## 1. Introduction

Quantum coherence and interference (Scully & Zubairy, 1997) are leading edge topics in quantum optics and laser physics, and have led to many important novel effects, such as coherent population trapping (CPT) (Arimondo & Orriols, 1976, Alzetta et al., 1976, Gray et al., 1978), lasing without inversion (LWI) (Harris, 1989, Scully et al., 1989, Padmabandu et al., 1996), electromagnetically induced transparency (EIT) (Boller et al., 1991, Harris, 1997, Ham et al., 1997, Phillips et al., 2003, Fleischhauer et al., 2005, Marangos, 1998), high refractive index without absorption (Scully, 1991, Scully & Zhu 1992, Harris et al., 1990), giant Kerr nonlinear effect (Schmidt & Imamoglu, 1996), and so on. In particular, EIT, which can dramatically modify the absorption and dispersion characteristics of an optical medium, plays an important role in quantum optics. In the last two decades, EIT has attracted great attention and has been successfully applied to ultraslow and stopping light (Kasapi et al., 1995, Hau et al., 1999, Kocharovskaya et al., 2001, Liu et al., 2001, Turukhin et al., 2002), quantum switching (Ham & Hemmer, 2000), quantum memory (Ham et al., 1997; ibid., 1998), quantum entanglement generation (Lukin & Imamoglu, 2000), and quantum computing (Lukin & Imamoglu, 2001).

It is well established that light is the fastest information carrier in nature. However, controlling light for localized application is very difficult. Thus, manipulation of light velocity becomes a crucial task in optical and quantum information processing (Nielsen & Chuang, 2000). Recently light localization using EIT has been demonstrated for stationary light (Bajcsy et al., 2003). Stationary light gives novel effects to nonlinear quantum optics in the context of lengthening light-matter interaction time. Compared with ultraslow light, where the medium’s length is a limiting factor, stationary light is free from spatial constraint. For example, the interaction time using ultraslow light in a semiconductor quantum dot, whose spatial dimension is less than a few tens of nanometers, is much less than nanosecond. By using a stationary light technique, however, we can enormously increase the interaction time of the light with such a nano optical medium. In this chapter, we discuss stationary light based on the EIT-induced ultraslow light phenomenon. We theoretically investigate how to dynamically manipulate multicolor (MC) stationary light in the multi double lambda-type system by simply changing the parameters of control fields, and demonstrate ultralong trapping of light, which is different from the conventional quantum mapping phenomenon. Quantum coherent control of the stationary light has potential applications to various quantum optical processing such as quantum nondemolition measurement and quantum wavelength conversion.

This chapter is organized as follows. In section 2, quantum coherent control of two-color stationary light is described. In section 3, quantum manipulation of MC stationary light is presented. In section 4, we give the results of MC stationary light, with discussions. Finally, section 5 offers conclusions.

## 2. Quantum coherent control of light

### 2.1. Model and theory

In this section, we present dynamic control of two-color stationary light in a double-lambda type four-level system using EIT. Figure 1 shows the energy level diagram. We assume that initially one weak probe quantum field _{+} which couples the transition |1> |3> enters the medium in a forward propagation direction k_{+}, where all the atoms are in the ground state |1>. Under the influence of classical control field _{}, which is resonant with the transition |2> |3> and propagates in the same direction _{}, the probe field propagates at an ultraslow group velocity. After the probe pulse enters the medium completely, we adiabatically switch on the second control field _{}, which is resonant with the transition |2> |4> and propagates in the backward direction _{}. The probe pulse will be almost stationary if the stationary light condition is satisfied. The propagation direction k_{}of the field _{-} is determined by phase matching with the Bragg condition: k_{}= k_{+}_{} +_{}. Here _{}and _{} are the Rabi frequencies of two control fields. Below we will study the evolution of the stationary light when (1) _{}is switched off after a certain time while keeping the control filed _{} on, and (2) _{}is switched off after a certain time while keeping the control filed _{} on (see Figure 2).

For theoretical analytical purposes, we introduce a quantum field

where

Under the typical adiabatic condition for the slow light propagations (Fleischhauer & Lukin, 2000, Zibrov et al., 2002), we introduce new field operators

where

Using spatial Fourier transformation

where

where * l* of

_{+}and E

_{-}can be determined by Eq. (6).

