state has a set of rotational states, labelled by the rotational angular momentum, *R*, and the parity *p*. Rotational transitions are in the microwave regime. Each rotational state usually has a set of hyperfine states with total angular momentum *F* determined by the electronic angular momentum and the nuclear spin. In the example shown, both are 1/2. The inset shows the laser cooling scheme discussed in the text. Arrows show transitions driven by lasers, and the weights of the dotted lines indicate the relative branching ratios of the decay channels

#### *5.2 Doppler and Sub-Doppler Cooling*

Despite the complexity of the molecular structure outlined above, and the need to drive many transitions using several laser frequencies, the basic principles of laser cooling can be understood by focussing on just two or three levels. Figure 4a illustrates the principle of Doppler cooling applied in one dimension to a hypothetical two-level molecule. A molecule moving to the right with speed v interacts with a pair of identical, counter-propagating laser beams with wavevector *k*. The frequency of the light, ω, is slightly smaller than the molecular transition frequency, ω0. The laser beam from the right is Doppler shifted closer to resonance, so the molecule scatters more photons from this beam and slows down as a result of this imbalanced radiation pressure. The force on the molecule due to each of the beams is

**Fig. 4** Principles of **a** Doppler cooling, where a negative detuning is used and **b** sub-Doppler cooling, where a positive detuning is used

$$F\_{\pm} = \pm \frac{\hbar k \,\Gamma}{2} \frac{I/I\_s}{(1 + I/I\_s + 4(\delta\_0 \mp kv)^2/\Gamma^2)},\tag{2}$$

where δ<sup>0</sup> = ω − ω<sup>0</sup> is the detuning, Γ is the natural linewidth of the transition, *I* is the laser intensity and *I*<sup>s</sup> is a characteristic intensity known as the saturation intensity. The graph in Fig. 4a shows these two forces as a function of v in the case where *I* = *I*<sup>s</sup> and δ<sup>0</sup> = −2Γ . The solid line is their sum and shows that there is a force driving the molecule towards zero velocity. In addition to this cooling force, there is heating due to the randomly-directed momentum kicks associated with the photon scattering events. When the heating and cooling rates are balanced, the molecule reaches its equilibrium temperature. The minimum temperature for Doppler cooling is known as the Doppler limit and is *<sup>T</sup>*<sup>D</sup>,min <sup>=</sup> -Γ /(2*k*B).

Figure 4b illustrates a method of sub-Doppler cooling. Here, we distinguish two Zeeman sub-levels of the ground state. Due to the choice of states and the angular momentum selection rules, laser light of a given polarization cannot drive transitions from one of the ground states. We call this the dark state because it does not couple to the light. The other state is called the bright state. The bright state has an ac Stark shift which is positive when the detuning is positive. The dark state has no ac Stark shift because it does not couple to the light. If the two counter-propagating laser beams have different polarizations, neither parallel nor orthogonal, both the intensity and

polarization of the light field will vary with position. This causes the ac Stark shift of the bright state to be modulated in position, setting up potential energy hills and valleys which the molecules move through. A molecule in the bright state will be excited by the light and optically pumped into the dark state. This is most likely to happen at positions where the intensity of the light field is high, which are also the positions where the ac Stark shift is largest, i.e. near the tops of the hills. A moving molecule can make a non-adiabatic transition back to the bright state because the polarization changes as it moves. This is most likely to happen where the energy gap between bright and dark states is smallest, i.e. near the bottom of the hills. As a result, molecules moving through the light field lose energy because they climb hills more often than they descend into valleys. The graph in Fig. 4b shows the typical force produced by this mechanism. It operates over a smaller range of velocities than Doppler cooling, and produces a smaller maximum force. Crucially however, the gradient near zero velocity, which is the damping constant, is substantially higher than for Doppler cooling. Furthermore, because the molecule spends much of its time in the dark state, there is less photon scattering, and thus a lower heating rate. Thus, for small velocities, the cooling rate is higher while the heating rate is lower, leading to much lower temperatures.

The Doppler cooling mechanism shown in Fig. 4a requires a negative detuning, while the sub-Doppler cooling mechanism requires a positive detuning.2 The two mechanisms often appear together, resulting in the typical velocity-dependent force illustrated in Fig. 5. A negative detuning is useful for capturing molecules with a wide range of initial velocities and cooling them to lower velocity. However, the lowest temperatures are not reached because the total force has the wrong sign at low

<sup>2</sup>Note that there are other methods of sub-Doppler cooling, commonly used to cool atoms, that work for negative detunings. For molecules, it appears that sub-Doppler cooling always requires a positive detuning.

velocity. Once molecules are slow enough, the frequency of the light can be switched to a positive detuning so that the sub-Doppler mechanism cools them further. In this way, molecules have been cooled to temperatures far below the Doppler limit.

