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2Physics Quote:
"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."
-- Markus Müller, Daniel Nigg (Read Full Article: "Quantum Computations on a Topologically Encoded Qubit" )

Sunday, September 14, 2014

Imaging the Dynamics of Free-Electron Landau States

Transmission electron microscope that was used in the experiment. Foreground: Michael Stöger-Pollach (left) and Peter Schattschneider.
From Left to Right: Th. Schachinger, S. Löffler, A. Steiger-Thirsfeld, K. Y. Bliokh, Franco Nori

Authors: P. Schattschneider1,2,3, Th. Schachinger1, M. Stöger-Pollach2, S. Löffler2, A. Steiger-Thirsfeld2, K. Y. Bliokh4,5, Franco Nori5,6

1Institute of Solid State Physics, Vienna University of Technology, Austria
2University Service Centre for Electron Microscopy, Vienna University of Technology, Austria
3LMSSMat (CNRS UMR 8579) Ecole Centrale Paris, France
4iTHES Research Group, RIKEN, Wako-shi, Saitama, Japan
5Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama, Japan
6Department of Physics, University of Michigan, Ann Arbor, USA.

Inspired by theoretical calculations [1,2] from RIKEN (Japan), the group at Vienna University of Technology devised a way to generate free-electron Landau states [3], a form of quantized states that electrons adopt when moving through a magnetic field. Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum Hall and related effects in condensed matter physics [4]. Landau states can be envisaged as electron vortices occurring naturally in the presence of magnetic fields. The magnetic field plays the same role for electrons as the earth's rotation plays for the creation of cyclones, but here on a nanometer scale, where quantum effects become important [5].

Past 2Physics articles by Konstantin Y. Bliokh:
January 17, 2009: "Optical Magnus Effect: Topological Monopole Deflects Spinning Light".

Classical electrons in a uniform magnetic field propagate freely along the field and form confined circular orbits in the plane perpendicular to the field. The angular velocity of such orbiting is constant and is known as the cyclotron frequency. But quantum mechanics calls for a counter-intuitive behaviour [2]. The researchers were able to induce intrinsic rotation in single moving electrons. It was observed [3] that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions [2].

States with different quantum numbers are produced using nanometre-sized electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasi-uniform magnetic field. Scanning the beams along the propagation direction [3], the researchers reconstructed the rotational dynamics of the Landau wave functions with angular frequency of the order of 100 GHz.

Figure 1: A holographic fork mask generates a row of vortex beams with different azimuthal indices m. These beams are focused by a magnetic lens and are studied in the region of maximal quasi-uniform magnetic field (red arrow on the left). The focal plane is shifted a few Rayleigh ranges below the observation plane z=0 to reduce the Gouy-phase rotation. A knife-edge stop is placed at zk, where it blocks half of each of the beams. Varying the position zk of the knife edge allows the observation of the spatial rotational dynamics of the cut beams propagating to the observation plane [3].

The focusing lenses of a transmission electron microscope were used [3] to reconfigure the vortices so that they almost perfectly resembled Landau states. In an electron vortex beam, the wave current swirls around a common center similar to air in a tornado [6]. Measuring the rotation can be compared to determining how many times a thin wire is wound around a rod. When looking at the wire directly, it is extremely difficult to count the number of windings. But when stretching it along the direction of the rod, the wire takes the form of a well-spaced spiral, for which it is easy to count the revolutions. This is precisely what happens with Landau states: they were 'elongated' to vortex beams. That way, their rotation could be measured [3] with very high accuracy.

This is a very exciting finding that will contribute to a better understanding of the fundamental quantum features of electrons in magnetic fields [3]. In addition to showing that the rotational dynamics of the electrons are more complex and intriguing than was once believed, the new findings could have practical implications for technology. Taking Landau states into free space, away from the solids where they normally play a key role [4,7], can inspire new ideas in materials science.