Now we study temporal dynamics of the two-color coupled fields. Initially the control field _{+} (_{-}=0) is turn on and a probe pulse A_{+} with a Gaussian shape (* T* are the amplitude and temporal duration of the field) enters the medium. The solution of Eq. (4) describes the propagation of the field

_{+}is transparent to the optically dense medium (

From Eq. (7), the stationary light condition (

Thus the group velocity of the coupled light can be easily controlled by manipulating the control field Rabi frequencies. If the stationary light condition is satisfied, the two coupled lights can be completely stopped. Ignoring small terms proportional to

where

### 2.2. Numerical simulations and discussions

Here we numerically demonstrate quantum manipulation of a traveling light pulse for the two-color stationary light by solving Eqs. (2) and (3). For simplicity we ignore the weak decay rate

Compared with the standing-wave grating case (Bajcsy et al., 2003, Andre et al., 2005), the group velocity _{+} and A_{-} are strongly coupled and move or stay together with the following amplitude ratio:_{-} can be generated by the frequency shift _{+} is tuned to

At time

Figure 3 shows numerical simulation of the two-color stationary light for case (a) mentioned above (also see Figure 2(1)). For simplification, we set

frequencies for the control fields. Figure 3(c) and Figure 3(d) show the top view of the fields A_{+} and A_{-} propagation, respectively. Figure 3(e) and Figure 3(f) show the temporal evolutions of the field amplitudes A_{+} and A_{-}, respectively. Figure 3(g) and Figure 3(h) show space-time evolution of the fields A_{+} and A_{-}, respectively. These figures show that when the control pulse _{-} disappears completely, and the original field A_{+} keeps propagation moving in the same direction at the same group velocity as it had for

Figure 4 shows numerical simulation of the two-color stationary light for case (b) (see Figure 2(2)). Similar to Figure 3, Figure 4(c) and Figure 4(d) show the top view of the coupled fields. Figure 4(e) and Figure 4(f) show the temporal evolutions of the coupled field amplitudes. Figure 4(g) and Figure 4(h) show space-time evolution of the coupled fields. As seen in Figure 4(c) and 4(d), when _{+} disappears at_{-} propagates in the backward direction with new carrier frequency_{+} and E_{-} can be manipulated for stationary light or frequency conversion, which has potential application for quantum nonlinear optics which needs a longer interaction time. Here we note that the maximum trapping time of stationary light is determined by the coherence decay rate

In summary, we have demonstrated two-color stationary light and quantum wavelength conversion using a quantum-mechanically reversible process between photons and atomic coherence in a double-

## 3. Quantum manipulation of MC stationary light

If we choose a multilevel system, MC stationary light should be possible. In this section, we show quantum coherent control of multiple travelling light pulses in an optically dense medium by generalizing the approach we used in the previous section. Figure 5 shows an energy level diagram of the present MC stationary light. The control fields with Rabi frequencies

We assume that initially all the atoms stay in the ground state

atoms are driven by one classical control field _{+}+M_{-} control fields.

For a theoretical analysis of MC stationary light based on the multi double lambda-type scheme in Figure 5, we introduce quantum fields* m* and

*are related to the signs*n

where

Using Eq. (12) and adding the relaxation constants and Langevin forces associated with the atomic relaxation processes, we derive the Heisenberg equations for the atomic operators

where

In Eqs. (13)-(15), we ignore the influences of the weak populations of excited states

where

We assume a typical adiabatic condition for slow light propagation

where the fluctuation forces are:

Eqs. (21) and (22) can be solved by using spatial Fourier transformation. Assuming that initial field

where

## 4. Results and discussion

In this section, we analyze quantum evolution of the MC field. Under the slow light condition and taking into account weak relaxation processes between the two ground levels

where

### 4.1. Traveling MC field

Ifwhere the spatial dispersion will be

### 4.2. Stationary MC field

Eg. (26) shows that all the laser fields propagate and evolve together with one group velocity

This is the MC stationary light condition. Obviously, this relation generalizes the results obtained for two-color stationary light. The amplitudes of these coupled fields can be manipulated by varying the Rabi frequencies of the corresponding control fields:

For a Gaussian shape of the initial input probe pulse, we have the following amplitudes

Eq. (32) shows that the amplitude

The maximum MC field stationary time is determined by relaxation constant

Using the conditions

by satisfying the particular relationship to the initial probe field

From Eqs. (34) and (35), we can know that the condition for minimum spreading is independent of the total number of the control fields M_{+} and M_{-}. Putting Eq. (31) into Eq. (27), we find the important relation for the minimum spatial spreading of the MC stationary light pulses:

### 4.3. MC-wavelength conversion

After optical trapping of the initial probe pulse by using MC stationary light, it is possible to generate an arbitrary forward traveling light field

The electric field amplitude of the m-th component in the MC light at time

with the temporal duration

## 5. Conclusion

We first demonstrated the two-color stationary light and quantum wavelength conversion using quantum coherence resulting from strongly coupled slow light through EIT in a double-lambda system. Then we generalized the approach to the MC light fields in the multi double Λ coherent atomic medium driven by the M_{+}+M_{-} control intensive laser fields and showed how to manipulate the MC light field within the adiabatic limit. The results show that the MC light fields can be controlled by simply adjusting the control fields’ parameters for (1) MC stationary light, (2) selection of propagation direction (forward or backward), and (3) MC wavelength conversion. The maximum stationary time and minimum spatial spreading of the MC field have also been discussed. On-demand quantum manipulation of the MC light field can greatly increase the interaction time of the light and medium, and holds promise for applications in optical buffer, controllable switching, and quantum optical information processing.

We acknowledge that this work was supported by the CRI program (Center for Photon Information Processing) of the Korean Ministry of Education, Science and Technology via National Research Foundation.

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