Evidence for Pre-formed Pairs in an Oxide Superconductor

From left to right: Mengcheng Huang, Shicheng Lu, Jeremy Levy, Guanglei Cheng, Michelle Tomczyk, Patrick Irvin. 

Authors: Guanglei Cheng, Michelle Tomczyk, Jeremy Levy

Affiliation:
Department of Physics and Astronomy, University of Pittsburgh, USA.
Pittsburgh Quantum Institute, Pittsburgh, Pennsylvania, USA.

Link to Levy Research Group >>

Strontium titanate (SrTiO₃ or simply STO) is the first and best-known superconducting semiconductor. Its many fascinating properties, especially superconductivity, motivated Georg Bednorz and Alex Müller to search for high-temperature (high-T_c) superconductors in perovskite oxides [1]. STO shares many features related to the cuprate high-T_c superconductors, including a dome-shaped phase diagram and a pseudogap phase [2]. However, STO has much lower temperature and carrier density than the high-T_c compounds. A longstanding question surrounds the nature of the pseudogap state: ‘To pair or not to pair?’ – Does the existence of a pseudogap indicate electron pairing in the absence of superconductivity?

In 1969, long before high-T_c superconductivity was discovered, D.M. Eagles predicted [3] that electrons could remain paired outside of the superconducting state in STO. Eagles predicted that electrons can form a dilute gas so that the size of pairs is very small compared to the inter-electron distance. Namely, electrons pair in real space and the superconductivity is a result of Bose-Einstein condensation (BEC), contrasting the weak pairing in momentum space described by Bardeen-Schrieffer-Cooper (BCS) theory, which is highly successful in explaining conventional superconductors. Eagles was the first to propose the concept of BEC-BCS crossover, which was later independently developed by Nobel laureate Anthony Leggett [4] and experimentally realized in an ultracold atomic gas [5]. The discovery of BEC-type superconductivity in solid state systems has been challenging.

The LaAlO₃/SrTiO₃ (LAO/STO) interface [6] has attracted numerous interests in the last decade. It hosts a two-dimensional conducting interface that possesses a wealth of strongly correlated phenomena including superconductivity, magnetism, metal-insulator transition (MIT) and spin-orbit interaction [7]. A few years ago, we developed a lithography technique that allows us to ‘write’ and ‘erase’ nanostructures at the LAO/STO interface by using a sharp conductive atomic force microscope (c-AFM) tip, thus effectively programming all the novel properties at the nanoscale [8,9]. The writing mechanism relies critically on the MIT, in which the interface becomes conducting above the critical 3 unit cell (uc) LAO thickness. While the 3uc LAO/STO interface is insulating, it is switchable by a voltage-biased c-AFM tip. Under the c-AFM tip, a series of nanoscale devices have been made available. The resulting nanowires are only a few nanometers wide, exhibit anomalously high mobility [10] and show potential for the development of new types of nanoelectronics.

To investigate electron pairing, we use a superconducting single-electron transistor (Figure 1a). The device consists of a main superconducting nanowire channel intersected with voltage leads. Two tunnel potential barriers are engineered through c-AFM ‘cutting’ procedures so that a nanoscale island is defined. Due to the nanoscale confinement, the energy levels inside the island are quantized, and carrier transport is only possible when the chemical levels of external electrodes (source and drain) are aligned with an island energy level, which is dependent on the side gate voltages.

Figure 1: (click on the figure to view with higher resolution) Device schematic and transport characteristics. (a), Device schematic written by c-AFM lithography. The nanowires are typically 5 nm wide, and the length between 2 barriers is 1 μm. (b) and (c) Differential conductance dependent on the source-drain bias and side gate voltages at B=0 T (b) and B=4 T (c). The number of diamonds is doubled in (c). Color scales are 0-80 µS. (d) Magnetic field dependence of the conductance peaks. The bifurcation of conductance peaks above B_p suggest electron pairing without superconductivity. Color scale: 0-40 µS.

Indeed, bias spectroscopy reveals [11] a series of conductance diamonds (Fig. 1b), reminiscent of so-called ‘Coulomb diamonds’ in conventional blockade physics. In the latter case, each sequential Coulomb diamond corresponds to the stability of one additional electron, and applying an external magnetic field merely changes the size of diamonds due to Zeeman Effect. Remarkably, when we increase the magnetic field, the diamonds initially remain insensitive to field, then bifurcate above a critical magnetic field B_p~2 T (Fig. 1b,c). Such behavior is clearly revealed when we track the magnetic field dependence of the zero-bias conductance peaks. We find that all of the peaks bifurcate above a ‘pairing field’ B_p (Fig. 1d), suggesting transport is dominated by electron pairs rather than single electrons below B_p. Electron pairing persists far above the critical temperature (T_c~0.3 K) and for magnetic fields far above the upper critical field (H_c2~0.2 T) for superconductivity in bulk STO.

The observed electron pairing without superconductivity is difficult to explain using BCS theory. Pair fluctuations in disordered BCS superconductor films may give signatures of pairing above T_c which is greatly suppressed by disorder. However, the corresponding pairing temperature will not exceed T_c in the clean limit. Here the pairing temperature we have observed is around several kelvin, one order of magnitude higher than the T_c in the bulk. These experimental signatures are captured by a phenomenological model that favors BEC pairing, consistent with D. M. Eagles’s theory proposed 46 years back.

References:
[1] J. Georg Bednorz and K. Alex Müller, "Perovskite-type oxides - the new approach to High Tc superconductivity", Nobel Lecture (1987).
[2] C. Richter, H. Boschker, W. Dietsche, E. Fillis-Tsirakis, R. Jany, F. Loder, L. F. Kourkoutis, D. A. Muller, J. R. Kirtley, C. W. Schneider, J. Mannhart, "Interface superconductor with gap behaviour like a high-temperature superconductor", Nature, 502, 528 (2013). Abstract.
[3] D.M. Eagles, "Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors", Physical Review, 186, 456 (1969). Abstract.
[4] Anthony J. Leggett, "A theoretical description of the new phases of liquid ³He", Reviews of Modern Physics,  47, 331 (1975). Abstract.
[5] M. W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck, W. Ketterle, "Vortices and superfluidity in a strongly interacting Fermi gas",  Nature, 435, 1047 (2005). Abstract.
[6] A. Ohtomo,  H.Y. Hwang, "A high-mobility electron gas at the LaAlO₃/SrTiO₃ heterointerface", Nature, 427, 423 (2004). Abstract.
[7] Joseph A. Sulpizio, Shahal Ilani, Patrick Irvin, Jeremy Levy, "Nanoscale Phenomena in Oxide Heterostructures", Annual Review of Materials Research, 44, 117 (2014). Abstract.
[8] C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C. S. Hellberg, J. Mannhart, and J. Levy, "Nanoscale control of an interfacial metal-insulator transition at room temperature", Nature Materials, 7, 298 (2008). Abstract.
[9] Cheng Cen, Stefan Thiel, Jochen Mannhart, Jeremy Levy, "Oxide nanoelectronics on demand", Science, 323, 1026 (2009). Abstract.
[10] Patrick Irvin, Joshua P. Veazey, Guanglei Cheng, Shicheng Lu, Chung-Wung Bark, Sangwoo Ryu, Chang-Beom Eom, Jeremy Levy, "Anomalous High Mobility in LaAlO₃/SrTiO₃ Nanowires", Nano Letters, 13, 364 (2013). Abstract.
[11] Guanglei Cheng, Michelle Tomczyk, Shicheng Lu, Joshua P. Veazey, Mengchen Huang, Patrick Irvin, Sangwoo Ryu, Hyungwoo Lee, Chang-Beom Eom, C. Stephen Hellberg, Jeremy Levy, "Electron pairing without superconductivity", Nature 521, 196 (2015). Abstract.

Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor

From Left to Right: (top row) Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen ; (bottom row) Sangjun Jeon, Allan H. MacDonald, B. Andrei Bernevig, Ali Yazdani.

Authors: Stevan Nadj-Perge¹, Ilya K. Drozdov¹, Jian Li¹, Hua Chen², Sangjun Jeon¹, Jungpil Seo¹, Allan H. MacDonald², B. Andrei Bernevig¹, Ali Yazdani¹.

Affiliation
¹Joseph Henry Laboratories and Dept of Physics, Princeton University, USA.
²Department of Physics, University of Texas at Austin, USA.

Link to Yazdani Lab >>
Link to Allan H. MacDonald's Group >>

In 1937 Italian scientist Ettore Majorana, one of the most promising theoretical physicists at that time, proposed a hypothetical fermionic excitation, now called Majorana fermion, which has a property that it is its own anti-particle [1]. Ever since significant efforts were invested in finding an elementary particle described by Majorana. While present for many years in particle physics community, it was only in 2001 that Alexei Kitaev suggested an intriguing possibility that a type of a quasi-particle, a condensed matter analog of the Majorana fermion could exist [2]. Such quasi-particle would emerge as a zero energy excitation localized at the boundary of a one dimensional topological superconductor. Following his seminal work various systems were proposed as a potential platform for realization of Majorana quasi-particles. Apart from fundamental scientific interest, the motivation for investigating Majorana bound states partly relies on their potential for use for quantum computing [3,4].

Past 2Physics article by Ali Yazdani :
August 25, 2013: "Visualizing Nodal Heavy Fermion Superconductivity"
by Brian Zhou, Shashank Misra, and Ali Yazdani.

Previous to our experiments, the most promising experimental route to realize these elusive bound states was based on semiconductor-superconductor interfaces in which Majorana fermions would appear as conductance peaks at zero energy [5,6]. Indeed experiments in 2012 reported zero energy conductance peaks suggesting presence of Majorana modes in these type of systems [7,8], however, alternative explanations related to disorder and Kondo phenomena proposed latter could not be fully ruled out. It is worth noting that in these interfaces spatial information about localized excitations is very hard to obtain.

Fig. 1: (A) Schematic of the proposal for realization and detection of Majorana states: A ferromagnetic atomic chain is placed on the surface of strongly spin-orbit–coupled superconductor and studied using STM. (B) Band structure of a linear suspended Fe chain before introducing spin-orbit coupling or superconductivity. The majority spin-up (red) and minority spin-down (blue) d-bands labeled by azimuthal angular momentum m are split by the exchange interaction J (degeneracy of each band is noted by the number of arrows). a, interatomic distance. (C) Regimes for trivial and topological superconducting phases are identified for the band structure shown in (B) as a function of exchange interaction in presence of SO coupling. The value J for Fe chains based on density functional calculations is noted. μ is the chemical potential.

Building upon previous proposal to realize Majorana modes in an array of magnetic nanoparticles [9], we proposed to use a chain of magnetic atoms coupled to a superconductor [10]. The key advantage of this platform is that the experimentally properties of this system can easily be studied using standard scanning tunneling microscopy (STM) technique. While ours and other follow-up initial proposals [11-16] consider specific orientation of the magnetic moments, the approach works also for ferromagnetic atomic chains as long as it is coupled to a superconductor with strong spin-orbit coupling (Fig. 1A) [17]. In this case, the large exchange interaction results in a band structure of the chains such that majority spin band is fully occupied while the Fermi level is in the minority spin bands. For example the electronic structure of a linear Iron (Fe) ferromagnetic atomic chain is shown in Fig. 1B. Considering only d-orbitals which are spin-polarized it is easy to show that many of the bandstructure degeneracies are lifted and that for large range of parameters, chemical potential and the exchange energy, such chains are in topologically non-trivial regime characterized by the odd number of crossing at the Fermi level (Fig 1C). When placed on the superconducting substrate with strong-spin orbit coupling the resulting superconductivity on the chain will necessarily be topological in nature resulting in zero energy Majorana bound states located at the chain ends.

Fig. 2: (A) Topograph of the Pb(110) surface after growth of Fe, showing Fe islands and chains indicated by white arrows and atomically clean terraces of Pb (regions with the same color) with size exceeding 1000 Å. (Lower-right inset) Anisotropic atomic structure of the Pb(110) surface (Upper-left insets) images of several atomic Fe chains and the islands from which they grow (scale bars, 50 Å). (B) Topography of the chain colorized by the conductance at H= ±1 T from low (dark blue) to high conductance (dark red). (C) Difference between conductance on and off the chain showing hysteresis behavior. (D and E) Atomic structure of the zigzag chain, as calculated using density functional theory. The Fe chain structure that has the lowest energy in the calculations matches the structural features in the STM measurements, a_B is the Bohr radius.

We have developed a way to grow iron atomic chains on the surface of lead (Pb) which, due to heavy atomic mass, is expected to have strong spin-orbit coupling. For this purpose we used Pb(110) crystallographic surface orientation which has characteristic anisotropy (Fig. 2(A) lower left inset). When a sub-monolayer of Fe is evaporated and slight annealing, the anisotropy of the substrate could trigger growth of one-dimensional atomic chains. We investigated the resulting structures by using STM at cryogenic temperatures (temperature was 1.4K in the experiment). On relatively large atomically ordered regions of the Pb(110) surface we observed self-assembled islands as well as single atom wide chains of Fe. Depending on growth conditions, we find Fe chains as long as 500 Å with ordered regions approaching 200 Å. In order to confirm ferromagnetic order on the chain we have performed spin-polarized measurements using bulk antiferromagnetic Chromium STM tips. Tunneling conductance (dI/dV) at a low bias voltage as a function of the out-of-plane magnetic field shows contrast for opposite fields (Fig. 2B) and a hysteresis behavior (Fig. 2C, note that no hysteresis is observed on the Pb substrate). The observed hysteresis loop corresponds to the tunneling of electrons between two magnets with the field switching only one of them at around 0.25 T.