This will certainly lead to novel insights and a better understanding of the delicate interaction between magnetic fields and matter, which might one day give rise to new and better technologies such as sensors, memory devices, or nanomanipulation.

[1] Konstantin Yu. Bliokh, Yury P. Bliokh, Sergey Savel’ev, Franco Nori, "Semiclassical dynamics of electron wave packet states with phase vortices". Physical Review Letters, 99, 190404 (2007). Abstract.
[2] Konstantin Y. Bliokh, Peter Schattschneider, Jo Verbeeck, Franco Nori, "Electron vortex beams in a magnetic field: a new twist on Landau levels and Aharonov-Bohm states". Physical Review X, 2, 041011 (2012). Abstract.
[3] P. Schattschneider, Th. Schachinger, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, K. Y. Bliokh, Franco Nori, "Imaging the dynamics of free-electron Landau states". Nature Communications, 5, 4586 (2014). Full Article.
[4] Daijiro Yoshioka, "The Quantum Hall Effect" (Springer, 2002).
[5] J. Verbeeck, H. Tian, P. Schattschneider, "Production and Application of Electron Vortex Beams". Nature, 467, 301 (2010). Abstract.
[6] Giulio Guzzinati, Peter Schattschneider, Konstantin Y. Bliokh, Franco Nori, Jo Verbeeck, "Observation of the Larmor and Gouy rotations with electron vortex beams". Physical Review Letters, 110, 093601 (2013). Abstract.
[7] David L. Miller, Kevin D. Kubista, Gregory M. Rutter, Ming Ruan, Walt A. de Heer, Markus Kindermann, Phillip N. First, Joseph A. Stroscio, "Real-space mapping of magnetically quantized graphene states". Nature Physics, 6, 811–817 (2010). Abstract.

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Sunday, September 07, 2014

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 

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

Link to Rydberg Quantum Optics, Emmy Noether Group >>


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].


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 |rg> 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 |rs>.
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 104 87Rb 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.


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 Ng,in = 1.04(3), we observe a switch contrast Ccoh = 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 Ccoh = 1 - exp(-Ng,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 Cexc = 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 = Ns,outno gate - Ns,outwith gate. In Fig. 3 (a), we plot the measured optical gain for an average input of gate photons Ng,in = 0.75(3). For this gate input, we observe a maximum optical gain G(Ng,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 Ng,in = 0 (Ng,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 Gst = 28(2).
FIG. 3: (a) Optical gain of our transistor, measured for coherent gate input Ng,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 Ng,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].

[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.

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Sunday, August 31, 2014

A True Randomness Generator Exploiting a Very Long and Turbulent Path

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

Davide G. Marangon, Giuseppe Vallone,  Paolo Villoresi

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.

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 nf  speckle centroids of the frame f are elaborated, an ordered sequence Sf = {s, s, . . . , snf } with s1 < s2 < · · · < snf  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 Sf  out of the
possible and equally likely sequences of nf centroids. Among all of combinations, a given Sf can be univocally identified with its lexicographic index I (Sf )
with 0 ≤ I (Sf ) ≤ Tf  - 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, Tf - 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.

[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.

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Sunday, August 24, 2014

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 Kotler1,2

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].

[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.

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Sunday, August 17, 2014

Comparing Matter Waves in Free Fall

[From Left to Right] J. Hartwig, D. Schlippert, E. M. Rasel

Authors: J. Hartwig, D. Schlippert, E. M. Rasel

Affiliation: Institut für Quantenoptik and Centre for Quantum Engineering and Space-Time Research (QUEST), Leibniz Universität Hannover, Germany

Introduction to Einstein’s Equivalence Principle

Einstein’s general relativity is based on three fundamental building blocks: local Lorentz invariance, the universality of the gravitational redshift and the universality of free fall. The enormous importance of general relativity in modern science and technology merits a continuous effort in improving experimental verification of these underlying principles.

