Ann. Phys. (Berlin) 18, No. 12, 844 – 848 (2009) / DOI 10.1002/andp.200910391 Anderson localization of matter waves Philippe Bouyer1,2∗ 1 2 Laboratoire Charles Fabry de l’Institut d’Optique, Campus Polytechnique, rd 128, 91127 Palaiseau Cedex, France Physics Department, Stanford University, 328, via Pueblo Mall, Room 236, CA 94305-4060, Stanford, USA Received 13 October 2009, accepted 27 October 2009 Published online 11 December 2009 Key words Quantum gases, Bose-Einstein condensate, Anderson localisation, matter waves. PACS 66.90.+r, 03.75.-b,03.75.Nt The transport of quantum particles in non ideal material media (eg the conduction of electrons in an imperfect crystal) is strongly affected by scattering from impurities of the medium. Even for a weak disorder, semi-classical theories, such as those based on the Boltzmann equation for matter-waves scattering from the impurities, often fail to describe transport properties and full quantum approaches are necessary. The properties of the quantum systems are of fundamental interest as they show intriguing and non-intuitive phenomena that are not yet fully understood such as Anderson localization [1,2], percolation [3], disorderdriven quantum phase transitions and the corresponding Bose-glass [4] or spin-glass [5] phases. Understanding quantum transport in amorphous solids is one of the main issues in this context, related to electric and thermal conductivities. The basic knowledge is that contrary to Bloch’s theory which predicts a (frictionless) transport of non-interacting particles [6] as a consequence of the extension of all eigenstates in a periodic crystal, localization effects in disordered potentials result in a strong suppression of the electronic transport in amorphous solids. In 1958, P.W. Anderson predicted the exponential localization [1] of electronic wave functions in disordered crystals and the resulting absence of diffusion. It has been realized later that Anderson localization (AL) is ubiquitous in wave physics [7] as it originates from the interference between multiple scattering paths and this has prompted an intense activity. Experimentally, localization has been reported in light waves [8], microwaves [9], sound waves [10] and electron gases [11]. Here we present the observation of Anderson localization [12, 13] of a Bose-Einstein condensate (BEC) released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle. In the experiment we observe the 1D Anderson Localization of an expanding BEC in presence of weak disorder potential created by laser speckle (Fig. 1). The 87 BEC is created in a hybrid optomagnetic trap [14] where the transverse confinement (ω⊥ /2π = 70) is given by an optical wave guide (Nd:YAG laser at 1064 nm) whereas a weak magnetic gradient ensures the longitudinal trapping (ωz /2π = 5.4) (see Fig. 2). When the magnetic field is switched off, the BEC starts to expand along the optical guide under the effect of the initial interactions. In the early time of the expansion these interactions decrease rapidly and the associated energy is converted into kinetic one. We also show that, in our one-dimensional speckle potential whose noise spectrum has a high spatial frequency cut-off, exponential localization occurs only when the de Broglie wavelengths of the atoms in the expanding BEC are larger than an effective mobility edge corresponding to that cut-off. In the opposite case, we find that the density profiles decay algebraically [15]. The random potential V is realized by focusing an optical speckle pattern, resulting from an Argon laser (wavelength λ =514 nm) passing through a diffusing plate [16], on the atoms. Since the speckle grain size (defined by the radius of the autocorrelation function c(z) = V (z)V (z + δz)) is directly related ∗ E-mail: philippe.bouyer@institutoptique.fr, Phone: +33 1 69 45 53 44, Fax: +33 1 72 70 35 76 c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Ann. Phys. (Berlin) 18, No. 12 (2009) 845 Fig. 1 (online colour at: www.ann-phys.org) Observation of Anderson localization in 1D with an expanding Bose-Einstein Condensate in the presence of a 1D speckle disorder. to the numerical aperture N A, we create an anisotropic speckle pattern by shining anisotropically the diffusive plate. With our numerical aperture (N A = 0.3), the speckle pattern has a very thin grain size of Δz = λ/(2N A) = 0.82 μm along the BEC propagation direction. It corresponds to the correlation length σR = Δz/π = 0.26 ± 0.03 μm. In the transverse directions, the typical speckle grain sizes (97 and 10 μm) are larger than the BEC dimension and the atoms feel an homogeneous potential. The disorder potential can then be considered