Quantum Newton's Cradle with Bose-Einstein Condensates

Roberto Franzosi (left) and Ruggero Vaia (right)

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

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

1. Introduction

Fig.1: Newton's cradle

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

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

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

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

Figure 2

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

2. Tonks–Girardeau regime: Fermionizing Bosons

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

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

3. The analogy

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

Video 1

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

Video 2

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

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

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

4. Bad and Good Quantum Newton’s Cradles

Uniform QNC

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

Figure 3

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

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

Perfect QNC

Figure 4

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

5. Two realistic schemes

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

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

Uniform array with a Gaussian trap

Figure 5

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

Figure 6

6. Conclusions

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

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

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

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

Atom Interferometry in a 10 Meter Atomic Fountain

[From left to right] Mark Kasevich; the 10 m fountain team in 2013: Alex Sugarbaker, Tim Kovachy, Jason Hogan, Susannah Dickerson, Sheng-wey Chiow; and Dave Johnson

Authors: Alex Sugarbaker, Susannah M. Dickerson, Jason M. Hogan, David M. S. Johnson, and Mark A. Kasevich

Affiliation: Department of Physics, Stanford University, USA

Link to Kasevich Group Website >>

The equivalence principle states that all objects fall with the same acceleration under the influence of gravity. It is the conceptual foundation of Einstein’s general relativity, but is it true exactly? If not, there are profound implications for our understanding of gravity and the nature of the universe. It is therefore important to continue to test the equivalence principle as precisely as we can.

Galileo reportedly tested it by dropping spheres from the Leaning Tower of Pisa. Apollo astronauts tested it by dropping a hammer and a feather on the moon. More recent measurements have shown that the accelerations of two falling objects differ by no more than one part in 10¹³ [1, 2]. We aim to test the equivalence principle to one part in 10¹⁵ by dropping atoms of two different isotopes of rubidium in a 10 meter tower.

We will precisely measure acceleration differences between the two isotopes using atom interferometry. According to quantum mechanics, atoms are waves. Just as in optical interferometry, it is possible to split and recombine them to form an interference pattern [3, 4, 5]. In our interferometer, we send each atom along two different paths through space – each is in two places at once. When the atom waves are brought back together, the interference pattern depends on the phase difference between the two paths taken.

This phase difference in turn depends sensitively on the forces that act differently on the two parts of the atom while they are separated. This sensitivity to forces is what makes atom interferometry so useful. Compact atom interferometers have been made that can precisely measure rotation and acceleration, which can aid in navigation, mineral exploration, and geophysics. Atom interferometers have also measured the gravitational and fine-structure constants [6, 7]. They could also be used to search for gravitational waves [8].

The sensitivity of an atom interferometer increases with longer interferometer durations. Therefore, as recently described in Physical Review Letters [9, 10, 11], we have built an atom interferometer in which ⁸⁷Rb atoms are separated for 2.3 seconds before being recombined and interfered (Fig. 1). Three times longer than previous records [12], this multiple-second duration is well into the range of macroscopic, human-perceivable timescales. Furthermore, the two halves of each atom are separated by 1.4 centimeters before recombination – that's enough for you to swing your hand between them!

Fig. 1 Photograph of the 10 meter atomic fountain in a pit in the basement of the physics building at Stanford University.

How do we make a long-duration atom interferometer? We prepare a cloud of atoms at the bottom of a 10 meter vacuum tower and then launch them to the top. The interferometry is performed while the atoms rise up and fall back down to the bottom of the tower. The atoms are in free-fall, isolated from the noisy environment.

The cloud of atoms used must be very cold – a few billionths of a degree above absolute zero. At room temperature, the atoms in a gas move at speeds of hundreds of meters per second. Room-temperature rubidium atoms would collide with the walls of our vacuum chamber long before they fell back to the bottom. We therefore cool a few million rubidium atoms to a few nanokelvin before launching them into the tower. (The cooling process builds upon the same techniques used to generate Bose-Einstein condensates [13].)

Even at a few nanokelvin, individual atoms follow slightly different trajectories through the interferometer (like the droplets in a fountain of water), experiencing different position- and velocity-dependent forces. This yields a spatially-dependent phase, which in turn yields a spatial variation in the output atom density distribution that we can observe directly with a CCD camera (Fig. 2). This might at first appear undesirable, but it actually reveals rich details about the forces that generate the spatial interference pattern. Similar spatial fringe patterns have been used to great benefit in optical interferometers for centuries, but it is only recently that the effect has been leveraged in atom interferometry.

Fig. 2 Atomic interference patterns observed at the output of the interferometer. The images are sorted by phase, which can be measured for each experimental shot.

The long drift time of our interferometer enables it to have an acceleration sensitivity of 7 X 10^-12 g for each experimental shot, a hundredfold improvement over previous limits [14]. This is roughly the same as the gravitational attraction you would feel towards a person 10 meters away from you. We have used the sensitive interferometer and the spatial fringe patterns mentioned above to make precise measurements of Earth's rotation [9, 10].

The high sensitivity of the interferometer also holds great promise for our design goal – testing the equivalence principle (as mentioned above). By averaging more measurements or implementing advanced interferometry techniques, we can achieve the desired 10^-15 g sensitivity. Adding a simultaneous ⁸⁵Rb interferometer and comparing the results for the two isotopes will then enable us to make a new precision test of the equivalence principle. This will probe the fundamental assumptions of our current theory of gravity.