The universality of free fall is one of the oldest mechanical theories originally proposed by Galileo. Testing can be done by so called free fall experiments, where two bodies with different composition are freely falling towards a third gravitating body.
Figure 1: Goddard Spaceflight Center Laser Ranging Facility. Source: NASA

Amongst the most sensitive measurements of this principle is the Lunar Laser Ranging experiment, which compares the free fall of earth and moon in the solar gravitational potential [1]. This measurement is only surpassed by torsion balance experiments based on the design of Eötvös [2]. In addition, exciting new insights are expected from the MICROSCOPE experiment [3] that is planned to launch in 2016.

Figure 2: Torsion balance experiment as used in the group of E. Adelberger, University of Washington. Source: Eöt-Wash-Group

The emergence of quantum physics and our improved understanding of the basic building blocks of matter increases the interest scientists have in the understanding of gravity. How do gravity and quantum mechanics interact? What`s the connection between different fundamental particles and their mass? Is there a deeper underlying principle combining our fundamental theories? To comprehensively approach these questions a wide array of parameters must be analyzed. The way how certain test materials may act under the influence of gravity can either be parametrized using a specific violation scenario, like the Dilaton scenario by T. Damour [4], or by using a test theory such as the extended Standard Model of particles (SME) [5]. Since the SME approach is not based on a specific mechanism of violating UFF it also does not predict a level to which a violation may occur. Instead, it delivers a model-independent approach to compare methodically different measurements and confine possible violation theories.

Table 1 states possible sensitivities for violations based on the SME framework for a variety of test masses and underlines the importance of complementary test mass choices are. Hence in comparison to classical tests, the use of atom interferometry opens up a new field of previously inaccessible test masses with perfect isotopic purity in a well-defined spin state. Quantum tests appear to differ from previous test also in a qualitative way. They allow to perform test with new states of matter, such as wave packets by Bose Einstein condensates being in a superposition state. The work presented here is just another early step in a quest to understand the deeper connections between the quantum and classical relativistic world.
Table 1: Sample violation strengths for different test masses linked to “Neutron excess” and the “total Baryon number” based on the Standard Model Extension formalism. The test mass pairs are chosen according to the best torsion balance experiment [6] and existing matter wave tests [7]. An anomalous acceleration would be proportional to the stated numerical coefficients. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014).

Measuring accelerations with atom interferometry

Measuring accelerations with a free fall experiment is always achieved by tracking the movement of an inertial mass in free fall in comparison to the lab frame of reference. This is even true when the inertial mass in question needs to be described by a matter wave operating on quantum mechanics. Falling corner cube interferometers operating on this principle are among the most accurate measurements of gravity with classical bodies. They use a continuous laser beam to track the change of velocity of a corner cube reflector due to gravity in a Michelson interferometer with the corner cube changing one of the arm lengths. Acceleration sensors based on free falling matter waves use a similar principle.
Figure 3: A side view of the experimental setup with the two-dimensional (left side) and three-dimensional (right side) magneto-optical traps employed in [Phys. Rev. Lett. 112, 203002 (2014)].

The first demonstration of a true quantum test of gravity with matter waves was performed 1975 with neutrons in the Famous COW experiment [8]. We will focus on atom interferometers using alkaline atoms, since they are most commonly used for inertial sensing and are also employed in the discussed experiment. Experiments of this kind were first used for acceleration measurements in 1992 [9] and have improved in their performance ever since. The first test of the equivalence principle comparing two different isotopes was then performed in 2004 [10]. Research on quantum tests is for example proposed at LENS in Italy [11], in Stanford in an already existing large fountain [12] and in the scope of the French ICE mission in a zero-g plane [13]. All these initiatives show the high interest of testing gravity phenomena with quantum matters as opposed to classical tests.

