Attempts of coherent of coherent control contro with ultracold atoms in optical lattices Martin Holthaus Institut für Physik Carl von Ossietzky Universität Oldenburg http://www.physik.uni-oldenburg.de/condmat Quo vadis BEC? at MPIPKS Dresden August 11, 2010
SF-MI transition with ultracold atoms in optical lattices Observation: absorption images after 15 ms time of flight a-h: V 0 /E r = 0, 3, 7, 10, 13, 14, 16, 20 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002)
Overwhelming response: 1655 citations since Jan. 2002 18.8 cit/m! Compare with BCS: 5488 citations since Dec. 1957 8.9 cit/m! Web of Science, May 01, 2009
Overwhelming response: 1655 citations since Jan. 2002 18.8 cit/m! Compare with BCS: 5488 citations since Dec. 1957 8.9 cit/m! Why all this excitement? After all, the essential physics of the underlying Bose Hubbard model is well understood since 1989: Fisher et al., PRB 40, 546 (1989). Web of Science, May 01, 2009
Phase diagram now is known with high precision 1 0.8 Freericks et al. Our data 0.6 µ/u 0.4 0.2 0 0 0.01 0.02 0.03 J/U Freericks et al.: Scaling theory [PRA 79, 053631 (2009)] Oldenburg group: HODPT [PRB 79, 224515 (2009)]
Accurate data available even for arbitrary filling factor g 1 0.8 µ/u-g+1 0.6 0.4 0.2 0 0 0.01 0.02 0.03 0.04 0.05 0.06 gj/u g=1 g=2 g=3 g=4 g=5 g=10 g=20 g=50 g=10000 QMC High-order diagrammatic many-body perturbation theory: N. Teichmann, D. Hinrichs, M.H., A. Eckardt, PRB 79, 100503(R) (2009)
Thus, the issue at stake here definitely is not the confirmation of known physics. What actually matters is that a difficult many-body Hamiltonian has been emulated.
Thus, the issue at stake here definitely is not the confirmation of known physics. What actually matters is that a difficult many-body Hamiltonian has been emulated. This brings about a new vision: The quantum simulator. (I. Bloch, Science 319, 1202 (2008)) For example, emulating the fermionic Hubbard model could yield information on high-t c -superfluidity. (Hofstetter et al., PRL 89, 220407 (2002))
But then, one would like to control the elements from which the quantum simulator is made, or even add new ones.
But then, one would like to control the elements from which the quantum simulator is made, or even add new ones. Electronic characteristic energies fall into the ev-range. Thus, for controlling electronic transitions one employs laser radiation. The characteristic energies of trapped ultracold atoms fall into a regime around 10 11 ev. Thus, for controlling ultracold atoms one has to employ the lower khz-regime.
0. Overview: 1. What this talk is about (over!) 2. Optical lattices: Basic facts 3. Dynamic localization and quasienergy band engineering 4. Coherent control of the SF-MI-transition 5. Dressed matter waves 6. Future directions
Optical lattices: Basic 1. Optical lattices: Basic facts Single-particle Hamiltonian: E H 0 = p2 2m + V 0 2 cos(2k Lx) Characteristic energy scale: V λ x E r = 2 k 2 L 2m x ( 87 Rb at λ L = 852 nm : E r = 1.31 10 11 ev) Stationary Schrödinger equation (z = k L x )... ( d 2 dz 2 + E E r V 0 2E r cos(2z) ) ψ(z) = 0... is the Mathieu equation with q = V 0 /4E r : ψ (z) + [ a 2q cos(2z) ] ψ(z) = 0
