Nonlinear optical spectroscopy of single excitonic states

Key words: ultrafast spectroscopy, semiconductor nanostructures, excitons, quantum computing, coherent control, four-wave mixing, heterodyne spectral interferometry, photon echo, Rabi oscillation

Motivation - Quantum computing

Quantum computers [1, 2] will be able to perform calculations that classical computers will never be able to do. The basic elements of quantum computer are Qbits that, as ordinary bits, have two states. However, in contrast to ordinary bits, qbits can exist in quantum superposition of two states. The state of a Qbit can be conveniently represented as a Bloch vector on the Bloch sphere (see Fig. 1). In order to operate classical bits one needs to switch between 0 and 1 state on demand. On the other hand, quantum computing requires the ability to drive the qbits from any state to any other in controlled manner. Furthermore, in order to perform more advanced operations one must be able to couple different qbits. Basic quantum computation operations have now been demonstrated for variety of systems [2].  In particular, solid state qbits can be realized by using localized excitons in semiconductor nanostructures..


Figure 1 - The bit is an unit of classical information. It can be only in two discrete states. The Qbit is an unit of quantum information. It can exist in a superposition of two states. Qbits are usually represented as a vector on the Bloch sphere.

Quantum dots and localized excitons

Excitons in semiconductors are electron-hole pairs bound by Coulomb attraction. By band gap engineering they can be spatially localized.  For instance, quantum well excitons are confined in the growth direction and move freely in plane of the structure. Yet, they can be trapped also in the plane of the quantum well, owing to the interface fluctuations [8]. As a result, a three-dimensional trapping potential is created. Spatially localized excitons can be naturally found in quantum dots. In these devices, a region of one material is surrounded by the other one, characterized by the larger bandgap. In this project both types of localized excitons are investigated.

Experimental technique

The influence of light pulses on the dynamics of localized excitons (treated as two-level system) is described by optical Bloch equations. In particular, the dynamics of the induced polarization in the ensemble is governed by the amplitude decay of the individual polarizations and the mutual phase coherence in the ensemble average. However, due to the presence of inhomognous broadening, the dephasing of individual transitions is difficult to be measured. Specifically, the third-order nonlinearity probed in four-wave mixing (FWM) can be used to determine homogeneous line-shapes of inhomogeneously broadened transitions. Since the size of localized excitonic states in semiconductors is typically much smaller than the wavelength of the resonant light, the light emitted by the polarization of an individual state propagates in all directions, so that the commonly used directional selection of the four-wave mixing signal is not effective.  In order to overcome this problem, a novel spectroscopic technique called heterodyne spectral interferometry has been developed [3, 4]. Conceptually it represents a combination of the previously used phase-stabilized two-dimensional femtosecond spectroscopy and phase-sensitive selection of the non-linear signal used in heterodyne-detected four-wave mixing or in phase-cycling detection. The scheme of the used experimental setup is presented in Fig. 2.

 

Figure 2 - Scheme of the experimental setup. Boxes: Acousto-optical modulators of the indicated frequency. MO: High numerical aperture (0.85) microscope objective, L1-L5 achromatic doublet lenses. Spectrometer: Imaging spectrometer of 15 µeV resolution. NF: near field of sample, FF: far-field of sample.


Recent advances

Setup and assessment of the technique of heterodyne-detected single quantum dot transient four-wave mixing (SQDFWM) [3]

Coherent control and polarization readout of individual excitonic states [5]

The related experiments were performed on excitons confined by interface fluctuations in an AlAs/GaAs/AlAs quantum well (QW) with a nominal GaAs thickness of 5 nm. Figure 3 (left bottom) shows a FWM spectrum from a small sample region which is dominated by a single localized exciton state. Coherent manipulation of the state vector is presented in figure 3 (right, top) . One can observe oscillations of the measured FWM intensity as a function of the pump pulse area, which due to Rabi oscillations of the state vector. The damping of the oscillations orginates from the transfer of the polarization towards near-resonant multiexcitonic states. Coherent control by the probe pulse on the excitonic transition  is presented in figure 3 (right, bottom). This result demonstrate the possiblity of placing state vector for the excitonic transition in any position of the Bloch sphere. The control of the polarization in amplitude and phase by detuning of the excitation pulse with respect to the transition frequency has been demonstrated as well [5].


