Laboratoire Kastler Brossel Collège de France, ENS, UPMC, CNRS. Ultracold quantum gases in optical lattices

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1 Laboratoire Kastler Brossel Collège de France, ENS, UPMC, CNRS Ultracold quantum gases in optical lattices solid-state physics with atoms in crystals made of light International school on phase transitions, Bavaria March 21, 2016

2 The context : ultracold quantum gases Laser cooling and trapping successes in the 80 s-90 s lead to ever colder and denser atomic samples [Chu, Cohen-Tannoudhi, Phillips : Nobel 1997] Quantum degenerate gases: Bose-Einstein condensation in 1995 [Cornell, Wieman, Ketterle : Nobel 2001] Degenerate Fermi gases in 2001 [JILA] Superfluid-Mott insulator transition for bosons in 2002 [Munich] Fabrice Gerbier (fabrice.gerbier@lkb.ens.fr )

3 Ultracold atomic gases as many-body systems Quantum degeneracy : phase space density nλ 3 db > 1 n :spatial density λ db = 2π 2 : thermal De Broglie wavelength mk B T From W. Ketterle group website, Interacting atoms, but dilute gas: na 3 1 a : scattering length for s wave interactions 8πa 2 : scattering cross-section (bosons) a n 1/3 λ db Typical values (BEC of 23 Na) : a 2 nm, n 1/3 100 nm, λ db 1 µm at T = 100 nk

4 Many-body physics with cold atomic gases Ultracold atomic gases as model systems for many-body physics : dilute but interacting gases tunability (trapping potential, interactions, density,...) and experimental flexibility microscopic properties well-characterized well-isolated from the external world Bose-Einstein condensates : Superfluid gas Atom laser Optical lattices : Superfluid-Mott insulator transition BEC-BCS crossover : Condensation of fermionic pairs JILA, MIT, Rice (1995) Munich 2002 Many other examples : gas of impenetrable bosons in 1D, non-equilibrium many-body dynamics, disordered systems,... JILA, MIT, ENS ( )

5 From one to three-dimensional optical lattices A coherent superposition of waves with different wavevectors results in interferences. The resulting interference pattern can be used to trap atoms in a periodic structure. Linear stack of 2D gases Planar array of 1D gases Cubic array Additional, weaker trapping potentials provide overall confinement of the atomic gas. Many more geometries are possible by playing with the interference patterns.

6 Scope of the lectures Why is this interesting? 1 Connection with solid-state physics (band structure and related phenomenon) 2 A tool for atom optics and atom interferometry: coherent manipulation of external degrees of freedom 3 Path to realize strongly correlated gases and new quantum phases of matter

7 1 Introduction 2 Band structure in one dimension 3 Thermodynamics of Bose gases in optical lattices 4 Dynamics of Bose-Einstein condensates in optical lattices 5 Bloch oscillations

8 Optical dipole traps Two-level atom interacting with a monochromatic laser field: Electric field : E = 1 2 E(r)e iωlt+iφ + c.c. Γ: transition linewidth d: electric dipole matrix element δ L = ω L ω eg: detuning from atomic resonance For large detuning, the atom-laser interaction can be pictured using the Lorentz model of an elastically bound electron. The electric field induces an oscillating electric dipole moment d E. The (time-averaged) potential energy of the induced dipole is V (r) = 1 2 d E = d2 E(r) 2 4 δ L δ L < 0 (red detuning) : V < 0, atoms attracted to intensity maxima δ L > 0 (blue detuning) : V > 0, atoms repelled from intensity maxima

9 Some numbers Potential energy : V (r) = d2 4 δ L E(r) 2 Take 87 Rb atoms for definiteness : Γ/2π 6 MHz, λ eg 780 nm [ω L /2π Hz] λ nm, δ L /2π Hz, Laser parameters : power P = 200 mw, beam size 100 µm] One finds a potential depth V h 20 khz k B 1 µk. Trapping atoms in such a potential requires sub-µk temperatures, or equivalently degenerate (or almost degenerate) gases. NB : trap depths up to mk are possible using tightly focused lasers.

10 One-dimensional optical lattices Superposition of mutually coherent plane waves : E(r) 2 2 = E ne ikn r = n n E n 2 + E n E n e i(k n kn) r n n Intensity (and dipole potential) modulations with wavevectors k n k n Simplest example : Standing wave with period d = π/k L Trapping potential of the form (red detuning): V (x) = 2V 0 (1 + cos(2k L x)) Atoms trapped near the antinodes where V 2V 0. N.B. : V 0 = d2 E δ L d: electric dipole matrix element, E 0 : electric field amplitude, Detuning from atomic resonance :δ L = ω L ω eg < 0.