References:
[1] S. Schlamminger, K.Y. Choi, T.A. Wagner, J.H. Gundlach, and E.G. Adelberger, “Test of the Equivalence Principle Using a Rotating Torsion Balance”, Physical Review Letters, 100, 041101 (2008). Abstract.
[2] 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.
[3] Mark Kasevich and Steven Chu, “Atomic interferometry using stimulated Raman transitions”, Physical Review Letters, 67, 181 (1991). Abstract.
[4] Alexander D. Cronin, Jörg Schmiedmayer, David E. Pritchard, “Optics and interferometry with atoms and molecules”, Reviews of Modern Physics, 81, 1051 (2009). Abstract.
[5] We focus on light-pulse atom interferometry, where pulses of laser light are used to split, recombine, and interfere the atoms.
[6] G. Lamporesi, A. Bertoldi, L. Cacciapuoti, M. Prevedelli, and G.M. Tino, “Determination of the Newtonian Gravitational Constant Using Atom Interferometry”, Physical Review Letters, 100, 050801 (2008). Abstract.
[7] Rym Bouchendira, Pierre Cladé, Saïda Guellati-Khélifa, François Nez, and François Biraben, “New Determination of the Fine Structure Constant and Test of the Quantum Electrodynamics”, Physical Review Letters, 106, 080801 (2011). Abstract.
[8] Jason Hogan, “A new method for detecting gravitational waves”, SPIE Newsroom, 6 May (2013). Article.
[9] 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.
[10] Alex Sugarbaker, Susannah M. Dickerson, Jason M. Hogan, David M. S. Johnson, Mark A. Kasevich, “Enhanced Atom Interferometer Readout through the Application of Phase Shear”, Physical Review Letters, 111, 113002 (2013). Abstract.
[11] P. Bouyer, “Viewpoint: A New Starting Point for Atom Interferometry”, Physics, 6, 92 (2013). Article.
[12] H. Müntinga, H. Ahlers, M. Krutzik, A. Wenzlawski, S. Arnold, D. Becker, K. Bongs, H. Dittus, H. Duncker, N. Gaaloul, C. Gherasim, E. Giese, C. Grzeschik, T. W. Hänsch, O. Hellmig, W. Herr, S. Herrmann, E. Kajari, S. Kleinert, C. Lämmerzahl, W. Lewoczko-Adamczyk, J. Malcolm, N. Meyer, R. Nolte, A. Peters, M. Popp, J. Reichel, A. Roura, J. Rudolph, M. Schiemangk, M. Schneider, S. T. Seidel, K. Sengstock, V. Tamma, T. Valenzuela, A. Vogel, R. Walser, T. Wendrich, P. Windpassinger, W. Zeller, T. van Zoest, W. Ertmer, W. P. Schleich, E. M. Rasel, “Interferometry with Bose-Einstein Condensates in Microgravity”, Physical Review Letters, 110, 093602 (2013). Abstract.
[13] “Bose-Einstein condensate”. Past 2Physics Article.
[14] Holger Müller, Sheng-wey Chiow, Sven Herrmann, Steven Chu, Keng-Yeow Chung, “Atom-Interferometry Tests of the Isotropy of Post-Newtonian Gravity”, Physical Review Letters, 100, 031101 (2008). Abstract.

Laser Cooling to Quantum Degeneracy

The SrBEC team in October 2012. From left to right: Slava M. Tzanova^1,2, Benjamin Pasquiou¹, Rudolf Grimm^1,2, Simon Stellmer^1,2,3 (author), Florian Vogl^1,2, Florian Schreck¹ (author), Alex Bayerle^1,2

Authors: Simon Stellmer^1,2,3 and Florian Schreck¹

Affiliation:
¹Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 6020 Innsbruck, Austria
²Institute for Experimental Physics and Center for Quantum Physics, University of Innsbruck, 6020 Innsbruck, Austria
³Institute for Atomic and Subatomic Physics, Vienna University of Technology, 1020 Vienna, Austria

Link to the Sr BEC project homepage >>           
Link to the “Ultracold Atoms and Quantum Gases” group in Innsbruck >>

Laser cooling is a very elegant and versatile technique, as it permits to cool atoms, molecules, ions, and even mechanical objects from room temperature down to temperatures as low as one millionth of a degree above absolute zero [1,2]. At these low temperatures, we can enter into the world of quantum mechanics. One of the fascinating phenomena associated with this ultracold regime is the appearance of quantum degeneracy in atomic gases [2]. Since the early days of laser cooling, the question has been asked if the quantum degenerate regime could be reached using laser cooling as the only cooling process. Despite significant experimental and theoretical effort to overcome the limitations of laser cooling this goal has been elusive.

Past 2Physics article by Florian Schreck:
November 29, 2009: "Bose-Einstein Condensation of Strontium"

In 1995, the combination of laser cooling with a subsequence stage of evaporative cooling led to the attainment of quantum degeneracy in bosonic alkali-metal gases [2,3]. These quantum gases, called “Bose-Einstein condensates” (BECs), have opened an extraordinary window for the exploration of the quantum world.

To create a BEC, the phase-space density of the gas has to be increased beyond a critical value by lowering the temperature and increasing the density of the gas. Laser cooling was so far unable to reach quantum degeneracy because the photons used to cool the gas have negative side effects, which limit the achievable density and destroy a BEC.

In our experiment we overcome these side effects and create a BEC of strontium by laser cooling [4]. Furthermore, our method creates a BEC immersed in a laser cooled cloud of atoms, which opens a simple path towards the construction of a truly continuous atom laser.

Our scheme relies on the combination of three techniques, favored by the properties of strontium. The first technique is called “narrow line cooling”. Strontium possesses a narrow laser cooling transition of only 7 kHz width. Operating a magneto-optical trap (MOT) on this narrow line, we can cool a gas of strontium atoms down to temperatures of about 800 nK. This is already more than an order of magnitude colder than conventional MOTs of alkali atoms!

The second technique is a separation of our cold gas into two spatial regions (see Fig. 1): one large region, in which about 10 million atoms are trapped in a “reservoir” optical dipole trap and continuously cooled by laser light, and a small region, in which about 1 million atoms are confined at a much higher density by a steep confining potential, often called a “dimple”. This is the region where the BEC will be created.

