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The few-body problems in complicated ultra-cold atom system

The few-body problems in complicated ultra-cold atom system (II) The dynamical theory of quantum Zeno and ant-Zeno effects in open system. Peng Zhang . Department of Physics, Renmin University of China. Collaborators. RUC: Wei Zhang Tao Yin Ren Zhang Chuan- zhou Zhu.

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The few-body problems in complicated ultra-cold atom system

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  1. The few-body problems in complicated ultra-cold atom system (II) The dynamical theory of quantum Zeno and ant-Zeno effects in open system Peng Zhang Department of Physics, Renmin University of China

  2. Collaborators RUC: Wei Zhang Tao Yin Ren Zhang Chuan-zhou Zhu Other institutes: Pascal Naidon Mashihito Ueda Chang-pu Sun Yong Li

  3. Outline • The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) • Long-range effect of p-wave magnetic Feshbach resonance (Phys. Rev. A 82, 062712 (2010)) • The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) • The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, 133202 (2009))

  4. Efimov state: universal3-body bound state identicalbosons k = sgn(E)√E 3 particles characteristic parameters: experimental observation: unstable 3-body recombination 1/a • scattering length a • 3-body parameter Λ • Cesium 133 (Innsbruck, 2006)‏ • 3-component Li6 (a12, a23, a31) (Max-Planck, 2009; University of Tokyo, 2010) • … dimer trimer trimer V. Efimov, Phys. Lett. 33, 563 (1970)

  5. Mixed dimensional system 1D+3D 2D+3D B B aeff (l ,a) D(xA,xB) D(xA,xB) A A scattering length in mixed dimensiton D(xA,xB)→0 G. Lamporesi, et. al., PRL 104, 153202 (2010) Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008)

  6. Stable many-body bound state stable 3-body bound state: no 3-body recombination Everything described by a1 and a2 Y. Nishida, Phys. Rev. A 82, 011605(R) (2010) rB z1 light atom B: 3D heavy atom A1 , A2 : 1D a2 a1 Our motivation: to investigate the many-body bound state with mB <<m1 , m2via Born-Oppenheimer approach Advantage: clear picture given by the A1–A2 interaction induced by B z2 BP boundary condition step1: wave function of B Veff: effective interaction between A1, A2 step2: wave function of A1, A2 3-body bound state: -E: binding energy T. Yin, Wei Zhang and Peng Zhang arXiv:1104.4352

  7. 1D-1D-3D system: a1=a2=a Effective potential rB z1 a1 a2 Veff (regularized) L z2 L L z1–z2 (L) Potential depth Binding energy new “resonance”condition: a=L L/a L/a

  8. 1D-1D-3D system: arbitrarya1and a2 rB 3-body binding energy z1 a1 a2 L/a2 L/a2 z2 L/a1 L/a1 • resonance occurs when a1=a2=L • non-trivial bound states (a1<0 or a2<0) exists

  9. 2D-2D-3D system a2 a1 3-body binding energy L/a2 L/a2 resonance occurs when a1=a2=L L/a1 L/a1

  10. Validity of Born-Oppenheimer approximation 1D-1D-3D 2D-2D-3D L/a L/a a1=a2=a exact solution: Y. Nishida and S. Tan, eprint-arXiv:1104.2387

  11. 4-body bound state: 1D-1D-1D-3D Light atom B can induce a 3-body interaction for the 3 heavy atoms a3 a1 a2 a1=a2=a3=L Veff (regularized ) /L /L

  12. 4-body bound state: 1D-1D-1D-3D Binding energy of 4-body bound state /L a1=a2=a3=L Depth of 4-body potential /L /L resonance condition: L1=L2=L /L

  13. Summary • Stable Efimov state exists in the mixed-dimensional system. • The Born-Oppenheimer approach leads to the effective potential between the trapped heavy atoms. • New “resonance” occurs when the mixed-dimensional scattering length equals to the distance between low-dimensional traps. • The method can be generalized to 4-body and multi-body system.

  14. The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) • Long-range effect of p-wave magnetic Feshbach resonance (Phys. Rev. A 82, 062712 (2010)) • The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) • The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, 133202 (2009))

  15. p-wave magnetic Feshbach resonance s-wave Feshbach resonance: Bose gas and two-component Fermi gas p-wave Feshbach resonance: single component Fermi gas 40K:C. A. Regal, et.al., Phys. Rev. Lett. 90, 053201 (2003); Kenneth GÄunter, et.al., Phys. Rev. Lett. 95, 230401 (2005); C. Ticknor, et.al., Phys. Rev. A 69, 042712 (2004). C. A. Regal, et. al., Nature 424, 47 (2003). J. P. Gaebler, et. al., Phys. Rev. Lett. 98, 200403 (2007). 6Li:J. Zhang,et. al., Phys. Rev. A 70, 030702(R)(2004) . C. H. Schunck, et. al., Phys. Rev. A 71, 045601 (2005). J. Fuchs, et.al., Phys. Rev. A 77, 053616 (2008). Y. Inada, Phys. Rev. Lett. 101, 100401 (2008). theory: F. Chevy, et.al., Phys. Rev. A, 71, 062710 (2005) p-wave BEC-BCS cross over T.-L. Ho and R. B. Diener, Phys. Rev. Lett. 94, 090402 (2005).

