Diffusive-Light Invisibility Cloaking

[From Left to Right] Robert Schittny, Muamer Kadic, Tiemo Bückmann, Martin Wegener

Authors:
Robert Schittny^1,2, Muamer Kadic^1,3, Tiemo Bückmann^1,2, Martin Wegener^1,2,3

Affiliations:
¹Institute of Applied Physics, Karlsruhe Institute of Technology (KIT), Germany, 
²DFG-Center for Functional Nanostructures (CFN), Karlsruhe Institute of Technology (KIT), Germany, 
³Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Germany.

In an invisibility cloak [1–5], light is guided on a detour around an object such that it emerges behind unchanged, thus making the object invisible to an outside observer. An ideal cloak should be macroscopic and work perfectly for any direction, polarization, and wavelength of the incoming light. To make up for the geometrical detour, light has to travel faster inside the cloak than outside, that is, faster than the vacuum speed of light for cloaking in air or vacuum. Furthermore, the absence of wavelength dependence means that energy velocity and phase velocity are strictly equal. However, general relativity forbids energy velocities higher than the vacuum speed of light. Thus, macroscopic, omnidirectional, and broadband invisibility cloaking is fundamentally impossible in air [4, 5]. Consistently, all experimental demonstrations of optical cloaking so far came with a drawback in terms of operation bandwidth, size, or both [6–10].

In contrast to this, more recently, we have demonstrated [11] close to ideal macroscopic and broadband invisibility cloaking in diffusive light scattering media at visible wavelengths.

Past 2Physics articles by this Group:
May 06, 2012: "A Cloak for Elastic Waves in Thin Polymer Plates"
     by Nicolas Stenger, Manfred Wilhelm, Martin Wegener
June 19, 2011:
"3D Polarization-Independent Invisibility Cloak at Visible Wavelengths"
     by Tolga Ergin, Joachim Fischer, Martin Wegener
April 11, 2010: "3D Invisibility Cloaking Device at Optical Wavelengths"
     by Tolga Ergin, Nicolas Stenger, Martin Wegener

Fig. 1 [11] illustrates the principle and results of invisibility cloaking in diffusive media. In such media, many scattering particles are randomly distributed, causing each photon to travel along a random path (see artistic illustration in the magnifying glass in Fig. 1). This effectively slows down light with respect to the vacuum speed of light, making perfect cloaking possible. In contrast to “ballistic” light propagation in vacuum or air as described by Maxwell’s equations, light propagation in such a medium can be described by diffusion of photons [12].

Figure 1 (Ref.[11]): Principle of diffusive-light cloaking. Computer-generated image of an illuminated cuboid diffusive medium with a zero-diffusivity obstacle (left-hand side) and a core-shell cloak (right-hand side). The magnifying glass shows an artistic illustration of a photon’s random walk inside the diffuse medium. The black streamline arrows are simulation results illustrating the photon current around obstacle and cloak. Corresponding measurement results are projected onto the front side of the cuboid volume, showing a diffuse shadow for the obstacle (left-hand side) and its elimination for the cloak (right-hand side). The euro coin illustrates the macroscopic dimensions of the cloak.

If a diffusive medium is illuminated from one side, any object with a different diffusivity inside this medium will cause perturbations of the photon flow. On the left-hand side of Fig. 1, a hollow cylinder with a diffusivity of exactly zero (the “obstacle”) suppresses any photon flow inside and casts a pronounced shadow, reducing the photon current on the downstream side (see black streamline arrows in Fig. 1). To compensate for this, a thin layer with a higher diffusivity than in the surrounding medium is added to the cylinder on the right-hand side of Fig. 1 (the “cloak”). Intuitively, a higher diffusivity (that is, a lower concentration of scattering particles) leads to an effectively higher light propagation speed and thus makes up for the geometrical detour the light has to take on its way around the obstacle. The black streamline arrows show that the photon current behind the cloak is unchanged. In other words, the shadow cast by the obstacle vanishes.

Such a core-shell cloak design can be thought of as the reduction of more complex multilayer designs based on transformation optics [1–3] to just two layers. It is known theoretically [13, 14] to work perfectly in the static case and for spatially constant gradients of the photon density across the cloak. Core-shell cloaks have been demonstrated before in magnetostatics [15], thermodynamics [16, 17], and elastostatics [18], recently even for non-constant gradients [16, 17].

For our experiments, we used a hollow aluminum cylinder as the obstacle, coated with a thin layer of white paint that acted as a diffusive reflector. For the cloaking shell, we coated the cylinder with a thin layer of a transparent silicone doped with dielectric microparticles. Obstacle and cloak are truly macroscopic, as indicated by the euro coin in Fig. 1 for comparison. We realized the diffusive background medium by mixing de-ionized water and white wall-paint. By changing the paint concentration, we could easily vary the surrounding’s diffusivity to find good cloaking performance. Other common examples of diffusive media are clouds, fog, paper or milk.

The samples were submerged in a Plexiglas tank filled with the water-paint mixture. The tank was illuminated from one side with white light coming from a computer monitor; photographs of the other side of the tank were taken with an optical camera. Two of these photographs are projected onto the front side of the cuboid volume shown in Fig. 1. The left-hand side shows the case with just the obstacle inside, exhibiting a pronounced diffuse shadow as expected from the discussion above. This shadow vanishes almost completely on the right-hand side, where the cloak is inside the tank. The yellowish tint of the photographs is caused by partial absorption of blue light in the water-paint mixture. Furthermore, we could trace the small remaining intensity variations for the cloaking case back to a finite absorption of light at the core-shell interface.