Figure 1: Three absorption images of 10 million strontium atoms trapped in a dipole trap and cooled to about 800nK by laser light. All images are taken on the narrow cooling transition. The left image shows the reservoir of atoms, which is used to dissipate heat. For the central image, we have applied the transparency beam, such that atoms located within this beam cannot absorb photons from the cooling light. The density in this region is greatly enhanced by a dimple beam, as can be seen on the right image, where the transparency beam has been turned off.

The third technique allows us to overcome the negative side effects of laser cooling photons in the dimple region. We protect atoms in this region from those photons by the help of an extra laser beam, which we call the “transparency beam”. This beam acts like a cap of invisibility, as it modifies the energy states of the atoms in the dimple region such that they cannot absorb laser cooling photons (see Fig. 1). Importantly, the atoms are not only transparent to the cooling laser beam, but also to laser cooling photons scattered towards the dimple region from atoms in the laser cooled reservoir.

Now we have two different regions: the outside “reservoir” region, in which atoms are gently cooled by laser light and a central dimple region, in which the BEC will form. A connection between the two regions is established through the elastic scattering between atoms: in this way, heat can be transferred very rapidly (on timescales of a few 10 ms) from the dimple into the reservoir, where the heat is dissipated. To maximize this heat transfer, we place the dimple right into the center of the reservoir, as can be seen in Fig. 1.

Once the system is prepared in this configuration, it takes only about 60 ms for a BEC to appear, and after a little over 100 ms, the BEC has reached its final size of about 100 000 atoms (see Fig. 2).

Figure 2: Absorption images, taken 24 ms after release from the trap. On the left image, the BEC is faintly visible as an elliptic density increase in the center. For the right image, we have removed all atoms from the reservoir just before the release from the trap, and the BEC stands out clearly.

An important property of our system is that laser cooling constantly provides strong dissipation, removing entropy from the gas. Even if we destroy our BEC by local heating of the dimple region, it will quickly reform after the heating process is switched off, as long as enough atoms are contained in the dipole trap.

We believe that our scheme can be adapted to other elements. We expect it to work with all species that possess a narrow cooling transition and have a reasonable scattering behavior. These criteria are fulfilled by a selection of elements, most prominently the lanthanides. The range of suitable candidates can be increased substantially by going one step further: sympathetic cooling between strontium and another element. This element would not need to feature a narrow cooling transition. Instead, it would be trapped in the dimple region and sympathetically cooled through collisions with the strontium atoms in the reservoir. We have recently implemented this sympathetic laser cooling scheme in a mixture of rubidium and strontium. The rubidium gas is cooled very efficiently by thermal contact with laser cooled strontium atoms, delivering ideal starting conditions for the creation of quantum degenerate Rb-Sr mixtures by evaporative cooling [5]. By using a Rb specific dipole trap as a dimple, it should also be possible to create a Rb BEC without evaporative cooling.

Our scheme also paves a relatively simple path towards a truly continuous atom laser. Here, a thermal source of atoms would be converted into a coherent beam of atoms, constantly outcoupled from the dimple region. The dimple would be continuously fed by the reservoir region, which in turn would be replenished by pre-cooled atoms from a MOT. Such truly continuous atom lasers are highly desired in various schemes of precision measurements.

References :
[1] Proceedings of the International School of Physics "Enrico Fermi", Course CXVIII, Varenna, 9-19 July 1991, Laser Manipulation of Atoms and Ions, edited by E. Arimondo, W. D. Phillips, and F. Strumia (North Holland, Amsterdam, 1992).
[2] Physics 2000, BEC homepage. Link.
[3] Proceedings of the International School of Physics ‘‘Enrico Fermi’’, Course CXI, Varenna, 7-17 July 1998, Bose-Einstein Condensation in Atomic Gases, edited by M. Inguscio, S. Stringari, and C. E. Wieman (North Holland, Amsterdam, 1999).
[4] Simon Stellmer, Benjamin Pasquiou, Rudolf Grimm, and Florian Schreck, "Laser Cooling to Quantum Degeneracy", Physical Review Letters, 110, 263003 (2013). Abstract.
[5] Benjamin Pasquiou, Alex Bayerle, Slava M. Tzanova, Simon Stellmer, Jacek Szczepkowski, Mark Parigger, Rudolf Grimm, and Florian Schreck, "Quantum degenerate mixtures of strontium and rubidium atoms", Physical Review A, 88, 023601 (2013). Abstract. 

Quantum Flutter: A Dance of an Impurity and a Hole in a Quantum Wire

[Clockwise from Top left]: Charles J. M. Mathy, Eugene Demler, Mikhail B. Zvonarev.

Authors: 
Charles J. M. Mathy^1,2, Mikhail B. Zvonarev^2,3,4, Eugene Demler²

Affiliation:
¹ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts, USA
³Université Paris-Sud, Laboratoire LPTMS, UMR8626, Orsay, France,
⁴CNRS, Orsay, France.

What happens when a particle moves through a medium at a velocity comparable to the speed of sound? The consequences lie at the heart of several striking phenomena in physics. In aerodynamics, for example, an object experiencing winds close to the speed of sound may experience a vibration that grows with time called flutter, which can ultimately have dramatic consequences such as the destruction of aeroplane wings or the iconic Tacoma Narrows bridge collapse. Other examples of physics induced by fast motion include acoustic shock waves and Cerenkov radiation. What we addressed in our work [1] was the effect of fast disturbances in strongly interacting quantum systems of many particles, in a case where the particles are effectively restricted to move in one dimension known as a quantum wire.

When the interactions between particles are weak, quantum systems can sometimes be described by a simple hydrodynamic equation. For example, the Gross-Pitaevskii Equation (GPE) describes the evolution of a weakly coupled gas of bosons at low temperatures when it forms a Bose-Einstein condensate. The GPE is analogous to equations found in hydrodynamics, which explains why one can see analogs of classical hydrodynamical effects such as shock waves and solitons in these systems [2,3]. But what if the interactions are too strong and such an approximation breaks down?

