Selective and efficient control of coherent Selective and efficient control of coherent populationpopulation

transfer with time-separated chirped pulsestransfer with time-separated chirped pulses

Xihua Yang (Xihua Yang ( 杨希华杨希华 ))

Department of Physics, Shanghai University, Shanghai 200444, P. R. ChinaDepartment of Physics, Shanghai University, Shanghai 200444, P. R. China

2010.05.032010.05.03

I) Background

II) Our work

Selective and efficient control of coherent population

transfer with time-separated chirped pulses

III) Conclusions

Outline

Coherently controlling population transfer from an initial state to a target state and creating an arbitrary coherent superposition between two states has attracted considerable interest in recent years.

Three main methods have been proposed and employed:

1) Stimulated Raman adiabatic passage (STIRAP) ;

2) Chirped adiabatic passage (CHIRAP) ;

3) Temporal coherent control (TCC)

I) Background

The simplest and most extensively-studied model is the -type three-level system.

p

2

S

31

STIRAP has proven to be an efficient and robust technique for selective and complete coherent population transfer between two discrete atomic or molecular states.

(K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys.70, 1003 (1998))( N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, Annu. Rev. Phys. Chem. 52, 763 (2001)) (X.H. Yang and S.Y. Zhu, Phys. Rev. A 77, 063822 (2008); Phys. Rev. A 78, 023818 (2008)).

S

p

Fig. 1. STIRAP technique.

CHIRAP employs rapid adiabatic passage to efficiently and selectively transfer population from one state to another with chirped pulse.

Loy and Grischkowsky first studied the population inversion by using rapid adiabatic passage technique in the optical domain.

Broers showed that efficient population transfer by two-photon absorption can be achieved in a Rb three-level ladder system with a short chirped pulse.

Chang implemented CHIRAP to selectively and robustly transfer population in a -type four-level system.(M.M.T. Loy, Phys. Rev. Lett. 32, 814 (1974); D. Grischkowsky, and M.M.T. Loy, Phys. Rev. A 12, 1117 (1975); M.M.T. Loy, Phys. Rev. Lett. 41, 473 (1978))

(B. Broers, H. B. van Linden van den Heuvell and L. D. Noordam, Phys. Rev. Lett. 69, 2062 (1992)).

(B. Y. Chang, I. R. Solá, V. S. Malinovsky, and J. Santamaria, Phys. Rev. A 64, 033420 (2001))

TCC uses temporal quantum interference to achieve flexible

and selective control of population transfer.

As experimentally demonstrated by Blanchet, the selective

excitation of the 7D doublet of Cs atom can be realized via

TCC by using a sequence of two identical ultrafast laser pulses. ( R.N. Zare, Science 279, 1875 (1998); S. Chu, Nature 416, 206 (2002))

(M.M. Salour, Rev. Mod. Phys. 50, 667 (1978))

(V. Blanchet, et al., Phys. Rev. Lett. 78, 2716 (1997))

(X.H. Yang, Z.R. Sun, Z.G. Wang, Phys. Rev. A 76, 043417 (2007))

Apart from exerting one of the above-mentioned methods, the combination of two of them has also been exploited.

Band presented a selective and efficient scheme for coherent control of population transfer in a -type or ladder-type four-level system by merging STIRAP and CHIRAP techniques.

Netz developed a method for executing selective control of population transfer between quantum states by using two identical time-delayed frequency-chirped pulses.

Kral proposed a way of combining STIRAP and TCC, termed as coherently controlled adiabatic passage, to achieve both selectivity and completeness of population transfer. (Y. B. Band and O. Magnes, Phys. Rev. A 50, 584 (1994); Y. B. Band, Phys. Rev. A 50, 5046 (1994))

(R. Netz, A. Nazarkin, and R. Sauerbrey, Phys. Rev. Lett. 90, 063001 (2003))

(P. Kral, and M. Shapiro, Rev. Mod. Phys. 79, 53 (2007))

We propose a selective and efficient way to realize control of coherent population transfer in a -type four-level system with the final state consisting of a closely-separated doublet with time-separated chirped pump pulse and Stokes pulse pair.

By merging STIRAP, TCC, and CHIRAP techniques, perfect population transfer from the initial state to either of the final states can be selectively and robustly realized . Moreover, an arbitrary coherent superposition between the final doublet, or between the intermediate state and either of the doublet can be created.

In this talk, we will discuss the selective and efficient control of coherent population transfer with time-separated chirped pulses .

Fig. 2. The -type four-level system driven by time-separated chirped pump pulse and chirped Stokes pulse pair (shown in the right part) with the time delay τ (and Δτ) between the pump and the first Stokes pulse (and between the two Stokes pulses).