While the illustration in Fig. 1 only shows results for homogeneous illumination, we also found excellent cloaking performance using an inhomogeneous line-like illumination pattern (not depicted). Furthermore, we also performed successful experiments with spherical samples (not depicted), proving that our cloak is truly three-dimensional and works for any polarization and any direction of incidence.

References:
[1] J. B. Pendry, D. Schurig, D. R. Smith, "Controlling electromagnetic fields". Science, 312, 1780 (2006). Abstract.
[2] Ulf Leonhardt, "Optical conformal mapping". Science, 312, 1777 (2006). Abstract.
[3] Vladimir M. Shalaev, "Transforming light". Science, 322, 384 (2008). Abstract.
[4] David A. B. Miller, "On perfect cloaking". Optics Express, 14, 12457 (2006). Full Article.
[5] Hila Hashemi, Baile Zhang, J. D. Joannopoulos, Steven G. Johnson, "Delay-bandwidth and delay-loss limitations for cloaking of large objects". Physical Review Letters, 104, 253903 (2010). Abstract.
[6] D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B. Pendry, A.F. Starr, D.R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies". Science, 314, 977 (2006). Abstract.
[7] R. Liu, C. Ji, J.J. Mock, J.Y. Chin, T.J. Cui, D.R. Smith, "Broadband ground-plane cloak". Science, 323, 366 (2009). Abstract.
[8] Jason Valentine, Jensen Li, Thomas Zentgraf, Guy Bartal, Xiang Zhang, "An optical cloak made of dielectrics". Nature Materials, 8, 568 (2009). Abstract.
[9] Lucas H. Gabrielli, Jaime Cardenas, Carl B. Poitras, Michal Lipson, "Silicon nanostructure cloak operating at optical frequencies". Nature Photonics, 3, 461 (2009). Abstract.
[10] Tolga Ergin, Nicolas Stenger, Patrice Brenner, John B. Pendry, Martin Wegener, "Three-dimensional invisibility cloak at optical wavelengths". Science, 328, 337 (2010). Abstract. 2Physics Article.
[11] Robert Schittny, Muamer Kadic, Tiemo Bückmann, Martin Wegener, "Invisibility Cloaking in a Diffusive Light Scattering Medium". Science, Published Online June 5 (2014). DOI:10.1126/science.1254524.
[12] C. M. Soukoulis, Ed., “Photonic Crystals and Light Localization in the 21st Century”, (Springer, 2001).
[13] Graeme W. Milton, “The Theory of Composites”, (Cambridge Univ. Press, 2002).
[14] Andrea Alù, Nader Engheta, "Achieving transparency with plasmonic and metamaterial coatings". Physical Review E, 72, 016623 (2005). Abstract.
[15] Fedor Gömöry, Mykola Solovyov, Ján Šouc, Carles Navau, Jordi Prat-Camps, Alvaro Sanchez, "Experimental realization of a magnetic cloak". Science, 335, 1466 (2012). Abstract.
[16] Hongyi Xu, Xihang Shi, Fei Gao, Handong Sun, Baile Zhang, "Ultrathin three-dimensional thermal cloak". Physical Review Letters, 112, 054301 (2014). Abstract.
[17] Tiancheng Han, Xue Bai, Dongliang Gao, John T. L. Thong, Baowen Li, Cheng-Wei Qiu, "Experimental demonstration of a bilayer thermal cloak". Physical Review Letters, 112, 054302 (2014). Abstract.
[18] T. Bückmann, M. Thiel, M. Kadic, R. Schittny, M. Wegener, "An elasto-mechanical unfeelability cloak made of pentamode metamaterials". Nature Communications, 5, 4130 (2014). Abstract.

2D Electronic-Vibrational Spectroscopy Technique Provides Unprecedented Look into Photochemical Reactions

From left to Right: Nicholas Lewis, Graham Fleming and Tom Oliver [photo courtesy: Lawrence Berkeley National Laboratory, USA].

From allowing our eyes to see, to enabling green plants to harvest energy from the sun, photochemical reactions – reactions triggered by light – are both ubiquitous and critical to nature. Photochemical reactions also play essential roles in high technology, from the creation of new nanomaterials to the development of more efficient solar energy systems. Using photochemical reactions to our best advantage requires a deep understanding of the interplay between the electrons and atomic nuclei within a molecular system after that system has been excited by light. A major advance towards acquiring this knowledge has been reported by a team of researchers with the U.S. Department of Energy (DOE)’s Lawrence Berkeley National Laboratory (Berkeley Lab) and the University of California (UC) Berkeley.

Graham Fleming, UC Berkeley’s Vice Chancellor for Research, a faculty senior scientist with Berkeley Lab’s Physical Biosciences Division, and member of the Kavli Energy NanoSciences Institute at Berkeley, led the development of a new experimental technique called two-dimensional electronic-vibrational spectroscopy (2D-EV). By combining the advantages of two well-established spectroscopy technologies – 2D-electronic and 2D-infrared – this technique is the first that can be used to simultaneously monitor electronic and molecular dynamics on a femtosecond (millionth of a billionth of a second) time-scale. The results show how the coupling of electronic states and nuclear vibrations affect the outcome of photochemical reactions.

