University of California at Berkeley Department of Chemistry Department of Computer Science Pennsylvania State University Department of Physics
Quantum Computing with Neutral Atoms in an Addressable Optical lattice
Jiri Vala Ashish V. Thapliyal, Simon Myrgren, Umesh Vazirani, David S. Weiss, K. Birgitta Whaley
J. Vala et al.: quant-ph/0307085, submitted (2003) D.S. Weiss et al.: submitted (2003) J. Vala et al.: in preparation (2003)
Scalability
physical resources ~ polynomial(# of qubits)
Tensor product structure of the Hilbert space H
individually addressable qubits with controllable interactions
1 2 3 n H2 H2 H2 H2
H2n
Physical implementations: atomic lattices, solid state and superconducting systems 133Cs Qubit 6s1, I = 7/2 5 F hyperfine structure 62P 4 (electron spin - nuclear spin 3/2 3 interaction) fine structure 2 (spin-orbit interaction) D2 line 852 nm 4 2 6 P1/2 1.17 GHz 3 splitting caused by external field
D1 line 894 nm
+4 mF 2F+1 sublevels 4 2 qubit 6 S1/2 9.19 GHz mF = 0 3 field-insensitive +3 states Lattice Potential electromagnetic standing wave field ... E = 2 E cos(k z) L L θ = 0
z
kL = 2π /λ 2 V0 = α |E| …plus matter polarizability - depends on atom
lattice potential 1 V(z,θ) = 2 V0 cos(θ ) cos(4π z /λ) Lattice Potential iθ/2 - -iθ/2 + EL= E[e cos(kLz-θ/2) σ - e cos(kLz+ θ/2) σ ] m = +1 F’ 2 6 P1/2 F’=4 6 15 15 2 6 P1/2 F’=3 * V(z) = EL (z) α EL (z) π
dipole 6 1 5 polarizability moment detuning
- + α = - ΣF’ PFdPF’dPF/∆F,F’ σ σ
PF = Σm |F,m> PF’ = Σm’ |F’,m’> V = α|E|2 V = α’|E|2 m /F 0 θ = 0 1 F linear lattice laser beams polarization θ = 0 mF > 0 mF < 0 V0 = V1 V0 > V1 V/V0 θ/π V0=100 µK λ/2 = 5.3 µm z/a 3D Addressable Optical Lattice 62P CO2 lattice 3/2 Blue-detuned lattice Winoto et al. λ λ PRA 59, R19 (1999) r,2 b,2 2 6 P1/2 λ λr,1 b,1 λr λb λ λ r,3 2 b,3 6 S1/2 a ~ 5 µm CO2 lattice parameters: 30 W per beam, 100 µm waist, 100 µK deep wells, -5 100 nm ground state, 10 Hz scattering rate ~ 8000 sites Quantum Computation Requirements Quantum Computation Initialization Processing Measurement - preparation of - implementation of - detection of fiducial qubit state universal operations a final qubit state Scalability physical resources ~ polynomial(# of qubits) Fault-tolerance error correction Initialization loading cooling/imaging compacting optical lattice, filling factor fi ~ 0.5 Motivation for compacting: No Mott insulator quantum phase transition to get fi ~ 1 Computing on an imperfect lattice: mapping of quantum algorithm onto the imperfect lattice • difficult with scalability and fault tolerance requirements • may be computationally hard to find such a map Qubit Preparation 1) loading the lattice with many atoms from an MOT 2) polarization gradient coolling in the lattice Winoto,DePue,Bramall,Weiss, PRA 59, R19 (1999) CCD Before cooling After cooling DePue,McCormick,Winoto,Oliver,Weiss, PRL 82, 2262 (1999) filling factor ~ 0.5 3) imaging the lattice occupation: the occupation map Scheunemann, Cataliotti, Hansch, Weitz, PRA 62, 051801 (2000) Raman Sideband Cooling 6P states 1. A Raman pulse transfers atoms from νÆν-1. 2. Optical pumping returns B the atoms to mF=3. In the CO lattice, mF=2 6S1/2,F=3 2 mF=3 rescattering is no With ions: Monroe et al. PRL 75 4011 (1995) problem Æ~100% to Atoms in a 2D FORL: Hamann et al. PRL 80 4149 (1998) the vibrational ground Atoms in a 1D FORL: Vuletic et al. PRL 81 5768 (1998) state Perrin et al. EuroPhys. Lett. 42 395 (1998) Atoms in a 3D FORL: Kerman et al. PRL 84 439 (2000) Han,Wolf,Oliver,DePue,Weiss. PRL 85 724 (2000) flip operation compacting scaling shift operation Timescales: flip operation ~ 100 µs ~10000 operations shift operation ~ 1ms before recooling decoherence ~10 s CO2 optical lattice with 8000 sites can be compacted in ~ 1s 1) State preparation 2) Shift 3) State flip perfect lattice Use off-resonant circularly polarized light to change 2 6 P3/2 the energy separation of the two qubit states at the target site. 2 6 P1/2 mobile F=4 B 133 microwave Cs 6S1/2 pulse ac Stark shift local on the lattice 2 6 S1/2 F=3 A stationary Perform single qubit operations with microwave pulses that are only resonant with the target atom. Flip Operation Other possibility: adiabatic rapid passage using chirped field 2 mF = 