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

(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

RUC:

Wei Zhang Tao YinRen ZhangChuan-zhou Zhu

Collaborators

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 lengths in multi-

component ultra-cold gas (PRL 103, 133202 (2009))

Efimov state: universal 3-body bound state

identical bosons

1/a

k = sgn(E)√E

dimer

3 particles

trimer

trimer

• scattering length a

• 3-body parameter Λ

characteristic parameters:

• Cesium 133 (Innsbruck, 2006)

• 3-component Li6 (a12, a23, a31)

(Max-Planck, 2009; University of

Tokyo, 2010)

• …

experimental observation:

V. Efimov, Phys. Lett. 33, 563 (1970)3-body recombination

unstable

aeff (l ,a)

Mixed dimensional system

Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008)

1D+3D 2D+3D

A

B B

A

G. Lamporesi, et. al., PRL 104, 153202 (2010)

D(xA,xB) D(xA,xB)

D(xA,xB)→0scattering length in mixed dimensiton

Stable many-body bound state

-E: binding energy

step2: wave function of A1, A2

3-body bound state:

step1: wave function of BBP boundary condition

Veff: effective interaction between A1, A2

stable 3-body bound state:no 3-body recombinationEverything described by a1 and a2Y. 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

Advantage: clear picture given by the A1–A2 interaction induced by B

light atom B: 3Dheavy atom A1 , A2 : 1D

T. Yin, Wei Zhang and Peng Zhang arXiv:1104.4352

rBz1

z2

a1 a2

z1–z2 (L)

LL

L Vef

f (re

gula

rized

)

Effective potential

Binding energy

L/a

1D-1D-3D system: a1=a2=a

new “resonance”condition: a=L

L/a

Potential depth

rBz1

z2

a1 a2

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

L/a1

L/a 2

L/a1

L/a 2

• resonance occurs when a1=a2=L

• non-trivial bound states (a1<0 or a2<0) exists

3-body binding energy

rBz1

z2

a1 a2

2D-2D-3D system

L/a1

L/a 2

L/a1

L/a 2

3-body binding energy

resonance occurs when a1=a2=L

a2

a1

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

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

Light atom B can induce a 3-body interaction for the 3 heavy atoms

/L /L

a1=a2=a3=L

Vef

f (re

gula

rized

)

a1 a2

a3

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

/L

/L

Depth of 4-body potential

/L

/L

Binding energy of 4-body bound state

resonance condition: L1=L2=L

a1=a2=a3=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.


(arXiv:1104.4352 )


A 82, 062712 (2010))





p-wave magnetic Feshbach resonance

p-wave Feshbach resonance:

Bose gas and two-component Fermi gas

s-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 overT.-L. Ho and R. B. Diener, Phys. Rev. Lett. 94, 090402 (2005).

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)2

0 11)(

kra

ikkf

eff

2

0 11)(

kra

ikkf

eff

Can we use effective range theory for van der Waals potential in p-wave case?

• p-wave (k→0)

RVkik

kf 111)(

2

1

RSkVkik

kf111

1)(2

1

Short-range potential (e.g. square well, Yukawa potential): effective-range theoryLong-rang potential (e.g. Van der Waals, dipole…): be careful!!

Low-energy scattering amplitude:

Long-range effect of p-wave magnetic Feshbach resonance

• two channel Hamiltonian

• back ground scattering amplitude

(bg)(bg)2(bg)

(bg)

1 1111)(

RkSkVik

kf

: background Jost function

• scattering amplitude in open channel

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) 1 the neglect of S (BEC side, B<B0; V, R have the same sign)

1

c) 1

1the 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.

• 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

3

6),( BkV eff

1

6

Fk

The condition r2<<1 and r3<<1

length der WaalsVan :6

• 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.