as 1D. The disorder amplitude, referred as VR in the following, is characterized by the standard deviation of the speckle potential σV . Since the random intensity is exponentially distributed, it is simply given by the mean value VR = σV = V , which is the quantity measured experimentally. Fig. 2 (online colour at: www.ann-phys.org) Schematic representation of the experimental set-up. The BEC is made in hybrid magneto-optical trap. The optical waveguide (transverse confinement) is made by far off resonance red-detuned Nd:YAG. The magnetic field (created by a ferro-magnet) is used for the longitudinal confinement. The speckle is shone perpendicularly to the propagation axis. The experiment starts with a small BEC in expansion from a loose hybrid opto-magnetic trap (ω⊥ /2π = 70 Hz and ωz /2π = 5.4 Hz). It is well described by the sum of non-interacting k-momentum waves, which momentum distribution (D(k)) has a maximum momentum kmax (directly related to the BEC che;ical potential μ). We measure1 kmax directly by monitoring the evolution of the BEC front edge in a flat 1Dpotential. kmax corresponds to the smallest de Broglie wavelength λdB = 2π/kmax of the propagating 1 Experimentally, we controlled kmax by varying the number of atoms and we achieved its smallest value by decreasing this number up to N = 1.7 104 , which corresponds to a chemical potential µ/h = 219 Hz. There the BEC front edge propagate at a velocity vmax = R˙z (t) = 1.7 mm s−1 that gives kmax = mRb vmax / = 2.47 ± .25 μm−1 . www.ann-phys.org c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 846 P. Bouyer: Anderson localization of matter waves atoms, it must be smaller than the speckle grain size Δz , in order to be able to observe Anderson localisation [15, 17]. In presence of weak disorder (VR μin ), the expansion is governed by the scattering of an almost non-interacting cloud. Each k-momentum wave will be scattered only if there is a corresponding momentum 2k (the Bragg condition) in the diffuser spatial spectrum c̃(2k) (the Fourier transform of the disorder correlation function c(z)). For speckle potential, the diffraction imposes that the spatial spectrum has a high frequency cut-off: it vanishes for k > 2kc = 2/σR . This special feature of speckle disorder imposes a limit (effective mobility edge) to observe an exponential profile : when kmax σR < 1, each k-waves of the expanding BEC is scattered at first order by the disorder potential and localizes exponentially with a k-dependent localization length L(k). In the stationary regime, the BEC localizes exponentially, with a localization length given by L(kmax ). On the contrary, when kmax σR > 1, the k-waves with kmax < k < kc are not scattered at first order. The localization scale is significantly increased and localization is observed through an algebraic profile, with a power law decay n1D ∝ 1/|z|β (with β = 2). Fig. 3 (online colour at: www.ann-phys.org) Stationary profile in a) linear and b) semi-logarithmic scale of the BEC one second after release in disorder potential. The initial BEC is made of 1.7 104 atoms (μin = 219 Hz) which corresponds to a measured value of kmax = 2.47 ± 0.25 μm−1 (kmax σR = 0.65 ± 0.09). The disorder potential amplitude is weak compared to the typical kinetic energy of the expanding BEC (VR /μin = 0.12). In the inset of b), we display the rms width of the profiles versus time in presence or absence of the disorder. This shows that a stationary regime is reached after 0.5 s. Straight lines in a) are exponential fits (exp(−2|z|/Lloc )) to the wings and correspond to the straight lines in b). The narrow central peak (pink) represents the trapped condensate before release (t = 0 s). Note that the profiles are obtained by averaging over five runs of the experiment with the same disorder realization. Left : Localization length Lloc versus the disorder amplitude VR for (kmax σR = 0.65 ± 0.09). The data are obtained with two different diffusing plates, inducing two different extensions for the disordered potential wR,z mm (diamond light blue) and wR,z = 5.3 (square dark blue). The dash-dot red curve shows the theoretical predictions for Lloc and the two straight lines represent the uncertainty associated with the evaluations of kmax and σR . For low amplitude values of VR (typically below 25 Hz), the smaller speckle realization gives much more reliable values as its extension remains much larger than the measured localization lengths. For intensity reasons, larger disorder amplitude were not accessible in the experiment. When kmax σR = 0.65 ± 0.09 < 1, (kmax = 2.47 μm−1 and σR = 0.26 μm), as soon as we switch off the longitudinal trapping, the BEC