  16. Long-range effect of p-wave magnetic Feshbach resonance Low-energy scattering amplitude: Short-range potential (e.g. square well, Yukawa potential): effective-range theory Long-rang potential (e.g. Van der Waals, dipole…): be careful!! • s-wave (k→0) Short range potential (effective-range theory) Van der Waals potential (V(r) ∝ r--6 ) • p-wave (k→0) Can we use effective range theory for van der Waals potential in p-wave case?

  17. Long-range effect of p-wave magnetic Feshbach resonance • two channel Hamiltonian • back ground scattering amplitude • scattering amplitude in open channel : background Jost function Seff is related to Veff

  18. The “effective range” approximation • The effective range theory is applicable if we can do the approximation • This condition can be summarized as a) the neglect of the k-dependence of V and R b) the neglect of S (BEC side, B<B0; V, R have the same sign) c) kF :Fermi momentum the neglect of S (BCS side, B>B0; V, R have different signs)

  19. The condition r1<<1 The Jost function can be obtained via quantum defect theory: the sufficient condition for r1<<1 would be • The background scattering is far away from the resonance or V(bg) is small. • The fermonic momentum is small enough.

  20. The condition r2<<1 and r3<<1 • Straightforward calculation yields • Then the condition r2<<1 and r3<<1 can be satisfied when • The effective scattering volume is large enough • The fermonic momentum is small enough

  21. Summary • The effective range theory can be used in the region near the p-wave Feshbach resonance when (r1,r2,r3<<1 ) • The background p-wave scattering is far away from resonance. • The B-field is close to the resonance point. • The Fermonic momentum is much smaller than the inverse of van der Waals length. • In most of the practical cases (Li6 or K40), the effective range theory is applicable in almost all the interested region. Short-range effect from open channel Long-range effect from open channel

  22. The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) • Long-range effect of p-wave magnetic Feshbach resonance (Phys. Rev. A 82, 062712 (2010)) • The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) • The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, 133202 (2009))

  23. Quantum Zeno effect: close system Proof based on wave packet collapse Misra, Sudarshan, J. Math. Phys. (N. Y.) 18, 756 (1977) t: total evolution time τ: measurement period n:number of measurements measurement t ≈ general dynamical theory D. Z. Xu, Qing Ai, and C. P. Sun, Phys. Rev. A 83, 022107 (2011)

  24. Quantum Zeno and anti-Zeno effect: open system Proof based on wave packet collapse A. G. Kofman & G. Kurizki, Nature, 405, 546 (2000) |e> measurement |g> decay rate two-level system heat bath survival probability • without measurements • With measurements • n→∞: Rmea →0: Zeno effect • “intermediate” n: Rmea > RGR : anti-Zeno effect general dynamical theory?

  25. Dynamical theory for QZE and QAZE in open system single measurement: decoherence factor: 2-level system total-Hamiltonian Interaction picture

  26. Short-time evolution: perturbation theory • initial state • finial state • survival probability • decay rate R= γ=0: R=Rmea (return to the result given by wave-function collapse) γ=1: phase modulation pulses

  27. Long-time evolution: rate equation • master of system and apparatus • rate equation of two-level system • effective time-correlation function gB : bare time-correlation function of heat bath gA : time-correlation of measurements

  28. Long-time evolution: rate equation • Coarse-Grained approximation: ReCG : short-time result • steady-state population:

  29. summary • We propose a general dynamical approach for QZE and QAZE in open system. • We show that in the long-time evolution the time-correlation function of the heat bath is effectively tuned by the measurements • Our approach can treat the quantum control processes via repeated measurements and phase modulation pulses uniformly.

  30. The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) • Long-range effect of p-wave magnetic Feshbach resonance (Phys. Rev. A 82, 062712 (2010)) • The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) • The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, 133202 (2009))

  31. Motivation: independent control of different scattering lengths two-component Fermi gas or single-component Bose gas Three-component Fermi gas,… |1> a12 a12 a13 |2> a32 |3> Independentcontrol of different scattering lengths control of single scattering length ? • Magnetic Feshbach resonance • … We propose a method for the independent control of two scattering lengths in a three-component Fermi gas. BEC-BCS crossover strong interacting gases in optical lattice … Efimov states new superfluid … Independent control of two scattering lengths  control of single scattering length with fixed B

