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Collective optical response from quantum dot molecules
ARTICLE IN PRESS Microelectronics Journal 40 (2009) 505– 506
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Collective optical response from quantum dot molecules Anna Sitek , Pawe! Machnikowski ´ skiego 27, 50-370 Wroc!aw, Poland Institute of Physics, Wroc!aw University of Technology, Wybrzez˙e Wyspian
a r t i c l e in fo
abstract
Available online 31 July 2008
We study theoretically selected aspects of the collective interaction between two coupled quantum dots (QDs) forming a quantum dot molecule (QDM) and the modes of the electromagnetic field, as manifested in the optical response of QDMs under various experimental conditions. Unlike atomic systems, the artificial QD structures show shape and size inhomogeneity, which has to be included in the theoretical description. Moreover, for tightly spaced QDMs, coupling between the QDs cannot be neglected. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Quantum dots Decoherence Four-wave mixing method
1. Introduction Recently, coupled quantum dots (QDs) have been widely investigated. They attract attention not only because of their possible application in quantum electronic devices [1], but also because of the richness of their physical properties, which are often not present in systems of individual QDs. Optical experiments on quantum dot molecules (QDMs) have shown increased pure dephasing effects, non-monoexponential decay of coherent polarization [2] and non-monotonous temperature dependence of the luminescence decay rate [3]. These effects are believed to result at least partly from collective interaction with the modes of electromagnetic field, that is, from the superradiance effect well known from atomic samples. Indeed, superradiance has been recently observed in an ensemble of closely spaced self-assembled QDs [4]. In this paper, we will show that signatures of the collective radiative coupling can be observed in theoretical response of a single QDM or of an ensemble of QDMs. In the luminescence of a single QDM, one observes a transition between the regimes of independent decay (with the usual decay rate) to superradiant decay (double rate) via intermediate cases with nonmonoexponential decay [5]. The same transition is present in the non-linear [four-wave mixing (FWM)] response of inhomogeneously broadened QDM ensembles.
2. The model of a QDM Each QDM will be modelled as a four-level system with the basis states |00S, |10S, |01S and |11S, corresponding to the Corresponding author. Tel.: +48 71 3204327/3202579; fax: +48 71 3283696.
E-mail address: [email protected] (A. Sitek). 0026-2692/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.06.017
ground state (empty dots), an exciton in the first and second QD and excitons in both QDs, respectively. Transition energies are assumed to be E1,2 ¼ E7D, where E is the average energy and D is the energy deviation from the average. We will describe the evolution in a frame rotating with the frequency E=_. In the rotating-wave approximation, the Hamiltonian of a single QDM is H ¼ HX+HI, where HI describes the interaction with electromagnetic modes in the dipole approximation and HX ¼ Dðj1ih1j I I j1ih1jÞ þ Vðj01ih10j þ j10ih01jÞ, where V is Fo¨rster (interband dipole) or tunnel coupling between the dots. The evolution of the reduced density matrix of the charge _ ¼ i½HX ; r þ subsystem is generated by the Lindblad equation r ðjÞ L½r with L½r ¼ G½S Sþ ð1=2ÞfSþ S ; rgþ , where S ¼ Sj s , sðjÞ are creation or annihilation operators of excitation in the jth QD and G is the spontaneous decay rate (throughout the paper, we set 1/G ¼ 1 ns).
3. Occupation decay in a single QDM First, we will consider the evolution of exciton occupation in a QDM built of two QDs pffiffiffi and prepared initially in a subradiant ðj01i j10iÞ= 2Þ or superradiant state ðjcð0Þi ¼ ðj01i ðjcð0Þi ¼ pffiffiffi þj10iÞ= 2Þ. Currently, it is possible to prepare QDs with the difference between the transition energies of the order of meV, so we assume D ¼ 1 meV. The results following from a numerical solution of the Lindblad equation are shown in Fig. 1. For V5D, both states show simple exponential decay with the rate G; in the opposite limit, VbD, the subradiant state becomes stable, while the superradiant state decays exponentially with a twice larger rate. In the intermediate range of parameters, the decay is non-exponential. In general, it is possible to show that the average number of excitons n(t) of a QDM initially prepared in superradiant state is described by the formula
ARTICLE IN PRESS 506
A. Sitek, P. Machnikowski / Microelectronics Journal 40 (2009) 505–506
Fig. 1. The exciton occupation for subradiant (a) and superradiant (b) states for D ¼ 1 meV. The inset of (b) shows the values of the occupation decay rates.
n(t) ¼ sin2(f) exp(2Re lt)+cos2(f) exp(2Rel+t), where Re l7 depend on the interaction as shown in the inset in Fig. 1b.