Band structure of cosine lattices 20 a n (q) / b n (q) 10 0 10 0 5 10 q = V 0 /(4E r ) Upper (b n ) and lower (a n ) band edges J. C. Slater, Phys. Rev. 87, 807 (1952)
Important: The n -th Fourier coefficient of the dispersion E(k), E(k) = E 0 + n=1 ( 1) n 2J n cos(nkd), is the matrix element of H 0 taken with Wannier functions w l separated by n lattice sites: ( 1) n J n = w 0 H 0 w n V 0 /E r J 1 /E r J 2 /J 1 3.0 0.11103 0.1011 5.0 0.06577 0.0516 10.0 0.01918 0.0118 15.0 0.00652 0.0035 20.0 0.00249 0.0012 25.0 0.00104 0.0005 Boers et al., PRA 75, 063404 (2007)
Keep in mind: A 1d optical cosine lattice, H 0 = p2 2m V 0 2 cos(2k Lx), realizes the simplistic 1d nearest-neigbor tight-binding model H nn = J + l= ( ) l l + 1 + l + 1 l to an accuracy of 1% or better if V 0 10E r. Add time-periodic forcing: H(t) = J l ( ) l l + 1 + l + 1 l + K cos(ωt) l l l l
Dynamic localization 2. Dynamic localization System actually considered here: H(t) = J ( ) l l + 1 + l + 1 l + ef cos(ωt) l ld l l l
H(t) is T -periodic in time, H(t) = H(t + T ) with T = 2π/ω. Hence, the time-dependent Schrödinger equation i ψ(t) = H(t) ψ(t) t has Floquet-type solutions ψ α (t) = u α (t) exp( iε α t/ ) with u α (t) = u α (t + T ). These are obtained by solving the eigenvalue equation [ H(t) i t ] uα (t) = ε α u α (t) in an extended Hilbert space of T -periodic functions, with scalar product ( time as a coordinate ) 1 T T 0 dt.
Observe: If u (n,0) (t) solves [ H(t) i t ] u(n,0) (t) = ε (n,0) u (n,0) (t), then u (n,m) (t) u (n,0) (t) exp(imωt) solves [ H(t) i t ] u(n,m) (t) = ε (n,m) u (n,m) (t) with ε (n,m) = ε (n,0) + m ω. ( Brillouin zone-structure of quasienergy spectrum )
Observe: Any wave function can be expanded in the form ψ(t) = n c n u (n,0) (t) exp( iε (n,0) t/ ) Standard concepts, such as stationary-state perturbation theory, can immediately be adapted. Floquet states are true analogues of stationary states!
Observe: When a spatially periodic lattice is driven periodically in time, Floquet states are characterized by both a quasimomentum k and a quasienergy ε. There are spatio-temporal Bloch waves with quasienergy bands ε(k)! For example, gives H(t) = J l ( ) l l + 1 + l + 1 l + K cos(ωt) l l l l ε(k) = 2 J J 0 (K/ ω) }{{} cos(kd) mod ω J eff Quasienergy band collapse leads to dynamic localization. M.H., PRL 69, 351 (1992)
Experimental realization with a BEC in an optical lattice: H lab = p2 2m + V [ 0 (2k 2 cos L x λ ]) t dτ ν(τ) 2 t 0 is unitarily equivalent to H = p2 2m + V 0 2 cos(2k Lx) + m λ 2 d ν(t) dt x Sinusoidal forcing ν(t) = ν max sin(ωt) gives ε(k) = 2 J J 0 (K 0 ) }{{} cos(kd) with K 0 = J eff π 2 2 ν max E r Square-wave forcing with amplitude ν max gives ε(k) = 2 J sinc(πk 0 /2) }{{} cos(kd) with K 0 = J eff π 2 ν max E r
In-situ-measurement of expansion rate ( black: square-wave, red: sinusoidal) Lignier et al., PRL 99, 220403 (2007); Eckardt et al., PRA 79, 013611 (2009).
Measurement of dephasing time by t.o.f.-imaging ( black: square-wave, red: sinusoidal) Lignier et al., PRL 99, 220403 (2007); Eckardt et al., PRA 79, 013611 (2009).
