Transcript 幻灯片 1
(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
2
Vk
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
| g k | 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