We also observed the variation of the spin-polarized STM signal along the chain which is likely due to its electronic and structural properties. Indeed, both topographic features and periodicity of signal variation could be very well explained by our theory collaborators who performed density functional theory modeling. Their calculations suggested that our chains have zig-zag structure which explains both topographic information obtained using STM and matches well with our spin-polarized measurements (Fig. 2D and 2E).

Fig. 3: (A) STM spectra measured on the atomic chain at locations corresponding to those indicated in (B) and (C). For clarity, the spectra are offset by 100 nS. The red spectrum shows the zero-bias peak at one end of the chain. The gray trace measured on the Pb substrate can be fitted using thermally broadened BCS superconducting density of states (dashed gray line, fit parameters Δs = 1.36 meV, T = 1.45 K). (B and C) Zoom-in topography of the upper (B) and lower end (C) of the chain and corresponding locations for spectra marked (1 to 7). Scale bars, 25 Å. (D and E) Spectra measured at marked locations, as in (B) and (C). (F) Spatial and energy-resolved conductance maps of another atomic chain close to its end, which shows similar features in point spectra as in (A). The conductance map at zero bias (middle panel) shows increased conductance close to the end of the chain. Scale bar, 10 Å.

After establishing basic properties of our chains we investigated low-energy excitations using spatial spectroscopic mapping, see Fig 3. While on the surface of bare Pb(110), there is clear structure of the superconducting gap on the Fe atomic chain, the presence of the in-gap states is predominant (Fig. 3A). Most notably a peak close to zero bias voltage is observed near the chain end together with asymmetric less-developed gap-like structure in the middle of the chain (Fig. 3D and Fig. 3E). Both spatially resolved spectra and the spectroscopic maps at low bias voltage show signatures expected from Majorana bound states (Fig. 3F). The ability to correlate the location of the zero bias conductance peak with the end of the atomic chains is one of the main experimental results of our work. This is one of the basic requirements for interpreting that this feature is associated with the predicted Majorana bound state of a topological superconductor. In addition to robust observation of the zero bias peaks in many chains, we have performed several control experiments to eliminate other potential effects which may give similar looking signatures. For example, when superconductivity is suppressed by applying small magnetic field, the spectrum on the chain becomes featureless in contrast to what would be expected for Kondo effect. Also for very short chains zero biased peaks were not observed, ruling out trivial effects related to the chain ends. Furthermore, in order to increase experimental resolution we took measurements with superconducting tip which also confirm the over picture consistent with Majorana bound states in this system.

The observed spectroscopic signatures are consistent with the existence of Majorana bound states in our system. An obvious extension of our experiments is to create two dimensional islands and search for propagating Majorana modes or, for example, investigate other systems with both even and odd number of band crossing at Fermi level in order to further test the concept behind our study. Ultimately the future experiments will focus on manipulation of Majorana bound states in this system [18].

References: 
[1] Ettore Majorana, "Teoria simmetrica dell’elettrone e del positrone", Il Nuovo Cimento, 171 (1937). Abstract.
[2] A. Yu. Kitaev, "Unpaired Majorana fermions in quantum wires". Physics Uspekhi, 44, 131 (2001). Full Article.
[3] A. Yu. Kitaev, "Fault-tolerant quantum computation by anyons". Annals of Physics. 303, 2 (2003). Abstract.
[4] Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, Matthew P. A. Fisher, "Non-Abelian statistics and topological quantum information processing in 1D wire networks". Nature Physics, 7, 412 (2011). Abstract.
[5] Yuval Oreg, Gil Refael, and Felix von Oppen, "Helical Liquids and Majorana Bound States in Quantum Wires", Physical Review Letters, 105, 177002 (2010). Abstract.
[6] Roman M. Lutchyn, Jay D. Sau, S. Das Sarma, "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures". Physical Review Letters, 105, 077001 (2010). Abstract.
[7] V. Mourik, K. Zuo, S.M. Frolov, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, "Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices", Science 336, 1003 (2012). Abstract. 2Physics Article.
[8] Anindya Das, Yuval Ronen, Yonatan Most, Yuval Oreg, Moty Heiblum, Hadas Shtrikman, "Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions". Nature Physics, 8, 887 (2012). Abstract. 2Physics Article.
[9] T. P. Choy, J. M. Edge, A. R. Akhmerov, C. W. J. Beenakker, "Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin-orbit coupling". Physical Review B, 84, 195442 (2011). Abstract.
[10] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, Ali Yazdani, "Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor". Physical Review B, 88, 020407 (2013). Abstract.
[11] Falko Pientka, Leonid I. Glazman, Felix von Oppen, "Topological superconducting phase in helical Shiba chains". Physical Review B, 88, 155420 (2013). Abstract.
[12] Jelena Klinovaja, Peter Stano, Ali Yazdani, Daniel Loss, "Topological Superconductivity and Majorana Fermions in RKKY Systems". Physical Review Letters, 111, 186805 (2013). Abstract.
[13] Bernd Braunecker, Pascal Simon, "Interplay between Classical Magnetic Moments and Superconductivity in Quantum One-Dimensional Conductors: Toward a Self-Sustained Topological Majorana Phase". Physical Review Letters, 111, 147202 (2013). Abstract.
[14] M. M. Vazifeh, M. Franz, "Self-Organized Topological State with Majorana Fermions". Physical Review Letters, 111, 206802 (2013). Abstract.
[15] Sho Nakosai, Yukio Tanaka, Naoto Nagaosa, "Two-dimensional superconducting states with magnetic moments on a conventional superconductor". Physical Review B, 88, 180503 (2013). Abstract.
[16] Younghyun Kim, Meng Cheng, Bela Bauer, Roman M. Lutchyn, S. Das Sarma, "Helical order in one-dimensional magnetic atom chains and possible emergence of Majorana bound states". Physical Review B, 90, 060401 (2014). Abstract.
[17] Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen, Sangjun Jeon, Jungpil Seo, Allan H. MacDonald, B. Andrei Bernevig, Ali Yazdani, "Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor". Science 346, 602-607 (2014). Abstract.
[18] Jian Li, Titus Neupert, B. Andrei Bernevig, Ali Yazdani, "Majorana zero modes on a necklace". arXiv:1404.4058 [cond-mat] (2014).

Enhancement of Long-Range Correlations in a 2D Vortex Lattice by an Incommensurate 1D Disorder Potential

(Left to Right) Top Row: Isabel Guillamón, Rosa Córdoba; Middle Row: Javier Sesé, José María De Teresa, M. Ricardo Ibarra; Bottom Row: Sebastián Vieira, Hermann Suderow.