11 Two and three-dimensional optical lattices Two dimensions : E(r) 2 2E 0 cos(k L x) 2 + 2E 0 cos(k L y) 2 Two-dimensional square potential Three-dimensional cubic potential Square or cubic lattices : V lat (r) = ν=1,,d Vν cos(kνxν)2 Separable potentials : sufficient to analyze the 1D case Other lattice geometries are realizable as well: Soltan-Panahi et al., Nature Phys. (2011) Jo et al., PRL (2012)

12 One-dimensional lattice Standing wave with period d = π/k L Trapping potential : V (x) = 2V 0 cos 2 (k L x) = 2V 0 (1 + cos(2k L x)) Natural units: lattice spacing d = λ L /2 = π/k L recoil momentum k L recoil energy E R = 2 k 2 L 2M a 87 Rb, λ L = 1064 nm: d 532 nm E R h 2 khz (100 nk)

13 Diffraction from a pulsed lattice Pulsing a lattice potential on a cloud of ultracold atoms (BEC) : Lattice axis 2 k L 0 µs 4 µs 8 µs 12 µs 16 µs 20 µs 24 µs 28 µs Pulse duration Diffraction of a matter wave from a light grating : nλ db = 2 p ( λl2 ) sin(θ n) θ Incoming : p x = 0 Outgoing : p x = p sin(θn) De Broglie wavelength : λ db = h p p x = 2n k L

14 Diffraction from a pulsed lattice : Kapitza-Dirac regime Pulsing a lattice potential on a cloud of ultracold atoms (BEC) : Lattice axis 2 k L Raman-Nath approximation: 0 µs 4 µs 8 µs BEC plane wave with k = 0 neglect atomic motion in the potential during the diffraction pulse : Ψ(x, t) e i V 0 cos(2k L x)t Ψ(x, 0) + p= J p(v 0 t/ )e i2pk Lx. Analogous to phase modulation of light wave by a thin phase grating 12 µs 16 µs 20 µs 24 µs 28 µs Pulse duration Relative populations p=0 p=1 p=2 Raman-Nath Pulse duration (µs) Raman-Nath approximation valid only for short times

15 1 Introduction 2 Band structure in one dimension 3 Thermodynamics of Bose gases in optical lattices 4 Dynamics of Bose-Einstein condensates in optical lattices 5 Bloch oscillations

16 Bloch theorem Hamiltonian : Ĥ = ˆp2 2M + V lat(ˆx), V lat (x) = V 0 sin 2 (k L x) Lattice translation operator : definition : ˆTd = exp (iˆpd/ ) x ˆT d φ = φ (x + d) for any φ [ ˆT d, Ĥ] = 0. Bloch theorem : Simultaneous eigenstates of Ĥ and ˆT d (Bloch waves) are of the form φ n,q (x) = e iqx u n,q (x), where the u n,q s (Bloch functions) are periodic in space with period d. q : quasi-momentum n : band index

17 Bloch theorem Bloch waves : φ n,q (x) = e iqx u n,q (x), where the u n,q s (Bloch functions) are periodic in space with period d. q : quasi-momentum n : band index Quasi-momentum is defined from the eigenvalue of ˆT d : ˆT d φ n,q (x) = e iqd φ n,q (x). For Q p = 2pk L with p integer (a vector of the reciprocal lattice), ˆT d φ n,q+qp (x) = e i(q+qp)d φ n,q+qp (x) = e iqd φ n,q+qp (x). To avoid double-counting, restrict q to the first Brillouin zone: BZ1 = [ k L, k L ].

18 Fourier decomposition of Bloch waves on plane waves Bloch waves : φ n,q (x) = e ikx u n,q (x) The Bloch function u n,q is periodic with period d : Fourier expansion with harmonics Q m = 2mk L of 2π/d = 2k L. u n,q (x) = m Z ũ n,q(m)e iqmx, V lat (x) = m Z Ṽ lat (m)e iqmx = V V 0 (e ) iq 1x + e iq 1x 4 the Bloch functions are superpositions of all harmonics of the fundamental momentum 2k L. the lattice potential couples momenta p and p ± 2k L. Useful to solve Schrödinger equation : reduction to band-diagonal matrix equation for the Fourier coefficients ũ n,q(m) (tridiagonal for sinusoidal potential)

