Single Photon Transistor Mediated by Rydberg Interaction

From Left to Right: Hannes Gorniaczyk, Christoph Tresp, Johannes Schmidt, Ivan Mirgorodskiy, Sebastian Hofferberth

Authors: Christoph Tresp, Ivan Mirgorodskiy, Hannes Gorniaczyk, Sebastian Hofferberth 

Affiliation:
Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Germany.

Link to Rydberg Quantum Optics, Emmy Noether Group >>

Introduction:

In analogy to their electronic counterparts, all-optical switches and transistors are required as basic building blocks for both classical and quantum optical information processing [1,2]. Reaching the fundamental limit of such devices, where a single gate photon modifies the transmission or phase accumulation of multiple source photons, requires strong effective interaction between individual photons. Engineering sufficiently strong optical nonlinearities to facilitate photon-photon interaction is one of the key goals of modern optics. Immense progress towards this goal has been made in a variety of systems in recent years. Most prominent so far are cavity QED experiments where a high finesse resonator enhances the interaction between light and atoms [3,4] or artificial atoms [5,6].

In this work, we demonstrate a free-space all-optical transistor operating on the single photon level using a novel approach to realize effective photon-photon interaction [7], which is based on mapping the strong interaction of Rydberg atoms [8] onto slowly travelling photons using electromagnetically induced transparency [9]. This technique has already been used to demonstrate highly efficient single-photon generation [10], attractive interaction between single photons [11], entanglement generation between light and atomic excitations [12], and most recently single-photon all-optical switching [13].

However, demonstration of amplification, that is, controlling many photons with a single one, has so far only been achieved in a cavity QED setup [14]. Gain > 1 is one of the key properties of the electric transistor that lies at the heart of its countless applications. In our experiment, we demonstrate an all-optical transistor with optical gain G > 10 [15]. Similar results have been obtained by the group of G. Rempe, their results have been published in parallel to ours [16].

Experiment:

The level scheme and geometry of our transistor are illustrated in Fig. 1 (a, b). Photons in the weak gate pulse are stored as Rydberg excitations in an atomic ensemble by coupling the ground state |g> to the Rydberg state |r_g> via the strong gate control field. After this storage process, a second weak pulse, the source pulse, is sent through the medium at reduced velocity due to EIT provided by the source control laser coupling to the Rydberg state |r_s>.

FIG. 1: (a) Level scheme, (b) simplified schematic, and (c) pulse sequence of our all-optical transistor. (d) The absorption spectrum for the source field (dots) over the full intermediate state absorption valley shows the EIT window on resonance; the gate field spectrum (circles) is taken around the two-photon resonance at Δ = 40 MHz. The solid lines are fits to the EIT spectra.

In the absence of the gate pulse, source photons travel through the transparent medium (Fig. 1d). If a gate photon has been stored, the strong interaction between the two Rydberg states destroys the EIT condition for the source photons in the medium, resulting in absorption. To observe this conditional switching, we record the number of transmitted source photons in a time interval tint after the gate excitation pulse, cf. Fig. 1 (c). For the experimental realization of this scheme, we prepare 2.5 X 10^{4 87}Rb atoms at a temperature of T = 40 µK in an optical dipole trap. All four lasers required for the transistor scheme are focused into this medium along a single direction (Fig. 1b). The weak gate and source pulses are recorded on single photon counters.

Results:

We first investigate the relative attenuation of a weak source pulse as a function of mean incident gate photons. In Fig. 2 (a) we plot the switch contrast in the source beam transmission as a function of the mean incoming gate photon number. For an average gate photon number of N_g,in = 1.04(3), we observe a switch contrast C_coh = 0.39(4). The switch contrast is mainly determined by the Poissonian statistics of our coherent gate photons, which sets a fundamental upper bound. In other words, a perfect switch with coherent gate photons has a switch contrast C_coh = 1 - exp(-N_g,in) (dashed line in Fig. 2 (a)). How close our switch approaches this fundamental limit depends on the gate photon storage efficiency and the source attenuation caused by a single gate excitation. In Fig. 2 (b) we plot the switch contrast versus the mean number of stored gate photons, which is smaller than the mean incident gate photon number due to not perfect gate photon storage. Finally, by again taking the Poissonian statistics of the input light into account, we extrapolate the switch contrast caused by a single stored excitation to be C_exc = 0.9.