#### **6 Laser Cooling of Molecules: Practice**

Figure 6 illustrates an apparatus for laser cooling and trapping of molecules. The experiments begin with a molecular beam, a testament to the experimental power of the method developed by Stern and subsequent researchers. The cryogenic buffer gas sources described in Sect. 2 are ideal for this application because they deliver the critical combination of a high flux of molecules with a low initial speed. The illustration shows how this beam is laser cooled in the transverse directions, decelerated to low speed using radiation pressure, and then captured and cooled in a magneto-optical trap. We discuss each of these steps in turn.

#### *6.1 Transverse Laser Cooling of a Molecular Beam*

The density in a molecular beam drops with distance from the source because the beam spreads out as it propagates. Laser cooling can reduce the transverse temperature enormously, resulting in an intense, highly collimated molecular beam. The pioneering work on laser cooling of molecules was done at Yale [58]. They worked with a beam of SrF molecules and showed how to cool the beam in one transverse direction using both Doppler cooling and sub-Doppler cooling. Several other diatomic and polyatomic molecular species have since been cooled using similar

**Fig. 6** An illustration of an apparatus for laser cooling and trapping of molecules. A beam of molecules from a cryogenic buffer gas source is cooled in the transverse directions, decelerated by the radiation pressure of a counter-propagating laser beam, and captured in a magneto-optical trap

**Fig. 7** Transverse laser cooling of YbF molecules. **a** Schematic of experiment. **b** Density distribution in one transverse direction following laser cooling in that direction. The lines are fits to a sum of four Gaussians. Adapted from [54]

methods [54, 59–64]. Figure 7a illustrates an experiment [54] to cool YbF molecules in one transverse direction, *x*. A beam of molecules from a cryogenic buffer gas source passes through a 20 cm long sheet of laser cooling light that forms a standing wave in the *x* direction. All molecules are then optically pumped to the lowest vibrational level by the clean-up light, and then detected by laser-induced fluorescence on a camera. Figure 7b shows the resulting density distribution of these molecules along *x*. When the detuning of the laser light is positive, a narrow peak appears at the centre of the distribution, corresponding to molecules that have been cooled to low temperature by sub-Doppler cooling. In this experiment, the transverse temperature of the cooled molecules was found to be below 100 μK. When the detuning is negative, a dip appears at the centre with broad wings on either side where the molecules have accumulated. The reason for this profile can be appreciated from the form of the force curve in Fig. 5. For negative detuning, the sub-Doppler mechanism forces molecules near zero velocity to a higher velocity, while Doppler cooling forces high velocity molecules towards lower velocities. As a result, molecules accumulate around the non-zero velocity where the force curve crosses zero.

#### *6.2 Slowing a Molecular Beam with Radiation Pressure*

Transverse laser cooling produces a highly collimated molecular beam, but the molecules still have a high forward speed. They can be decelerated using the radiation pressure of a laser beam propagating in the opposite direction. Here, it is essential to account for the changing Doppler shift as the molecules slow down. This can be done by chirping the frequency of the laser so that it follows the changing Doppler shift, or by broadening the frequency spectrum of the laser to cover the full range of Doppler shifts. Laser slowing of molecules was first demonstrated by the group at Yale using SrF [65], and similar methods have been applied to other molecules [66–69].

Figure 8 illustrates frequency-chirped laser slowing of a cryogenic buffer-gas beam of CaF molecules [69]. The black curves show the velocity distributions with no slowing applied, and the coloured curves show the distributions after slowing using various frequency chirps. The initial frequency of the laser is tuned to be resonant with molecules moving at about 180 m/s. When there is no chirp, the molecules are decelerated to about 100 m/s. They bunch up around this speed because the faster molecules are initially closer to resonance so are decelerated more than the slower ones. The distribution is shifted to lower velocities as the chirp increases, but the number of detected molecules drops at low velocities. This is because there is no transverse cooling in these experiments, so the beam diverges rapidly as it slows down, reducing the number of molecules that pass through the detector.

#### *6.3 Trapping the Molecules*

With the molecules slowed to low velocity, it becomes possible to trap them. Magneto-optical traps (MOTs) have been used to cool and trap atoms for decades [71]. In a MOT, counter-propagating laser beams result in a velocity-dependent force which cools the atoms, as described in Sect. 5.2. The detuning of the light is usually negative so that Doppler cooling, with its large capture velocity, is the dominant process. This alone does not trap the atoms because the force does not depend on position. To produce a position-dependent force, a magnetic field gradient is added,

typically by using a pair of coils as illustrated in Fig. 6. The current flows in opposite directions in the two coils so that the magnitude of the field is zero at the centre and increases linearly in all directions away from this point. For a stationary atom at the field-zero, there is no net force because the atom is equally likely to scatter photons from any of the laser beams. When the atom is displaced, the Zeeman effect shifts the frequencies of transitions with Δ*mF* = ±1 in opposite directions, one closer to resonance and the other further away. These two transitions are driven by circularly polarized light of opposite handedness. By choosing the handedness of the beams in the correct way, the transition closest to resonance is always driven by the beam that pushes the atom back towards the field-zero. This traps the atoms.