II) Our works

S2

p

S1

4

p

S

3

2

1

( X.H. Yang, et al., Phys. Rev. A 81, 035801 (2010))

We consider the case of linear chirped and temporally-delayed pump-Stokes laser pulses with the electric fields written as EP,S(t)=AP,Sexp(-jwP,St)+c.c., where wP,S and AP,S(t) are the central frequencies and the electric-field amplitudes of the pump and Stokes fields, respectively. The electric-field amplitudes AP,S(t) can be denoted as :

2 2 20( ) exp[ ( ) / 2 ( ) / 2].i i i i iA t E t t T j t t

So the Rabi frequencies of the pump and Stokes pulses for the transitions 1-2 and 2-3 (or 4) can be given by : 2 2 2

0( ) exp[ ( ) / 2 ( ) / 2]p p it t T j t 2 2 2

1 0 1( ) exp( / 2 / 2)S S it t T j t

2 2 22 0 2( ) exp( ( ) / 2 ( ) / 2)S S it t T j t

with the linear chirp rate , which represents a linear chirp with the instantaneous frequency .

i( ) ( )i i i it t t

In the rotating-wave approximation, the time-dependent Hamiltonian of the system can be written as

1

1 2

1 2

2 2 34

( ) 0 0

( ) 0 ( ) ( ).

0 ( ) 0

0 ( ) 0

p

p S S

S

S

t

t t tH

t

t

The time evolution of the system can be readily treated by resolving the time-dependent Schrodinger equation with the fourth-order Runge-Kutta integrator.

In what follows, we will consider how selective and efficient control of population transfer can be realized by varying the chirp rates, frequency detunings, and time delay between the Stokes pair.

0.0

0.2

0.4

0.6

0.8

1.0

-8 -4 0 4 80.0

0.2

0.4

0.6

0.8

1.0

-8 -4 0 4 8

(a)

(d)

(c)

(b)

P

opul

atio

ns

Time

Fig. 3. The time evolution of the populations in the four states 1 (dotted line), 2 (dotted-dashed line), 3 (dashed line), and 4 (solid line) with the central frequencies of the pump and Stokes laser fields tuned to resonance with the transitions 1-2 and 2-3 and the chirp rates of the pump and Stokes pulses being of the same sign under different time delay Δτ between the Stokes pair withτ=2, ΩP=2Ω0S1=2Ω0S2=40,w34=10, and w23=16000 , in corresponding units of T or T-1. (a) and (b) Δτ=0 , (c) and (d) Δτ=(1/2wS), (a) and (c) αP=αS=-5(nuits of T-2), and (b) and (d) α

P=αS=5 （ units of T-2).

-40 -20 0 20 40

0.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

Pop

ualti

ons

(c)

(b)

(a)

Fig. 4. The final populations in the four states 1 (dotted line), 2 (dotted-dashed line), 3 (dashed line), and 4 (solid line) as a function of the chirp rates α=αP=αS of the pump and Stokes lasers with Δτ=0, (a) wS=(w23+w24)/2, (b) wS=w23, and (c) wS=w2

4, and the other parameters are the same as those in Fig. 3.

-40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

P

opul

atio

ns

(b)

(a)

Fig. 5. The final populations in the four states 1 (dotted line), 2 (dotted-dashed line), 3 (dashed line), and 4 (solid line) as a function of the detuning Δ2 of the Stokes field with Δτ=0, (a) αP=αS=-5, (b) αP=αS=5, and the other parameters are the same as those in Fig. 3.

-0.00008 -0.00004 0.00000 0.00004 0.000080.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

Pop

ualti

ons

(b)

(a)

Fig. 6. The final populations in the four states 1 (dotted line), 2 (dotted-dashed line), 3 (dashed line), and 4 (solid line) as a function of the time delay Δτ between the Stokes pulse pair, (a) αP=αS=5 , (b) αP=αS=-5, and the other parameters are the same as those in Fig. 3.

III) Conclusions:III) Conclusions:We proposed a selective, efficient, and robust way to realize control of

coherent population transfer in a -type four-level scheme by merging STIRAP, TCC, and CHIRAP techniques.

An arbitrary coherent superposition between the intermediate state and either of the final doublet, or between the final doublet can be created by varying the pulses chirping rates, or by tuning the time delay or one-photon detuning of the Stokes pulses .

This method combines the properties of efficiency and robustness of STIRAP, selectivity and robustness of CHIRAP, and flexibility and selectivity of TCC, and holds the ability to “control with control”, which has potential applications in coherent control of chemical reactions and quantum information processing.

Thank you !Thank you !