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

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

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Outline 1.The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) 2.Long-range effect of p-wave magnetic Feshbach resonance ( Phys. Rev. A 82, 062712 (2010)) 3.The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) 4.The independent control of different scattering lengths in multi- component ultra-cold gas (PRL 103, 133202 (2009))

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Efimov state: universal 3-body bound state identical bosons 1/a k = sgn(E)√E dimer 3 particles trimer scattering length a 3-body parameter Λ characteristic parameters: Cesium 133 (Innsbruck, 2006) 3-component Li6 (a 12, a 23, a 31 ) (Max-Planck, 2009; University of Tokyo, 2010) … experimental observation: V. Efimov, Phys. Lett. 33, 563 (1970) 3-body recombination unstable

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

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Stable many-body bound state -E: binding energy step2: wave function of A 1, A 2 3-body bound state: step1: wave function of B BP boundary condition V eff : effective interaction between A 1, A 2 stable 3-body bound state: no 3-body recombination Everything described by a 1 and a 2 Y. Nishida, Phys. Rev. A 82, 011605(R) (2010) Our motivation: to investigate the many-body bound state with m B <

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z 1 –z 2 (L) L L L V eff (regularized) Effective potential Binding energy L/a 1D-1D-3D system: a 1 =a 2 =a new “resonance” condition: a=L L/a Potential depth rBrB z1z1 z2z2 a1a1 a2a2

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1D-1D-3D system: arbitrary a 1 and a 2 L/a 1 L/a 2 L/a 1 L/a 2 resonance occurs when a 1 =a 2 =L non-trivial bound states (a 1 <0 or a 2 <0) exists 3-body binding energy rBrB z1z1 z2z2 a1a1 a2a2

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2D-2D-3D system L/a 1 L/a 2 L/a 1 L/a 2 3-body binding energy resonance occurs when a 1 =a 2 =L a2a2 a1a1

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

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4-body bound state: 1D-1D-1D-3D Light atom B can induce a 3-body interaction for the 3 heavy atoms /L a 1 =a 2 =a 3 =L V eff (regularized ) a1a1 a2a2 a3a3

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4-body bound state: 1D-1D-1D-3D /L Depth of 4-body potential /L Binding energy of 4-body bound state resonance condition: L 1 =L 2 =L a 1 =a 2 =a 3 =L

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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.

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1.The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) 2.Long-range effect of p-wave magnetic Feshbach resonance ( Phys. Rev. A 82, 062712 (2010)) 3.The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) 4.The independent control of different scattering lengths in multi- component ultra-cold gas (PRL 103, 133202 (2009))

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p-wave magnetic Feshbach resonance p-wave Feshbach resonance: Bose gas and two-component Fermi gas s-wave Feshbach resonance: single component Fermi gas 40 K: 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). 6 Li: 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).

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

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Long-range effect of p-wave magnetic Feshbach resonance two channel Hamiltonian back ground scattering amplitude : background Jost function scattering amplitude in open channel S eff is related to V eff

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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**B 0 ; V, R have different signs) k F :Fermi momentum
**

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

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Straightforward calculation yields Then the condition r 2 <<1 and r 3 <<1 can be satisfied when The effective scattering volume is large enough The fermonic momentum is small enough The condition r 2 <<1 and r 3 <<1

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The effective range theory can be used in the region near the p-wave Feshbach resonance when (r 1,r 2,r 3 <<1 ) a.The background p-wave scattering is far away from resonance. b.The B-field is close to the resonance point. c.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. Summary Long-range effect from open channel Short-range effect from open channel

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1.The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) 2.Long-range effect of p-wave magnetic Feshbach resonance ( Phys. Rev. A 82, 062712 (2010)) 3.The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) 4.The independent control of different scattering lengths in multi- component ultra-cold gas (PRL 103, 133202 (2009))

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

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

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

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

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Long-time evolution: rate equation master of system and apparatus rate equation of two-level system effective time-correlation function g B : bare time-correlation function of heat bath g A : time-correlation of measurements

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Long-time evolution: rate equation Coarse-Grained approximation: R e CG : short-time result steady-state population:

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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.