**Fig. 9** Damped oscillations of CaF molecules in a magneto-optical trap. **a** Fluorescence images of the trapped molecules at various times after giving the molecules a radial push. **b** Red points: position of the centre of the cloud versus time. Blue line: fit to a damped harmonic oscillator model. Figure taken from [70]

The first three-dimensional magneto-optical trap of molecules was demonstrated in 2014 by the Yale group [72]. They captured a few hundred SrF molecules for about 50 ms at a density of 6 × 10<sup>2</sup> cm−3, and cooled them to 2.3 mK. They went on to investigate several other trapping configurations [73–75] and were soon able to trap about 10<sup>4</sup> molecules for 500 ms at a density of 2.5 × 10<sup>5</sup> cm−<sup>3</sup> and temperatures down to 250 μK. MOTs of CaF [70, 76, 77] and YO [78] molecules have also been produced, with steadily increasing number densities. Figure 9a is a sequence of pictures showing CaF molecules trapped in a MOT. Each picture is made by imaging the fluorescence of the trapped molecules onto a camera. The molecules were given a sudden push in the horizontal direction at time T=0 and the subsequent images show them oscillating in the trap. Figure 9b shows the displacement of the cloud versus T together with a fit to the motion of a damped harmonic oscillator. The results show that the MOT exhibits both a restoring force and a damping force, as expected.

Once molecules have been trapped in a MOT, they can be cooled to much lower temperature using the sub-Doppler cooling method described in Sect. 5.2. Typically, the magnetic field is turned off and the detuning of the lasers is switched from negative to positive. The molecules typically cool below the Doppler limit [76, 79, 80] in about 1 ms, and temperatures as low as 4 μK have been reached in this way [81–83]. At this point the cooling can be turned off and the molecules stored in conservative traps where their quantum states can be controlled and preserved for long periods. Ensembles of laser-cooled molecules have recently been confined in pure magnetic traps [80, 84] and in optical dipole traps [79], and single molecules have been held in tightly-focussed tweezer traps [85]. Coherent control of the hyperfine and rotational states of molecules has been studied [10, 84] and rotational superpositions with long coherence times have been demonstrated for trapped molecules [86].

#### **7 Applications of Laser-Cooled Molecules**

The ultracold molecules produced by direct laser cooling are well suited to a wide variety of exciting applications. Many of these applications require molecules trapped for long periods, long-lived coherences, and control over all degrees of freedom at the quantum level, often including the motional degree of freedom. Figure 10 illustrates four experimental approaches that satisfy some, or all, of these requirements. The molecules may be launched into a molecular fountain so that they are in free fall throughout a measurement [50, 87], or they could be trapped near the surface of a chip that integrates microscopic traps with superconducting microwave resonators [12]. Small, reconfigurable arrays of molecules can be produced using optical tweezer traps [85], while larger arrays can be made using optical lattices in one, two or three dimensions [88]. Here, we give an overview of future research directions using these platforms.

#### *7.1 Testing Fundamental Physics*

Ultracold molecules provide several avenues to constrain new theories or discover new physics [5]. As discussed in Sects. 3 and 4, the use of molecules to measure fundamental electric dipole moments is an amazingly powerful probe of symmetryviolating physics beyond the Standard Model. Other kinds of symmetry tests can also be done using molecules. For example, they can be used to explore parity violation in nuclei with unprecedented sensitivity [89]. Of particular interest is the measurement of nuclear anapole moments which arise from weak interactions within nuclei. A recent experiment with a beam of BaF has demonstrated the exceptional sensitivity achievable [90]. It is also of fundamental interest to measure the parity-violating energy difference between left- and right-handed chiral molecules, which is predicted but has not yet been observed [91]. The recent extension of laser cooling to polyatomic symmetric top molecules [63] shows that quite complex molecules can be cooled to sub-millikelvin temperature, making a parity-violation measurement using a laser cooled chiral species feasible in the future.

Precise measurements of molecular transition frequencies can also be used to test the idea that the fundamental constants may actually vary in time or space, or according to the local density of matter. Such variations are predicted by theories that aim to unify gravity with the other forces, and by some theories of dark energy [92, 93]. The frequencies of molecular transitions depend primarily on two fundamental constants, the fine-structure constant α and the proton-to-electron mass ratio μ = *mp*/*me*. The rotational and vibrational frequencies of molecules scale as μ−<sup>1</sup> and μ−1/<sup>2</sup> respectively, a direct dependence that an electronic transition in an atom does not have. Moreover, certain transitions have enhanced sensitivity to α or μ [94], sometimes because the transition energy results from a near cancellation between two large contributions of different origin. Astrophysical observations of atomic and molec-

**Fig. 10** Techniques for controlling ultracold molecules. **a** Molecules are launched into a fountain. **b** Molecules are stored on the surface of a chip using either electric or magnetic potentials created by planar electrodes or current-carrying wires. **c** Molecules are trapped in a 1D, 2D, or 3D lattice created by interfering counter-propagating lasers. **d** Molecules are trapped using an array of optical tweezers