We have found a model that shows interesting physics induced by supersonic motion which goes beyond a hydrodynamical description [1]. The system is a one-dimensional gas of hardcore bosons known as a Tonks-Girardeau gas (TG) [4]. We start the system in its ground state and inject a supersonic impurity that interacts repulsively with the background particles. We obtained exact results on what happens next using an approach from mathematical physics called the Bethe Ansatz approach, coupled with large-scale computing resources [5]. Thus we track the impurity velocity as a function of time and find two main surprising features. Firstly, the impurity does not come to a complete stop, instead it only sheds part of its momentum and keeps on propagating at a reduced velocity forever (Fig 1a). Secondly, the impurity velocity oscillates a function of time, a phenomenon we call quantum flutter as it arises from nonlinear interactions of a fast particle with its environment (see Fig. 1b).

Figure 1: Impurity momentum evolution and quantum flutter:
a, Schematic picture of our setup. Top: We start with a one-dimensional gas of hardcore bosons of mass m known as a Tonks Girardeau (TG) gas (red arrows), in its ground state. We then inject an impurity also of mass m with finite momentum Q (green arrow). Middle: The impurity loses part of its momentum by creating a hole around itself (sphere) and emitting a sound wave in the background gas (blue arrow). However it retains a finite momentum Q_sat after this process and carries on propagating without dissipation. Bottom: legend of the different characters in the story.
b, Time evolution of the expected momentum of the impurity, <P̂_↓(t)>. The momentum decays to a finite value Q_sat, and shows oscillations around Q_sat at a frequency we call ω_osc. The background gas has density ρ_↑, and we define a Fermi momentum k_F = πρ_↑, a Fermi energy E_F = k_F²/(2m) where m is the mass of the particles, and a Fermi time t_F = 1/E_F. Inset: zoom into the plot of <P̂_↓(t)> showing the oscillations we call quantum flutter.
c, Time evolution of the density of the background gas in the impurity frame. More precisely, shown is the density-density correlation function G_↓↑(x,t) = <ρ_↓(0,t) ρ_↑(x,t)> in units of ρ_↑/L where L is the system size, and the position along the wire is written in units of the interparticle distance ρ_↑^-1 in the background gas. Here ρ_↑ is the density of the background gas, and ρ_↓ the density of the impurity. G_↓↑(x,t) is effectively the density of the background gas with respect to the impurity position. We see the formation of the correlation hole around x ρ_↑ = 0 (blue valley), and the emission of the sound wave (red ridge). Underneath a schematic illustration of the dynamics is given: the blue arrow represents the emitted sound wave, the sphere is the hole, and the green arrow the impurity (see a). Inside the correlation hole the impurity and hole are dancing, meaning that they are oscillating with respect to each other, the phenomenon we denote as quantum flutter.

Using the exact methods just mentioned we were able to look in detail at the dynamical processes underlying quantum flutter. The time evolution of the impurity in the gas of bosons can be broken down into several steps. First the impurity carves out a depletion of the gas around itself, called a correlation hole. It expels the background density into a sound wave that carries away a large part of the momentum of the impurity, but not of all it (fig 1c). In fact the impurity retains part of its momentum and no longer sheds momentum because of kinematic constraints: there are no sound waves it can emit in the background gas while conserving momentum and energy.

After formation of the correlation hole, the impurity momentum starts to oscillate. When the dynamics of a quantum system shows a feature that is periodic in time, typically the frequency of the feature corresponds to an energy difference between two states of the system. Examples include light emission of an atom, or spin precession in response to a magnetic field, which underlies Nuclear Magnetic Resonance. In our case, the two states are an exciton and a polaron. The exciton corresponds to the impurity binding to a hole, since if the impurity repels the background gas, it is attracted to a hole (i.e. a missing particle in the background). The polaron is an impurity dressed due to interactions with the background particles, which affects its properties such as its effective mass: it becomes heavier as it carries a cloud of displaced background particles around it [6]. Thus we arrive at the following picture, as shown schematically in Fig. 2: first the impurity causes the emission of a sound wave in the background gas and creation of a hole close to it. It can bind to this hole and form an exciton, or not bind to it and form a polaron instead. In fact the impurity does both in the sense that it forms a quantum superposition of a polaron and an exciton. This quantum superposition leads to oscillations in the impurity velocity, a phenomenon called quantum beating, which is analogous to Larmor precession of a spin in a magnetic field. The difference here is that the two states that are beating, the exciton and polaron, are strongly entangled many-particle states. That we observe long-lived quantum coherence effects in a system composed of infinitely many particles is surprising. Namely, typically such systems exhibit decoherence, such that if one puts a particle in a quantum superposition of two states, the superposition decays because of interactions with other particles.

Figure 2: Origin of quantum flutter:
a, The quantum flutter oscillations originate from the formation of a superposition of entangled states of the impurity with its environment. After the impurity is injected in the system is creates a hole around itself. It can then bind to this hole and form an exciton, or not bind to it and form a state that is dressed with its environment called a polaron. In fact the system forms a coherent superposition of these two possibilities, which then leads a quantum beating and oscillations in the impurity momentum with a frequency given by the energy difference between these two possibilities.
b, Comparison between the frequency ω_osc of oscillations in the impurity momentum, and the energy difference between the polaron E(Pol(0)) and the exciton E(Exc(0)) (the zero between brackets refers to the exciton and polaron having momentum zero). The x-axis denotes the interaction strength between the impurity and the background particles: the interaction between a background particle at position x_i and the impurity at position x_↓ is a contact interaction of the form g δ(x_i - x_↓), and one defines the dimensionless interaction parameter γ = m g/ρ_↑. ℏω_osc and E(Pol(0))-E(Exc(0)) are in quantitative agreement, which motivates the interpretation of quantum flutter as quantum beating between exciton and polaron.