FIG. 2: Switch contrast (red) as function of (a) mean number of incident gate photons and (b) mean number of stored photons. The dashed line indicates the fundamental limit set by the photon statistics of the coherent gate input. Black data points represent the calculated switch contrast expected for (a) one-, two- and three-photon Fock input states or (b) deterministic single and two stored gate excitations.

Next, we investigate how many source photons can be switched by our system. To quantify the gate-induced change in source transmission, we consider the optical gain
G = N_s,out^{no gate} - N_s,out^{with gate}. In Fig. 3 (a), we plot the measured optical gain for an average input of gate photons N_g,in = 0.75(3). For this gate input, we observe a maximum optical gain G(N_g,in = 0.75) = 10(1). Further increase of the optical gain at fixed gate input is limited by the self-blockade of the source beam, which results in nonlinear source transmission even in the absence of gate photons [7, 17]. The red (blue) data points in Fig. 3 (b) show the source photon transfer function when N_g,in = 0 (N_g,in = 0.75(3)). For the given integration time the source transmission saturates at 46 photons, which limits the maximum gain we can observe. On the other hand, the self-nonlinearity of the source light does not affect the transistor performance, we observe a constant switch contrast of C = 0.22(3), consistent with the mean gate input, even for incoming source photons up to ~250. Based on this robustness, we can again extrapolate the transistor performance for a true single photon gate input (Fig. 3 green line) and a single stored excitation (grey line). For a single excitation, we calculate the maximally achievable optical gain of our current system as G_st = 28(2).

FIG. 3: (a) Optical gain of our transistor, measured for coherent gate input N_g,in = 0.75(3) (blue data), and extrapolated to single photon Fock state input (green line), and single stored excitation (black line). (b) Source photon transfer function without (red) and with coherent gate input N_g,in = 0.75(3) (blue). We observe a constant switch contrast between the two data sets over the whole source input range. The green (black) solid line are again the estimated behavior of the system for a single-photon Fock input state (a single stored excitation). Shaded regions are error estimates.

Discussion and outlook:

In summary, we have demonstrated a free-space single photon transistor based on two-color Rydberg interaction. Further improvements of our system could enable a high optical gain, high efficiency optical transistor, so far only realized in a cavity QED setup [14]. One approach to overcome the self-nonlinearity of the source photons has already been demonstrated by the Rempe group, who employ a two-color Förster resonance in their transistor scheme [16].

A key step towards turning our transistor into device which can perform quantum operations on single or few photons is the retrieval of gate photon(s) after the switch process, which could enable multi-photon entanglement protocols and creation of non-classical light-states with large photon numbers. Finally, our system is a highly sensitive probe for studying Rydberg interaction on the few-particle level [18]. In particular, the combination of two independently controlled Rydberg-EIT schemes enables novel fields of study, such as the interplay between slow light propagation and Rydberg exchange interaction [19], or realization of a two-photon phase gate based on Rydberg-polariton collision [20].