19 Spectrum and a few Bloch states, V 0 = 0E R Vlat [ER] q [k L ] Free particle spectrum : ɛ n(q) = 2 (q+2nk L ) 2 2M un,q(x) un,q(x) Bloch function n = 0, q = x [d] Bloch function n = 1, q = x [d] Momentum : k = q + 2nk L Degeneracy at the edges of the Brillouin zone : E n(±k L ) = E n+1 (±k L )

20 Spectrum and a few Bloch states, V 0 = 1E R Vlat [ER] q [k L ] Gap opening near the edges of the Brillouin zones (q ±k L ) un,q(x) un,q(x) Bloch function n = 0, q = x [d] Bloch function n = 1, q = x [d] Lifting of free particle degeneracy by the periodic potential

21 Spectrum and a few Bloch states, V 0 = 4E R Vlat [ER] q [k L ] un,q(x) un,q(x) Bloch function n = 0, q = x [d] Bloch function n = 1, q = x [d]

22 Spectrum and a few Bloch states, V 0 = 10E R Vlat [ER] q [k L ] un,q(x) un,q(x) Bloch function n = 0, q = x [d] Bloch function n = 1, q = x [d]

23 Very deep lattices In a deep lattice potential, atoms are tightly trapped around the potential minima. Harmonic approximation for each well : V lat (x x i ) 1 2 Mω2 lat (x x i) 2, ω lat = 2 V 0 E R. The bands are centered around the energy E n (n + 1/2) ω lat. Vlat(x) First correction : quantum tunneling across the potential barriers, x/d as in tight-binding methods used in solid-state physics (Linear Combination of Atomic Orbitals)

24 Diffraction from a pulsed lattice from band theory Pulsing a lattice potential on a cloud of ultracold atoms (BEC) : Lattice axis 2 k L Bloch wave treatment: φ n,q = Initial state : m= 0 µs 4 µs 8 µs ũ n,q(m) q + 2mk L Ψ(t = 0) = k = 0 = [ũ n,q=0 (m)] φ n,q=0 Evolution in lattice potential : Ψ(t) = [ũ n,q=0 (m)] e i E n,q=0 t φ n,q=0 12 µs 16 µs 20 µs 24 µs 28 µs Pulse duration Relative populations p=0 p=1 p=2 Bloch waves Raman-Nath Pulse duration (µs) Raman-Nath approximation valid only for short times

25 1 Introduction 2 Band structure in one dimension 3 Thermodynamics of Bose gases in optical lattices 4 Dynamics of Bose-Einstein condensates in optical lattices 5 Bloch oscillations

26 Bose-Einstein condensation in a box The total population of excited states is bounded from above : N N max = 1 e ɛ i >ɛ β(ɛ i ɛ min ) 1 min Saturation of excited states and macroscopic accumulation of particles in the lowest energy state when N N max for fixed T For a uniform gas : ( ) L 3 N max λ th 2π λ th = 2 : thermal De Broglie wavelength Mk B T 1.0 Fixing n = N/V : T c,box 2π 2 M ( n ) 2/ N /N N 0/N t = T/T c,box

27 Thermodynamics of an ideal Bose gas in a cubic lattice We do the same calculation (numerically) for a cubic lattice. We introduce the filling fraction n = N = ρd N 3, i.e. the mean number of atoms per lattice site. s T c /E R T c/e R T c/j Fraction in lowest band T c /J f V 0 /E R V 0 /E R Here J energy with of the lowest Bloch band. T c decreases quickly as V 0 increases, Atoms accumulate quickly in the lowest band as V 0 increases V 0 /E R tight-binding limit : k B T c 6J for n = 1, reached for V 0 10 E R,

28 How to prepare cold atoms in optical lattices Principle of evaporative cooling : Atoms trapped in a potential of depth U 0, undergoing collisions : E two atoms with energy close to U 0 collide result: one cold atom and a hot one with energy > U 0 rethermalization of the N 1 atoms remaining in the trap results in a lower mean energy per atom. U 0 x Experimental procedure to prepare a cold atomic gas in a lattice : prepare a quantum gas using evaporation in an auxiliary trap, transfer it to the lattice by increasing the lattice potential from zero and simultaneously removing the auxiliary trap. Why not cool atomic gases directly in the periodic potential?. evaporative cooling no longer works due to the band structure as soon as V 0 a few E R. The best one can do is to transfer the gas adiabatically, i.e. at constant entropy.