To see quantum flutter in the laboratory directly, one can use methods from the field of ultracold atoms, in which neutral atoms are cooled and trapped using a combination of lasers and magnetic fields. The trapping potential can be chosen to restrict the atoms to move along 1D tubes, and effectively behave like a TG gas [7,8]. The interaction between the particles can be tuned using a Feshbach resonance. Impurity physics in one-dimensional TG gases has already been studied [9,10,11]. The only added ingredient needed for quantum flutter is to create impurities at finite velocities, which can be done using two-photon Raman processes. Quantum flutter can be observed by measuring the expected impurity velocity as a function of time. Thus cold atom experiments could confirm our predictions, and one could vary different parameters of the model so see how robust quantum flutter is. Our preliminary calculations suggest that quantum flutter survives within a certain window of varying all the parameters in the theory such as the interaction between background particles, the relative mass of the impurity and the background particles, and the form of the interactions.

In summary, we have found an example of a system of many particles where injecting a supersonic impurity leads to the spontaneous formation of a long-lived quantum superposition state which travels through the system at a finite velocity. The question of which systems allow transport of quantum coherent states is important for quantum computing applications [12], and has surfaced in recent studies of quantum effects in biology [13]. Thanks to the advent of exact methods and the development of precise experiments in the study of many-particle quantum dynamics, we expect to see progress being made on this question in the near future.

References
[1] Charles J. M. Mathy, Mikhail B. Zvonarev, Eugene Demler. "Quantum flutter of supersonic particles in one-dimensional quantum liquids". Nature Physics, 8, 881 (2012). Abstract.
[2] A.M. Kamchatnov and L.P. Pitaevskii. "Stabilization of solitons generated by a supersonic flow of a bose-einstein condensate past an obstacle". Physical Review Letters, 100, 160402 (2008). Abstract.
[3] I. Carusotto, S.X. Hu, L.A. Collins, and A. Smerzi. "Stabilization of solitons generated by a supersonic flow of a bose-einstein condensate past an obstacle". Physical Review Letters, 97, 260403 (2006). Abstract.
[4] M. Girardeau. "Relationship between systems of impenetrable bosons and fermions in one dimension". Journal of Mathematical Physics, 1, 516 (1960). Abstract.
[5] Jean-Sébastien Caux. "Correlation functions of integrable models: a description of the abacus algorithm". Journal of Mathematical Physics, 50, 095214 (2009). Abstract.
[6] A.S. Alexandrov, S. Devreeze, and T. Jozef. "Advances in Polaron Physics". Springer Series in Solid-State Sciences, Vol. 159 (2010).
[7]  Toshiya Kinoshita, Trevor Wenger and David S. Weiss. "A quantum newton's cradle". Nature, 440, 900 (2006). Abstract.
[8] Toshiya Kinoshita, Trevor Wenger and David S. Weiss. "Observation of a one-dimensional tonks-girardeau gas". Science, 305, 1125 (2004). Abstract.
[9] Stefan Palzer, Christoph Zipkes, Carlo Sias, Michael Köhl. "Quantum transport through a tonks-girardeau gas". Physical Review Letters, 103, 150601 (2009). Abstract.
[10] P. Wicke, S. Whitlock, and N.J. van Druten. "Controlling spin motion and interactions in a one-dimensional bose gas". ArXiv:1010.4545 [cond-mat.quant-gas] (2010).
[11] J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio, F. Minardi, A. Kantian, and T. Giamarchi. "Quantum dynamics of impurities in a one-dimensional bose gas". Physical Review A, 85, 023623 (2012). Abstract.
[12] D.V. Averin, B. Ruggiero, and P. Silvestrini. "Macroscopic Quantum Coherence and Quantum Computing". Plenum Publishers, New York (2000).
[13] Gregory S. Engel, Tessa R. Calhoun, Elizabeth L. Read, Tae-Kyu Ahn, Tomá Manal, Yuan-Chung Cheng, Robert E. Blankenship, Graham R. Fleming. "Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems". Nature, 446, 782 (2007). Abstract.

First Demonstration of Spin-Orbit Coupling in Ultracold Fermi Gases

Photo: Jing Zhang of Shanxi University

Authors: Hui Zhai¹ and Jing Zhang²
Affiliation:
¹Institute for Advanced Study, Tsinghua University, Beijing, China
²State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, China

Photo: Hui Zhai of Tsinghua University

In 1995, scientists successfully cooled bosons to Bose-Einstein condensate, and later on in 1999, quantum degenerate Fermi gas could also be demonstrated in experiment. These degenerate atomic gases have the advantage that both their single particle motion and the interaction between atoms can be manipulated and well controlled by light or magnetic field. Utilizing these advantages -- during the last decade -- many exciting experiments have been carried out, which either simulate interesting models for condensed matter systems or reveal interesting new many-body quantum phases or quantum phenomena.

However, in all these studies, an important ingredient has not been included until very recently, that is, the coupling between the atomic spin degree of freedom and its orbital motion. This is because for neutral atoms, unlike charged electrons, there is no intrinsic spin-orbit coupling. Nevertheless, in reality, spin-orbit coupling plays a very important role in determining electron structure in solid and the nuclear structure. Spin-orbit coupling is also the key ingredient giving birth to some new states of matters such as topological insulators and topological superfluids. Therefore, for the purposes of both quantum simulation and discovering new state of matter, it is desirable to introduce spin-orbit coupling into the field of ultracold quantum gases.