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1.The universal many-body bound states in mixed dimensional system (arXiv:1104.4352 ) 2.Long-range effect of p-wave magnetic Feshbach resonance ( Phys. Rev. A 82, 062712 (2010)) 3.The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv:1104.4640) 4.The independent control of different scattering lengths in multi- component ultra-cold gas (PRL 103, 133202 (2009))

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Motivation: independent control of different scattering lengths two-component Fermi gas or single-component Bose gas a 12 control of single scattering length Magnetic Feshbach resonance … ? a 12 a 32 a 13 |1> |2> |3> Three-component Fermi gas,… Independent control of different scattering lengths Efimov states new superfluid … BEC-BCS crossover strong interacting gases in optical lattice … We propose a method for the independent control of two scattering lengths in a three-component Fermi gas. Independent control of two scattering lengths control of single scattering length with fixed B

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The control of a single scattering length with fixed B-field r:inter-atomic distance |f 2 >|g> W |Φ res ) |c> |a> HF relaxation |f 1 >|g> Ω Δ |l>|g> |h>|g> scattering length of the dressed states can be controlled by the single- atom coupling parameters (Ω,Δ) under a fixed magnetic field energy of |l> is determined by E l (Ω,Δ) |f 2 > |f 1 > (Ω,Δ)(Ω,Δ) |l> |h> |f 1 > |f 2 > |e>: excited electronic state g1g1 g2g2 D a lg =a bg lg -2π 2 Λ ll – ζ e 2iη Λ al D-i2π 2 χ 1/2 Λ aa D=-E l (Ω,Δ)+E c (B)+Re(Ф res |W + G bg W|Ф res ) D: control Re[a lg ] through (Ω,Δ) Λ al and Λ aa : the loss or Im[a lg ]

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The independent control of two scattering lengths: method I a lg a dl a dg |g> |l> |d> |f 2 > |f 1 > (Ω,Δ)(Ω,Δ) |l> |h> Step 1: control a dg Magentic Feshbach resonance, and fix B Step 2:control a lg with our trick condition: two close magnetic Feshbach resonances for |d>|g> and |f 2 >|g>

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

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|f 2 >|g> |f 1 >|g> W |Φ res ) |c> The independent control of two scattering lengths: 40 K– 6 Li mixture numerical illustration: square-well model a 0 -V 2 |f 1 >|g> -V c -V 1 |f 2 >|g> |c> A. D. Lange et. al., Phys. Rev. A 79 013622 (2009) a lg (a 0 ) a is determined by the van der Waals length the parameters Vc, V 2 and V 1 … are determined by the realistic scattering lengths of 40 K- 6 Li mixture Ω=40MHz

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

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(Ω,Δ)(Ω,Δ) { |h> |l> The independent control of two scattering lengths: 40 K gas B |g>=| 40 K1> |f 2 >=| 40 K2> |f’ 2 >=| 40 K3> |f’ 1 >=| 40 K4> |f’ 1 >=| 40 K17> } (Ω’, Δ’) { |h’> |l’> magnetic Feshbach resonance: |g>|f 2 >: 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).

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The independent control of two scattering lengths: 40 K gas |9/2,7/2>| 9/2,5/2> |9/2,9/2>| 9/2,3/2> results given by square-well model hyperfine relaxation B. DeMarco, Ph.D. thesis, University of Colorado, 2001. The source of the hyperfine relaxation: unstable channels |f 1 >|g> and |f’ 1 >|g> In our simulation, we take the background hyperfine relaxation rate to be 10 -14 cm 3 /s Ω’=2MHz Δ’(MHz) a l’g (a 0 ) Ω=2MHz

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Another approach: Light induced shift of Feshbach resonance point r |Φ1>|Φ1> open channel a |1S>|1S> (incident channel): W1W1 excited channel : l1S>|2P> |Φ2>|Φ2> U :laser Δ Ω close channel : ground hyperfine level Shifting the energy of bound state |Φ 1 > via laser-induced coupling between |Φ 1 > and |Φ 2 > The Feshbach resonance point can be shifted for 10 -1 Gauss-10 1 Gauss Extra loss can be induced by the spontaneous decay of |Φ 2 > Easy to be generalized to the multi-component case Dominik M. Bauer, et. al., Phys. Rev. A, 79, 062713 (2009). D. M.Bauer et al., Nat. Phys. 5, 339 (2009). Peng Zhang, Pascal Naidon and Masahito Ueda, in preparation

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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 6 Li- 40 K mixture or 40 K atom. The scheme is possible to be generalized to the control of more than two scattering lengths or the gas of Boson-Fermion mixture ( 40 K- 87 Rb). The shortcoming of our scheme: a. the dressed state |l> b. possible hyperfine loss

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