4. FWM response from a QDM ensemble A FWM experiment consists in exciting an ensemble of QDMs (in our case prepared in r(0) ¼ |00S/00| state) with two ultrashort laser pulses arriving at t1 ¼ t and t2 ¼ 0. If the durations of the pulses are much shorter than both _=D and _=V, the action of each of them corresponds to the independent rotation of the state of each QD, that is, to the unitary ~i U ~ i , where U ~ i ¼ cosðai =2ÞI i sinðai =2Þ transformation U i ¼ U ½expðiðji þ Eti ÞÞj0ih1j þ h:c: and ai is the pulse area. Between the pulses and after the second pulse, the system evolution is generated by the Lindblad equation. The optical polarization of a single QDM under consideration is proportional to PðtÞ ¼ h11jrðtÞðj01i þ j10iÞ þ ðh01j þ h10jÞrðtÞ j00i þ c:c: In order to extract the FWM polarization, we pick out the terms containing the factor exp(i(2j2j1)). The total response from the sample is obtained by summing up the contributions from individual QDMs with a Gaussian distribution weight factor, which reflects the distribution of transition energies in the sample (we neglect a possible variation of dipole moments and assume equal energy variances of transition energy of both QDs). The detection of weak signals originating from QDs is based on a heterodyne technique [6]: the response P(t) is superposed ¯ t 0 ÞÞ þ c:c:, onto a reference pulse Eref ðtÞ ¼ f ref ðt t 0 Þ expðiEðt where fref(tt0) is a Gaussian envelope. The measured FWM signal R is proportional to FWMðt 0 ; tÞ ¼ j dtPðþÞ ðtÞEðÞ ðtÞj, where P(+)(t) ref are the positive frequency part of the polarization and the and EðÞ ref negative frequency part of the reference pulse, respectively. Timeintegrated signal is obtained by summing up contributions for all R times t0, it is TIFWMðtÞ ¼ dt 0 FWMðt 0 ; tÞ. For the calculations, we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fix V 2 þ D2 ¼ 11 meV, the energy variance 8 meV and the duration of the reference pulse 43 fs [2]. The results are plotted in Fig. 2. For non-interacting QDs, the TIFWM signal decays exponentially with the rate G. For vanishing energy mismatch, the signal for t4100 ps may be approximated by an exponential with the rate 2G. In the intermediate range, the decay is not exponential, but it cannot be approximated by a sum of two exponential decays, as in the previous case. For non-vanishing interaction, strong oscillation is seen for picosecond delay times
Fig. 2. TIFWM signal as a function of the delay for vanishing interaction (dashed), for V ¼ 2, 5 meV (solid) (also in the inset) and for vanishing energy mismatch (dotted).
(inset in Fig. 2), which nearly instantaneously reduces the amplitude of the response by a few percent whenever the dots are coupled. Thus, we have identified another fingerprint of the coupling between the dots forming a QDM. In a real system, this signal drop will add to the phonon-induced pure dephasing effect, which may explain the increased initial dephasing observed in the experiment [2]. 5. Conclusions In the luminescence of a single QDM, one observes a transition between the regimes of independent decay (with the usual decay rate) to superradiant decay (double rate) via intermediate cases with non-monoexponential decay [5]. Our results show that the same transition is present in the non-linear (FWM) response of inhomogeneously broadened QDM ensembles. In both cases, the transition is driven by the interplay of the energy mismatch between the QDs forming the molecule and the coupling (tunneling or dipole) between them. In the case of the FWM response, the strength of the coupling between the dots and correlation between their sizes is also reflected in the shape of the time-resolved signal [7]. References [1] O. Gywat, G. Burkard, D. Loss, Phys. Rev. B 65 (2002) 205329. [2] P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, P. Hawrylak, Phys. Rev. Lett. 91 (2003) 267401. [3] C. Bardot, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, P. Hawrylak, Phys. Rev. B 72 (2005) 035314. [4] M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, D. Hommel, Nat. Phys. 3 (2007) 106. [5] A. Sitek, P. Machnikowski, Phys. Rev. B 75 (2007) 035328. [6] P. Borri, W. Langbein, J. Mørk, J.M. Hvam, F. Heinrichsdorff, M.H. Mao, D. Bimberg, Phys. Rev. B 60 (1999) 7784. [7] A. Sitek, P. Machnikowski, Acta Phys. Pol. A 112 (2007) 167.