References:
[1] H. John Caulfield and Shlomi Dolev, "Why future supercomputing requires optics". Nature Photonics, 4, 261 (2010). Abstract.
[2] Jeremy L. O'Brien, Akira Furusawa, Jelena Vuckovic, "Photonic quantum technologies". Nature Photonics, 3, 687 (2009). Abstract.
[3] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom". Nature, 436, 87 (2005). Abstract.
[4] Tatjana Wilk, Simon C. Webster, Axel Kuhn, Gerhard Rempe, "Single-Atom Single-Photon Quantum Interface". Science, 317, 488 (2007). Abstract.
[5] P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, A. Imamoglu, "A Quantum Dot Single-Photon Turnstile Device". Science, 290, 2282 (2000). Abstract.
[6] Dirk Englund, Andrei Faraon, Ilya Fushman, Nick Stoltz, Pierre Petroff, Jelena Vuckovic, "Controlling cavity reflectivity with a single quantum dot". Nature, 450, 857 (2007). Abstract.
[7] J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, C. S. Adams, "Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble". Physical Review Letters, 105, 193603 (2010). Abstract.
[8] M. Saffman, T. G. Walker, K. Mølmer, "Quantum information with Rydberg atoms". Review of Modern Physics, 82, 2313 (2010). Abstract.
[9] Michael Fleischhauer, Atac Imamoglu, Jonathan P. Marangos, "Electromagnetically induced transparency: Optics in coherent media". Review of Modern Physics, 77, 633 (2005). Abstract.
[10] Y. O. Dudin and A. Kuzmich, "Strongly Interacting Rydberg Excitations of a Cold Atomic Gas". Science 336, 887 (2012). Abstract.
[11] Ofer Firstenberg, Thibault Peyronel, Qi-Yu Liang, Alexey V. Gorshkov, Mikhail D. Lukin, Vladan Vuletić, "Attractive photons in a quantum nonlinear medium". Nature, 502, 71 (2013). Abstract.
[12] L. Li, Y. O. Dudin, and A. Kuzmich, "Entanglement between light and an optical atomic excitation". Nature, 498, 466 (2013). Abstract.
[13] Simon Baur, Daniel Tiarks, Gerhard Rempe, Stephan Dürr, "Single-Photon Switch Based on Rydberg Blockade". Physical Review Letters, 112, 073901 (2014). Abstract.
[14] Wenlan Chen, Kristin M. Beck, Robert Bücker, Michael Gullans, Mikhail D. Lukin, Haruka Tanji-Suzuki, Vladan Vuletić, "All-Optical Switch and Transistor Gated by One Stored Photon". Science 341, 768 (2013). Abstract.
[15] H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, S. Hofferberth, "Single-Photon Transistor Mediated by Interstate Rydberg Interactions". Physical Review Letters, 113, 053601 (2014). Abstract.
[16] Daniel Tiarks, Simon Baur, Katharina Schneider, Stephan Dürr, Gerhard Rempe, "Single-Photon Transistor Using a Förster Resonance". Physical Review Letters, 113, 053602 (2014). Abstract.
[17] Thibault Peyronel, Ofer Firstenberg, Qi-Yu Liang, Sebastian Hofferberth, Alexey V. Gorshkov, Thomas Pohl, Mikhail D. Lukin, Vladan Vuletić, "Quantum nonlinear optics with single photons enabled by strongly interacting atoms". Nature, 488, 57 (2012). Abstract.
[18] L. Béguin, A. Vernier, R. Chicireanu, T. Lahaye, A. Browaeys, "Direct Measurement of the van der Waals Interaction between Two Rydberg Atoms". Physical Review Letters, 110, 263201 (2013). Abstract.
[19] Weibin Li, Daniel Viscor, Sebastian Hofferberth, Igor Lesanovsky, "Electromagnetically Induced Transparency in an Entangled Medium". Physical Review Letters, 112, 243601 (2014). Abstract.
[20] Alexey V. Gorshkov, Johannes Otterbach, Michael Fleischhauer, Thomas Pohl, Mikhail D. Lukin, "Photon-Photon Interactions via Rydberg Blockade". Physical Review Letters, 107, 133602 (2011). Abstract.

A True Randomness Generator Exploiting a Very Long and Turbulent Path

From Left to Right: Paolo Villoresi,  Davide Marangon, Giuseppe Vallone

Authors:
Davide G. Marangon, Giuseppe Vallone,  Paolo Villoresi

Affiliation:
Department of Information Engineering, University of Padova, Italy.

Random numbers are the main ingredients of cryptographic protocols for both Classical and Quantum Information. However, it is well known that to rely on random numbers produced with deterministic algorithms can be very risky and it is of fundamental priority to discover physical processes to generate "pure" random numbers. Usually True Random Number Generators (TRNG) are implemented by exploiting classical or quantum microscopical processes. However it can be shown that random numbers can be extracted from macroscopic physical systems.

Past 2Physics articles by Paolo Villoresi:
November 24, 2013: "How to Realize Quantum Key Distribution with a Limited and Noisy Link".
May 19, 2008: "The Frontier of Quantum Communication is the Space".

In the 60s, the famous "Butterfly Effect" captured the idea that when one deals with the terrestrial atmosphere, very tiny perturbations such as the air moved by the tail strokes of a butterfly can lead to very huge consequences as a hurricane in some other place in the world. Terrestrial atmosphere indeed may be seen as a physical system ruled by a chaotic dynamic. Moreover, while statistical models are available for average trends, the prediction of the instantaneous motion of the air mass in a spot is out of reach.