29 Isentropic loading Increase lattice depth from 0 to 10 E R a constant entropy. The three curves indicate the entropy vs lattice depth curves for V 0 = 0, 5, 10 E R. Isentropic path goes horizontally from the blue curve to the red one. Red dots mark the location of T c for each case. A: adiabatic cooling path B: adiabatic heating path

30 Isentropic loading Increase lattice depth from 0 to 10 E R a constant entropy. The three curves indicate the entropy vs lattice depth curves for V 0 = 0, 5, 10 E R. Isentropic path goes horizontally from the blue curve to the red one. Red dots mark the location of T c for each case. C: adiabatic decondensation

31 1 Introduction 2 Band structure in one dimension 3 Thermodynamics of Bose gases in optical lattices 4 Dynamics of Bose-Einstein condensates in optical lattices 5 Bloch oscillations

32 Adiabatic theorem Quantum adiabatic theorem : Slowly evolving quantum system, with Hamiltonian Ĥ(t). Instantaneous eigenbasis of Ĥ: Ĥ(t) φ n (t) = ε n (t) φ n (t). Time-dependent wave function in the { φ n(t) } basis: Ψ(t) = n a n(t)e i t0 ε n(t )dt φ n(t), From Schrödinger equation, one gets [ω mn = ε m ε n] : ȧ n = φ n φ n a n(t) e i t0 ω mn(t )dt φ n φ m a m(t), m n Berry phase : φ n φ n = iγ B is a pure phase. Wavefunction unchanged up to a phase evolution after a cyclic change. The adiabatic theorem : for arbitrarily slow evolution starting from a particular state n 0 (a n(0) = δ n,n0 ), and in the absence of level crossings, a n(t) δ n,n0 (up to a global phase).

33 Adiabatic loading of a condensate Adiabatic approximation for slow evolutions and initial condition a n(0) = δ n,n0 : Validity criterion : φ n φ m = φn Ḣ φm ε m ε n = Time-dependent lattice potential : a n(t) e iφ(t) δ n,n0 V lat = V 0 (t) α V 0 (t) increases from 0 to some final value. ) 2 (ε m ε n φ n Ḣ φm. sin(k αx α) 2 Quasi-momentum = good quantum number: only band-changing transitions 10 V 0 =0E R 10 V 0 =0.5E R 10 V 0 =2E R 10 V 0 =5E R Energy (Er) 6 4 Energy (Er) 6 4 Energy (Er) 6 4 Energy (Er) quasi momentum (2 /d) quasi momentum (2 /d) quasi momentum (2 /d) Adiabaticity criterion for a system prepared in a Bloch state (n, q): V ) 2 0 (ε m(q) ε n(q) quasi momentum (2 /d)

34 Adiabatic loading of a condensate Adiabaticity criterion for a system prepared in a Bloch state (n, q): V lat = V 0 (t) α sin(k αx α) 2 Quasi-momentum = good quantum number: V ) 2 0 (ε m(q) ε n(q) Sodium atoms, E R /h 20 khz [Denschlag et al., J. Phys. B 2002]: Adiabaticity most sensitive for small depths : Near band center n = 0, q = 0: ε 1 ε 0 4E R Near band edge n = 0, q α = π/d: ε 1 ε 0 0 Caution: for real systems interactions and tunneling within the lowest band are the limiting factors, not the band structure. Adiabaticity requires ramp-up times in excess of 100 ms.

35 Band mapping : adiabatic release from the lattice Thermal Bose gas, J 0 k B T ω lat : almost uniform quasi-momentum distribution. Mapping by releasing slowly the band structure before time of flight (instead of suddenly) typically a few ms. Qualitative value only if band edges matter. Greiner et al., PRL 2001

36 Time-of-flight experiment : sudden release from the lattice Time-of-flight experiment : suddenly switch off the trap potential at t = 0 and let the cloud expand for a time t. Time of flight (tof) expansion reveals momentum distribution (if interactions can be neglected). Classical version : r(t) = r(0) + p(0)t M Quantum version: wave-function after tof mirrors the initial momentum distribution P 0 (p) with p = Mr t. n tof (r, t) = ψ(r, t) 2 ( M t ) 3 P 0 ( p = Mr t ) Analogous to Fraunhofer regime of optical diffraction, requires p 0t M x 0 with x 0, p 0 the spread of ψ 0 in real and in momentum space. for a condensate : N atoms behaving identically, density profile n tof (r, t) N ψ(p, t) 2 with ψ the Fourier transform of the condensate wavefunction.