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Collective optical response from quantum dot molecules Anna Sitek , Pawe! Machnikowski ´ skiego 27, 50-370 Wroc!aw, Poland Institute of Physics, Wroc!aw University of Technology, Wybrzez˙e Wyspian
a r t i c l e in fo
abstract
Available online 31 July 2008
We study theoretically selected aspects of the collective interaction between two coupled quantum dots (QDs) forming a quantum dot molecule (QDM) and the modes of the electromagnetic field, as manifested in the optical response of QDMs under various experimental conditions. Unlike atomic systems, the artificial QD structures show shape and size inhomogeneity, which has to be included in the theoretical description. Moreover, for tightly spaced QDMs, coupling between the QDs cannot be neglected. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Quantum dots Decoherence Four-wave mixing method
1. Introduction Recently, coupled quantum dots (QDs) have been widely investigated. They attract attention not only because of their possible application in quantum electronic devices [1], but also because of the richness of their physical properties, which are often not present in systems of individual QDs. Optical experiments on quantum dot molecules (QDMs) have shown increased pure dephasing effects, non-monoexponential decay of coherent polarization [2] and non-monotonous temperature dependence of the luminescence decay rate [3]. These effects are believed to result at least partly from collective interaction with the modes of electromagnetic field, that is, from the superradiance effect well known from atomic samples. Indeed, superradiance has been recently observed in an ensemble of closely spaced self-assembled QDs [4]. In this paper, we will show that signatures of the collective radiative coupling can be observed in theoretical response of a single QDM or of an ensemble of QDMs. In the luminescence of a single QDM, one observes a transition between the regimes of independent decay (with the usual decay rate) to superradiant decay (double rate) via intermediate cases with nonmonoexponential decay [5]. The same transition is present in the non-linear [four-wave mixing (FWM)] response of inhomogeneously broadened QDM ensembles.
2. The model of a QDM Each QDM will be modelled as a four-level system with the basis states |00S, |10S, |01S and |11S, corresponding to the Corresponding author. Tel.: +48 71 3204327/3202579; fax: +48 71 3283696.
E-mail address: [email protected] (A. Sitek). 0026-2692/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.06.017
ground state (empty dots), an exciton in the first and second QD and excitons in both QDs, respectively. Transition energies are assumed to be E1,2 ¼ E7D, where E is the average energy and D is the energy deviation from the average. We will describe the evolution in a frame rotating with the frequency E=_. In the rotating-wave approximation, the Hamiltonian of a single QDM is H ¼ HX+HI, where HI describes the interaction with electromagnetic modes in the dipole approximation and HX ¼ Dðj1ih1j I I j1ih1jÞ þ Vðj01ih10j þ j10ih01jÞ, where V is Fo¨rster (interband dipole) or tunnel coupling between the dots. The evolution of the reduced density matrix of the charge _ ¼ i½HX ; r þ subsystem is generated by the Lindblad equation r ðjÞ L½r with L½r ¼ G½S Sþ ð1=2ÞfSþ S ; rgþ , where S ¼ Sj s , sðjÞ are creation or annihilation operators of excitation in the jth QD and G is the spontaneous decay rate (throughout the paper, we set 1/G ¼ 1 ns).
3. Occupation decay in a single QDM First, we will consider the evolution of exciton occupation in a QDM built of two QDs pffiffiffi and prepared initially in a subradiant ðj01i j10iÞ= 2Þ or superradiant state ðjcð0Þi ¼ ðj01i ðjcð0Þi ¼ pffiffiffi þj10iÞ= 2Þ. Currently, it is possible to prepare QDs with the difference between the transition energies of the order of meV, so we assume D ¼ 1 meV. The results following from a numerical solution of the Lindblad equation are shown in Fig. 1. For V5D, both states show simple exponential decay with the rate G; in the opposite limit, VbD, the subradiant state becomes stable, while the superradiant state decays exponentially with a twice larger rate. In the intermediate range of parameters, the decay is non-exponential. In general, it is possible to show that the average number of excitons n(t) of a QDM initially prepared in superradiant state is described by the formula
ARTICLE IN PRESS 506
A. Sitek, P. Machnikowski / Microelectronics Journal 40 (2009) 505–506
Fig. 1. The exciton occupation for subradiant (a) and superradiant (b) states for D ¼ 1 meV. The inset of (b) shows the values of the occupation decay rates.
n(t) ¼ sin2(f) exp(2Re lt)+cos2(f) exp(2Rel+t), where Re l7 depend on the interaction as shown in the inset in Fig. 1b.