From the textbooks we know that the propagation of light through an inhomogeneous medium is strongly influenced by the refractive index distribution. We experimentally investigated this phenomenon with the purpose of realizing if such propagation along a free-space path may induce a useful randomness. An intuition of such effect manifested during the campaigns for the experiments we carried out at the Canarias on the quantum Communications along extremely long links [1, 2]. The atmospheric turbulence in the path is very strong, preventing for example the direct application of interferometry [3]. However, the effect of turbulence is crucial there for the application of a method that exploits brief moments of high transmissivity for good communication [2]. We tried to turn it here instead into a useful resource for randomness.

FIG. 1. The experiment was set up between the islands of La Palma and Tenerife where a laser beam (with λ = 810 nm) was exchanged between the two islands. After propagating across a 143 km Free Space Optical link, the wavefront of the beam features a randomly composed speckle pattern as consequence of the distortions induced by the atmosphere.

The experiment we describe here was set between the two islands of La Palma and Tenerife: on the rooftop of the Jacobus Kaptein Telescope (JKT) building at an altitude of 2360 m; our transmitting telescope for the Quantum Communication was aimed to send a continuous laser beam towards the ESA Optical Ground Station (OGS) 143 km far away, at Izana, near the mount Teide, see Figure 1. The telescope -- that was designed and realized in Padova -- is a refractor based on a 230 mm aspheric singlet. In the path, the turbulent atmosphere is comparable to a dynamic volumetric scatterer and the electromagnetic field is subjected to phase delays and amplitude fluctuations, induced by the inhomogeneities of the refractive index of the air [4–6]. The receiver then observes a beam profile which does not feature the typical intensity Gaussian distribution, rather a collection of clear and dark spots of irregular shape, the so-called speckle pattern. The speckle pattern evolves according the unpredictable dynamic of the turbulence as consequence of the random walks the electromagnetic field suffers while propagating. Therefore at the receiving plane a continuously and randomly evolving distribution of speckles was acquired with a CMOS camera and for every frame one has a variable number of spots randomly taking different spatial configurations [7].

Randomness is then extracted by using the geometrical complexity of the frames evaluating the centers of mass, the so-called centroids, of those speckle areas with the same intensity. For the implementation of the method the relevant pixels in CCD are labelled sequentially with an index s, s ∈ {1, . . . , N}, the n_f  speckle centroids of the frame f are elaborated, an ordered sequence S_f = {s₁ , s₂ , . . . , s_nf } with s₁ < s₂ < · · · < s_nf  is formed, by considering then the pixels where a centroid falls in. The pixel grid can be regarded as the classical collection of urns where the turbulence randomly throws balls (the centroids) in, see Figure 2. Because of the random nature of the process, the centroids visit every part of the grid with the same probability. A given frame f  “freezes” one S_f  out of the

possible and equally likely sequences of n_f centroids. Among all of combinations, a given S_f can be univocally identified with its lexicographic index I (S_f )

with 0 ≤ I (S_f ) ≤ T_f  - 1. Basically, (2) enumerates all the possible arrangements which succeed a given centroids configuration. As an uniform RNG is supposed to yield numbers identically and independently distributed (i.i.d.) in a range [X,Y ], as this method generates a random integer in the range [0, T_f - 1]. In order then to optimize the conversion from integer to random bits without introducing any bias, an efficient algorithmic procedure was applied to the bits [8].

FIG. 2. (click on the figure to view higher resolution) The figure represents a scheme of the mechanism employed to extract randomness from the frames of the captured video. Every frame features a different spatial disposition of centroids (the yellow crosses). To every centroid configuration, a univocal lexicographic index is associated. The lexicographic index then is converted in random bits.

In this proof of principle, a generation rate of 400 kbit/s was achieved but it can be easily enhanced by using cameras with higher resolutions. Another point, worth to be stressed, is that this method does not rely on sensitive and hardly detectable processes which require extremely tuned hardware: indeed unavoidable hardware non-idealities can induce bits dependencies and bias. In addition, from the theoretical point of view, the strength of the method lies in the fact the dynamic of turbulent atmosphere on such a long link represents a physical process which is practically impossible to be predicted, both analytically (at the present time only statical models are given) and numerically (it would require an unbounded computational power).