0 6 P1/2 Effective two-level B adiabatic transfer 133 Cs A B laser chirped laser pulse pulse time σ+ σ - F=4 6S1/2 A B F=3 mF = -1 mF = +1 Compacting linear A B polarization θ = 0 adiabatic shift A B θ = π flip A B θ = π shift back A B θ = 0 flip Shift Operation: Coherent Wavepacket Dynamics Realistic Potential V0 > V1 θ = r t V(z,θ) = 1 V cos(θ)cos(4π z /λ) - 1 V sin(θ) sin(4π z /λ) 2 0 2 1 1) the moving potential first 2) the vibrational frequency changes over the course of the shift operation accelerates and then decelerates ∂2Vzt(,) ω ∂z2 4π zz= 0 0 ()t = λ m V/V0 V0=100 µK θ/π z/a determined by atomic states | F, mF > two heating mechanisms 1) due to moving reference frame - can be eliminated by timing condition 2) due to acceleration and deceleration Coherent Wavepacket Dynamics Simulation Methodology R. Kosloff, in Fourier grid representation Dynamics of Molecules and ψ(x) ~ Σ a θ (x ) Chemical Reactions, by k k k ψ(x) R.E. Wyatt and J.Z.H. Zhang. Hψ(x) ~ (1/2N) Σm exp(i2πmp∆x/L) T(p) Σn exp(-i2πnp∆x/L) ψ(x) + U(x) ψ(x) Advantages: arbitrary accuracy and fast convergence rate Chebyshev Propagator R. Kosloff, Ann. Rev. Phys. Chem. U(∆t) |ψ> = Σk bk (∆t) Tk(H’) 45, 145 (1994) Bessel functions Chebyshev Polynomials of the order k T0 = |ψ> T = H’ T Advantages: optimal sampling 1 0 high accuracy T2 = 2H’ T1 -T0 etc. intrinsically finite expansion applicable to arbitrary Hamiltonian Compacting Algorithm and Complexity a) Balancing Flip Shift Flip Shift Row compacting A B b) Flip Shift Flip Shift Flip Two dimensional lattice Æ 5/2 N1/2 (worst case) 1/3 scaling linear in Three dimensional lattice Æ 4 N (worst case) the number of sites N is the number of sites in the lattice per dimension! Maxwell Demon with a Memory Maxwell demon’s memory entropy production step recording acting erasing non-demolition elementary measurement operations ? Optical lattice initialization Φ = 0 0 1 Non-demolition measurement H. Mabuchi, J. Ye, H.J. Kimble, Appl. Phys. B 68, 1095 (1999) Single-qubit Operations 2 6 P3/2 Use off-resonant circularly polarized light to change the energy separation of the two qubit states at 2 6 P1/2 the target site. F=4 |1> 133 microwave Cs 6S1/2 pulse ac Stark shift local on the lattice 2 |0> 6 S1/2 F=3 Tranfer qubit from the field-insensitive states to the field sensitive states. Perform single qubit operations with microwave pulses that are only resonant with the target atom. Return back to the field-insensitive states. Two-qubit Operations Jaksch et al., 1 µ1 . µ2 (µ1 . r) (µ2 . r) Vd(r) = 3 -3 5 PRL 85, 2208 (2000) 4 π ε0[]|r| |r| |RR> qubit 1 qubit 2 qubit 1 Vd π 2ππ n ~ 25 - 50 t |1R> |R1> |0R> |R0> |00> |00> |11> |01> - |01> |10> - |10> |01> |10> |11>eiδ |11> |00> Qubit Measurement 2 Fluorescent detection e.g. mF=5 = 5 6 P3/2 in succesive image planes CCD CCD σ+ e.g. mF=4 = 4 2 6 S1/2 mF=4 = 0 |1> Alternative: mF=3 = 0 |0> ionization & ion detection Decoherence |qubit> = c0 |0> + c1 |1> Errors: i? Pure dephasing |qubit> = c0 |0> + e c1 |1> - elastic collisions with the background gas particles - fluctuation of the laser fields Relaxation |qubit> = ? |0> + ? |1> |?|2 + |?|2 = 1 - spontaneous emission Leakage - inelastic collisions with the background gas particles - spontaneous emission Qubit loss - inelastic collisions with the background gas particles Qubit Loss and its Error Correction background gas ? 123 4 1 no error collision non-demolition insertion no error induces measurement of a new after standard an error identifies atom quantum error the qubit loss correction the net quantum error after a new atom is inserted, is equivalent to amplitude damping Error Detection Non-demolition measurement of presence of an atom 0 1 Φ = 0 2 high-finesse cavity 6 P3/2 2 6 P1/2 0 1 Φ = 0 2 6 S1/2 H. Mabuchi, J. Ye, H.J. Kimble, Appl. Phys. B 68, 1095 (1999) Error Correction conveyor C conveyor computer initialization lattice chamber chamber adiabatic translation dipole trap conveyor computer lattice lattice Initialization - compacting Maxwell demon elementary complexity operations imaging Quantum Computation ~ 4000 qubits initialization processing measurement operation ~ 100 µs scalable architecture decoherence ~ 10 s error correction fault-tolerant architecture Happy Day !