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Quantum Phase Transition from a Super¯Uid to a Mott Insulator in a Gas of Ultracold Atoms

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Quantum Phase Transition from a Super¯Uid to a Mott Insulator in a Gas of Ultracold Atoms articles Quantum phase transition from a super¯uid to a Mott insulator in a gas of ultracold atoms Markus Greiner*, Olaf Mandel*, Tilman Esslinger², Theodor W. HaÈnsch* & Immanuel Bloch* * Sektion Physik, Ludwig-Maximilians-UniversitaÈt, Schellingstrasse 4/III, D-80799 Munich, Germany, and Max-Planck-Institut fuÈr Quantenoptik, D-85748 Garching, Germany ² Quantenelektronik, ETH ZuÈrich, 8093 Zurich, Switzerland ............................................................................................................................................................................................................................................................................ For a system at a temperature of absolute zero, all thermal ¯uctuations are frozen out, while quantum ¯uctuations prevail. These microscopic quantum ¯uctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose±Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a super¯uid to a Mott insulator phase. In the super¯uid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system. A physical system that crosses the boundary between two phases (here between kinetic and interaction energy) is fundamental to changes its properties in a fundamental way. It may, for example, quantum phase transitions4 and inherently different from normal melt or freeze. This macroscopic change is driven by microscopic phase transitions, which are usually driven by the competition ¯uctuations. When the temperature of the system approaches zero, between inner energy and entropy. all thermal ¯uctuations die out. This prohibits phase transitions in The physics of the above-described system is captured by the classical systems at zero temperature, as their opportunity to change Bose±Hubbard model1, which describes an interacting boson gas in has vanished. However, their quantum mechanical counterparts can a lattice potential. The hamiltonian in second quantized form reads: show fundamentally different behaviour. In a quantum system, 1 ¯uctuations are present even at zero temperature, due to Heisen- H 2 J aòaà e nà U nà nà 2 1 1 berg's uncertainty relation. These quantum ¯uctuations may be ^ i j ^ i i 2 ^ i i hi;ji i i strong enough to drive a transition from one phase to another, ² bringing about a macroscopic change. Here aÃi and aÃi correspond to the bosonic annihilation and ² A prominent example of such a quantum phase transition is the creation operators of atoms on the ith lattice site, nÃi aÃi aÃi is change from the super¯uid phase to the Mott insulator phase in a the atomic number operator counting the number of atoms on system consisting of bosonic particles with repulsive interactions the ith lattice site, and ei denotes the energy offset of the ith hopping through a lattice potential. This system was ®rst studied lattice site due to an external harmonic con®nement of the theoretically in the context of super¯uid-to-insulator transitions in atoms2. The strength of the tunnelling term in the hamiltonian liquid helium1. Recently, Jaksch et al.2 have proposed that such a is characterized by the hopping matrix element between adja- 3 2 2 transition might be observable when an ultracold gas of atoms with cent sites i,j J 2ed xw x 2 xi 2~ = =2m V lat xw x 2 xj, repulsive interactions is trapped in a periodic potential. To illustrate where w x 2 xi is a single particle Wannier function localized to this idea, we consider an atomic gas of bosons at low enough the ith lattice site (as long as ni < O 1), Vlat(x) indicates the optical temperatures that a Bose±Einstein condensate is formed. The lattice potential and m is the mass of a single atom. The repulsion condensate is a super¯uid, and is described by a wavefunction between two atoms on a single lattice site is quanti®ed by the on-site that exhibits long-range phase coherence3. An intriguing situation interaction matrix element U 4p~2a=mejw xj4d3x, with a appears when the condensate is subjected to a lattice potential in being the scattering length of an atom. In our case the interaction which the bosons can move from one lattice site to the next only by energy is very well described by the single parameter U, due to the tunnel coupling. If the lattice potential is turned on smoothly, the short range of the interactions, which is much smaller than the system remains in the super¯uid phase as long as the atom±atom lattice spacing. interactions are