“We think that 2D-EV, by providing unprecedented details about photochemical reaction dynamics, has the potential to answer many currently inaccessible questions about photochemical and photobiological systems,” says Fleming, a physical chemist and internationally acclaimed leader in spectroscopic studies of events that take place on the femtosecond time-scale. “We anticipate its adoption by leading laboratories across the globe.”

Fleming is the corresponding author of a paper in the Proceedings of the National Academy of Sciences (PNAS) titled “Correlating the motion of electrons and nuclei with two-dimensional electronic–vibrational spectroscopy” [1]. Co-authors are Thomas Oliver and Nicholas Lewis, both members of Fleming’s research group.

Fleming and his research group were one of the key developers of 2D electronic spectroscopy (2D-ES), which enables scientists to follow the flow of light-induced excitation energy through molecular systems with femtosecond temporal resolution. Since its introduction in 2007, 2D-ES has become an essential tool for investigating the electronic relaxation and energy transfer dynamics of molecules, molecular systems and nanomaterials following photoexcitation. 2D infrared spectroscopy is the go-to tool for studying nuclear vibrational couplings and ground-state structures of chemical and complex biological systems.

“Combining these two techniques into 2D-EV tells us how photoexcitation affects the coupling of electronic and vibrational degrees of freedom,” says Oliver. “This coupling is essential to understanding how all molecules, molecular systems and nanomaterials function.”

In 2D-EV, a sample is sequentially flashed with three femtosecond pulses of laser light. The first two pulses are visible light that create excited electronic states in the sample. The third pulse is mid-infrared light that probes the vibrational quantum state of the excited system. This unique combination of visible excitation and mid-infrared probe pulses enables researchers to correlate the initial electronic absorption of light with the subsequent evolution of nuclear vibrations.

“2D-EV’s ability to correlate the initial excitation of the electronic–vibrational manifold with the subsequent evolution of high-frequency vibrational modes, which until now have not been explored, opens many potential avenues of fruitful study, especially in systems where electronic–vibrational coupling is important to the functionality of a system,” Fleming says.

As a demonstration, Oliver, Lewis and Fleming used their 2D-EV spectroscopy technique to study the excited-state relaxation dynamics of DCM dye dissolved in a deuterated solvent. DCM is considered a model “push-pull” emitter – meaning it contains both electron donor and acceptor groups – but with a long-standing question as to how it fluoresces back to the ground energy state.

Figure 1: 2D-EV spectral data tells researchers how photoexcitation of a molecular system affects the coupling of electronic and nuclear vibrations that is essential to understanding how the system functions.

“From 2D-EV spectra, we elucidate a ballistic mechanism on the excited state potential energy surface whereby molecules are almost instantaneously projected uphill in energy toward a transition state between locally excited and charge-transfer states before emission,” Oliver says. “The underlying electronic dynamics, which occur on the hundreds of femtoseconds time-scale, drive the far slower ensuing nuclear motions on the excited state potential surface, and serve as an excellent illustration for the unprecedented detail that 2D-EV will afford to photochemical reaction dynamics.”

One example of how 2D-EV might be applied is in the study of rhodopsin, the pigment protein in the retina of the eye that is the primary light detector for vision, and carotenoids, the family of pigment proteins, such as chlorophyll, found in green plants and certain bacteria that absorb light for photosynthesis.

“The nonradiative energy transfer in rhodopsin and carotenoids is thought to involve the breakdown of one of the most widely used approximations of quantum mechanics, the Born-Oppenheimer approximation, which states that since motion of electrons are far faster than nuclei, as represented by vibrational motion, the nuclei respond to changes in electronic states,” Oliver says. “With 2D-EV, we will be able to directly correlate the degrees of electronic and vibrational freedom and track their evolution as a function of time. It’s a chicken and egg kind of problem: Do the electrons or nuclei move first? 2D-EV will give us insight into whether or not the Born-Oppenheimer approximation is still valid in these cases.”

For nanomaterials, 2D-EV should be able to shed much needed light on how the coupling of phonons – atomic soundwaves – with electrons impacts the properties of carbon nanotubes and other nanosystems. 2D-EV can also be used to investigate the barriers to electron transfer between donor and acceptor states in photovoltaic systems.

“We are continuing to refine the 2D-EV technology and make it more widely applicable so that it can be used to study lower frequency motions that are of great scientific interest,” Oliver says.

This research was funded by the DOE Office of Science and the National Science Foundation.

References:
[1] Thomas A. A. Oliver, Nicholas H. C. Lewis, Graham R. Fleming, "Correlating the motion of electrons and nuclei with two-dimensional electronic–vibrational spectroscopy". Proceedings of the National Academy of Sciences of the United States of America, published May 16 (2016). doi: 10.1073/pnas.1409207111.

[The text is authored by Lynn Yaris of Lawrence Berkeley National Laoratory]

Light that is Advanced

Paul Lett 
(photo courtesy: University of Maryland at College Park)

Michael Lewis’s bestselling book Flash Boys describes how some brokers, engaging in high frequency trading, exploit fast telecommunications to gain fraction-of-a-second advantage in the buying and selling of stocks. But you don’t need to have billions of dollars riding on this-second securities transactions to appreciate the importance of fast signal processing. From internet to video streaming, we want things fast.