starts expanding, but in the presence of weak disorder the expansion rapidly stops, in stark contrast with the free expansion case (see inset of Fig. 3b showing the evolution of the rms width of the observed profiles). A plot of the density profile, in linear and semi logarithmic coordinates (Figs. 3a,b), then shows clear exponential wings, a signature of Anderson Localization. In addition, we verified that we rapidly reach (in a couple of seconds) a stationary situation when the exponential profile no longer evolves. An exponential fit to the wings of the density profiles yields the localization length Lloc , which as well no longer evolves when the stationary situation is reached. We can then compare the measured localization length with the theoretical value [15]. c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 18, No. 12 (2009) 847 We have also investigated the regime where the initial interaction energy is large enough that a fraction of the atoms have a k-vector larger than 1/σR by repeating the experiment with a BEC containing a larger number of atoms. In this regime, the localization of the BEC becomes algebraic for the scales accessible experimentally. Indeed, the part of the BEC wave function, corresponding to the waves with momenta in the range 1/σR < k will expand further before it eventually localizes at much longer scales. This plays the role of an effective mobility edge where a significant change in the wavefunction behavior is expected. Fig. 4a shows the observed density profile in such a situation (kmax σR = 1.16 ± 0.13). The log-log plot suggests a power law decrease in the wings, with an exponent of 1.95 ± 0.1, in agreement with the theoretical prediction of wings decreasing as 1/|z|2 . In this regime, where no localization length can be extracted, we verified that the algebraic decay does not depend on the amplitude of the disorder (Fig. 4b). Fig. 4 (online colour at: www.ann-phys.org) a) Stationary profiles in the algebraic localization regime for different disorder amplitudes VR . The initial BEC is made of 1.7 105 atoms (μin = 520 Hz) which corresponds to a measured value of kmax = 4.47 ± 0.3 μm−1 (kmax σR = 1.16 ± 0.14). The straight lines are the fits of the wings with a power law decay (n1D ∝ 1/|z|β ). The different values obtained for β are shown in b) (blue square). In addition the red circles correspond to values found with higher number of atoms (2.5 105 ) with kmax σR = 1.3. To conclude, our experiments reveals that direct imaging of atomic quantum gases in controlled optical disordered potentials is a promising technique to investigate this variety of open questions. Firstly, as in other problems of condensed matter simulated with ultra-cold atoms, direct imaging of atomic matterwaves offers unprecedented possibilities to measure important properties, such as the localization length in this problem. Secondly, our experiment can be extended to quantum gases with controlled interactions where localization of quasi-particles Bose glass Lifshits glass are expected, as well as to Fermi gases and to Bose-Fermi mixtures where rich phase diagrams have been predicted. The reasonable quantitative agreement between our measurements and the theory of 1D Anderson localization in a speckle potential demonstrates the high degree of control in our set-up. It opens the path to the realization of “real” quantum simulators for investigating Anderson localization in a wider variety of models. Extending the technique to two and three dimensions, and better controlling interactions, it might be possible to better understand the behavior of real materials. We could experience situations that theory can not currently precisely predict and even, in the long run, use these simulators to improve semi-conductors devices, such as amorphous silicon-based electronic devices, for example. Acknowledgements The data presented in this article have been obtained in the atom optics group in Palaiseau, France, with the contribution of Juliette Billy, Vincent Josse, Zhanchun Zuo, Alain Bernard, Ben Hambrecht, Pierre Lugan, David Clément, Laurent Sanchez-Palencia, and Alain Aspect. www.ann-phys.org c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 848 P. Bouyer: Anderson localization of matter waves References [1] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [2] Anderson Localization. In: Springer Series in Solid State Sciences 39, edited by Y. Nagaoka and H. Fukuyama. (Springer, Berlin, 1982); Anderson Localization. In: Springer Proceedings in Physics 28, edited by T. Ando and H. Fukuyama. (Springer, Berlin, 1988). [3] A. Aharony and D. 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