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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 Peng Zhang Department of Physics, Renmin University of China Collaborators RUC: Wei Zhang Tao Yin Ren Zhang Chuan-zhou Zhu Other institutes: Pascal Naidon Mashihito Ueda Chang-pu Sun Yong Li 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 component ultra-cold gas (PRL 103, 133202 (2009)) lengths in multi- Efimov state: universal 3-body bound state identical bosons k = sgn(E)√E characteristic parameters: •scattering length a •3-body parameter Λ 3 particles 1/a experimental observation: •Cesium 133 (Innsbruck, 2006) trimer dimer •3-component Li6 (a12, a23, a31) (Max-Planck, 2009; University of Tokyo, 2010) trimer •… unstable 3-body recombination V. Efimov, Phys. Lett. 33, 563 (1970) Mixed dimensional system 1D+3D 2D+3D B B D(xA,xB) D(xA,xB) A A scattering length in mixed dimensiton D(xA,xB)→0 aeff (l ,a) Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008) G. Lamporesi, et. al., PRL 104, 153202 (2010) Stable many-body bound state light atom B: 3D heavy atom A1 , A2 : 1D rB z1 a1 a2 stable 3-body bound state: no 3-body recombination Everything described by a1 and a2 Y. Nishida, Phys. Rev. A 82, 011605(R) (2010) Our motivation: to investigate the many-body bound state with mB <<m1 , m2 via Born-Oppenheimer approach z2 Advantage: clear picture given by the A1–A2 interaction induced by B BP boundary condition step1: wave function of B step2: wave function of A1, A2 3-body bound state: Veff: effective interaction between A1, A2 -E: binding energy T. Yin, Wei Zhang and Peng Zhang arXiv:1104.4352 1D-1D-3D system: a1=a2=a Effective potential rB a1 a2 z2 Veff (regularized) z1 L L L z1–z2 (L) Potential depth Binding energy new “resonance” condition: a=L L/a L/a 1D-1D-3D system: arbitrary a1 and a2 L/a2 L/a2 3-body binding energy L/a1 z1 L/a1 rB a1 a2 •resonance occurs when a1=a2=L z2 •non-trivial bound states (a1<0 or a2<0) exists 2D-2D-3D system a2 a1 L/a2 L/a2 3-body binding energy resonance occurs when a1=a2=L L/a1 L/a1 Validity of Born-Oppenheimer approximation 1D-1D-3D 2D-2D-3D L/a a1=a2=a exact solution: Y. Nishida and S. Tan, eprint-arXiv:1104.2387 L/a 4-body bound state: 1D-1D-1D-3D Light atom B can induce a 3-body interaction for the 3 heavy atoms a3 a2 a1=a2=a3=L Veff (regularized ) a1 /L /L 4-body bound state: 1D-1D-1D-3D Binding energy of 4-body bound state /L /L Depth of 4-body potential a1=a2=a3=L /L resonance condition: L1=L2=L /L 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. 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 component ultra-cold gas (PRL 103, 133202 (2009)) lengths in multi- 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). 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!! Short range potential (effective-range theory) •s-wave (k→0) f (k ) 0 1 1 ik r k a Van der Waals potential (V(r) ∝ r--6 ) f (k ) 2 0 eff •p-wave (k→0) f (k ) 1 1 1 1 ik Vk R 2 1 1 ik r k a 2 eff f1 (k ) 1 ik 1 1 1 Vk 2 Sk R Can we use effective range theory for van der Waals potential in p-wave case? Long-range effect of p-wave magnetic Feshbach resonance •two channel Hamiltonian •back ground scattering amplitude f (bg) 1 (k ) 1 V k S k R •scattering amplitude in open channel 1 ik 1 (bg) 2 1 (bg) (bg) : background Jost function Seff is related to Veff The “effective range” approximation •The effective range theory is applicable if we can do the approximation •This condition can be summarized as a) b) c) 1 1 1 1 the neglect of the k-dependence of V and R the neglect of S (BEC side, B<B0; V, R have the same sign) the neglect of S (BCS side, B>B0; V, R have different signs) kF :Fermi momentum 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. 