In addition to a sound knowledge of the physical process employed, it is necessary to apply statistical tests in order to exclude the presence of defects caused by a faulty hardware. This has been done by applying the most stringent test batteries for randomness such as the Alphabit and Rabbit batteries belonging to the TESTU01, the NIST SP-800-22 suite and the AIS31 suite. All the tests were successfully passed.

The presented procedure then could be an efficient method to generate random numbers to be employed in long range QC setups. More in detail, bits generated in this way could be used in connection with other protocols involving Quantum Random Number Generator: for example in the first well known experiment of randomness expansion by means of non-locality [9] the initial seed was obtained by mixing numbers obtained with several generators including atmospheric radio electromagnetic noise. Finally, the extraction algorithm can be easily adapted to other paradigms involving spatial random complex patterns.

References:
[1] Ivan Capraro, Andrea Tomaello, Alberto Dall’Arche, Francesca Gerlin, Ruper Ursin, Giuseppe Vallone, Paolo Villoresi, "Impact of turbulence in long range quantum and classical communication". Physical Review Letters, 109, 200502, (2012). Abstract.
[2] Giuseppe Vallone, Davide Marangon, Matteo Canale, Ilaria Savorgnan, Davide Bacco, Mauro Barbieri, Simon Calimani, Cesare Barbieri, Nicola Laurenti, Paolo Villoresi, "Turbulence as a Resource for Quantum Key Distribution in Long Distance Free-Space Links". arXiv:1404.1272 [quant-ph] (2014).
[3] Cristian Bonato, Alexander V. Sergienko, Bahaa E. A. Saleh, Stefano Bonora, Paolo Villoresi, "Even-Order Aberration Cancellation in Quantum Interferometry". Physical Review Letters, 101, 233603 (2008). Abstract.
[4] Larry C. Andrews and Ronald L. Phillips, "Laser beam propagation through random media", volume 152 (SPIE press, 2005). 
[5] R. L. Fante, "Electromagnetic beam propagation in turbulent media". Proceedings of the IEEE, 63, 1669,(1975). Abstract.
[6] R. L. Fante, "Electromagnetic beam propagation in turbulent media - An update". Proceedings of the IEEE, 68, 1424 (1980). Abstract.
[7] Davide G. Marangon, Giuseppe Vallone, Paolo Villoresi, "Random bits, true and unbiased, from atmospheric turbulence". Scientific Reports, 4 : 5490 (2014). Full Article.
[8] Peter Elias. "The efficient construction of an unbiased random sequence". Annals of Mathematical Statistics, 43, 865 (1972). Full Article.
[9] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D.N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T.A. Manning,  C. Monroe. "Random numbers certified by Bell’s theorem". Nature, 464, 1021 (2010). Abstract.

Quantum Information in the Service of Metrology

The spin-spin measurement team at Weizmann Institute of Science, Israel. From left-to-right: Nir Navon, Nitzan Akerman, Roee Ozeri, Shlomi Kotler and Yinnon Glickman

Authors: Roee Ozeri¹, Shlomi Kotler^1,2

Affiliation:
¹Department of Physics of Complex Systems, Weizmann Institute of Science, Israel.
²Current address: Physical measurement Laboratory, National Institute of Standards and Technology, Boulder, USA.

Quantum systems have been extensively studied in the past few years as possible ultra-efficient computers. Such systems have to be as isolated as possible from their environment in order to prevent it from “measuring” the state of the quantum computer, a process which would render the computer classical. Candidate systems that were experimentally studied thus had to be sufficiently isolated from their environment while allowing for a high degree of controllability.

In addition, active methods were developed, in which special quantum states and control techniques were engineered and helped mitigate the effect of noise. Many such techniques, such as dynamic-decoupling or decoherence-free subspaces, were experimentally implemented with great success, increasing the coherence times of quantum systems by many orders of magnitude. Other methods, such as Quantum error-correction codes, have had proof-of principle demonstrations but hold the promise of being able to reject noise in a way where quantum super-positions of very large systems will be maintained coherent for as long as necessary – one of the key requirement from a quantum computer.

