Okinawa School in Physics 2017 Coherent Quantum Dynamics. Cold Rydberg gases

Transcription

1 Okinawa School in Physics 2017 Coherent Quantum Dynamics Cold ydberg gases 1. Basics of ydberg atoms 2. ydberg atoms in external fields. ydberg-ydberg interaction Wenhui Li Centre for Quantum Technologies Department of Physics National University of Singapore

2 Outline ydberg-ydberg interaction Long-range dipole dipole interaction between ydberg atoms. Some current research topics with interacting ydberg atoms/ensembles

3 Long-range dipole dipole interaction between ydberg atoms

4 Interaction of two classical static dipoles V dd 1 2 Some special cases: μ 1 μ 2 Attractive epulsive

5 Permanent Dipole of ydberg Stark states in E-field Permanent Dipoles in E fields esonant dipole-dipole (almost like classical interaction between static dipoles) μ 2 The resulting of linear Stark effects have permanent electric dipoles: ψ ± r ψ ± 0 while ψ ± are superposition of different l states. μ 1 E

6 Interaction of two synchronously oscillating classical dipoles 1 z1 cost z cost 2 2 μ 2 V dd μ 1 The interaction is essentially the same but reduced by a factor of two due to the time averaging of the dipoles. NOTE We always assume that we are in the near field regime. 2c and we ignore the radiation field.

7 Dipole dipole interaction of two ydberg atoms The quantum version of oscillating dipoles is the interaction between transition dipole moments. One example Atomic states np ns ns, np μ 1μ 2 and ns, np V dd np, ns 0 np, ns = ns μ 1 np np μ 2 ns 0 E( ns, np ) = E( np, ns ) States of two atoms (Pair states or Diatomic states) Atom 1 ns, np Pair states d-d interaction or nsnp np, ns Atom 2 npns H V dd nsnp npns 12 0 n 4, E ± = ± μ 1μ 2

8 Another example Atomic states np ns, ns V dd np, n 1 p 0 Atom 1 ns, ns Pair states or ns d-d interaction np, n 1 p Atom 2 (n-1)p or n 1 p, np ns, ns μ 1μ 2 np, (n 1)p = ns μ 1 np ns μ 2 (n 1)p 0 E( ns, ns ) - E( np, n 1 p ) = Δ H nsns 0 12 np(n - 1)p 12 nsns Δ np(n-1)p (n-1)pnp 12 Vdd ~ 1/ n n 4,

9 esonance d-d interaction vs Van der Waals interaction resonance dipole-dipole interactions van der Waals (off-resonance d-d) 20 S + 20 S 19 P + 20 P Hamiltonian Potential Curves 20p 20s H 20s20s p20p Sodium 19p 12 Vdd ~ 1/ n n 4, 4 n ~ n (d - d) 11 ~ 6 (vdw) Safinya et al., PL 47, 2405 (1981) Huge interaction, ~ GHz ( n ~ 40, ~ 1 m) Easily tuned by: external E field (), n (V dd, ), or (V dd )

10 esonance d-d interaction vs Van der Waals interaction resonance dipole-dipole interactions van der Waals (off-resonance d-d) If V dd (at large ) H Hamiltonian 20s20s Vdd ~ 1/ n 19p20p 12 n 4, Potential Curves 4 n ~ 12 n (d - d) 11 ~ 6 (vdw) W = V dd 2 W = ± V dd 2, + V 2 dd ~ n11 6 So van der Waals interaction, C 6 6 Where C 6 n 11 If V dd (at small ) W = ±V dd ~ n4 Dipole-dipole interaction

11 Energy Energy Tuning the dipole-dipole interaction with E-field Energy 2 E np (F 0 ) = E ns (F 0 ) + E (n+1)s (F 0 ) E (n+1)s (F) E np (F) E ns (F) (n+1)s (n-1)d 5/2 np /2 ns H V dd ss' pp n 21 2 F 4 E pp pp ss' 2 ss' V 2 dd F 0 (n-1)d 5/2 (n-1)d 5/2 2V dd + For n = 42 Pour n 42 ns (n 1)s - ss np /2 np /2 F F 0 F

12 Some details of dipole-dipole interaction Consider the nsnp npns pair states, which are coupled by the dipole-dipole coupling ns, np V dd np, ns = µ 2 E( ns, np ) = E( np, ns ) µ nsnp V dd npns nsnp 1 2 npns There are two natural choices for an axis of quantization The internuclear axis The direction suggested by a field, which might produce the dipoles, messy but possible.