Authors:
Isabel Guillamón^1,2, Rosa Córdoba^3,*, Javier Sesé³, José María De Teresa^3,4, M. Ricardo Ibarra³, Sebastián Vieira¹, Hermann Suderow¹

Affiliations:
¹Laboratorio de Bajas Temperaturas, Departamento de Física de la Materia Condensada, Instituto de Ciencia de Materiales Nicolás Cabrera, Condensed Matter Physics Center, Unidad Asociada de Bajas Temperaturas y Altos Campos Magnéticos, Universidad Autónoma de Madrid, Spain
²H.H. Wills Physics Laboratory, University of Bristol, UK
³Laboratorio de Microscopías Avanzadas (LMA), Instituto de Nanociencia de Aragón (INA), Universidad de Zaragoza, Spain
⁴Instituto de Ciencia de Materiales de Aragón (ICMA), CSIC-Universidad de Zaragoza, Spain
^*Present address: Department of Applied Physics, Eindhoven University of Technology, The Netherlands.

Recently we studied the effect of random disorder in a particularly simple two dimensional system and found a new recipe to enhance order. We used a thin superconductor under magnetic fields[1]. The Cooper pairs in the superconductor turn and form quantum vortices. Each one is like a small tornado, but they are all very tiny and, what is most important, they all look exactly the same and repel each other. As a result, they form a perfect hexagonal lattice, which can be imaged with a surprising accuracy.

Past 2Physics article by this group:
March 10, 2013: "Nanostructuring Improves Vortex Pinning in Superconductors at Elevated Temperatures and Magnetic Fields" by R. Córdoba, T. I. Baturina, J. Sesé, A. Yu. Mironov, J. M. De Teresa, M. R. Ibarra, D. A. Nasimov, A. K. Gutakovskii, A.V. Latyshev, I. Guillamón, H. Suderow, S.Vieira, M. R. Baklanov, J. J. Palacios, V.M.Vinokur.

To introduce random disorder in a controlled way, we made a one-dimensional nanostructure and showed that, under certain conditions, the hexagonal lattice floats over the nanostructure (Fig.1 right panel). The floating lattice disorders when increasing the vortex density. Random disorder appears because both lattices are incommensurate. We find that the transition into the disordered state occurs far above present expectations from random field theory. Our conclusion is that the weak one-dimensional correlations inhibit the effect of random disorder.

Figure 1: Vortices in a hexagonal lattice are shown as red points. A linear modulation acts on the vortex positions. When the vortex density is low, vortices are confined to the minima of the linear modulation, and the interaction among them is essentially confined to the lines along the potential minima, as schematically shown in the left panel. However, when the vortex density increases, one can accommodate many vortices within one modulation’s wavelength. In the right panel we schematically show this situation (for clarity, we only draw a small amount of vortices). In that case, the lattice and 1D potential modulation have incommensurate spatial periods and the orientation of the lattice no longer follows the linear modulation. The resulting vortex landscape is quasi-random. The vortex positions are slightly displaced with respect to the ordered lattice. In this work, we show that the displacement grows logarithmically with distance, a feature which demonstrates that the disorder created by this situation is scale-invariant.

To see this in more detail, let us think of many identical particles that repel each other forming a single layer on a corrugated surface and at zero temperature. Let the position of the particles depend on the size of the corrugation. If the surface shows a periodic one-dimensional pattern, as the wrinkles you create on a carpet when you lay it down on your floor, you may obtain hexagonal order commensurate to the wrinkles (left panel of Fig.1, [2]). The question is, what happens if you produce a fully random pattern which looks the same at all scales and increase the size of the corrugation? To answer this question (posed in several theory papers some time ago, see e.g.Refs. [3,4]) we first need to create such a random pattern. Think of a carpet with many small (much smaller than the particle size) and stiff yarns of many different lengths. To make such a carpet, you have to prune it yarn by yarn choosing each time a completely random height. This looks difficult.

Instead, we preferred to use the one-dimensional modulation [5]. The wavelength of the modulation is incommensurate to the wavelength of the ordered particle lattice. Disorder in the particle positions appears automatically, simply because both lattices do not match.

Then, as the second step, we decrease the interaction strength among particles. This is easy for vortices in superconductors [6]. We just have to increase the magnetic field, and we effectively increase the size of the corrugation, with respect to the strength of the interaction among particles.

The experiment directly showing the disorder transition in the vortex lattice was made using a scanning tunneling microscope effectively at absolute zero in the low temperature laboratory of the Universidad Autonóma de Madrid [7]. The thin superconductor with a weak one-dimensional modulation was fabricated in the Laboratory of advanced Microscopies & Nanoscience Institute of Aragón [8]. We eliminated the effect of temperature altogether and produced a very well controlled linear nanostructure. We imaged up to several thousands of vortices in Madrid. This allowed us to characterize the critical exponents of the transition very accurately.

Transitions are often measured in macroscopic experiments, where the obtained information is the result of the average of particles’ positions and the interaction among them. Even microscopic experiments, as neutron scattering, provide an average over many particles. In our work, we were able to show the macroscopic behavior by imaging lots of vortices one by one. This allowed us to characterize precisely the transition. The order-disorder transition goes on by particle displacements producing topological defects (dislocations and disclinations) in an otherwise ordered lattice. The experiment shows how random quenched disorder progresses in the lattice when increasing the magnetic field. We find a logarithmic increase of displacements with respect to the ordered lattice at low fields and density fluctuations in the high field disordered phase. Both effects nicely agree with random field theories.

But we also find that the size of the random potential at which the transition takes places does not agree with random field theory. On rather universal arguments, theory finds that those topological defects should arise at a critical disorder strength of 1/8 [3,4]. The experiment gives a much higher critical value, of 1/2. The lattice stays ordered over a significantly larger range of magnetic fields than expected. What is favoring order? The answer lies in the correlations produced by the one-dimensional modulation. This was discussed theoretically for spin systems [9], but had not been fully addressed by experiments before. Although the one-dimensional modulation does not influence the orientation of the lattice nor directly determines the precise positions of the particles in the lattice, correlations give consistency to the two dimensional hexagonal order.

Figure 2: Image of the vortex lattice in the high field disordered phase at 5.5 T taken with our microscope. Background color represents the intensity of vortex density fluctuations. The lattice is shown by its Delaunay triangulation. Disclinations –five or seven coordinated vortices, green and orange dots, respectively– and disclination pairs –dislocations– are present over the whole surface producing a very disordered vortex lattice. The 1D thickness modulation generating the symmetry breaking disorder is also shown.

The result that you can help supporting an ordered hexagonal two-dimensional lattice in a random medium by introducing a tiny amount of symmetry breaking correlations is both useful and unexpected. It adds now a new looking-glass for vortex physics in nanostructured superconductors.

Our to-do list includes studying vortex dynamics in presence of the disorder potential. By applying a current, we want to see when vortices start moving and study the stability of superconductivity to disorder. The careful observation of vortex lattices in superconductors will be useful to understand more about the influence of disorder in macroscopic quantum coherence.