In 2011, Ian Spielman’s group in NIST first generated spin-orbit coupling in Bose gases [1]. In that experiment, two counter-propagating laser beams are applied to the Bose condensate of Rubidium atoms. One of the two lasers is linearly polarized and the other is circularly polarized. Thus, when an atom absorbs a photon from one beam and then emits a photon to the other laser beam, their spin is flipped and also their momentum is also changed by the momentum difference of these two lasers. That is to say, the spin flip process is always accompanied by the change of momentum. In this way, the spin and their motion are locked together and a synthetic spin-orbit coupling is added into the motion of atoms. Such a coupling changes the single particle spectrum dramatically and the energy minimum is shifted from zero-momentum to finite momentum, which gives rise to unconventional condensate with intriguing phase or density patterns [2].

In our recent paper [3] we have for the first time applied the similar scheme to create spin-orbit coupling and demonstrate its effect in ultracold Fermi gases of Potassium-40. Fermi gas differs from Bose gas because fermions should obey Pauli exclusion principle, and therefore they have to occupy different momentum states and form a Fermi surface at low temperature. This also leads to the major difference for the manifestation of the spin-orbit coupling effects in a Fermi gas compared to in a Bose condensate. We have demonstrated several effects in our work.

Fig. 1: Spin oscillation under spin-orbit coupling. Different curves represent Fermi gases with different density.

First, if one starts with a fully polarized Fermi gas and applies a pulse of Raman coupling, the whole system will start Rabi oscillation. If all atoms oscillate with the same frequency, the Rabi oscillation will remain coherent for long time. However, in this system, atoms occupy different momenta and because of spin-orbit coupling, atoms with different momenta have different energies. Thus, different atoms oscillate with different periods, which leads to strong dephasing. This simulates spin-orbit coupling induced spin diffusion process of a spin polarized current in semiconductors. In our work, we also provide strong evidence for the topology change of Fermi surface. Using the momentum resolved radio-frequency spectroscopy, the single dispersion is also mapped out, where the effects of spin-orbit coupling is clearly demonstrated. Later on, MIT group led by Martin Zwierlein also studied spin-orbit coupled Fermi gas with lithium-6 atoms, and they measured spin-resolved single particle dispersion using spin-injection spectroscopy [4].

Fig 2: single particle dispersion measured by momentum resolved radio-frequency spectroscopy

In the near future, we plan to bring the system nearby a magnetic Feshbach resonance, and utilize the strong attraction there to create a fermion superfluid in the presence of spin-orbit coupling. Such a superfluid, when confined into one-dimensional geometry by optical lattices, becomes topological and displays Majorana edge mode, as discovered in nanowire recently [5]. Realizing such a topological phase in cold atom setup will allow us to study its properties in a more controllable way.

To reach this goal, we also need to overcome several challenges. One major challenge is the heating due to spontaneous mission in the Raman process. For instance, for our experiment with Potassium-40, the temperature of the Fermi gases increases from around 0.2 of Fermi temperature to around 0.5 of Fermi temperature after Raman laser is turn on for around 100 ms. The heating is more profound for light atoms like Lithium. Such a problem may be overcome by further cooling fermions with very low temperature boson bath or by choosing other atoms like Yb or Dy, which have excited level with very narrow linewidth, and the spontaneous mission rate can be greatly suppressed.

The spin-orbit coupling generated in current experiment is a special type, which can be viewed as equal weight of Rashba and Dresselhaus. Another direction for future studies is to generate more complicated spin-orbit coupling, and one of the most interesting forms is pure Rashba because of the higher symmetry of this type of coupling. Such a coupling increases the single particle ground state degeneracy and the low-energy density-of-state, and thus it leads to many profound many-body quantum phenomena, as predicated by many of recent theoretical studies [6]. It is exciting to discover them in experiments.

References:
[1] Y. J. Lin, K. Jimenez-Garcia and I. B. Spielman, "Spin–orbit-coupled Bose–Einstein condensates", Nature, 471, 83 (2011). Abstract. 2Physics Article.
[2] Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai, "Spin-Orbit Coupled Spinor Bose-Einstein Condensates", Physical Review Letters, 105, 160403 (2010). Abstract;   Tin-Lun Ho and Shizhong Zhang, "Bose-Einstein Condensates with Spin-Orbit Interaction", Physical Review Letters, 107, 150403 (2011). Abstract.
[3] Pengjun Wang, Zeng-Qiang Yu, Zhengkun Fu, Jiao Miao, Lianghui Huang, Shijie Chai, Hui Zhai, and Jing Zhang, "Spin-Orbit Coupled Degenerate Fermi Gases", Physical Review Letters, 109, 095301 (2012). Abstract.
[4] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadzibabic, Tarik Yefsah, Waseem S. Bakr, and Martin W. Zwierlein, "Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas", Physical Review Letters, 109, 095302 (2012). Abstract.
[5] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, "Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices", Science, 336, 1003 (2012). Abstract. 2Physics Article.
[6] For a review, see Hui Zhai, "Spin-orbit coupled quantum gases", International Journal of Modern Physics, 26, 1230001 (2012). Full Article.

Superfluidity in Two Dimensions

Christof Weitenberg

Author: Christof Weitenberg 
Affiliation: Laboratoire Kastler Brossel, Ecole Normale Supérieure, UPMC, CNRS, Paris, France. 

Link to the Research Group on 'Bose-Einstein Condensates' >>

In our daily life experience, flow is always accompanied by friction. But in quantum physics, some materials can form a superfluid, in which the friction vanishes completely. This flow without friction was first observed in liquid helium in 1938 [1,2], when cooled below the lambda temperature. Also the electron gas in a metal can form a superfluid, which leads to the phenomenon of superconductivity, i.e. electric current without resistance.

Superfluid flow with respect to an external body is a metastable state. The ground state would be the fluid at rest, but the decay of the superflow is protected by an energy barrier. The superflow can decay via collective excitations such as phonons or vortices, which are only activated if the flow surpasses a critical velocity. Equivalently, an obstacle moving in a superfluid at rest can only create excitations if it surpasses the critical velocity.