Figure 3 - Coherent control and polarization readout of individual excitonic states


 Photon echo formation in groups of individual excitonic transitions [4, 6]

For positive delay times the FWM signal shows photon echo. Due to the specific non-linearity probed at this frequency of the FWM, the FWM phase at a time "tau" after the arrival of the second pulse does not depend on the eigenfrequency of the two-level system, and thus a macroscopic polarization is created even in presence of an ensemble of transitions with different transition energies (inhomogeneous broadening). This mechanism is illustrated in the left part of Fig. 4. In a representative measurement, shown in the right part of this figure the formation of the echo in a finite ensemble is observed as intensity enhancement at the time of the echo. At this time the FWM fields from all two-level systems are in phase, and therefore interfere constructively. This results in a signal amplitude N times larger than individual FWM amplitude of a single transition. In the limit of a large number of systems in the ensemble, the signal at t = "tau" is thus far larger than at other times, and is called a photon echo. The formation of the photon echo with increasing number of excitonic transitions is presented in Fig. 5. Photon echos originating from a single transition was also measured with heterodyne-spectral interferometry technique [4, 7].  This is possible due to spectral wandering of the individual transition.



Figure 4 - Left: Scheme of the echo formation in transient FWM. Right: Measured time-resolved FWM intensity from a group of localized exciton states. Excitation pulses 1,2 (black) and time-resolved FWM intensity (red line) for a delay time of "tau" = 20 ps are shown.


Figure 5 - Measured FWM intensity spectrally- (left-) and time-resolved (right side) for exciton state ensambles of different size.

Multidimensional spectroscopy in the optical frequency range [3, 4, 7]

In Fig. 6 we present delay-time dependent FWM spectra for a localized exciton system. The strong intensity modulation versus delay time of some peaks indicate the presence of a coherent coupling between them. Since we measure the signal in amplitude and phase for each delay time tau, we can not only look at the intensity beating, which is non-straightforward to interpret, but we can also Fourier-transform the delay-time dependence of the signal for tau > 0. The result of such a operation is presented in Fig. 6b. The presence of the off-diagonal peaks demonstrates the coherent coupling between different localized excitons.
 

Figure 6 - Two dimensional FWM measured with heterodyne spectral interferometry technique.


Present developments

  1. Demonstration of the coherent coupling between spatially separated localized excitons.
  2. Development of a high-resolution optical pulse shaper creating from an incoming laser pulse four seperate optical pulses, which are independently shaped in amplitude and phase on a femtosecond to picosecond time scale.
  3. Use of coherent optical control of the excitonic states by the shaped optical pulses to demonstrate simple quantum computational operations on coupled exciton states.

References:

  1. Charles et al. Nature  404, 247 (2000)
  2. http://en.wikipedia.org/wiki/Quantum_computer, http://www.cs.caltech.edu/~westside/quantum-intro.html#power, http://www.quantiki.org/wiki/index.php/Main_Page
  3. W. Langbein and B. Patton, Optics Letters 31, 1151, (2006)
  4. W. Langbein and B. Patton, J. Phys.: Condens. Matter, in press (2007)
  5. B. Patton et al., Phys. Rev. Lett. 95, 266401 (2005)
  6. W. Langbein and B. Patton, Phys. Rev. Lett. 95, 017403 (2005)
  7. B. Patton et al., Phys. Rev. B 73, 235354 (2006)
  8. V. Savona and W. Langbein, Phys. Rev. B 74, 075311 (2006)

Project Members

EPSRC
Created by J.Kasprzak and W.Langbein, 03/2007