Paul Lett and his colleagues at the Joint Quantum Institute (JQI, jointly operated by the National Institute of Standards and Technology in Gaithersburg, MD and the University of Maryland in College Park) specialize in producing modulated beams of light for encoding information. They haven’t found a way to move data faster than c, the speed of light in a vacuum, but in a new experiment they have looked at how light traveling through so called “fast-light” materials does seem to advance faster than c, at least in one limited sense. They report their results (published online May 25, 2014) in the journal 'Nature Photonics'.

Seeing how light can be manipulated in this way requires a look at several key concepts, such as entanglement, mutual information, and anomalous dispersion. At the end we’ll arrive at a forefront result.

Continuous Variable Entanglement :

Much research at JQI is devoted to the processing of quantum information, information coded in the form of qubits. Qubits, in turn are tiny quantum systems---sometimes electrons trapped in a semiconductor, sometimes atoms or ions held in a trap---maintained in a superposition of states. The utility of qubits increases when two or more of them can be yoked into a larger quantum arrangement, a process called entanglement. Two entangled photons are not really sovereign particles but parts of a single quantum entity.

The basis of entanglement is often a discrete variable, such as electron spin (whose value can be up or down) or photon polarization (say, horizontal or vertical). The essence of entanglement is this: while the polarization of each photon is indeterminate until a measurement is made, once you measure the polarization of one of the pair of entangled photons, you automatically know the other photon’s polarization too.

But the mode of entanglement can also be vested in a continuous variable. In Lett’s lab, for instance, two whole light beams can be entangled. Here the operative variable is not polarization but phase (how far along in the cycle of the wave you are) or intensity (how many photons are in the beam). For a light beam, phase and intensity are not discrete (up or down) but continuous in variability.

Quantum Mutual Information:

Biologists examining the un-seamed strands of DNA can (courtesy of the correlated nature of nucleic acid constituents) deduce the sequence of bases along one strand by examining the sequence of the other strand. So it is with entangled beams. A slight fluctuation of the instantaneous intensity of one beam (such fluctuations are inevitable because of the Heisenberg uncertainty principle) will be matched by a comparable fluctuation in the other beam.

Lett and his colleagues make entangled beams in a process called four-wave mixing. A laser beam (pump beam) enters a vapor-filled cell. Here two photons from the pump beam are converted into two daughter photons proceeding onwards with different energies and directions. These photons constitute beams in their own right, one called the probe beam, the other called the conjugate beam. Both of these beams are too weak to measure directly. Instead each beam enters a beam splitter (yellow disk in the drawing below) where its light can be combined with light from a local oscillator (which also serves as a phase reference). The ensuing interference patterns provide aggregate phase or intensity information for the two beams.

When the beam entanglement is perfect, the mutual correlation is 1. When studying the intensity fluctuations of one beam tells you nothing about those of the other beam, then the mutual correlation is 0.

Fast-Light Material:

In a famous experiment, Isaac Newton showed how incoming sunlight split apart into a spectrum of colors when it passed through a prism. The degree of wavelength-dependent dispersion for a material that causes this splitting of colors is referred to as its index of refraction.

In most materials the index is larger than 1. For plain window glass, it is about 1.4; for water it is 1.33 for visible light, and gradually increases as the frequency of the light goes up. At much higher frequency (equivalent to shorter wavelength), though, the index can change its value abruptly and go down. For glass, that occurs at ultraviolet wavelengths so you don’t ordinarily see this “anomalous dispersion” effect. In a warm vapor of rubidium atoms, however, (and especially when modified with laser light) the effect can occur at infrared wavelengths, and here is where the JQI experiment looks.

Figure 1: Experimental setup for studying fast light. Pump beams (purple) create correlated probe (turquoise) and conjugate (gold) beams. Each of these beams is aimed at a beam splitter (yellow disks). A local oscillator (LO) also sends a laser beam into each of the beam splitters. The resulting interference pattern---registered in a spectrum analyzer, SA---for the probe and conjugate arms are compared [Image courtesy: Paul Lett, JQI]

In Figure 1  notice that the conjugate beam is sent through a second cell, filled with rubidium vapor. Here the beam is subject to dispersion. The JQI experiment aims to study how the entanglement of this conjugate beam with the probe beam (subject to no dispersion) holds up.

When the refraction is “normal”---that is, when index of refraction causes ordinary dispersion---the light signal is slowed in comparison with the beam which doesn’t undergo dispersion. For this set of conditions, the cell is referred to as a “slow-light” material. When, however, the frequency is just right, the conjugate beam will undergo anomalous dispersion. When the different frequency components that constitute a pulse or intensity fluctuation reformulate themselves as they emerge from the cell, they will now be just slightly ahead of a pulse that hadn’t gone through the cell. (To make a proper measurement of delay one needs two entangled beams---beams whose fluctuations are related.)

Causality:

No, the JQI researchers are not saying that any information is traveling faster than c. Figure 2 shows that the peak for the mutual information for the fast-light-material is indeed ahead of the comparable peaks for an unscattered beam or for a beam emerging from a slow-light material. It turns out that the cost of achieving anomalous dispersion at all has been that additional gain (amplification) is needed, and this amplification imposes noise onto the signal.

Figure 2. The mutual information of the two beams (how much we know about one beam if we know the fluctuation of the other beam) peaks at different times depending on whether the conjugate beam passes through a fast-light medium (red), a slow-light medium (green), or no medium at all (black) [Image courtesy: Paul Lett, JQI].

This inherent limitation in extracting useful information from an incoming light beam is even more pronounced with beams containing (on average) one or less-than-one photon. Such dilute beams are desirable in many quantum experiments where measurement control or the storage or delay of quantum information is important.