This is how it looks like... (Courtesy of O. Morsch, Pisa)
Interband effects: Much more involved... M.H., D.W. Hone, Phil. Mag. B 74, 105 (1996); K. Drese, M.H., J. Phys.: Condens. Matter 8, 1193 (1996)
Keep in mind: A condensate in a strongly driven optical lattice occupies a single Floquet state and remains phase-coherent Extension to photon-assisted tunneling straightforward (Th: Eckardt et al., PRL 95, 200401 (2005); Exp: Sias et al., PRL 100, 040404 (2008)) Design of forcing allows quasienergy band engineering, possibly including interband effects But all this is essentially single-particle physics!
control of th 3. Coherent control of the SF-MI-transition N interacting ultracold atoms in a 1d optical lattice... Ĥ 0 = J M 1 l=1 (â lâl+1 + â l+1âl ) + U 2 M l=1 ˆn l (ˆn l 1)... subjected to an oscillating bias: Ĥ 1 (t) = K cos(ωt) ) M l=1 lˆn l SF-MI-transition governed by U/J eff?
Regular quasienergies for high frequencies ω U, J : ( U/J = 3, ω/j = 14, N = M = 5 )
Digression: Adiabatic following Standard adiabatic theorem: Assume H P ϕ P n = EP n ϕp n with ϕ P n P ϕ P n = 0. Let parameter P vary slowly in time; ψ(t=0) = ϕ P (t=0) n Then ψ(t) = ϕ P n (t) exp ( i t 0 dt E P (t ) n ) solves i t ψ(t) = H P (t) ψ(t) under suitable conditions (energy gap!)
Transfer to time-periodic systems: Solve ( H P (t) i t ) u P α (t) = ε P α u P α (t) with u P α P u P α = 0 and lift to extended Hilbert space: Adiabatic solution Ψ(τ, t) = uα P (τ) (t) exp ( i τ dτ ε P (τ ) α 0 ) Project back: ψ(t) = u P α (t) (t) exp ( i t 0 dt ε P (t ) α ) Quasienergy gap? Resonances? Formulation: H.-P. Breuer, M.H., Z. Phys. D 11, 1 (1989) Phys. Lett. A 140, 507 (1989) Technicalities: K. Drese, M.H., Eur. Phys. J. D 5, 119 (1999)
Signature of transition: Quasimomentum distribution A. Eckardt, C. Weiss, M.H., PRL 95, 260404 (2005)
The Pisa setup: Shaken 3d optical lattices Zenesini et al., PRL 102, 100403 (2009)
SF-MI transition (and back) induced by ac forcing Zenesini et al., PRL 102, 100403 (2009)
Keep in mind: Many-body Floquet states obey an adiabatic principle, even though there is no adiabatic limit in most cases Adiabatic following of such dressed matter waves works under carefully chosen (nonresonant) conditions, despite the dense quasienergy spectrum Coherent control of the SF-MI transition then is a natural consequence But what about multiphoton-like resonances?
matter wave 4. Dressed matter waves Atoms get dressed by electromagnetic fields do matter waves get dressed by shaking? S. Haroche, C. Cohen-Tannoudji, et al., PRL 24, 861 (1970)
Quasienergy spectrum of the dressed Bose-Hubbard system ( ω/j = 20, K/ ω = 2, N = M = 5 )
Multiphoton-like resonances translate into anticrossings! A. Eckardt, M.H., PRL 101, 245302 (2008)
Resonances can be used for switching... ( ω/j = 20, UT/J = 0.3,..., 0.001, N = M = 7 )
... and resonances can be disabled... ( ω/j = 20, UT/J = 0.3,..., 0.001, N = M = 7 )
... because their strength depends on the amplitude K! A. Eckardt, M.H., PRL 101, 245302 (2008)
Keep in mind: Multiphoton resonances translate into anticrossings of quasienergies, or even of entire quasienergy bands Resonance strengths can be controlled by adjusting the driving amplitude The concept of LZ-transitions can be adapted to manybody Floquet states This should allow one to steer a macroscopic matter wave into a variety of prescribed target states
The next steps... 5. The next steps...... should involve avoided-level-crossing spectroscopy: Exp.: Search for large isolated multiphoton resonances (signature: sudden breakdown of adiabaticity) Exp.: Traverse such resonances with varying speed / analyze final states Exp./Th.: Creation of, e.g., highly entangled states feasible? Th./Exp.: Apply techniques from optimal control theory (... )