Non-exponential decay of Feshbach molecules Saverio Pascazio - - PowerPoint PPT Presentation

Non-exponential decay of Feshbach molecules

CUAS, Padova, 27 September 2013

Saverio Pascazio Dipartimento di Fisica and INFN Bari, Italy

in collaboration with P . Facchi and F . Pepe

preliminaries: the survival probability

• f a decaying system

(Zeno) (exponential) (power)

survival probability time 1 1 − t2 τ 2

Z

Ze−γt t−α

in QM and QFT

P(t) = Z exp (−γt) + additional contributions

survival probability wave function renormalization second order in coupling constant

exponential decay modified at short (Zeno effect) and long times (power law)

in QM and QFT

P(t) = Z exp (−γt) + additional contributions

survival probability wave function renormalization second order in coupling constant

exponential decay modified at short (Zeno effect) and long times (power law)

OBSERVE in EXPT?

von Neumann,1932 Beskow and Nilsson,1967 Khalfin 1968 Friedman 1972 Misra and Sudarshan, 1977

History

von Neumann,1932 Beskow and Nilsson,1967 Khalfin 1968 Friedman 1972 Misra and Sudarshan, 1977

(main) Experiments

(Cook 1988) Itano, Heinzen, Bollinger, and Wineland 1990 Nagels, Hermans, and Chapovsky 1997 Wunderlich, Balzer, and Toschek, 2001 Fischer, Gutierrez-Medina, Raizen, 2001 Streed, Mun, Boyd, Campbell, Medley, Ketterle, Pritchard, 2006 Bernu, Sayrin, Kuhr, Dotsenko, Brune, Raimond, Haroche 2008

History

von Neumann,1932 Beskow and Nilsson,1967 Khalfin 1968 Friedman 1972 Misra and Sudarshan, 1977

(main) Experiments

(Cook 1988) Itano, Heinzen, Bollinger, and Wineland 1990 Nagels, Hermans, and Chapovsky 1997 Wunderlich, Balzer, and Toschek, 2001 Fischer, Gutierrez-Medina, Raizen, 2001 Streed, Mun, Boyd, Campbell, Medley, Ketterle, Pritchard, 2006 Bernu, Sayrin, Kuhr, Dotsenko, Brune, Raimond, Haroche 2008

Theory and interesting Mathematics History

experiment on “unstable” system

Wilkinson, Bharucha, Fischer, Madison, Niu, Sundaram, and Raizen, Nature 1997

a more recent experiment

           

 

    

 

  





 

      

a more recent experiment

           

 

    

 

  





 

      

Lörch, Pepe, Lignier, Ciampini, Mannella, Morsch, Arimondo, Facchi, Florio, Pascazio and Wimberger, PRA 2012

0.5 0.6 0.8 1 0.7 0.9

P(t), PZ(t)

5 4 3 2 1

t / TB

0.2 0.3 0.4 0.5 0.6 0.8 1 0.7

P(t), PZ(t)

2.5 2.0 1.5 1.0 0.5 0.0

t / TB

(a) (b)

a more recent experiment

           

 

    

 

  





 

      

Lörch, Pepe, Lignier, Ciampini, Mannella, Morsch, Arimondo, Facchi, Florio, Pascazio and Wimberger, PRA 2012

0.5 0.6 0.8 1 0.7 0.9

P(t), PZ(t)

5 4 3 2 1

t / TB

0.2 0.3 0.4 0.5 0.6 0.8 1 0.7

P(t), PZ(t)

2.5 2.0 1.5 1.0 0.5 0.0

t / TB

(a) (b)

wfr Z: Facchi, Nakazato and P ., PRL 2001

Z Z

the ideas to be discussed today

stable vs unstable non exponential decay wave function renormalization BEC-BCS

H = H0 + HAM + HF H0 = X

p

X

σ=",#

p2 2mc†

p,σcp,σ +

X

q

✓ q2 4m + EB ◆ b†

qbq

HAM = X

K,p

⇣ G(p) b†

Kcp+K/2,#cp+K/2," + h.c.