In the case of atom interferometers, coherent beam splitting is performed by absorption and stimulated emission of photons. Which atomic transitions are used is dependent on the specific application but, in the case of alkaline atoms two photon transitions coupling either two hyperfine and respective momentum states (Raman transitions) or just momentum states (Bragg transitions) are employed. The point of reference for the measurement is then given by a mirror reflecting the laser beams used to coherently manipulate the atoms, since the electromagnetic field is vanishing at the mirror surface. This results in a reliable phase reference of the light fields and constitutes the laboratory frame. The role of the retroreflecting mirror is similar to the one of the mirror at rest in in the case of the falling corner cube experiment.

The atomic cloud acts as the test mass, which in an ideal case, is falling freely without any influence by the laboratory, except during interaction with the light fields employed as beam splitters or mirrors. During the interaction, the light fields drive Rabi-oscillations in the atoms between the two interferometer states |g> and |e> with a time 2τ needed for a full oscillation. This allows for the realization of beam splitters with a τ/2 pulse length resulting in an equal superposition of |g> and |e>. Mirrors can be realized the same way by applying the beam splitter light fields for a time of τ which leads to an inversion of the atomic state. These pulses are generally called π/2 (for the superposition) and π (for the inversion pulses) in accordance with the Rabi-oscillation phase. The simplest geometry used to measure acceleration is a Mach- Zehnder-like geometry. This is produced by applying a π/2-π-π/2 sequence with free evolution times T placed between pulses. The resulting geometry can be seen in Figure 4.

Figure 4: Space-time diagram of a Mach-Zehnder-like atom interferometer. An atomic ensemble is brought into a coherent superposition of two momentum states by a stimulated Raman transition (π/2 pulse). The two paths I+II propagate separated, are reflected by a pi-pulse after a time T and superimposed and brought to interference with a final π/2 pulse after time 2T. The phase difference is encoded in the population difference of the two output states.

During the interaction with the light fields, the lattice formed by the two light fields imprints its local phase onto the atoms. This results in an overall phase scaling with the relative movement between the atomic cloud and the lattice. Calculating the overall phase imprinted on the atoms results in first order term, Φ=a*T2*keff, where keff is the effective wave vector of the lattice and a is the relative acceleration between lattice and atoms. This immediately shows the main feature of free fall atom interferometry: the T2 scaling of the resulting phase. This is of particular interest for future experiments aiming for much higher free evolution times than currently possible. The phase Φ also shows another key feature. As the acceleration between atoms and lattice approaches zero, the phase also goes to zero, independently of the interferometry time T. This yields a simple way to determine the absolute acceleration of the atomic sample by accelerating the lattice until the lattice motion is in the same inertial reference frame as the freely falling atoms.

Lattice acceleration is achieved by chirping the frequency difference between the two laser beams used for the two photon transition. This transforms the measurement of a relatively large phase, spanning many thousand radians, to a null measurement. The signal produced is the population difference between the interferometer states |g> and |e> as a function of lattice acceleration and thus frequency sweep rate, α. The sweep rate corresponding to a vanishing phase directly leads to the acceleration experienced by the atoms according to lattice acceleration formula a=α/keff. Taking into account Earth’s gravitational field and a lattice wavelength of 780/2 nm (the factor of ½ is introduced due to the use of a two-photon transition) this leads to a sweep rate of around 25 MHz/s. The advantage of this method is that the acceleration measurement is now directly coupled to measurement of the wavelength of the light fields and frequencies in the microwave regime, which are easily accessible.

Our data
Figure 5: Determination of the differential acceleration of rubidium and potassium. The main systematic bias contributions do not change their sign when changing the direction of momentum transfer. Hence, the mean acceleration of upward and downward momentum transfer direction greatly suppresses the aforementioned biases. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)