A different quantum technology which has seen great progress in recent years is that of quantum metrology. At face value, quantum sensors seem to be exactly antipodal to quantum computers. Here, quantum systems do have to couple to their environment in order to sense some aspect of it. However, as in many seemingly contradicting concepts there can also be a lot of common ground. For example, environmental noise (i.e. that part of the environment which you don’t want to measure!) is a foe of both quantum sensors and quantum computers. Can we therefore use the techniques that were developed to help quantum computers overcome the harmful effects of noise to improve on the measurement precision of quantum sensors? The answer is yes, and with great success! Along these lines, different dynamic modulations schemes, originally used in dynamic decoupling were used for the measurement of alternating signals. This way quantum lock-in amplifiers as well as quantum noise spectrum analyzers were demonstrated. In a recent experiment in our lab at the Weizmann Institute of Science, we used the powerful technique of decoherence-free subspaces in order to measure the very weak magnetic interaction between two electrons that were separated by more than two microns [1].

Figure 1: An artist impression of the spin-spin experiment. Two electrons are placed two microns away from each other. The magnetic field emanating from one electron interacts with the spin of the other electron, resulting in a change of the spin-orientations.

Electrons, like many fundamental particles have an intrinsic magnetic dipole moment which is aligned with their spin. These tiny magnets have a magnetic field that decays as the cube of the distance from the electron. To illustrate, the magnetic field of a single electron two microns away from it, is as small as the earths’ magnetic field at 10 times the distance to the moon. When two electrons feel each other’s magnetic fields their spins interact as magnets do: their opposite poles will attract, their identical poles repel in a way that torques will be applied and the two spins will respectively rotate due to this interaction.

The magnetic interaction between two electrons was never directly observed before. At short atomic distances, such as that between the two electrons of a Helium atom, the magnetic interaction is large enough to be easily measured. Unfortunately, at these distances, it is overwhelmed by the much larger exchange interaction between them which is the result of the interplay between the strong Coulomb interaction between the electron charges and Fermi’s exclusion principle. At large distances, where the exchange interaction is negligible, the magnetic interaction between the electrons is also very small. At a distance of two microns for example, the rotation rate the two spins impose on each other is on the order of one rotation every four minutes. This interaction is way too small to be measured due to typical magnetic noise in labs.

Figure 2: An image of the trap in which the ions were trapped for the duration of the measurement. The image is taken through one of the ultra-high vacuum chamber view-ports.

This is where techniques, borrowed from quantum computing science come to the rescue. We have placed the spins of two trapped Sr+ ions in a decoherence-free subspace that was completely immune to the effect of magnetic field noise. While being immune to noise, this subspace still allowed for the slow and gentle two-spin correlated dance to be performed without interference. Under the protection provided by this technique we could allow the electronic spins to rotate coherently for 15 seconds, after which we measured their collective rotation of more than 20^◦. We also changed the distance between the electrons and verified that the interaction between them varies inverse cubical with their separation. Thus, almost a 100 years after the discovery of the electronic spin, we were able to cleanly observe the interaction between two such tiny magnets.

This measurement bears importance that reaches beyond its demonstrative nature. This is because some hypothetical anomalous spin forces are speculated to modify the interaction between electronic spins at large distances. The motivation for the introduction of these anomalous forces is due to their ability to explain the weakness with which certain symmetries are broken in nature. The experimental bound on the strength and range of these hypothetical fields is therefore important.

The use of quantum error-suppression schemes for the benefit of precision measurements is a fast developing area of research. With the advent of experimental quantum error-correction codes, another opportunity will emerge to apply these codes towards the detection of small and highly correlated signals [2-5].

References:
[1] Shlomi Kotler, Nitzan Akerman, Nir Navon, Yinnon Glickman, Roee Ozeri, "Measurement of the magnetic interaction between two bound electrons of two separate ions". Nature, 510, 376 (2014). Abstract.
[2] Roee Ozeri, "Heisenberg limited metrology using Quantum Error-Correction Codes". arXiv:1310.3432 [quant-ph].
[3] G. Arrad, Y. Vinkler, D. Aharonov, A. Retzker, "Increasing Sensing Resolution with Error Correction". Physical Review Letters, 112, 150801 (2014). Abstract.
[4] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, M. D. Lukin, "Quantum Error Correction for Metrology". Physical Review Letters, 112, 150802 (2014). Abstract.
[5] W. Dür, M. Skotiniotis, F. Fröwis, B. Kraus, "Improved Quantum Metrology Using Quantum Error Correction". Physical Review Letters, 112, 080801 (2014). Abstract.