“We did these experiments not to try to violate causality, said Paul Lett, “but because we wanted to see the fundamental way that quantum noise “enforces” causality, and working near the limits of quantum noise also lets us examine the somewhat surprising differences between slow and fast light materials when it comes to the transport of information.”

Reference:
[1] Jeremy B. Clark, Ryan T. Glasser, Quentin Glorieux, Ulrich Vogl, Tian Li, Kevin M. Jones, Paul D. Lett, "Quantum mutual information of an entangled state propagating through a fast-light medium". Nature Photonics, published online May 25th (2014). doi:10.1038/nphoton.2014.112.

Excited Efimov State Observed

Rudolf Grimm (Photo by: C. Lackner)

Author: Rudolf Grimm

Affiliations:
Institute of Experimental Physics, University of Innsbruck, Austria,
and
Institute of Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Innsbruck, Austria.



Introduction:

In 1970, the Russian physicist Vitaly Efimov made a prediction that today represents one of the most bizarre and fascinating results of quantum mechanics [1]. In the context of nuclear physics, he considered the elementary situation of three bosons with pairwise interactions near a scattering resonance. He found an infinite ladder of three-body bound states, even existing under conditions where the interaction is too weak to support two-bound bound states. Until 2006, these Efimov trimer states have remained a theoretical curiosity without any experimental confirmation. Then the situation changed with the first observations in ultracold atomic systems, and Efimov’s scenario turned into a hot topic with many experimental investigations [2].

Efimov’s factor 22.7 :

Inherent to Efimov’s scenario is a discrete scaling law, expressing the self-similarity of all these trimer states. Discrete scaling means that, if a three-body state exists, then another state must also exist, being just a factor of 22.7 larger. Similarly the binding energy is by factor 22.7² smaller. This leads to an infinite series of states, as illustrated in Fig. 1. In practice, by the usual standards of molecular physics, already the first state is a very large quantum object (a so-called “halo state”), as it is a hundred times larger than conventional trimers and it is extremely weakly bound. In our earlier experiments on cesium atoms [3], we found the first Efimov state with a size of roughly 50 nm.

The second state in the series, also referred to as the first excited Efimov trimer, is expected to be 22.7 larger and about 500 times more weakly bound than the lowest Efimov state. This makes it an extreme molecular quantum object, 1 µm in size and exceptionally weakly bound.

Fig. 1: Efimov’s scenario for three interacting particles. The energy of the Efimov states (green solid lines) is plotted versus the inverse scattering length. In the grey-shaded region (E>0), the system consists of unbound atoms. In the blue-shaded region, atoms coexist with dimers. In the green-shaded region, the trimer states occur. For illustration purposes, the discrete scaling factor is artificially reduced from 22.7 to 2.

Ultracold atomic gases:

In our experiments, an ultracold atomic gas of cesium atoms is prepared by methods of laser cooling and subsequent evaporative cooling. The ensemble of a few ten thousand atoms is then kept by the weak attractive forces of an infrared laser beam in a volume of about 50µm diameter. They atoms collide with each other in the trap and the ensemble is observed over a few seconds.

A key feature is the possibility to tune the quantum-mechanical interaction between pairs of atoms in a well-controlled way. Experimentalists take advantage of so-called Feshbach resonances [4], which arise as a consequence of the coupling of two colliding atoms to a molecular state. The collision sensitively depends on a magnetic field and -- at specific field strengths -- resonances occur. Very accurate knowledge of these resonances is an essential prerequisite for the experiments.

The experimental signature of the formation of an Efimov state is the loss of atoms from the trap [5]. At specific magnetic fields, loss is observed to be much faster than usual. This marks the situation when three atoms couple to an Efimov state. The Efimov state here is an object that can live for a few milliseconds, before it undergoes a fragmentation into a deeply bound molecule and a free atom, with the energetic fragments leaving the trap.

Three challenges:

For observing the excited Efimov trimer [6] we were confronted with three major challenges: (1) We needed an extremely cold gas, colder than everything else we had prepared in our lab before. We extended our previous methods with an additional stage, where we expanded the ultracold ample into an ultraweak electromagnetic trap made of a combination of magnetic and optical fields. In this way we reached a temperature as low as 7 nK. (2) We had to control the interaction properties near a resonance at an unprecedented accuracy level. Here we benefitted from the special properties of cesium and our long-standing expertise with this particular species [7]. (3) We had to understand the role of the finite temperatures in our sample, as even the 7nK reached is a relatively high temperature on the energy scale of the excited Efimov state. Thanks to a theoretical model developed by a group at the École Normale Supérieure, Paris [8], this was finally possible.

Observing the long-sought second Efimov resonance:

We show the main result of our experiments in Fig. 2. The loss of atoms out of the trap is quantified by a rate coefficient L3, which is plotted as a function of the inverse scattering length 1/a. In the center, where 1/a is zero, the strongest interaction is realized. The strongest loss does not occur there, but somewhat shifted away from this center. From a detailed analysis of this peak we could obtain the exact position where the Efimov state couples to three atoms colliding in the limit of zero energy. It appears when the scattering length has a value 20000(000) larger than Bohr’s radius (size of a hydrogen atom in the ground state). This result also characterizes the size of the Efimov state.