References:
[1] I. Guillamón, R. Córdoba, J. Sesé, J. M. De Teresa, M. R. Ibarra, S. Vieira, H. Suderow, "Enhancement of long-range correlations in a 2D vortex lattice by an incommensurate 1D disorder potential". Nature Physics, 10, 851 (2014). Abstract.
[2] Piero Martinoli, "Static and dynamic interaction of superconducting vortices with a periodic pinning potential". Physical Review B, 17, 1175–1194 (1978). Abstract.
[3] Thomas Nattermann, Stefan Scheidl, Sergey E. Korshunov, Mai Suan Li, "Absence of reentrance in the two-dimensional XY-model with random phase shift". Journal of Physics I France, 5, 565–572 (1995). Abstract.
[4] David Carpentier, Pierre Le Doussal, "Melting of two dimensional solids on disordered substrates". Physical Review Letters, 81, 1881 (1998). Abstract.
[5] Laurent Sanchez-Palencia, Maciej Lewenstein, "Disordered quantum gases under control". Nature Physics, 6, 87 (2010). Abstract.
[6] Ernst Helmut Brandt, "Vortex-vortex interaction in thin superconducting films". Physical Review B, 79, 134529 (2009). Abstract.
[7] http://lbtuam.es
[8] http://ina.unizar.es/, http://www.icma.unizar-csic.es/ICMAportal/, http://www.unizar.es 
[9] J. Wehr, A. Niederberger, L. Sanchez-Palencia, M. Lewenstein, "Disorder versus the Mermin–Wagner–Hohenberg effect: From classical spin systems to ultracold atomic gases". Physical Review B, 74, 224448 (2006). Abstract.

Optomechanical Coupling between a Multilayer Graphene Mechanical Resonator and a Superconducting Microwave Cavity

Left to Right: (top row) V. Singh, S. J. Bosman, B. H. Schneider, (bottom row) Y. M. Blanter, A. Castellanos-Gomez, G. A. Steele.

Authors: 
V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, 
G. A. Steele

Affiliation:
Kavli Institute of NanoScience, Delft University of Technology, The Netherlands.

Introduction:

Mechanical resonators made from two dimensional exfoliated crystals offer very low mass, low stress, and high quality factor due to their crystalline structure [1]. These properties make them very attractive for application in mass sensing, force sensing, and exploring the quantum regime of motion by providing large quantum zero-point fluctuations over a small bandwidth. The most studied exfoliated crystal so far is graphene, where a considerable progress has been made in exploring its properties for mass sensing, study of nonlinear mechanics, and voltage tunable oscillators [2-9]. These properties also make graphene attractive for exploring the quantum regime of motion.

Past 2Physics articles by Andres Castellanos-Gomez and Gary A. Steele:

July 20, 2014: "Few-layer Black Phosphorus Phototransistors for Fast and Broadband Photodetection" by Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, Gary A. Steele, Herre S.J. van der Zant, Andres Castellanos-Gomez.

A possible route towards exploring the quantum regime of graphene motion is cavity optomechanics [10]. It has shown exquisite position sensitivity, enabled the preparation and detection of mechanical systems in the quantum ground state with conventional top-down superconducting mechanical resonators [11-18]. Therefore, a natural candidate for implementing cavity optomechanics with graphene resonator is to couple it to a high Q superconducting microwave cavity. However, coupling graphene resonators with superconducting cavities in such a way that both retain their excellent properties (such as their high quality factors) is technologically challenging. Using a deterministic dry transfer technique [19], we combine a multilayer graphene resonator to a high quality factor microwave cavity [20]. Although multilayer graphene has a higher mass than a mono-layer, it could be advantageous for coupling to a superconducting cavity because of its lower electrical resistance.

Results:

Device

To fabricate the superconducting cavities in coplanar waveguide geometry, we use an alloy of molybdenum and rhenium with superconducting transition temperature of 8.1 K. Using the dry transfer technique, we place a few layer thick graphene mechanical resonator near the coupler forming coupling capacitor for the cavity. Figure 1(a) shows a false color scanning electron microscope image of a device with a 10 nm thick multilayer graphene resonator coupled to a superconducting cavity. Figure 1(b) shows an equivalent schematic diagram with graphene resonator acting as a capacitor (C_g ) between the superconducting cavity (formed by L_sc and C_sc ) and the external microwave source. By cooling these cavities to very low temperatures (14 mK), we measured internal quality factor as high as 107,000.

FIG. 1: Coupling of a multilayer graphene mechanical resonator to a superconducting cavity. (a) A tilted angle scanning electron micrograph (false color) near the coupler showing 4 μm diameter multilayer (10 nm thick) graphene resonator (cyan) suspended 150 nm above the gate. (b) Schematic lumped element representation of the device with the equivalent lumped parameters as C_sc ≈ 415 fF and L_sc ≈ 1.75 nH.

Mechanical motion readout sensitivity

To the first order, the superconducting microwave cavity can be thought simply as motional transducer for the graphene resonator. To readout the motion of the graphene resonator, we inject a microwave near the cavity frequency given by

                                       
The motion of graphene resonator modulates the cavity frequency and hence its displacement gets imprinted on the phase of the reflected microwave signal from the cavity. By measuring the phase of the reflected signal (technically known as the homodyne detection), one can directly read the mechanical motion of the resonator [11]. The large quality factor of our cavity and its ability to sustain superconductivity with large number of the microwave photons enable us to measure the thermo-mechanical motion of the graphene resonator down to temperatures of 96 mK and a displacement sensitivity as low as 17 fm/√Hz.

Optomechanical coupling

In addition to detecting the motion of the graphene drum, we can also exert a force on the mechanical drum by using the radiation pressure of microwave photons trapped in the superconducting cavity. This force comes from the fact that light carries momentum: shining light from a flashlight at a piece of paper would in principle apply a force to it, pushing it away from the light source. The radiation pressure force that light exerts, however, is usually far too small to detect. Due to the tiny mass of the graphene sheet and the ability to detect small displacement, we could see the graphene sheet shaking in response to a "beat" set by the microwave light sent into the cavity.

By sending two microwave signals, a probe signal ω_p (near the cavity resonance frequency ω_c ) and another signal at ω_d (detuned by mechanical frequency ω_m , such that ω_d = ω_c +ω_m ), one can apply a a radiation pressure force on the mechanical resonator. This radiation pressure force beats at the mechanical resonance frequency, leading to coherent driven motion of the mechanical resonator, as shown schematically by process 1 in Figure 2(a). In presence of the significant optomechanical coupling, this coherent drive of the mechanical resonator down-converts the detuned drive photons exactly at the probe frequency (pink arrow) shown by process 2 in Figure 2(a). These two signals at probe frequency interfere with each other leading to a transmission window, appearing as a sharp peak in the cavity response, shown in Figure 2(b). This phenomena is known as "optomechanically induced transparency" (OMIT) and is a signature of the optomechanical coupling between the graphene mechanical resonator and the superconducting cavity [21-23]. As this effect rely on the coherent driven motion of the graphene mechanical resonator, the width of the transparency window is set by the mechanical resonator's linewidth as shown in the inset of Figure 2(b). Using the radiation pressure force driving, we measure the quality factor of the graphene resonator as high as 220,000.