Quantum Computations on a Topologically Encoded Qubit

From Left to Right: (top row) Daniel Nigg, Markus Müller, Esteban Martínez, Philipp Schindler, (bottom row) Markus Hennrich, Thomas Monz, Miguel Angel Martín-Delgado, Rainer Blatt.

Authors: Markus Müller¹ and Daniel Nigg²

Affiliation:
¹Departamento de Física Teórica I, Universidad Complutense, Spain.
²Institut für Experimentalphysik, Universität Innsbruck, Austria.

Email: mueller@ucm.es, daniel.nigg@uibk.ac.at

Even computers are error-prone. The slightest disturbances may alter saved information and falsify the results of calculations. To overcome these problems, computers use specific routines to continuously detect and correct errors. This also holds true for a future quantum computer, which will also require procedures for error correction. Whereas general quantum states can not be simply copied, fragile quantum information can still be protected from errors during storage and information processing by using quantum error correcting codes. Here, quantum states are encoded in entangled states that are distributed over several physical particles.

A quantum bit encoded in seven ions

In the experiment realized at the University of Innsbruck, Austria [1], we confined seven calcium ions in an ion trap, with one qubit stored in each of the ions. In our setup, we use lasers to cool the ion string to almost absolute zero temperature and to precisely control their quantum properties. We used the register of seven physical qubits to encode quantum states of one logical qubit in entangled states of these particles. The topological quantum error-correcting code employed in the experiment provided the program for this encoding process, and was proposed and developed in the theory group at the Universidad Complutense in Madrid, Spain. The code arranges the qubits on a two-dimensional lattice structure where they interact with the neighboring particles. The encoding of the logical qubit in the seven physical qubits was the experimentally most challenging step. It required a long sequence of laser pulses to effectively realize three entangling gate operations, each acting on subsets of four neighboring qubits belonging to one plaquette.

Figure 1: Schematics of the string of 7 ions stored in a linear Paul trap, with each ion hosting one physical qubit. One logical qubit is encoded in entangled states of these 7 physical qubits, by using a quantum error correcting code which arranges the qubits on a two-dimensional triangular lattice of three plaquettes.

Detection of arbitrary errors and logical quantum gate operations

After the encoding step, once the atoms are entangled in this specific way, the quantum correlations provide a resource for subsequent error correction and quantum computations on the encoded logical qubit. Using the available set of laser pulses we induced at purpose all types of single-qubit errors that can occur on any of the seven physical qubits. Our measurements demonstrate that the quantum code is indeed able to independently detect phase flip errors, bit flip errors as well as combinations of both, regardless on which of the qubits these occur.

Figure 2: Schematics of error detection by the 7-qubit code: Arbitrary errors (in the shown example a phase flip error Z on qubit 5) manifests itself as excitations on one or several plaquettes (black filled circle on the blue plaquette) and by its characteristic signature, the error syndrome. The latter allows one to deduce the type, i.e. whether a bit flip, phase flip or combined error of both has occurred, as well as the location of the error in the qubit register.

Next, we applied logical quantum gate operations onto the encoded logical qubit. The 7-qubit quantum code we used allowed us to implement individual operations and longer sequences of gate operations (the single-qubit Clifford group) on the logical qubit in a transversal way, i.e. by applying the corresponding operations bitwise on each of the 7 physical qubits.

Towards a fault-tolerant quantum computer

The 7-ion system we used for encoding one logical quantum bit can serve as a building block for larger quantum systems. Storing and processing logical quantum information in larger lattice systems with more physical qubits is predicted to further increase the robustness with respect to noise and errors. The required technology in the form of two-dimensional ion trap arrays, which would enable the storage and manipulation of larger numbers of qubits, are currently developed and tested at the University of Innsbruck as well as in other laboratories worldwide. Together with further theoretical progress and optimized quantum error correcting codes, the result of these developments might be a quantum computer that could reliably perform arbitrarily long quantum computations without being impeded by errors.

For further background information and explication, please watch this video:



Funding:
The researchers are financially supported by the Spanish Ministry of Science, the Austrian Science Fund, the U.S. Government, the European Commission and the Federation of Austrian Industries Tyrol.

Reference:
[1]  Daniel Nigg, Markus Müller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel Angel Martin-Delgado, Rainer Blatt, "Quantum Computations on a Topologically Encoded Qubit". Science, 345, 302 (2014). Abstract.

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.