  32. The control of a single scattering length with fixed B-field |c> |e>: excited electronic state D |h> |h>|g> g1 g2 Δ |f2> |f1>|g> Λll –ζe2iηΛal |f2>|g> W |f1> Ω alg=abglg-2π2 D-i2π2χ1/2Λaa |Φres) |l> |l>|g> |f1> (Ω,Δ) |f2> D=-El(Ω,Δ)+Ec(B)+Re(Фres|W+GbgW|Фres) D: control Re[alg] through (Ω,Δ) Λal and Λaa: the loss or Im[alg] energy of |l> is determined by El(Ω,Δ) r:inter-atomic distance |a> HF relaxation scattering length of the dressed states can be controlled by the single-atom coupling parameters (Ω,Δ) under a fixed magnetic field

  33. The independent control of two scattering lengths: method I Step 2:control alg with our trick Step 1: control adgMagenticFeshbach resonance, and fix B |f1> |g> (Ω,Δ) alg |h> |f2> adg |l> adl |d> |l> condition: two close magnetic Feshbach resonances for |d>|g> and |f2>|g>

  34. The independent control of two scattering lengths: 40K–6Li mixture hyperfine levels of 40K and 6Li 6Li |g> alg F=3/2 adg 40K |l> E E adl 40K |d> 1/2 6Li Efimov states of two heavy and one light atom? B B } { |f1>=|40K3> |h> (Ω, Δ) |f2>=|40K2> |l> |g>=|6Li1> |d>=|40K1> |g>|d> |g>|f2> magnetic Feshbach resonance: |g>|d>: B=157.6G |g>|f2>: B=159.5G B(10G) E. Wille et. al., Phys. Rev. Lett. 100, 053201 (2008). no hyperfine relaxation

  35. The independent control of two scattering lengths: 40K–6Li mixture numerical illustration: square-well model |c> |f1>|g> |f2>|g> -Vc |f2>|g> |f1>|g> -V2 -V1 0 A. D. Lange et. al., Phys. Rev. A 79 013622 (2009) |c> a • a is determined by the van der Waals length • the parameters Vc, V2 and V1… are determined by the realistic scattering lengths of 40K-6Li mixture alg(a0) W Ω=40MHz |Φres)

  36. The independent control of two scattering lengths: method II |h’> |h> |f’1> |f1> |g> (Ω,Δ) (Ω’,Δ’) al’g |l> |f2> |l’> alg |f’2> |l’> |g> adl |l> alg : controlled by the coupling parameters (Ω,Δ) al’g :controlled by the coupling parameters (Ω’,Δ’) condition: two close magnetic Feshbach resonances for |f2>|g> and |f’2>|g> disadvantage: possible hyperfine relaxation

  37. The independent control of two scattering lengths: 40K gas |f’1>=|40K17> { |h> (Ω,Δ) |l> } { B |h’> |f’1>=|40K4> (Ω’, Δ’) |f’2>=|40K3> |l’> |f2>=|40K2> |g>=|40K1> magnetic Feshbach resonance: |g>|f2>: B=202.1G C. A. Regal, et. al., Phys. Rev. Lett. 92, 083201 (2004). |g>|f’2>: B=224.2G C. A. Regal and D. S. Jin, Phys. Rev. Lett. 90, 230404 (2003).

  38. The independent control of two scattering lengths: 40K gas hyperfine relaxation • The source of the hyperfine relaxation: unstable channels |f1>|g> and |f’1>|g> • In our simulation, we take the background hyperfine relaxation rate to be 10-14cm3/s |9/2,7/2>| 9/2,5/2>|9/2,9/2>| 9/2,3/2> B. DeMarco, Ph.D. thesis, University of Colorado, 2001. results given by square-well model Ω’=2MHz al’g(a0) Ω=2MHz Δ’(MHz)

  39. Another approach: Light induced shift of Feshbach resonance point excited channel : l1S>|2P> |Φ2> Δ Ω U :laser close channel : ground hyperfine level |Φ1> W1 open channel a |1S>|1S> (incident channel): r Dominik M. Bauer, et. al., Phys. Rev. A, 79, 062713 (2009). D. M.Bauer et al., Nat. Phys. 5, 339 (2009). • Shifting the energy of bound state |Φ1> via laser-induced coupling between |Φ1> and |Φ2> • The Feshbach resonance point can be shifted for 10-1Gauss-101Gauss • Extra loss can be induced by the spontaneous decay of |Φ2> • Easy to be generalized to the multi-component case Peng Zhang, Pascal Naidon and Masahito Ueda, in preparation

  40. summary • We propose a method for the independent control of (at least) two scattering lengths in the multi-component gases, such as the three-component gases of 6Li-40K mixture or 40K atom. • The scheme is possible to be generalized to the control of more than two scattering lengths or the gas of Boson-Fermion mixture (40K-87Rb). • The shortcoming of our scheme: • a. the dressed state |l> • b. possible hyperfine loss

  41. Thank you!

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