FIG. 2: Optomechanically induced transparency (OMIT). (a) Schematic illustrate OMIT features in terms of the interference of the probe field (black arrow) with the microwave photons that are cyclically down- and then up- converted by the optomechanical interaction (pink arrow). (b) Measurement of the cavity reflection |S₁₁| in presence of sideband detuned drive tone. A detuned drive at ω_c+ω_m results in a window of optomechanically induced reflection (OMIR) in the cavity response. Inset: Zoom of the OMIR window. (c) Measurement of the cavity reflection |S₁₁| with a stronger detuned drive. At the center of the cavity response, the reflection coefficient exceeds 1, corresponding to mechanical microwave amplification of 17 dB by the graphene resonator.

By increasing the drive signal amplitude further, one can increase the strength of the optomechanical coupling. Using this, we make an amplifier in which microwave signals are amplified by the mechanical motion of the graphene resonator [16]. With a stronger detuned drive, we observed a microwave gain of 17 dB (equivalent to a photon gain of 50) as shown in Figure 2(c), before the nonlinear effects from the mechanical resonators come into play. Similarly, a different "beat" of the microwave photons (having ω_d = ω_c - ω_m ) allows one to store microwave photons into the mechanical motion of the resonator [24]. To this end we show a storage time up to 10 millisecond, which is equivalent to delay from a few hundreds of kilometer long coaxial cable.

The phenomena of OMIT also allow one to directly extract a quantity called "cooperativity" C without any fi t parameters. The quantity C is an important fi gure of merit in characterizing the optomechanical systems. For example, in sideband resolved limit (when mechanical frequency exceeds the cavity linewidth), the criteria for quantum-coherent regime can be simply written as C + 1 > n_th , where n_th is the average number of thermal phonon in the mechanical resonator. In our experiment, we have been able to achieve C = 8 close to the expected number of thermal phonon in the mechanical resonator at 14 mK, bringing this system close to the quantum coherent regime.

Summary and outlook:

In our work, we demonstrated the potential of exfoliated graphene crystal applied to form an optomechanical device, which so far have been realized using top-down technology. This opens up a new dimension to explore exfoliated two-dimensional crystals in optomechanical systems, and harnessing their unique properties such as extremely low mass and high quality factors. For future devices, two-dimensional superconducting exfoliated flakes could be of great interest for such applications. Superconducting cavity in our work is a very good detector for mechanical displacement with a bandwidth three orders of magnitude larger than the mechanical line-width. This would provide a new tool to study nonlinear restoring forces, nonlinear damping, and mode coupling in mechanical resonators from twodimensional crystals. The characterization of our device shows that in future by making little larger area mechanical resonators, devices operating in quantum regime can be easily realized, which can possibly be used as a memory element in a quantum computer. As many of the 2D crystals can be grown by chemical processes in large areas, they also hold the promise of scalability.

References:
[1] Andres Castellanos-Gomez, Vibhor Singh, Herre S.J. van der Zant, Gary A. Steele, "Mechanics of freely-suspended ultrathin layered materials". arXiv:1409.1173 [cond-mat] (2014).
[2] J. Scott Bunch, Arend M. van der Zande, Scott S. Verbridge, Ian W. Frank, David M. Tanenbaum, Jeevak M. Parpia, Harold G. Craighead, Paul L. McEuen, "Electromechanical resonators from graphene sheets". Science, 315, 490-493 (2007). Abstract.
[3] Changyao Chen, Sami Rosenblatt, Kirill I. Bolotin, William Kalb, Philip Kim, Ioannis Kymissis, Horst L. Stormer, Tony F. Heinz, James Hone, "Performance of monolayer graphene nanomechanical resonators with electrical readout". Nature Nanotechnology, 4, 861-867 (2009). Abstract.
[4] Vibhor Singh, Shamashis Sengupta, Hari S Solanki, Rohan Dhall, Adrien Allain, Sajal Dhara, Prita Pant, Mandar M Deshmukh, "Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators". Nanotechnology, 21, 165204 (2010). Abstract.
[5] Robert A. Barton, B. Ilic, Arend M. van der Zande, William S. Whitney, Paul L. McEuen, Jeevak M. Parpia, Harold G. Craighead, "High, size-dependent quality factor in an array of graphene mechanical resonators". Nano Letters, 11, 1232{1236 (2011). Abstract.
[6] A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, A. Bachtold, "Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene". Nature Nanotechnology, 6, 339-342 (2011). Abstract.
[7] Xuefeng Song, Mika Oksanen, Mika A. Sillanpää, H. G. Craighead, J. M. Parpia, Pertti J. Hakonen, "Stamp transferred suspended graphene mechanical resonators for radio frequency electrical readout". Nano Letters, 12, 198-202 (2012). Abstract.
[8] Robert A. Barton, Isaac R. Storch, Vivekananda P. Adiga, Reyu Sakakibara, Benjamin R. Cipriany, B. Ilic, Si Ping Wang, Peijie Ong, Paul L. McEuen, Jeevak M. Parpia, Harold G. Craighead, "Photothermal self-oscillation and laser cooling of graphene optomechanical systems". Nano Letters, 12, 4681-4686 (2012). Abstract.
[9] Changyao Chen, Sunwoo Lee, Vikram V. Deshpande, Gwan-Hyoung Lee, Michael Lekas, Kenneth Shepard, James Hone, "Graphene mechanical oscillators with tunable frequency". Nature Nanotechnology 8, 923{927 (2013). Abstract.
[10] Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt, "Cavity optomechanics". arXiv:1303.0733 [cond-mat.mes-hall] (2013).
[11] C. A. Regal, J. D. Teufel, K. W. Lehnert, "Measuring nanomechanical motion with a mi- crowave cavity interferometer". Nature Physics, 4, 555-560 (2008). Abstract.
[12] J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, K. W. Lehnert, "Nanomechanical motion measured with an imprecision below that at the standard quantum limit". Nature Nanotechnology 4, 820-823 (2009). Abstract.
[13] T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, K. C. Schwab, "Preparation and detection of a mechanical resonator near the ground state of motion". Nature, 463, 72-75 (2010). Abstract.
[14] J. D. Teufel, Dale Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, R. W. Simmonds, "Circuit cavity electromechanics in the strong-coupling regime". Nature, 471, 204-208 (2011). Abstract.
[15] J. D. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, R. W. Simmonds, "Sideband cooling of micromechanical motion to the quantum ground state". Nature, 475, 359-363 (2011). Abstract.
[16] F. Massel, T.T. Heikkilä, J.-M. Pirkkalainen, S.U. Cho, H. Saloniemi, P.J. Hakonen, M.A. Sillanpää, "Microwave ampli fication with nanomechanical resonators". Nature, 480, 351-354 (2011). Abstract.
[17] Fredrik Hocke, Xiaoqing Zhou, Albert Schliesser, Tobias J Kippenberg, Hans Huebl, Rudolf Gross, "Electromechanically induced absorption in a circuit nano-electromechanical system". New Journal of Physics, 14, 123037 (2012). Abstract.
[18] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, K. W. Lehnert, "Entangling mechanical motion with microwave fields". Science, 342, 710-713 (2013). Abstract.
[19] Andres Castellanos-Gomez, Michele Buscema, Rianda Molenaar, Vibhor Singh, Laurens Janssen, Herre S J van der Zant, Gary A Steele, "Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping". 2D Materials, 1, 011002 (2014). Abstract.
[20] V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, G. A. Steele, "Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity". Nature Nanotechnology 9, 820–824 (2014). Abstract.
[21] G. S. Agarwal, Sumei Huang, "Electromagnetically induced transparency in mechanical eff ects of light". Physical Review A, 81, 041803 (2010). Abstract.
[22] Stefan Weis, Rémi Rivière, Samuel Deléglise, Emanuel Gavartin, Olivier Arcizet, Albert Schliesser, Tobias J. Kippenberg, "Optomechanically induced transparency". Science, 330, 1520-1523 (2010). Abstract.
[23] A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, O. Painter, "Electromagnetically induced transparency and slow light with optomechanics". Nature, 472, 69-73 (2011). Abstract.
[24] X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, T. J. Kippenberg, "Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics". Nature Physics, 9, 179-184 (2013). Abstract.