⌘ HF = X

pp0q

U(p, p0)c†

p+q/2,"c† p+q/2,#cp+q/2,#cp+q/2,"

G(p) = hψM,K|Hint|K/2 + p ", K/2 p #i

Hamiltonian

H = H0 + HAM + HF H0 = X

p

X

σ=",#

p2 2mc†

p,σcp,σ +

X

q

✓ q2 4m + EB ◆ b†

qbq

HAM = X

K,p

⇣ G(p) b†

Kcp+K/2,#cp+K/2," + h.c.

⌘ HF = X

pp0q

U(p, p0)c†

p+q/2,"c† p+q/2,#cp+q/2,#cp+q/2,"

G(p) = hψM,K|Hint|K/2 + p ", K/2 p #i

Hamiltonian

|ψM,Ki = b†

K|0i

c† c†

|ψM,0i = b†

0|0i

|Ψ0i = (b†

0)N

p N! |0i !

initial state

focus on s-wave Feshbach resonance of

6Li (m ' 10−25 Kg) at B = 543.25 G

atom-molecule interaction

20 40 60 80 100 r @aoD

• 4000
• 2000

2000 4000 6000 8000

c HrL

rmax

survival amplitude and propagator self-energy function and spectral function

second Riemann sheet

I

pole

E ω II E

second Riemann sheet

I

pole

E ω II E

Re Epole Im Epole

B=Bres B=B1=Bres+2.64â10-5 G SIIIHEL SIIHEL

so that

I

pole

E ω II E

due to cut

lifetime

0.1 0.2 0.3 0.4 0.5 B-Bres @GD 0.5â105 1.0â105 1.5â105 2.0â105 2.5â105

g @s-1D

B1-Bres 1â10-4 2â103 4â103

wave function renormalization

0.2â10-2 0.4â10-2 0.6â10-2 0.8â10-2 1.0â10-2

B-Bres @GD

1â10-2 2â10-2 3â10-2

H»Z»L2-1

wave function renormalization

0.2â10-2 0.4â10-2 0.6â10-2 0.8â10-2 1.0â10-2

B-Bres @GD

1â10-2 2â10-2 3â10-2

H»Z»L2-1

!!

examples of survival probability

1â10-5 2â10-5 3â10-5 4â10-5 5â10-5

t

0.2 0.4 0.6 0.8 1.0

PHtL

2â10-5 4â10-5

t

• 0.03

0.01

PHtL-e-g t

0.5â10-4 1.0â10-4 1.5â10-4 2.0â10-4 2.5â10-4

t

0.2 0.4 0.6 0.8 1.0

PHtL

1â10-4 2â10-4

t

• 0.05
• 0.10

0.02

PHtL-e-g t

(a)B − Bres = 1.2 × 10−2 G (b)B − Bres = 9.2 × 10−4 G

examples of survival probability

1â10-5 2â10-5 3â10-5 4â10-5 5â10-5

t

0.2 0.4 0.6 0.8 1.0

PHtL

2â10-5 4â10-5

t

• 0.03

0.01

PHtL-e-g t

0.5â10-4 1.0â10-4 1.5â10-4 2.0â10-4 2.5â10-4

t

0.2 0.4 0.6 0.8 1.0

PHtL

1â10-4 2â10-4

t

• 0.05
• 0.10

0.02

PHtL-e-g t

(a)B − Bres = 1.2 × 10−2 G (b)B − Bres = 9.2 × 10−4 G

notice how deviations from exp are enhanced for B close to resonance

1â10-4 2â10-4 3â10-4 4â10-4

t

0.5 1.0 1.5 2.0

@logHPHtLLD2

finally: a surprise and open questions

1â10-4 2â10-4 3â10-4 4â10-4

t

0.5 1.0 1.5 2.0

@logHPHtLLD2

finally: a surprise and open questions

Re Epole Im Epole

B=Bres B=B1=Bres+2.64â10-5 G SIIIHEL SIIHEL