I The fewbody problems in complicated ultracold atom
(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 (ar. Xiv: 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 (ar. Xiv: 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 Li 6 (a 12, a 23, a 31) (Max -Planck, 2009; University of Tokyo, 2010) trimer • … unstable 3 -body recombination V. Efimov, Phys. Lett. 33, 563 (1970)
Mixed dimensional system 1 D+3 D 2 D+3 D B B D(x. A, x. B) A A scattering length in mixed dimensiton D(x. A, x. B)→ 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: 3 D heavy atom A 1 , A 2 : 1 D r. B z 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 <<m 1 , m 2 via Born-Oppenheimer approach z 2 Advantage: clear picture given by the A 1–A 2 interaction induced by B BP boundary condition step 1: wave function of B step 2: wave function of A 1, A 2 3 -body bound state: Veff: effective interaction between A 1, A 2 -E: binding energy T. Yin, Wei Zhang and Peng Zhang ar. Xiv: 1104. 4352
1 D-1 D-3 D system: a 1=a 2=a r. B a 1 a 2 z 2 Veff (regularized) z 1 Effective potential L L L z 1–z 2 (L) Potential depth Binding energy new “resonance” condition: a=L L/a
1 D-1 D-3 D system: arbitrary a 1 and a 2 L/a 2 3 -body binding energy L/a 1 z 1 L/a 1 r. B a 1 a 2 • resonance occurs when a 1=a 2=L z 2 • non-trivial bound states (a 1<0 or a 2<0) exists
2 D-2 D-3 D system a 2 a 1 L/a 2 3 -body binding energy resonance occurs when a 1=a 2=L L/a 1
Validity of Born-Oppenheimer approximation 1 D-1 D-3 D 2 D-2 D-3 D L/a a 1=a 2=a exact solution: Y. Nishida and S. Tan, eprint-ar. Xiv: 1104. 2387 L/a
4 -body bound state: 1 D-1 D-1 D-3 D Light atom B can induce a 3 -body interaction for the 3 heavy atoms a 3 a 2 a 1=a 2=a 3=L Veff (regularized ) a 1 /L /L
4 -body bound state: 1 D-1 D-1 D-3 D Binding energy of 4 -body bound state /L /L Depth of 4 -body potential a 1=a 2=a 3=L /L resonance condition: L 1=L 2=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 (ar. Xiv: 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 (ar. Xiv: 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 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).
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?
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
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 the same sign) c) the neglect of S (BCS side, B>B 0; V, R have different signs) k. F : Fermi momentum
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.
The condition r 2<<1 and r 3<<1 • 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
Summary • 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 (Li 6 or K 40), 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 (ar. Xiv: 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 (ar. Xiv: 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> two-level system heat bath decay rate survival probability • without measurements • With measurements • 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 g. B : bare time-correlation function of heat bath g. A : 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 (ar. Xiv: 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 (ar. Xiv: 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> a 12 a 13 control of single scattering length |2> |3> a 32 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 |f 1>|g> g 1 Ω W |Φres) |f 2>|g> HF relaxation |e>: excited electronic state g 2 |f 2> |l>|g> |f 1> |h> |f 1> |a> (Ω, Δ) |f 2> Λal ll –ζe |l> energy of 2|l>Λis bg algdetermined =a lg-2π by E (Ω, Δ) l 2χ1/2Λ D-i 2π 2 iη 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 1: control adg Magentic Feshbach resonance, and fix B Step 2: control alg with our trick |g> alg adg |l> |h> |f 1> (Ω, Δ) |f 2> |d> adl condition: two close magnetic Feshbach resonances for |d>|g> and |f 2>|g> |l>
The independent control of two scattering lengths: 40 K– 6 Li mixture hyperfine levels of 40 K and 6 Li |g> alg F=3/2 40 K |l> E E adg adl 40 K 1/2 6 Li B |d> Efimov states of two heavy and one light atom? B |f 1>=|40 K 3> |f 2>=|40 K 2> |g>=|6 Li 1> |d>=|40 K 1> } { (Ω, Δ) |h> |l> |g>|d> magnetic Feshbach resonance: |g>|d>: B=157. 6 G |g>|f 2>: B=159. 5 G E. Wille et. al. , Phys. Rev. Lett. 100, 053201 (2008). no hyperfine relaxation B(10 G) |g>|f 2>
The independent control of two scattering lengths: 40 K– 6 Li mixture numerical illustration: square-well model |c> |f 1>|g> |f 2>|g> -Vc |f 2>|g> W |f 1>|g> -V 2 |Φres) -V 1 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, V 2 and V 1… are determined by the realistic scattering lengths of 40 K-6 Li mixture alg(a 0) Ω=40 MHz
The independent control of two scattering lengths: method II |h’> |f’ 1> (Ω’, Δ’) |l’> |f’ 2> |f 1> |h> al’g (Ω, Δ) |f 2> |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 |f 2>|g> and |f’ 2>|g> disadvantage: possible hyperfine relaxation adl
The independent control of two scattering lengths: 40 K |f’ 1>=|40 K 17> (Ω, Δ) B |f’ 1>=|40 K 4> |f’ 2>=|40 K 3> |f 2>=|40 K 2> } { (Ω’, Δ’) |h’> |l’> |g>=|40 K 1> magnetic Feshbach resonance: |g>|f 2>: B=202. 1 G C. A. Regal, et. al. , Phys. Rev. Lett. 92, 083201 (2004). |g>|f’ 2>: B=224. 2 G C. A. Regal and D. S. Jin, Phys. Rev. Lett. 90, 230404 (2003). { |h> |l> gas
The independent control of two scattering lengths: 40 K 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 |f 1>|g> and |f’ 1>|g> • In our simulation, we take the background hyperfine relaxation rate to be 10 -14 cm 3/s B. De. Marco, Ph. D. thesis, University of Colorado, 2001. results given by square-well model al’g(a 0) Ω’=2 MHz Ω=2 MHz Δ’(MHz) gas
Another approach: Light induced shift of Feshbach resonance point excited channel : l 1 S>|2 P> |Φ 2> Δ Ω |Φ 1> W 1 U : laser close channel : ground hyperfine level open channel a |1 S> (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 |Φ 1> via laser-induced coupling between |Φ 1> and |Φ 2> • The Feshbach resonance point can be shifted for 10 -1 Gauss-101 Gauss • Extra loss can be induced by the spontaneous decay of |Φ 2> • 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 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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