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.

Orbital Fluctuations of Electron Distributions: A New Path to Superconductivity

Brookhaven Lab scientists (from left) Lijun Wu, Yimei Zhu, Chris Homes, and Weiguo Yin stand by the electron microscope used to reveal the multi-orbital distributions with a technique called quantitative convergent beam electron diffraction (CBED).

Armed with just the right atomic arrangements, superconductors allow electricity to flow without loss and radically enhance energy generation, delivery, and storage. Scientists tweak these superconductor recipes by swapping out elements or manipulating the valence electrons in an atom's outermost orbital shell to strike the perfect conductive balance. Most high-temperature superconductors contain atoms with only one orbital impacting performance—but what about mixing those elements with more complex configurations?

Now, researchers at the U.S. Department of Energy's Brookhaven National Laboratory have combined atoms with multiple orbitals and precisely pinned down their electron distributions. Using advanced electron diffraction techniques, the scientists discovered that orbital fluctuations in iron-based compounds induce strongly coupled polarizations that can enhance electron pairing—the essential mechanism behind superconductivity. The study, published in the journal Physical Review Letters [1], provides a breakthrough method for exploring and improving superconductivity in a wide range of new materials.

"For the first time, we obtained direct experimental evidence of the subtle changes in electron orbitals by comparing an unaltered, non-superconducting material with its doped, superconducting twin," said Brookhaven Lab physicist and project leader Yimei Zhu.

While the effect of doping the multi-orbital barium iron arsenic—customizing its crucial outer electron count by adding cobalt—mirrors the emergence of high-temperature superconductivity in simpler systems, the mechanism itself may be entirely different.

"Now superconductor theory can incorporate proof of strong coupling between iron and arsenic in these dense electron cloud interactions," said Brookhaven Lab physicist and study coauthor Weiguo Yin. "This unexpected discovery brings together both orbital fluctuation theory and the 50-year-old 'excitonic' theory for high-temperature superconductivity, opening a new frontier for condensed matter physics."

Atomic Jungle Gym :

Imagine a child playing inside a jungle gym, weaving through holes in the multicolored metal matrix in much the same way that electricity flows through materials. This particular kid happens to be wearing a powerful magnetic belt that repels the metal bars as she climbs. This causes the jungle gym's grid-like structure to transform into an open tunnel, allowing the child to slide along effortlessly. The real bonus, however, is that this action attracts any nearby belt-wearing children, who can then blaze through that perfect path.

Flowing electricity can have a similar effect on the atomic lattices of superconductors, repelling the negatively charged valence electrons in the surrounding atoms. In the right material, that repulsion actually creates a positively charged pocket, drawing in other electrons as part of the pairing mechanism that enables the loss-free flow of current—the so-called excitonic mechanism. To design an atomic jungle gym that warps just enough to form a channel, scientists audition different combinations of elements and tweak their quantum properties.

"High-temperature copper-oxide superconductors, or cuprates, contain in effect a single orbital and lack the degree of freedom to accommodate strong enough interactions between electricity and the lattice," Yin said. "But the barium iron arsenic we tested has multi-orbital electrons that push and pull the lattice in much more flexible and complex ways, for example by inter-orbital electron redistribution. This feature is especially promising because electricity can shift arsenic's electron cloud much more easily than oxygen's."

In the case of the atomic jungle gym, this complexity demands new theoretical models and experimental data, considering that even a simple lattice made of north-south bar magnets can become a multidimensional dance of attraction and repulsion. To control the doping effects and flow of electricity, scientists needed a window into the orbital interactions.

Tracking Orbits :

"Consider measuring waves crashing across the ocean's surface," Zhu said. "We needed to pinpoint those complex fluctuations without having the data obscured by the deep water underneath. The waves represent the all-important electrons in the outer orbital shells, which are barely distinguishable from the layers of inner electrons. For example, each barium atom alone has 56 electrons, but we're only concerned with the two in the outermost layer."

The Brookhaven researchers used a technique called quantitative convergent beam electron diffraction (CBED) to reveal the orbital clouds with subatomic precision. After an electron beam strikes the sample, it bounces off the charged particles to reveal the configuration of the atomic lattice, or the exact arrays of nuclei orbited by electrons. The scientists took thousands of these measurements, subtracted the inner electrons, and converted the data into probabilities—balloon-shaped areas where the valence electrons were most likely to be found.

Shape-Shifting Atoms :

The researchers first examined the electron clouds of non-superconducting samples of barium iron arsenic. The CBED data revealed that the arsenic atoms—placed above and below the iron in a sandwich-like shape (see Figure 1) exhibited little shift or polarization of valence electrons. However, when the scientists transformed the compound into a superconductor by doping it with cobalt, the electron distribution radically changed.

Figure 1: These images show the distribution of the valence electrons in the samples explored by the Brookhaven Lab collaboration—both feature a central iron layer sandwiched between arsenic atoms. The tiny red clouds (more electrons) in the undoped sample on the left (BaFe₂As₂) reveal the weak charge quadrupole of the iron atom, while the blue clouds (fewer electrons) around the outer arsenic ions show weak polarization. The superconducting sample on the right (doped with cobalt atoms), however, exhibits a strong quadrupole in the center and the pronounced polarization of the arsenic atoms, as evidenced by the large, red balloons.