4. FWM response from a QDM ensemble A FWM experiment consists in exciting an ensemble of QDMs (in our case prepared in r(0) ¼ |00S/00| state) with two ultrashort laser pulses arriving at t1 ¼ t and t2 ¼ 0. If the durations of the pulses are much shorter than both _=D and _=V, the action of each of them corresponds to the independent rotation of the state of each QD, that is, to the unitary ~i U ~ i , where U ~ i ¼ cosðai =2ÞI i sinðai =2Þ transformation U i ¼ U ½expðiðji þ Eti ÞÞj0ih1j þ h:c: and ai is the pulse area. Between the pulses and after the second pulse, the system evolution is generated by the Lindblad equation. The optical polarization of a single QDM under consideration is proportional to PðtÞ ¼ h11jrðtÞðj01i þ j10iÞ þ ðh01j þ h10jÞrðtÞ j00i þ c:c: In order to extract the FWM polarization, we pick out the terms containing the factor exp(i(2j2j1)). The total response from the sample is obtained by summing up the contributions from individual QDMs with a Gaussian distribution weight factor, which reflects the distribution of transition energies in the sample (we neglect a possible variation of dipole moments and assume equal energy variances of transition energy of both QDs). The detection of weak signals originating from QDs is based on a heterodyne technique [6]: the response P(t) is superposed ¯ t 0 ÞÞ þ c:c:, onto a reference pulse Eref ðtÞ ¼ f ref ðt t 0 Þ expðiEðt where fref(tt0) is a Gaussian envelope. The measured FWM signal R is proportional to FWMðt 0 ; tÞ ¼ j dtPðþÞ ðtÞEðÞ ðtÞj, where P(+)(t) ref are the positive frequency part of the polarization and the and EðÞ ref negative frequency part of the reference pulse, respectively. Timeintegrated signal is obtained by summing up contributions for all R times t0, it is TIFWMðtÞ ¼ dt 0 FWMðt 0 ; tÞ. For the calculations, we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fix V 2 þ D2 ¼ 11 meV, the energy variance 8 meV and the duration of the reference pulse 43 fs [2]. The results are plotted in Fig. 2. For non-interacting QDs, the TIFWM signal decays exponentially with the rate G. For vanishing energy mismatch, the signal for t4100 ps may be approximated by an exponential with the rate 2G. In the intermediate range, the decay is not exponential, but it cannot be approximated by a sum of two exponential decays, as in the previous case. For non-vanishing interaction, strong oscillation is seen for picosecond delay times
Fig. 2. TIFWM signal as a function of the delay for vanishing interaction (dashed), for V ¼ 2, 5 meV (solid) (also in the inset) and for vanishing energy mismatch (dotted).
(inset in Fig. 2), which nearly instantaneously reduces the amplitude of the response by a few percent whenever the dots are coupled. Thus, we have identified another fingerprint of the coupling between the dots forming a QDM. In a real system, this signal drop will add to the phonon-induced pure dephasing effect, which may explain the increased initial dephasing observed in the experiment [2]. 5. Conclusions In the luminescence of a single QDM, one observes a transition between the regimes of independent decay (with the usual decay rate) to superradiant decay (double rate) via intermediate cases with non-monoexponential decay [5]. Our results show that the same transition is present in the non-linear (FWM) response of inhomogeneously broadened QDM ensembles. In both cases, the transition is driven by the interplay of the energy mismatch between the QDs forming the molecule and the coupling (tunneling or dipole) between them. In the case of the FWM response, the strength of the coupling between the dots and correlation between their sizes is also reflected in the shape of the time-resolved signal [7]. References [1] O. Gywat, G. Burkard, D. Loss, Phys. Rev. B 65 (2002) 205329. [2] P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, P. Hawrylak, Phys. Rev. Lett. 91 (2003) 267401. [3] C. Bardot, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, P. Hawrylak, Phys. Rev. B 72 (2005) 035314. [4] M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, D. Hommel, Nat. Phys. 3 (2007) 106. [5] A. Sitek, P. Machnikowski, Phys. Rev. B 75 (2007) 035328. [6] P. Borri, W. Langbein, J. Mørk, J.M. Hvam, F. Heinrichsdorff, M.H. Mao, D. Bimberg, Phys. Rev. B 60 (1999) 7784. [7] A. Sitek, P. Machnikowski, Acta Phys. Pol. A 112 (2007) 167.