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**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

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**Collaborators RUC: Wei Zhang Tao Yin Ren Zhang Chuan-zhou Zhu**

Other institutes: Pascal Naidon Mashihito Ueda Chang-pu Sun Yong Li

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

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**Efimov state: universal 3-body bound state**

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

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**Mixed dimensional system**

B B D(xA,xB) D(xA,xB) A A G. Lamporesi, et. al., PRL 104, (2010) D(xA,xB)→0 scattering length in mixed dimensiton aeff (l ,a) Y. Nishida and S. Tan, Phys. Rev. Lett. 101, (2008)

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**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, (R) (2010) rB Our motivation: to investigate the many-body bound state with mB <<m1 , m2 via Born-Oppenheimer approach Advantage: clear picture given by the A1–A2 interaction induced by B light atom B: 3D heavy atom A1 , A2 : 1D z1 a1 a2 z2 step1: wave function of B BP boundary condition Veff: effective interaction between A1, A2 -E: binding energy step2: wave function of A1, A2 3-body bound state: T. Yin, Wei Zhang and Peng Zhang arXiv:

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**1D-1D-3D system: a1=a2=a rB z1 z2 a1 a2 Effective potential**

z1–z2 (L) L Veff (regularized) Effective potential L/a Potential depth Binding energy L/a new “resonance”condition: a=L

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**1D-1D-3D system: arbitrary a1 and a2**

3-body binding energy L/a1 L/a2 rB z1 z2 a1 a2 resonance occurs when a1=a2=L non-trivial bound states (a1<0 or a2<0) exists

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**2D-2D-3D system a2 a1 3-body binding energy**

L/a2 L/a2 resonance occurs when a1=a2=L L/a1 L/a1

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**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:

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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 a1 a2 a3 a1=a2=a3=L Veff (regularized ) /L /L

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**4-body bound state: 1D-1D-1D-3D**

/L Depth of 4-body potential Binding energy of 4-body bound state /L a1=a2=a3=L /L resonance condition: L1=L2=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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**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, (2010)) The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv: ) The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, (2009))

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**s-wave Feshbach resonance: p-wave Feshbach resonance:**

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, (2003); Kenneth GÄunter, et.al., Phys. Rev. Lett. 95, (2005); C. Ticknor, et.al., Phys. Rev. A 69, (2004). C. A. Regal, et. al., Nature 424, 47 (2003). J. P. Gaebler, et. al., Phys. Rev. Lett. 98, (2007). 6Li: J. Zhang,et. al., Phys. Rev. A 70, (R)(2004) . C. H. Schunck, et. al., Phys. Rev. A 71, (2005). J. Fuchs, et.al., Phys. Rev. A 77, (2008). Y. Inada, Phys. Rev. Lett. 101, (2008). theory: F. Chevy, et.al., Phys. Rev. A, 71, (2005) p-wave BEC-BCS cross over T.-L. Ho and R. B. Diener, Phys. Rev. Lett. 94, (2005).

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**(effective-range theory) Van der Waals potential**

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) Van der Waals potential (V(r) ∝ r--6 ) s-wave (k→0) p-wave (k→0) Can we use effective range theory for van der Waals potential in p-wave case?

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**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

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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<B0; V, R have the same sign) c) kF :Fermi momentum the neglect of S (BCS side, B>B0; V, R have different signs)

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**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.

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**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

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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

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**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, (2010)) The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv: ) The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, (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) 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, (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) measurement |e> |g> heat bath two-level system decay rate survival probability without measurements With measurements n→∞: Rmea →0: Zeno effect “intermediate” n: Rmea > RGR : anti-Zeno effect 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 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

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**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

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**Long-time evolution: rate equation**

Coarse-Grained approximation: Re 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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**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, (2010)) The dynamical theory for quantum Zeno and anti-Zeno effects in open system (arXiv: ) The independent control of different scattering lengths in multi-component ultra-cold gas (PRL 103, (2009))

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**? Motivation: independent control of different scattering lengths**

two-component Fermi gas or single-component Bose gas a12 a32 a13 |1> |2> |3> Three-component Fermi gas,… Independent control of different scattering lengths a12 control of single scattering length We propose a method for the independent control of two scattering lengths in a three-component Fermi gas. ? Magnetic Feshbach resonance … 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

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**The control of a single scattering length with fixed B-field**

energy of |l> is determined by El(Ω,Δ) |f2> |f1> (Ω,Δ) |l> |h> |e>: excited electronic state g1 g2 D |c> |l>|g> |h>|g> |f1>|g> Ω Δ |a> HF relaxation |f2>|g> W |Φres) alg=abglg-2π2 Λll –ζe2iηΛal D-i2π2χ1/2Λaa 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] r:inter-atomic distance scattering length of the dressed states can be controlled by the single-atom coupling parameters (Ω,Δ) under a fixed magnetic field

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**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 alg adl adg |g> |l> |d> |f2> |f1> (Ω,Δ) |l> |h> condition: two close magnetic Feshbach resonances for |d>|g> and |f2>|g>

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**} { The independent control of two scattering lengths: 40K–6Li mixture**

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

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**The independent control of two scattering lengths: 40K–6Li mixture**

-V2 |f1>|g> -Vc -V1 |f2>|g> |c> A. D. Lange et. al., Phys. Rev. A (2009) numerical illustration: square-well model |f2>|g> |f1>|g> W |Φres) |c> 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

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**The independent control of two scattering lengths: method II**

(Ω,Δ) |l> |h> |f’2> |f’1> (Ω’,Δ’) |h’> |l’> al’g adl alg |g> |l’> |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

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**{ } { The independent control of two scattering lengths: 40K gas (Ω,Δ)**

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

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**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 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 B. DeMarco, Ph.D. thesis, University of Colorado, 2001. results given by square-well model Ω’=2MHz al’g(a0) Ω=2MHz Δ’(MHz)

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**Another approach: Light induced shift of Feshbach resonance point**

excited channel : l1S>|2P> |Φ2> U :laser Δ Ω close channel : ground hyperfine level |Φ1> W1 open channel a |1S>|1S> (incident channel): r Dominik M. Bauer, et. al., Phys. Rev. A, 79, (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

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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 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

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