Figure 6:  the wave nature of 87Rb and 39
atoms and their interference are exploited 
to measure the gravitational acceleration.
In order to test the universality of free fall, we simultaneously chirp the Raman frequencies to compensate for the accelerations a(Rb,±)(g) and a(K,±)(g) experienced by rubidium and potassium that were previously identified (Figure 6). Here, the observed phase shift exhibits contributions due to additional perturbations, such as magnetic field gradients. We make use of a measurement protocol based on reversing the transferred momentum (upward and downward directions ±). This technique makes use of the fact that many crucial perturbations do not depend on the direction of momentum transfer. Thus, by computing the half-difference of the phase differences determined in a single momentum direction, phase shifts induced by, e.g., the AC-Stark effect or Zeeman effect, can be strongly suppressed [14].
Figure 7: Allan deviations of the single species interferometer signals and the derived Eötvös ratio. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)

The data presented in this work [15] was acquired in a data run that was ~4 hours long. By acquiring 10 data points per direction of momentum transfer, and species and then switching to the opposite direction, we were able to determine the Eötvös ratio of rubidium and potassium to a statistical uncertainty of 5.4 x 10-7 after 4096s; the technical noise affecting the potassium interferometer is the dominant noise source.

Taking into account all systematic effects, our measurement yields η(Rb,K)=(0.3 ± 5.4) x 10-7.


In our measurement, the performance was limited both by technical noise and the limited free evolution time T. In order to improve these parameters, we are currently extending the free fall time in our experiment. Furthermore, in an attempt to increase the contrast of our interferometers and thus the signal-to-noise ratio, we are working on implementing state preparation schemes for both species.

We expect to constrain our uncertainty budget (which currently is on the 10 ppb level for the Eötvös ratio) on the ppb level and below through the use of a common optical dipole trap applied to both species. By using Bose-Einstein-condensed atoms, we gain the ability to precisely calculate the ensembles, as well as carefully control the input state. This technique will also be able to reduce uncertainty factors linked to the transverse motion of the cloud, in addition to spatial magnetic field and gravitational field gradients.

Improving the precision of a true quantum test into the sub-ppb regime is the focus of current research. For example we are currently planning a 10m very long baseline atom interferometer (VLBAI) in Hannover [16]. In the framework of projects funded by the German Space Agency (DLR), we moreover develop experiments that are suitable for microgravity operation in the ZARM drop tower in Bremen and on sounding rocket missions [17].

Parallel to the development done in the LUH and at a national level, we are also involved in projects on an international level looking into extending the frontier of atom interferometry and especially the test of the equivalence principle. A major project investigating the feasibility of a space borne mission is the STE-Quest Satellite Mission proposed by a European consortium including nearly all major research institutions working in the field of inertial sensing with atom interferometry, as well as a variety of specialist of other fields [18]. This mission is aimed towards doing a simultaneous test of the equivalence principle with two rubidium isotopes and a clock comparison with several ground based optical clocks, pushing the sensitivity to the Eötvös ratio into the 10-15 regime.