small compared to the tunnel coupling. In this In the limit where the tunnelling term dominates the hamilto- regime a delocalized wavefunction minimizes the dominant kinetic nian, the ground-state energy is minimized if the single-particle energy, and therefore also minimizes the total energy of the many- wavefunctions of N atoms are spread out over the entire lattice with body system. In the opposite limit, when the repulsive atom±atom M lattice sites. The many-body ground state for a homogeneous interactions are large compared to the tunnel coupling, the total system (ei const:) is then given by: energy is minimized when each lattice site is ®lled with the same !N number of atoms. The reduction of ¯uctuations in the atom M jª i ~ aò j0i 2 number on each site leads to increased ¯uctuations in the phase. SF U0 ^ i Thus in the state with a ®xed atom number per site phase coherence i1 is lost. In addition, a gap in the excitation spectrum appears. The Here all atoms occupy the identical extended Bloch state. An competition between two terms in the underlying hamiltonian important feature of this state is that the probability distribution NATURE | VOL 415 | 3 JANUARY 2002 | www.nature.com © 2002 Macmillan Magazines Ltd 39 articles for the local occupation ni of atoms on a single lattice site is period of 500 ms to nrad 24 Hz such that a spherically symmetric poissonian, that is, its variance is given by Var nihnÃii. Further- Bose±Einstein condensate with a Thomas±Fermi diameter of more, this state is well described by a macroscopic wavefunction 26 mm is present in the magnetic trapping potential. with long-range phase coherence throughout the lattice. In order to form the three-dimensional lattice potential, three If interactions dominate the hamiltonian, the ¯uctuations in optical standing waves are aligned orthogonal to each other, with atom number of a Poisson distribution become energetically very their crossing point positioned at the centre of the Bose±Einstein costly and the ground state of the system will instead consist of condensate. Each standing wave laser ®eld is created by focusing a localized atomic wavefunctions with a ®xed number of atoms per laser beam to a waist of 125 mm at the position of the condensate. A site that minimize the interaction energy. The many-body ground second lens and a mirror are then used to re¯ect the laser beam back state is then a product of local Fock states for each lattice site. In this onto itself, creating the standing wave interference pattern. The limit, the ground state of the many-body system for a commensu- lattice beams are derived from an injection seeded tapered ampli®er rate ®lling of n atoms per lattice site in the homogeneous case is and a laser diode operating at a wavelength of l 852 nm. All given by: beams are spatially ®ltered and guided to the experiment using M optical ®bres. Acousto-optical modulators are used to control the ² n jªMIiJ0 ~ & aÃi j0i 3 intensity of the lattice beams and introduce a frequency difference of i1 about 30 MHz between different standing wave laser ®elds. The This Mott insulator state cannot be described by a macroscopic polarization of a standing wave laser ®eld is chosen to be linear and wavefunction like in a Bose condensed phase, and thus is not orthogonal polarized to all other standing waves. Due to the amenable to a treatment via the Gross-Pitaevskii equation or different frequencies in each standing wave, any residual interfer- Bogoliubov's theory of weakly interacting bosons. In this state no ence between beams propagating along orthogonal directions is phase coherence is prevalent in the system, but perfect correlations time-averaged to zero and therefore not seen by the atoms. The in the atom number exist between lattice sites. resulting three-dimensional optical potential (see ref. 20 and As the strength of the interaction term relative to the tunnelling references therein) for the atoms is then proportional to the sum term in the Bose±Hubbard hamiltonian is changed, the system of the intensities of the three standing waves, which leads to a simple reaches a quantum critical point in the ratio of U/J, for which the cubic type geometry of the lattice: system will undergo a quantum phase transition from the super¯uid V x; y; zV sin2 kxsin2 kysin2 kz 4 state to the Mott insulator state. In three dimensions, the phase 0 transition for an average number of one atom per lattice site is Here k 2p=l denotes the wavevector of the laser light and V0 is expected to occur at U=J z 3 5:8 (see refs 1, 5, 6, 7), with z being the maximum potential depth of a single standing wave laser ®eld.
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