Fig. 2: The resonance that results from the excited Efimov state. The loss rate coefficient L3 is plotted as a function of the inverse scattering length. Two data sets have been recorded under similar conditions (sets A and B). The vertical dashed line indicates the expected position in the limit of zero temperature. The resonance is slightly upshifted because of the finite temperature of the sample. The solid line is a theoretical prediction based on the properties of the ground-state Efimov resonance, characterized in earlier work, with the grey-shaded region indicating the uncertainty range.

The final result for the Efimov period:

We can finally compare our observation for the first excited Efimov state to earlier measurements on the Efimov ground state in the same system [9]. The final result is that the second state is by a factor of 21.0 (with an uncertainty of ±1.3) larger than the first Efimov state. With some probability this result may still be consistent with the ideal factor of 22.7. However, we believe that the small deviation is due to the fact that the lowest Efimov state deviates somewhat from Efimov’s idealized case. This is theoretically expected, but not fully understood.

Future trends:

This exactly points to the intriguing question: How Efimov states appear in real systems and in how far the ideal scenario holds predictive power to describe the real situation. Many theoretical and experimental investigations currently pursue research along these lines.

The Efimov scenario is a paradigm for few-body physics, and it is just a tip of an iceberg of many more phenomena related to a few interacting particles in the quantum world. Many more phenomena will be discovered (see Ref. [10] for a recent example from the world of fermions). A particularly interesting question is how few-body interactions will affect the properties of macroscopic many-body quantum states, like in new superconducting materials.

Team and funding:

Bo Huang (graduate student) and Leonid Sidorenkov (postdoctoral researcher) carried out the work in the laboratory of the author at the University of Innsbruck. Theoretical support came from Jeremy M. Hutson, professor of physics and chemistry at the Univ. of Durham, United Kingdom. The experiment was funded by the Austrian Science Fund FWF.

The team: The main picture (photo M. Knabl/IQOQI) shows the experimental team Leonid Sidorenkov, Rudolf Grimm and Bo Huang on the background of the mountains Patscherkofel (right) and Glungezer (left) near Innsbruck, Austria. The inset shows Jeremy M. Hutson from the University of Durham, United Kingdom.

References
[1] V. Efimov, "Energy levels arising from resonant two-body forces in a three-body system". Physics Letters B, 33, 563 (1970). Abstract.
[2] Francesca Ferlaino, Rudolf Grimm, "Trend: Forty years of Efimov physics: How a bizarre prediction turned into a hot topic". Physics, 3, 9 (2010). Full Article.
[3] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, R. Grimm, "Evidence for Efimov quantum states in an ultracold gas of caesium atoms". Nature, 440, 315 (2006). Abstract.
[4] Cheng Chin, Rudolf Grimm, Paul Julienne, Eite Tiesinga, "Feshbach resonances in ultracold gases". Review of Modern Physics, 82, 1225 (2010). Abstract.
[5] B. D. Esry, Chris H. Greene, and James P. Burke, Jr., "Recombination of Three Atoms in the Ultracold Limit". Physical Review Letters, 83, 1751 (1999). Abstract.
[6] Bo Huang, Leonid A. Sidorenkov, Rudolf Grimm, Jeremy M. Hutson, "Observation of the Second Triatomic Resonance in Efimov’s Scenario". Physical Review Letters, 112, 190401 (2014). Abstract.
[7] Martin Berninger, Alessandro Zenesini, Bo Huang, Walter Harm, Hanns-Christoph Nägerl, Francesca Ferlaino, Rudolf Grimm, Paul S. Julienne, and Jeremy M. Hutson, "Feshbach resonances, weakly bound molecular states, and coupled-channel potentials for cesium at high magnetic fields". Physical Review, A 87, 032517 (2013). Abstract.
[8] B. S. Rem, A. T. Grier, I. Ferrier-Barbut, U. Eismann, T. Langen, N. Navon, L. Khaykovich, F. Werner, D. S. Petrov, F. Chevy, C. Salomon, "Lifetime of the Bose Gas with Resonant Interactions". Physical Review Letters, 110, 163202 (2013). Abstract.
[9] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, J. M. Hutson, "Universality of the Three-Body Parameter for Efimov States in Ultracold Cesium". Physical Review Letters, 107, 120401 (2011). Abstract.
[10] Michael Jag, Matteo Zaccanti, Marko Cetina, Rianne S. Lous, Florian Schreck, Rudolf Grimm, Dmitry S. Petrov, Jesper Levinsen, "Observation of a Strong Atom-Dimer Attraction in a Mass-Imbalanced Fermi-Fermi Mixture". Physical Review Letters, 112, 075302 (2014). Abstract.

Quantum Newton's Cradle with Bose-Einstein Condensates

Roberto Franzosi (left) and Ruggero Vaia (right)

Authors: Roberto Franzosi^1,2 and Ruggero Vaia^2,3

Affiliation:
¹QSTAR and Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Firenze, Italy,
²Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Sesto Fiorentino (FI), Italy,
³Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Sesto Fiorentino (FI), Italy.

1. Introduction

Fig.1: Newton's cradle

Newton’s cradle (Fig. 1) is a valuable paradigm of how physical mechanisms are concealed into nature. It is a device based on classical mechanics that demonstrates the conservation of momentum and energy. On the other hand, Quantum Mechanics has been shown to be one of most prolific sources of unexpected and hard-to-understand phenomena. Therefore, achieving a machine which is a paradigm for the quantum nature of a system is an engrossing challenge.