Summary

Long-range effect from open channel

Short-range effect from open channel


(arXiv:1104.4352 )


A 82, 062712 (2010))





≈

Quantum Zeno effect: close system

Proof based on wave packet collapseMisra, Sudarshan, J. Math. Phys. (N. Y.) 18, 756 (1977)

measurement

t: total evolution timeτ: measurement periodn:number of measurements

t

general dynamical theoryD. Z. Xu, Qing Ai, and C. P. Sun, Phys. Rev. A 83, 022107 (2011)

Quantum Zeno and anti-Zeno effect: open systemProof based on wave packet collapseA. G. Kofman & G. Kurizki, Nature, 405, 546 (2000)

|e>

|g> heat bathtwo-level system

measurement

• n→∞: Rmea →0: Zeno effect

• “intermediate” n: Rmea > RGR : anti-Zeno effect

survival probability expn

GR eP t P Rt

decay rate

• without measurements

• With measurements

22 | |GR egk kk

R g

general dynamical theory?

2 2sinc2

| | eg kmea k

k

tR gn

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

• decay rate

R=

γ=0: R=Rmea (return to the result given by wave-function collapse)γ=1: phase modulation pulses

• survival probability

• finial state

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.


(arXiv:1104.4352 )


A 82, 062712 (2010))





Motivation: independent control of different scattering lengths

two-component Fermi gasor single-component Bose gas

a12

control of single scattering length

• Magnetic Feshbach resonance• … ?

a12

a32

a13

|1>

|2>

|3>

Three-component Fermi gas,…

Independent control of

different scattering lengths

Efimov statesnew superfluid…

BEC-BCS crossoverstrong 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

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

r:inter-atomic distance

|f2>|g>W|Φres)

|c>

|a>HF relaxation

|f1>|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 El(Ω,Δ)

|f2>

|f1>

(Ω,Δ)

|l>

|h>|f1>

|f2>

|e>: excitedelectronic state

g1 g2

D

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]

The independent control of two scattering lengths: method I

alg

adl

adg

|g>

|l>

|d>

|f2>

|f1>

(Ω,Δ)

|l>

|h>

Step 1: control adg Magentic Feshbach resonance, and fix B

Step 2:control alg with our trick

condition:two close magnetic Feshbach resonances for |d>|g> and |f2>|g>

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

B(10G)

B

hyperfine levels of 40K and 6Li

6Li

F=3/2

1/2

B

E E

|g>=|6Li1>|d>=|40K1>

|f2>=|40K2>

|f1>=|40K3>

alg

adl

adg

6Li |g>

40K |l>

40K |d>

magnetic Feshbach resonance:|g>|d>: B=157.6G|g>|f2>: 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>|f2>

|f2>|g>

|f1>|g>

W|Φres)

|c>

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

numerical illustration: square-well model

a0

-V2

|f1>|g>-Vc

-V1

|f2>|g>

|c>

A. D. Lange et. al., Phys. Rev. A 79 013622 (2009)

alg(a0)

• 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

Ω=40MHz

The independent control of two scattering lengths: method II

|g>

|f2>

|f1>

(Ω,Δ)

|l>

|h>

|f’2>

|f’1>

(Ω’,Δ’)

|h’>

|l’>

alg : controlled by the coupling parameters (Ω,Δ)

al’g :controlled by the coupling parameters (Ω’,Δ’)

al’g

adl

alg

|g>

|l’>

|l>

condition:two close magnetic Feshbach resonances for |f2>|g> and |f’2>|g>

disadvantage:possible hyperfine relaxation

(Ω,Δ){|h>

|l>

The independent control of two scattering lengths: 40K gas

B

|g>=|40K1>

|f2>=|40K2>|f’2>=|40K3>

|f’1>=|40K4>

|f’1>=|40K17>

}(Ω’, Δ’) {|h’>

|l’>

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).

The independent control of two scattering lengths: 40K 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 |f1>|g> and |f’1>|g>

• In our simulation, we take the background hyperfine relaxation rate to be 10-14cm3/s

Ω’=2MHz

Δ’(MHz)

al’g(a0)

Ω=2MHz

Another approach: Light induced shift of Feshbach resonance point

r

|Φ1>

open channel a |1S>|1S>(incident channel):

W1

excited channel : l1S>|2P>|Φ2>

U :laser

Δ

Ωclose channel : ground hyperfine level

• 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

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

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

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

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