[1] James G. Williams, Slava G. Turyshev, Dale H. Boggs, "Progress in Lunar Laser Ranging Tests of Relativistic Gravity". Physical Review Letters, 93, 261101 (2004). Abstract.
[2] S. Schlamminger, K.-Y. Choi, T. A. Wagner, J. H. Gundlach, E. G. Adelberger, "Test of the Equivalence Principle Using a Rotating Torsion Balance". Physical Review Letters, 100, 041101 (2008). Abstract.
[3] P. Touboul, G. Métris, V. Lebat, A Robert, "The MICROSCOPE experiment, ready for the in-orbit test of the equivalence principle". Classical and Quantum Gravity, 29, 184010 (2012). Abstract.
[4] Thibault Damour, "Theoretical aspects of the equivalence principle". Classical Quantum Gravity, 29, 184001 (2012). Abstract.
[5] M.A. Hohensee, H. Müller, R.B. Wiringa, "Equivalence Principle and Bound Kinetic Energy". Physical Review Letters, 111, 151102 (2013). Abstract.
[6] S. Schlamminger, K.-Y. Choi, T.A. Wagner, J.H. Gundlach, E.G. Adelberger, "Test of the Equivalence Principle Using a Rotating Torsion Balance". Physical Review Letters, 100, 041101 (2008). Abstract.
[7] A. Bonnin, N. Zahzam, Y. Bidel, A. Bresson, "Simultaneous dual-species matter-wave accelerometer". Physical Review A, 88, 043615 (2013). Abstract ; S. Fray, C. Alvarez Diez, T. W. Hänsch, M. Weitz, "Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle". Physical Review Letters, 93, 240404 (2004). Abstract ; M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, G. M. Tino, "Test of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling Effects". Physical Review Letters, 113, 023005 (2014). Abstract.
[8] R. Colella, A. W. Overhauser, S. A. Werner, "Observation of Gravitationally Induced Quantum Interference". Physical Review Letters, 34, 1472 (1975). Abstract.
[9] M. Kasevich, S. Chu, "Measurement of the gravitational acceleration of an atom with a light-pulse atom interferometer". Applied Physics B, 54, 321–332 (1992). Abstract.
[10] Sebastian Fray, Cristina Alvarez Diez, Theodor W. Hänsch, Martin Weitz, "Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle". Physical Review Letters, 93, 240404 (2004). Abstract.
[11] M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, G. M. Tino, "Test of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling Effects". Physical Review Letters, 113, 023005 (2014). Abstract.
[12] Susannah M. Dickerson, Jason M. Hogan, Alex Sugarbaker, David M. S. Johnson, Mark A. Kasevich, "Multiaxis Inertial Sensing with Long-Time Point Source Atom Interferometry". Physical Review Letters, 111, 083001 (2013). Abstract. 2Physics Article.
[13] G Varoquaux, R A Nyman, R Geiger, P Cheinet, A Landragin, P Bouyer, "How to estimate the differential acceleration in a two-species atom interferometer to test the equivalence principle". New Journal of Physics, 11, 113010 (2009). Full Article.
[14] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and,M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry". Physical Review A, 65, 033608 (2002). Abstract; Anne Louchet-Chauvet, Tristan Farah, Quentin Bodart, André Clairon, Arnaud Landragin, Sébastien Merlet, Franck Pereira Dos Santos, "The influence of transverse motion within an atomic gravimeter". New Journal of Physics, 13, 065025 (2011). Full Article.
[15] D. Schlippert, J. Hartwig, H. Albers, L. L. Richardson, C. Schubert, A. Roura, W. P. Schleich, W. Ertmer, E. M. Rasel, "Quantum Test of the Universality of Free Fall". Physical Review Letters, 112, 203002 (2014). Abstract.
[16] http://www.geoq.uni-hannover.de/350.html
[17] http://www.iqo.uni-hannover.de/quantus.html
[18] D N Aguilera, H Ahlers, B Battelier, A Bawamia, A Bertoldi, R Bondarescu, K Bongs, P Bouyer, C Braxmaier, L Cacciapuoti, C Chaloner, M Chwalla, W Ertmer, M Franz, N Gaalou, M Gehler, D Gerardi, L Gesa, N Gürlebeck, J Hartwig, M Hauth, O Hellmig, W Herr, S Herrmann, A Heske, A Hinton, P Ireland, P Jetzer, U Johann, M Krutzik, A Kubelka, C Lämmerzah, A Landragin, I Lloro, D Massonnet, I Mateos, A Milke, M Nofrarias, M Oswald, A Peters, K Posso-Trujillo, E Rase, E Rocco, A Roura, J Rudolph, W Schleich, C Schubert, T Schuldt, S Seide, K Sengstock, C F Sopuerta, F Sorrentino, D Summers, G M Tino, C Trenkel, N Uzunoglu, W von Klitzing, R Walser, T Wendrich, A Wenzlawski, P Weßels, A Wicht, E Wille, M Williams, P Windpassinger, N Zahzam,"STE-QUEST—test of the universality of free fall using cold atom interferometry". Classical Quantum Gravity, 31, 115010 (2014), Abstract.

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