The physical origin of superfluidity is intimately related to Bose-Einstein condensation, which is the macroscopic occupation of the lowest energy state and which occurs below a critical temperature. The system can then be described by a macroscopic wavefunction, which implies irrotational flow and a long-ranged phase coherence between different parts of the system. A three-dimensional (3D) Bose gas at low temperatures is both a Bose-Einstein condensate (BEC) and a superfluid.

Things get more involved in lower dimensions. Here the increased role of thermal fluctuations prohibits a conventional phase transition to a state with long-ranged order. In particular, the uniform two-dimensional (2D) Bose gas does not form a BEC. Now the question arises whether superfluidity does survive the reduction to lower dimensions despite the absence of BEC. There is indeed a phase transition to a superfluid state with quasi-long-ranged order via the Berezinskii-Kosterlitz-Thouless (BKT) mechanism.

The original system to study superfluidity was liquid helium, and it was studied also in 2D films [3]. With the achievement of BEC in dilute atomic gases [4], there is a new highly-controllable superfluid system at hand. It has weak isotropic interactions, greatly facilitating the comparison with theory. Cold gases confined to two dimensions have been used to study the phase coherence and the microscopic origin of the BKT transition [5-7], but the direct observation of the superfluidity of the low temperature phase was so far missing. In our recent results published in Nature Physics [8], we present the first direct observation of superfluidity in a 2D atomic Bose gas.

To create the 2D gas, we tightly confine the atoms in the vertical direction. When the energy of the first excited state in this direction is larger than the thermal energy and the chemical potential, then the motion in this direction is essentially frozen and the system can be described as 2D. In the horizontal plane, the atoms are also trapped, but with a much weaker confinement, such that the overall shape of the atom cloud resembles a pancake.

Figure 1: Schematics of the experiment. The pancake-shaped atom cloud is stirred by a tightly focused laser beam, which acts as a repulsive obstacle. By stirring in circles, one can probe at a fixed density.

Following the method developed by the MIT group in 3D systems [9], we stir the cloud with an obstacle formed by a tightly focused laser beam (Figure 1). We observe the resulting heating as a function of the stirring velocity and find no dissipation below a critical velocity (see Figure 2). Only above this velocity can excitations occur, which lead to a heating of the cloud. This threshold behavior is the signature of superfluidity.

Figure 2: Measuring the critical velocity. The curve shows the temperature T of the cloud for varying stirring velocities v of the laser beam. There is no heating below a critical velocity v_c, indicating the superfluid response of the atom cloud.

The phase transition to the superfluid state occurs above a critical phase space density, i.e. when the temperature is sufficiently low and the density sufficiently high. Because the atomic gas is in equilibrium, the temperature is constant over the cloud. The density, however, varies across the cloud, being largest in the center. Therefore we expect the gas to be in the superfluid state in the center of the cloud and in the normal state in the wings. The position of the boundary between the two states depends on the total atom number and temperature of the cloud.

In the experiments, we stir in circles centered on the cloud. In this way, we can probe at a fixed density. By varying the stirring radius as well as the atom number and temperature, we can map out the density and temperature dependence of the transition between the superfluid and the normal state (see Figure 3).

Figure 3: Mapping out the BKT phase transition. The non-zero critical velocities v_c indicate the superfluid state, which occurs above a critical ratio of the local chemical potential µ_loc at the stirring radius and the temperature T of the cloud.

Our results complete the thermodynamic picture of the atomic 2D Bose gas after the measurement of the equation of state [10, 11]. The 2D superfluidity is also interesting in the context of the not-yet-understood cuprate high T_c superconductors, which are based on superfluidity whithin 2D layers of the cuprate oxide material [12].

References
[1] P. Kapitza, "Viscosity of Liquid Helium below the λ-Point", Nature 141, 74 (1938). Abstract. Full Article.
[2] J.F. Allen, A.D. Misener, "Flow of Liquid Helium II", Nature 141, 75 (1938). Abstract. Full Article.
[3] D. J. Bishop and J. D. Reppy, "Study of the Superfluid Transition in Two-Dimensional ⁴He Films", Physical Review Letters, 40, 1727 (1978). Abstract.
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, "Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor", Science 269, 198–201 (1995). Abstract.
[5] Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Battelier and Jean Dalibard, "Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas", Nature, 441, 1118–1121 (2006). Abstract.
[6] P. Cladé, C. Ryu, A. Ramanathan, K. Helmerson, and W. D. Phillips, "Observation of a 2D Bose Gas: From Thermal to Quasicondensate to Superfluid", Physical Review Letters, 102, 170401 (2009). Abstract.
[7] S. Tung, G. Lamporesi, D. Lobser, L. Xia, and E. A. Cornell, "Observation of the Presuperfluid Regime in a Two-Dimensional Bose Gas", Physical Review Letters, 105, 230408 (2010). Abstract.
[8] Rémi Desbuquois, Lauriane Chomaz, Tarik Yefsah, Julian Léonard, Jérôme Beugnon, Christof Weitenberg, Jean Dalibard, "Superfluid behaviour of a two-dimensional Bose gas", Nature Physics, 8 (Published online July 29, 2012). Abstract.
[9] C. Raman, M. Köhl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ketterle, "Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas", Physical Review Letters, 83, 2502–2505 (1999). Abstract.
[10] Chen-Lung Hung, Xibo Zhang, Nathan Gemelke, Cheng Chin, "Observation of scale invariance and universality in two-dimensional Bose gases", Nature 470, 236-239 (2011). Abstract.
[11] Tarik Yefsah, Rémi Desbuquois, Lauriane Chomaz, Kenneth J. Günter, and Jean Dalibard, "Exploring the Thermodynamics of a Two-Dimensional Bose Gas", Physical Review Letters, 107, 130401 (2011). Abstract.
[12] P.W. Anderson, "The Theory of Superconductivity in the High Tc Cuprates" (Princeton University Press, Princeton, 1997).