In the work of Ref.[1] we propose a possible experimental realization of a quantum analogue of Newton's cradle (NC). With that aim we ask for the system to be:
    (i) a one-dimensional array,
    (ii) made of individual quantum objects, representing the spheres in the NC,
    (iii) with a nearest-neighbour interaction between the individual quantum systems,
         modelling the contacts between the spheres.

The above requirements, which are necessary for realizing a quantum NC (QNC), can be achieved with a system of cold atoms trapped in a one-dimensional periodic potential. This system can be built by confining a Bose-Einstein condensate into a one-dimensional tube using an optical potential that constrains it to a strict Tonks-Girardeau regime. The first achievement of this regime in Bose-Einstein condensates has been reached in the remarkable experiment by Paredes et al. [2], with a set-up closely similar to the one considered here. A further optical potential of moderate amplitude, is superimposed along the longitudinal direction, so that it generates an optical lattice that fulfills condition (i). The dynamics of this system is effectively described by a one dimensional Bose-Hubbard model [3] where, due to the Tonks-Girardeau regime, the strong repulsive interaction between the atoms prevents the double occupancy of lattice sites [4]. In our proposal the condensate is made of atoms with two possible internal states, say |0> and |1>. Accordingly, each potential well hosts an effective two-state system (ii) and the wave-function at each lattice site is a superposition of these internal states.

The tunnelling interaction between nearby wells can be globally tuned by the intensity of the optical lattice beam, and provides the required coupling which meets condition (iii). We have shown that a local perturbation generated at one end of such a lattice propagates back and forth between the lattice ends in a way very similar to that in which an initial momentum pulse is periodically exchanged between the endpoint spheres of the classical NC. In fact, in the QNC the role of the classical momentum Δp transferred between the chain ends, is played by the wave-function disturbance ΔΨ which is transmitted through the system.

Figure 2

We start with the lattice prepared with all sites in (say) the |0> state, and the initial disturbance ΔΨ consists in changing the first site to the |1> state: the disturbance will propagate through the `sea' of |0> states (the analogy is shown in Fig. 2).

2. Tonks–Girardeau regime: Fermionizing Bosons

The system of atoms with two internal states has to be subjected to a strong transverse trapping potential and to a further standing-wave laser beam that creates a periodic potential in the longitudinal direction. At low temperatures and for sufficiently strong transversal and longitudinal potentials the system excitations are confined to the lowest Bloch band. The low-energy Hamiltonian is then given (see [4]) by the Bose-Hubbard model for two boson species labeled by α=0,1. In one dimension the homogeneous Bose–Hubbard model, has two remarkable limits: i) the case of a vanishing repulsion, the model reproduces two independent ideal Bose gases on a lattice, and ii) the case of strong repulsive interaction, that we consider here with a number of atoms equal to the number of sites (filling one). In the Tonks-Girardeau regime, an ideal Fermi gas is found. In fact, very high values of repulsion entail such a high amount of energy for accumulating more than one atom in a given site, that no site can be doubly occupied. Therefore, the only observable states are those where the occupancy of any site is equal to one.

The two possible one-atom states at a site j are |0>_j, and |1>_j, and correspond to the jth atom in the internal state 0 or 1, respectively. In this way the dynamics is ruled by only internal states and an effective Pauli exclusion is realized.

3. The analogy

During an oscillation of the classical NC there are several spheres at rest and in contact with each other, and some moving spheres. When a moving sphere hits a sphere at rest, the latter keeps being at rest and exchanges its momentum with the nearby sphere (see video 1).

Video 1

In the quantum analogue of the NC the role of the spheres’ momenta is played by the wave-functions at each site. Rather than the transfer of mechanical momentum, in the quantum system there is a transmission along the lattice of a disturbance of the wave-function. This is represented in video 2. Furthermore, in the place of the spheres oscillating at the boundaries of the chain, we expect to observe the oscillation of the wave-function amplitude on the lattice ends due to the disturbance that runs forward and back.

Video 2

The system’s wave-function at each lattice site j can be a superposition of the two atomic internal states |0>_j and |1>_j. Under the analogy we propose, one can for instance associate to the spheres at rest the states |0>_j. Accordingly, a moving sphere, let us say the first one, corresponds to a state a0|0>+a1|1>, a superposition of the two internal states. In terms of atoms this amounts to considering all sites initially populated by a species-0 atom, but for (a partial superposition with) a species-1 atom in the first site.

This setup triggers oscillations whose dynamics essentially consists in the disturbance travelling along the lattice: the solitary species-1 atom propagates through the chain of species-0 atoms and migrates until the opposite end, where it is reflected back -- thus determining the NC effect (see video 2).

Remarkably, this analogue of the classical propagation is described in terms of fermions: the most ‘non-classical’ particles.

4. Bad and Good Quantum Newton’s Cradles

Uniform QNC

In the simplest case all tunneling interactions are equal and the chain is uniform.

Figure 3

In Fig. 3 it is clearly shown that the initial disturbance of the wave-function travels along the chain in the form of a wave-packet, which reaches the opposite end of the chain and is reflected backward. However, one can clearly see a significant attenuation of the transmitted signal, an effect essentially due to the destructive interference of the wave-function components. In other words, after a few bounces the initial state evolves to a state where the species-1 atom is delocalized along the chain. This is the situation that occurs in a dispersive system: the wave-function spreads over the lattice during time and the initial wave-packet is rapidly lost. A similar phenomenon also occurs in the classical NC if the masses of the spheres are not identical, i.e., in the non-uniform case.