13 The internuclear axis as the quantization axis The nsnp npns pair states with the internuclear axis chosen as the quantization axis, in the z direction. The dipole dipole matrix element becomes = x1x 2 + y1 y2-2z1z2 nsnp Vdd npns = nsnp npns nsnp V npns = ns np V np ns dd dd ns x np np x ns + ns y np np y ns - 2 ns z np np z ns Angular momentum is conserved, so we construct states of total angular momentum M along the quantization axis using M = m1+ m2

14 Using basis states of the form nsnp there are six states, two each of M=0, 1, and -1. For example the M=0 states are nsnp 0 and np 0 ns, with the Hamiltonian matrix H nsnp 0 V dd 0 V np Where the matrix element is given explicitly by 0 0 dd ns The two M=0 eigenvalues and eigen states are 2 W Vdd nsnp npns 2 nsnp npns 2,, One of the eigenstates is symmetric in the interchange of the two atoms, and one is antisymmetric, W W V dd 2 2 Vdd antisymmetric V dd = - 2 ns z np np z ns symmetric

15 There are analogous 2x2 matrices for the M=1 and M=-1 states, which also give symmetric and antisymmetric states symmetric antisymmetric Solid lines are M=0 states Broken lines are M=1 and -1 states. There are two M=1 and two M=-1 states.

16 eturn to the case of the nsns np(n-1)p pairs M=0 for example Δ nsns np(n-1)p (n-1)pnp The Hamiltonian matrix of the three M=0 states where H nsns 0 V V dd dd np 0 (n-1)p 0 V dd 0 (n-1)p 0 np 0 V dd 0 V dd = - 2 ns z np np z ns The three eigen energies are W W V dd

17 The energy levels Δ nsns np(n1)p (n-1)pnp There is a state with no shift degeneracy of the p pairs The shift changes from 1/ 6 to 1/ when V~Δ, from a van der Waals to a dipole-dipole interaction

18 Fine, but quite real details The spin orbit splittings must be taken into account Often only one spin orbit coupled state is important. In b and Cs np /2 np /2 Is resonant with ns(n+1)s but np 1/2 np 1/2 is not.. There are smaller matrix elements. There are more states. The Zeeman degeneracy leads to states with no shift. -Kiffner, Walker and Saffman We have already seen an example of this.

19 Calculated van der Waals shifts from Walker and Saffman (2007) Note these are calculated with the internuclear axis as the quantization axis

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21 (GHz) Dipole Blockade and Collective Excitation Energy n ~ 40 r,r r,r g,r, r,g g,g Interatomic Spacing (m) r g Single Atom + = 1/2 ( g,r + r,g) + 2 Two Atoms 2 g,g - Blockade: a deterministic way to make entanglement abi Oscillation / Time

22 Dipole blockade in mesoscopic ensembles (principle) Only one ydberg atom can be excited inside the blockade sphere.

23 Dipole blockade in mesoscopic ensembles (possible applications) If the radius is smaller than the blockade radius, the only accessible state is the symmetric Dicke state with 1 ydberg excitation. Generalization of the 2 atoms case to N atoms Collective excitation with abi frequency: Ω coll = NΩ Ψ coll = 1 N eik.r 1 rgg g + e ik.r 2 grg g + + e ik.rn ggg r r 1 g N 1 All the N-1 other states with one excitation are uncoupled to ground state because of symmetry properties r 1 g N 1 NΩ g N

24 Fast Quantum Gate D. Jaksch, J.I. Cirac, P. Zoller, S.L. olston,. Cote, M.D. Lukin, Fast quantum gates for neutral atoms, Phys. ev. Lett. 85, 2208 (2000). 1 st 2 r,1 r,0 V dd All pulses: 0 r r 1,1 0,1 1,0 0, nd r,1 1,r r,0 r,r V dd No excitation Due to blockade - two q-bit states, 0 and 1 coupled differently to ydberg state r two atoms individually addressible 1,1 0,1 1,0 0,0 2 1 st r,1 r, ,1 0,1 1,0 0,0 V dd Phase Gate 1,1 1,1 0,1-0,1 1,0-1,0 0,0-0,0