37 Time-of-flight interferences Non-interacting condensate: Atoms condense in the lowest band n = 0 at quasi-momentum q = 0: φ 0,0 (p) ũ 0,0 (m)δ(p Q m), m Z 3 Q m = 2k L m (m Z 3 ) is a vector of the reciprocal lattice. Time-of-flight distribution: comb structure with peaks mirroring the reciprocal lattice ( Bragg peaks ).

38 1 Introduction 2 Band structure in one dimension 3 Thermodynamics of Bose gases in optical lattices 4 Dynamics of Bose-Einstein condensates in optical lattices 5 Bloch oscillations

39 Bloch oscillations Uniformly accelerated lattice : V lat [x x 0 (t)] with x 0 = F t2 2m Lab frame: Moving frame: H lab = p2 2m + V lat[x x 0 (t)] V (x) 1.2 ẋ x/d Unitary transformation H mov = p2 2m + V lat[x] F x V (x) x/d Bloch theorem still applies : H lab invariant by lattice translations Upon acceleration (moving frame): n, q 0 n, q(t) : q(t) = q 0 mẋ 0 = q 0 + F t Quasi-momentum scans linearly accross the Brillouin zone.

40 Bloch oscillations Quasi-momentum scan accross the Brillouin zone : q(t) = q 0 mẋ 0 = q 0 + F t When q = +k L, either non-adiabatic transfer to higher bands or, if adiabatic, Bragg reflection to q = k L. Bloch oscillations of quasi-momentum with period T B = 2 k L F Experimental observation with cold Cs atoms [Ben Dahan et al., PRL 1995, also in Raizen s group at UT Austin]: Moving frame ω L, +k L ω L + δ, k L Accelerated lattice : V lat = V 0 cos 2 (k L x δt)

41 Atomic wavepacket in a moving lattice Consider an atomic wavepacket narrow in momentum space (typical velocities v k L /M). Accelerated lattice : V lat = V 0 cos 2 (k L x δt) Standing wave traveling at velocity v = δ/k L ωl, +kl ωl + δ, kl Quasi-momentum remains a good quantum number Lattice frame : q q mv For slow (adiabatic) acceleration, a Bloch state q evolves to q mv. A wavepacket built from Bloch states propagates with the group velocity : v g = dε(q) dq q= mv = M M v, with M = d2 ε(q) dq 2 the effective mass (for v k L /M). ) Lab frame : group velocity : v BEC = v + v g = (1 M M v shallow lattice, V 0 E R : M M, atoms stand still deep lattice, V 0 E R : M M, atoms dragged by the moving lattice

42 Bloch oscillations with a BEC Quasi-momentum scan accross the Brillouin zone : q(t) = q 0 mẋ 0 = q 0 + F t Bloch oscillations of quasi-momentum with period T B = 2 k L F Experimental observation with non-interacting BEC [Gustavsson et al., PRL 2008]: N.B.: Assume atoms are prepared in a given band n = 0, and do not make a transition to higher bands (adiabatic approximation).

43 Application to precision measurements : g Bloch oscillations in a vertical optical lattice : T B = 2 k L Mg Poli et al., PRL 2011, Tino group at LENS (Florence) approx periods Measurement uncertainty : 10 6 g Further application : measurement of G at the 10 5 level [Rosi et al., Nature 2014, LENS Florence]

44 Application to precision measurements : /M Measurement of the fine structure constant α : α 2 = 4πR c M m e M R : Rydberg constant m e: electron mass M: atomic mass possible window on physics beyond QED : interactions with hadrons and muons, constraints on theories postulating an internal structure of the electron,... Measurement of M : Experiment in the group of F. Biraben (LKB, Paris) e q R = k 1 + k 2 Doppler-sensitive Raman spectroscopy : E hf k 1 k 2 g 1 g 2 ω res = E + 2 2M (p i + k + q R ) 2 = M = ωres(p i + k) ω res(p i ) q R k

45 Application to precision measurements : /M Large momentum beamsplitter using Bloch oscillations : After N Bloch oscillations, momentum transfer of k = 2N k L to the atoms in the lab frame. Measurement of M : N 10 3 : Comparable uncertainty as current best measurement (anomalous magnetic moment of the electron Gabrielse group, Harvard). Independent of QED calculations This transfer is perfectly coherent and enables beamsplitters where part of the wavepacket remains at rest while the other part is accelerated. [Bouchendira et al., PRL 2011] Other applications in precision measurements: measurement of weak forces, e.g. Casimir-Polder [Beaufils et al., PRL 2011].