Evidently, in the quantum analogue, the uniformity of the system causes dispersion: therefore, it is important to identify under which conditions such attenuation can be minimized.

Perfect QNC

Figure 4

The dynamic decoherence of the uniform case, can be not only reduced but even eliminated by letting the tunneling amplitudes to vary along the chain with well-defined nonuniform values. In fact, in the case of a system of M lattice sites, a dispersionless end-to-end quantum-state transmission can be obtained, when the Hamiltonian has nearest-neighbour couplings given by τ_j ∝[j(M-j)]^1/2. In this case a perfect QNC is realized, whose behaviour is illustrated in Fig. 4. One has to observe that the accurate tuning of each tunnel coupling, is experimentally hard.

5. Two realistic schemes

We are going to show here that it is possible to minimally modify the least demanding uniform lattice in order to strongly improve the cradle’s performance.

Quasi-uniform array
A simple way exists for the actual realization of a high-quality QNC in an essentially uniform chain, such that the need for engineering is small. A natural strategy is that of weakening the extremal τ_js. Indeed, keeping the requirement of a mirror-symmetric chain, one can minimally modify a uniform chain taking equal couplings, τ_j=τ, but for the ones at the edges, τ₁=τ_M-1=xτ, with x<1, and look for the best transfer conditions. In Ref.[5] it is shown that the optimal coupling results x≈1.03M^-1/6. As a matter of fact, taking into play also the second bonds τ₂=τ_M-2=yτ allows one to guarantee a response larger than 0.987 (i.e., the transmitted amplitude deteriorates of only 1.3%) when the coupling are tuned as x≈2M^-1/3 & y≈2^3/4M^-1/6, see Ref.[6].

Uniform array with a Gaussian trap

Figure 5

The last configuration we propose can also be implemented in an experiment. Besides the uniform one-dimensional optical potential, we add a trapping potential that generates a site-dependent energy-offset with a Gaussian profile (see Fig. 5). Furthermore, we choose as initial state a Gaussian wave-packet along the lattice. Such a setup appears to be the most realistic compared with the previous ones. In fact, in the schemes we illustrated so far, the bounce of the disturbance of the wave-function at the lattice ends is caused by the open-boundary conditions, while in the present setup, the wave-packet oscillates inside the trapping potential and its speed inversion is caused by the forces generated by the trapping potential. In Fig.6 it is evident that the packet never reaches the lattice ends: when the wave-packet moves towards a lattice end, it is slowed down by the trapping potential, until the motion is inverted and the packet is accelerated in the opposite direction.

Figure 6

6. Conclusions

We have investigated an experimental framework that could realize a quantum analogue of Newton's cradle, starting from a Bose–Einstein condensate of two atomic species in an optical lattice. We have shown that the tunneling between sites makes the system equivalent to a free-Fermion gas on a finite lattice. In these conditions, one can trigger at one lattice-end a disturbance that starts bouncing back and forth between the ends, just like the extremal spheres in the classical Newton cradle: the analogy associates the propagation of a wave-function disturbance with the transmission of mechanical momentum.

However, in the quantum system the travelling wave undergoes decoherence, a phenomenon that makes a uniform lattice almost useless. On the contrary, it is known that a special arrangement of the tunneling amplitudes can even lead to a virtually perpetual cyclic bouncing.

That's why we looked for compromises that minimized the required experimental adaptation of the interactions and, gave `almost' perfect quantum Newton cradles. Of course, the possibility of obtaining quantum systems that allow high-quality quantum-wave transmission is not only relevant from the speculative point of view, but also in the field of the realization of quantum devices like atomic interferometers, quantum memories, and quantum channels. Nevertheless, realizing the quantum Newton cradle we proposed would be stirring by itself for the insight it would give into the entangled beauty of quantum mechanics.

References:
[1] Roberto Franzosi and Ruggero Vaia, "Newton's cradle analogue with Bose–Einstein condensates". Journal of Physics B: Atomic, Molecular and Optical Physics, 47, 095303 (2014). Abstract.
[2] Belén Paredes, Artur Widera, Valentin Murg, Olaf Mandel, Simon Fölling, Ignacio Cirac, Gora V. Shlyapnikov, Theodor W. Hänsch, Immanuel Bloch, "Tonks–Girardeau gas of ultracold atoms in an optical lattice". Nature, 429, 277 (2004). Abstract.
[3] Roberto Franzosi, Vittorio Penna, Riccardo Zecchina, "Quantum dynamics of coupled bosonic wells within the Bose-Hubbard picture". International Journal of Modern Physics B, 14, 943-961 (2000). Abstract; Roberto Franzosi and Vittorio Penna, "Spectral properties of coupled Bose-Einstein condensates". Physical Review A, 63, 043609 (2001). Abstract.
[4] A. B. Kuklov and B. V. Svistunov, "Counterflow Superfluidity of Two-Species Ultracold Atoms in a Commensurate Optical Lattice". Physical Review Letters, 90, 100401 (2003). Abstract.
[5] L. Banchi, T. J. G. Apollaro, A. Cuccoli, R. Vaia and P. Verrucchi, "Long quantum channels for high-quality entanglement transfer". New Journal of Physics, 13, 123006 (2011). Abstract.
[6] T. J. G. Apollaro, L. Banchi, A. Cuccoli, R. Vaia, and P. Verrucchi, "99%-fidelity ballistic quantum-state transfer through long uniform channels". Physical Review A, 85, 052319 (2012). Abstract.