25 Frozen ydberg gases and dipole-dipole energy transfer

26 P P P P P P P S S P P P P P P S P P P P S P P S P P P S S P P P S S S M M Many-body effects Dipole-dipole allowed ppss ps s p pssp Many-body effects in the frozen ydberg gas Frozen ydberg gas: atoms move only a few percent of internuclear distance in a MOT

27 Early experimental studies Observing Förster resonance in cold atomic ensemble in magneto-optical tramp

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29 954 Principle : Absorption imaging in presence of coupling light ydberg atoms Ground state atoms

30 Energy (n-1)d 5/2 (n-1)d 5/2 Pour n For 42 n = 7 ns (n 1)s np /2 np /2 F 0 F

31 a) b) The light-harvesting apparatus of green sulphur bacteria and the Fenna-Matthews-Olson (FMO) protein. The excitations from absorbed light are transported through the FMO protein with detailed shown on the right. b) The pictorial representation of Förster energy transfer in the FMO protein. M. Sarovar, A. Ishizaki, G.. Fleming, and K. B. Whaley, Quantum entanglement in photosynthetic light-harvesting complexes, Nature Physic, 6, 462 (2010); E. Collini, Spectroscopic signatures of quantum-coherent energy transfer, Chem. Soc. ev. 42, 492 (201)

32 Excitation blockade and collective excitations

33 Early experimental evidence When the volume size is larger than the blockade radius, several atoms can excited to the ydberg states in the atomic cloud. However, there is rapid saturation of the excitation. Number of excited atoms ~ V sample /V blockade Blockade sphere Proper definition of dipole blockade radius b is: Ω N b = C p b p where N b is the number of atoms contained in the blockade sphere.

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37 Dipole blockade in mesoscopic ensembles Hundreds of atoms! Ω N Ω

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39 Single photon filter and deterministic single photon source 2 AUGUST 2012 VOL 488 NATUE 57 Only one photon can be present at a time in the atomic sample polaritons propagate one by one! The time difference between output single photons is determined by the time traveled by each polariton inside the medium. Transmission of the probe beam versus probe beam frequency for different incoming photon rates (Attenuation length l a = 1/(n a σ 0 ) of a few microns when one polariton is already excited) Blockade condition is b > l a, τ d = 00 ns between output pulses Dimensions σ r σ ax = 10 4 μm 2

40 Long range molecules

41 Ultralong-range ydberg Molecules Ground state atom ydberg atom Covalently bond molecules Macrodimers b(nl) + b(5s) Trilobite-like

42 Non-polar molecule Trilobite-like ydberg Molecules b(nl) + b(5s) (l 2) Polar molecule b(nl) + b(5s) (l >2) C.H. Greene et al. PL 85, 2458 (2000) Dipole moment ~ 1KD

43 Trilobite-like ydberg Molecules Bendkowsky V. et al. Nature 458, (2009) M.A. Bellos et al. PL 111, (201) Excitation to Trilobite-like 5s+5p Photoassociation Population in vibrational levels (FI detection) Spontaneous decay 5s+5s Population in vibrational levels (FI detection)

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47 Spontaneous evolution of a cold ydberg gas into a cold plasma

48 Dipolar forces Attractive epulsive

49 Spontaneous formation of Ultra-cold plasmas Spontaneous ionization g BB e - Potential energy Black-body Collisions with background gas Dipole-dipole collisions Trapping of electrons in space charge Direct photoionization? Expansion k B T electron 511 nm Ionization threshold ydberg n~0 6p /2 Trapping laser 6s 1/2

50 Ionization due to dipolar forces Cs( np ) Cs( np ) Cs Cs( n ' l ') n' n / 2 / 2 Ultracoldplasma No plasma

51 Ultracold neutral plasmas Direct photoionization of a gas of cold atoms Strongly-coupled plasmas? coupling parameter: =Coulomb Energy/Thermal Energy e a /k B T 1 Wigner Crystallization of neutral plasma? Killian, Science