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 V (k , B) eff 3 6 • The fermonic momentum is small enough k F 1 6 : Van der Waals length 6 Summary •The effective range theory can be used in the region near the p-wave Feshbach resonance when (r1,r2,r3<<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. Short-range effect from open channel Long-range effect from open channel 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 component ultra-cold gas (PRL 103, 133202 (2009)) lengths in multi- 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) Quantum Zeno and anti-Zeno effect: open system Proof based on wave packet collapse A. G. Kofman & G. Kurizki, Nature, 405, 546 (2000) measurement |e> |g> heat bath two-level system survival probability decay rate PGR t Pe exp Rt n •without measurements RGR 2 | gk | eg k 2 k t 2 2 eg | gk | sinc n k 2 k •With measurements Rmea • n→∞: Rmea →0: Zeno effect • “intermediate” n: Rmea > RGR : anti-Zeno effect general dynamical theory? Dynamical theory for QZE and QAZE in open system 2-level system single measurement: decoherence factor: total-Hamiltonian Interaction picture 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 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 Long-time evolution: rate equation •Coarse-Grained approximation: Re CG : short-time result • steady-state population: 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. 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 component ultra-cold gas (PRL 103, 133202 (2009)) lengths in multi- 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> control of single scattering length |3> a32 Independent control of different scattering lengths •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 The control of a single scattering length with fixed B-field |c> Δ |h>|g> D |f1>|g> g1 Ω W |Φres) |f2>|g> HF relaxation |e>: excited electronic state g2 |f2> |l>|g> |f1> |h> |f1> |a> (Ω,Δ) |f2> Λal ll –ζe |l> energy of2|l>Λis bg algdetermined =a lg-2π by El(Ω,Δ) D-i2π2χ1/2Λ 2iη r:inter-atomic distance aa D=-El(Ω,Δ)+Ec(B)+Re(Фres|W+Gbg W|Фres) D: control Re[alg] through (Ω,Δ) Λal and Λaa: the loss or Im[alg] scattering length of the dressed states can be controlled by the singleatom coupling parameters (Ω,Δ) under a fixed magnetic field The independent control of two scattering lengths: method I Step 2:control alg with our trick Step 1: control adg Magentic Feshbach resonance, and fix B |g> alg adg |h> |f1> |l> (Ω,Δ) |f2> |d> adl condition: two close magnetic Feshbach resonances for |d>|g> and |f2>|g> |l> The independent control of two scattering lengths: 40K–6Li mixture hyperfine levels of 40K and 6Li 6Li |g> alg F=3/2 40K |l> E E adg adl 40K 1/2 6Li |d> Efimov states of two heavy and one light atom? B B |f1>=|40K3> |f2>=|40K2> |g>=|6Li1> |d>=|40K1> } { (Ω, Δ) |h> |l> |g>|d> magnetic Feshbach resonance: |g>|d>: B=157.6G |g>|f2>: B=159.5G E. Wille et. al., Phys. Rev. Lett. 100, 053201 (2008). no hyperfine relaxation B(10G) |g>|f2> The independent control of two scattering lengths: 40K–6Li mixture numerical illustration: square-well model |c> |c> |f1>|g> |f2>|g> -Vc |f2>|g> W |f1>|g> -V2 |Φres) -V1 0 a A. D. Lange et. al., Phys. Rev. A 79 013622 (2009) •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) Ω=40MHz The independent control of two scattering lengths: method II |h’> |f’1> (Ω’,Δ’) |l’> |f’2> |f1> |h> al’g (Ω,Δ) |f2> |g> alg |l’> |l> |g> |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 adl The independent control of two scattering lengths: 40K gas |f’1>=|40K17> (Ω,Δ) |f’1>=|40K4> B |f’2>=|40K3> |f2>=|40K2> } { (Ω’, Δ’) |h’> |l’> |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). { |h> |l> The independent control of two scattering lengths: 40K gas hyperfine relaxation |9/2,7/2>| 9/2,5/2> |9/2,9/2>| 9/2,3/2> •The source of the relaxation: unstable |f1>|g> and |f’1>|g> hyperfine channels •In our simulation, we take the background hyperfine relaxation rate to be 10-14cm3/s B. DeMarco, Ph.D. thesis, University of Colorado, 2001. results given by square-well model al’g(a0) Ω’=2MHz Ω=2MHz Δ’(MHz) Another approach: Light induced shift of Feshbach resonance point excited channel : l1S>|2P> |Φ2> Δ Ω |Φ1> W1 U :laser close channel : ground hyperfine level 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 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