1. 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

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

3. 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))

4. 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)

5. Mixed dimensional systemB 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)

6. 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:

7. 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

8. 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

9. 2D-2D-3D system a2 a1 3-body binding energy
L/a2
L/a2
resonance occurs when a1=a2=L
L/a1
L/a1

10. 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:

11. 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

12. 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

13. 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.

14. 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))

15. 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).

16. (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?

17. 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

18. 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)

19. 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.

20. 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

21. 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

22. 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))

23. 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)

24. 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?

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

26. 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

27. 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

28. Long-time evolution: rate equation
Coarse-Grained approximation:
Re CG : short-time result
steady-state population:

29. 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.

30. 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))

31. ? 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

32. 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

33. 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>

34. } { 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

35. 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

36. 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

37. { } { 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).

38. 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)

39. 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

40. 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

41. Thank you!