"Cobalt doping pushed the orbital electrons in the arsenic outward, concentrating the negative charge on the outside of the 'sandwich' and creating a positively charged pocket closer to the central layer of iron," Zhu said. "We created very precise electronic and atomic displacement that might actually drive the critical temperature of these superconductors higher."

Added Yin, "What's really exciting is that this electron polarization exhibits strong coupling. The quadrupole polarization of the iron, which indicates the orbital fluctuation, couples intimately with the arsenic dipole polarization—this mechanism may be key to the emergence of high-temperature superconductivity in these iron-based compounds. And our results may guide the design of new materials."

This study explored the orbital fluctuations at room temperature under static conditions, but future experiments will apply dynamic diffraction methods to super-cold samples and explore alternative material compositions.

The experimental work at Brookhaven Lab was supported by the Office of Science at Department of Energy (DOE). The materials synthesis was carried out at the Chinese Academy of Sciences' Institute of Physics.

References:
[1] Chao Ma, Lijun Wu, Wei-Guo Yin, Huaixin Yang, Honglong Shi, Zhiwei Wang, Jianqi Li, C. C. Homes, Yimei Zhu, "Strong Coupling of the Iron-Quadrupole and Anion-Dipole Polarizations in Ba(Fe_1−xCo_x)₂As₂". Physical Review Letters, 112, 077001 (published on February 20th, 2014). Abstract.

Deterministic Quantum Teleportation with Feed-Forward in a Solid State System

[From left to right] Andreas Wallraff, Christopher Eichler, Yves Salathe, Markus Oppliger, Philipp Kurpiers, Lars Steffen

Authors: Lars Steffen, Yves Salathe, Markus Oppliger, Philipp Kurpiers, Matthias Baur^*, Christian Lang, Christopher Eichler, Gabriel Puebla-Hellmann, Arkady Fedorov^*, Andreas Wallraff

Affiliation: Department of Physics, ETH Zurich, Switzerland

^*Present address: ARC Centre for Engineered Quantum Systems, University of Queensland, Brisbane, Australia

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Transferring the state of an information carrier between two parties is an essential primitive in both classical and quantum communication and information processing. Quantum teleportation [1] describes the concept of transferring an unknown quantum state from a sender to a physically separated receiver without transmitting the physical carrier of information itself. To do so, teleportation makes use of the non-local correlations provided by an entangled pair shared between the sender and the receiver and the exchange of classical information.

In our recent publication [2] we report the first teleportation of information in a solid state system. We use a chip-based superconducting circuit architecture [3, 4] (Fig. 1) with three superconducting transmon qubits [5] and three superconducting coplanar waveguide resonators [6] which serve as a quantum bus [7, 8] and allow to perform qubit state readout [9, 10]. We have realized the full deterministic quantum teleportation protocol using quantum-limited parametric amplifiers [11] for single-shot readout, a crossed quantum bus technology and flexible real-time digital electronics.

Figure 1: In our chip-based superconducting circuit three qubits are coupled to three resonators to realize deterministic quantum teleportation.

The success of the teleportation protocol in every instance with unit fidelity is counterintuitive from a classical point of view [12]. The receiver’s quantum bit (qubit) does not interact with any other qubit after it has been entangled with one of two qubits in the sender’s possession. The input state (|ψ_in>) is prepared at a later time at the sender. The classical information sent by the sender is not sufficient to recreate |ψ_in> perfectly at the receiver. Indeed, assuming no entanglement between sender and receiver one can replicate the sender’s state at best with a process fidelity of 1/2. To always recover the original state |ψ_in> the sender performs a measurement in the basis of the Bell states, which projects the two qubits in the sender’s possession randomly onto one of the four Bell states. As a consequence the receiver’s qubit is projected instantaneously into a state related to |ψ_in> without ever having interacted with the sender’s qubit. The receiver’s qubit only differs from the input state by a single-qubit rotation which depends on the four possible measurement results. In the final step, the sender communicates the Bell measurement result as two bits of classical information via a classical channel and therefore the receiver can always obtain the original input state |ψ_in>. This final step is frequently referred to as feed-forward.

Figure 2: a) Standard protocol of quantum teleportation. The protocol starts with the preparation of a Bell state between Q2 and Q3 (blue box) followed by the preparation of an arbitrary state |ψ_in> (green box) and a Bell state measurement of Q1 and Q2 (red box). The classical information extracted by the measurement of Q1 and Q2 is transferred to the receiver to perform local gates conditioned on the measurement outcomes. At the end of the protocol Q3 is in a state |ψ_out> which ideally is identical to |ψ_in> (also colored in green). Here, H is the Hadamard gate, X and Z are Pauli matrices σ_x and σ_z, respectively. The cnot-gate is represented by a vertical line between the control qubit (•) and the target qubit (⊕). b) The protocol implemented in the experiment presented here uses controlled-phase gates indicated by vertical lines between the relevant qubits (•), and single qubit rotations R^θ_±y of angle θ about the ±y-axis. To finalize the teleportation we either post-select on any single one of the four measurement outcomes (00, 01, 10 and 11) acquired in a single shot, or we deterministically use all four outcomes, which we then may use to implement feed-forward. The feed-forward operators R^π_x and R^π_y are applied to Q3 conditioned on the four measurement outcomes according to the table presented in the framed box.

Our implementation of the protocol (Fig. 2b) uses single qubit rotations and controlled-phase gates [13, 14] and is equivalent to the original protocol shown in Fig. 2a. The teleportation process succeeds with order unit probability for any input state, as we reliably prepare entangled states as a resource and are able to distinguish all four maximally entangled Bell states in a single measurement.

Figure 3: The solid bars show the experimentally obtained process matrix describing the state transfer from the sender to the receiver. The ideal process matrix, which is the identity operation, is shown in wire frames.

To identify simultaneously the four outcomes of the Bell-state measurement in a deterministic way we use two-qubit single-shot read-out [15, 16] for which we achieve a success probability of (81.8±0.5) %. In the teleportation protocol we analyze the Bell-state measurement in real time by using fast electronics based on a field programmable gate array (FPGA) and realize the feed-forward step in about 500 ns. We have achieved an average process fidelity of (62.2±0.3) % for the full quantum teleportation algorithm, which is clearly above the classical threshold (Fig. 3). We have demonstrated teleportation at a rate of 10,000 quantum bits per second between two macroscopic systems separated by 6 mm.

The low transmission loss of superconducting waveguides is likely to enable the range of this and other schemes to be extended to significantly larger distances, enabling tests of non-locality and the realization of elements for quantum communication at microwave frequencies. The demonstrated feed-forward may also find application in error correction schemes.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels". Physical Review Letters, 70, 1895 (1993). Abstract.
[2] L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur, C. Lang, C. Eichler, G. Puebla-Hellmann, A. Fedorov, and A. Wallraff. "Deterministic quantum teleportation with feed-forward in a solid state system". Nature, 500, 319 (2013). Abstract.
[3] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf. "Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics". Nature, 431, 162 (2004). Abstract.
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