Imperfections, Disorder and Quantum Coherence

Steve Rolston [Image courtesy: University of Maryland, USA]

A new experiment conducted at the Joint Quantum Institute (JQI, operated jointly by the National Institute of Standards and Technology in Gaithersburg, MD and the University of Maryland in College Park, USA) examines the relationship between quantum coherence, an important aspect of certain materials kept at low temperature, and the imperfections in those materials. These findings should be useful in forging a better understanding of disorder, and in turn in developing better quantum-based devices, such as superconducting magnets. The new results are published in the New Journal of Physics [1].

Most things in nature are imperfect at some level. Fortunately, imperfections---a departure, say, from an orderly array of atoms in a crystalline solid---are often advantageous. For example, copper wire, which carries so much of the world’s electricity, conducts much better if at least some impurity atoms are present.

In other words, a pinch of disorder is good. But there can be too much of this good thing. The issue of disorder is so important in condensed matter physics, and so difficult to understand directly, that some scientists have been trying for some years to simulate with thin vapors of cold atoms the behavior of electrons flowing through solids trillions of times more dense. With their ability to control the local forces over these atoms, physicists hope to shed light on more complicated case of solids.

That’s where the JQI experiment comes in. Specifically, Steve Rolston and his colleagues have set up an optical lattice of rubidium atoms held at temperature close to absolute zero. In such a lattice atoms in space are held in orderly proximity not by natural inter-atomic forces but by the forces exerted by an array of laser beams. These atoms, moreover, constitute a Bose Einstein condensate (BEC), a special condition in which they all belong to a single quantum state.

This is appropriate since the atoms are meant to be a proxy for the electrons flowing through a solid superconductor. In some so called high temperature superconductors (HTSC), the electrons move in planes of copper and oxygen atoms. These HTSC materials work, however, only if a fillip of impurity atoms, such as barium or yttrium, is present. Theorists have not adequately explained why this bit of disorder in the underlying material should be necessary for attaining superconductivity.

The JQI experiment has tried to supply palpable data that can illuminate the issue of disorder. In solids, atoms are a fraction of a nanometer (billionth of a meter) apart. At JQI the atoms are about a micron (a millionth of a meter) apart. Actually, the JQI atom swarm consists of a 2-dimensional disk. “Disorder” in this disk consists not of impurity atoms but of “speckle.” When a laser beam strikes a rough surface, such as a cinderblock wall, it is scattered in a haphazard pattern. This visible speckle effect is what is used to slightly disorganize the otherwise perfect arrangement of Rb atoms in the JQI sample.

In superconductors, the slight disorder in the form of impurities ensures a very orderly “coherence” of the supercurrent. That is, the electrons moving through the solid flow as a single coordinated train of waves and retain their cohesiveness even in the midst of impurity atoms.

In the rubidium vapor, analogously, the slight disorder supplied by the speckle laser ensures that the Rb atoms retain their coordinated participation in the unified (BEC) quantum wave structure. But only up to a point. If too much disorder is added---if the speckle is too large---then the quantum coherence can go away. Probing this transition numerically was the object of the JQI experiment. The setup is illustrated in figure 1.

Figure 1: Two thin planes of cold atoms are held in an optical lattice by an array of laser beams. Still another laser beam, passed through a diffusing material, adds an element of disorder to the atoms in the form of a speckle pattern. [Image courtesy: Matthew Beeler]

And how do you know when you’ve gone too far with the disorder? How do you know that quantum coherence has been lost? By making coherence visible.

The JQI scientists cleverly pry their disk-shaped gas of atoms into two parallel sheets, looking like two thin crepes, one on top of each other. Thereafter, if all the laser beams are turned off, the two planes will collide like miniature galaxies. If the atoms were in a coherent condition, their collision will result in a crisp interference pattern showing up on a video screen as a series of high-contrast dark and light stripes.

If, however, the imposed disorder had been too high, resulting in a loss of coherence among the atoms, then the interference pattern will be washed out. Figure 2 shows this effect at work. Frames b and c respectively show what happens when the degree of disorder is just right and when it is too much.

Figure 2: Interference patterns resulting when the two planes of atoms are allowed to collide. In (b) the amount of disorder is just right and the pattern is crisp. In (c) too much disorder has begun to wash out the pattern. In (a) the pattern is complicated by the presence of vortices in the among the atoms, vortices which are hard to see in this image taken from the side. [Image courtesy: Matthew Beeler]

“Disorder figures in about half of all condensed matter physics,” says Steve Rolston. “What we’re doing is mimicking the movement of electrons in 3-dimensional solids using cold atoms in a 2-dimensional gas. Since there don’t seem to be any theoretical predictions to help us understand what we’re seeing we’ve moved into new experimental territory.”

Where does the JQI work go next? Well, in figure 2a you can see that the interference pattern is still visible but somewhat garbled. That arises from the fact that for this amount of disorder several vortices---miniature whirlpools of atoms---have sprouted within the gas. Exactly such vortices among electrons emerge in superconductivity, limiting their ability to maintain a coherent state.

Another of the JQI scientists, Matthew Beeler, underscores the importance of understanding the transition from the coherent state to incoherent state owing to the fluctuations introduced by disorder: “This paper is the first direct observation of disorder causing these phase fluctuations. To the extent that our system of cold atoms is like a HTSC superconductor, this is a direct connection between disorder and a mechanism which drives the system from superconductor to insulator.”

Reference:
[1] M C Beeler, M E W Reed, T Hong, and S L Rolston, "Disorder-driven loss of phase coherence in a quasi-2D cold atom system", New Journal of Physics, 14, 073024 doi:10.1088/1367-2630/14/7/073024 (2012). Abstract. Full Article.