Theory for strongly coupled quantum dot cavity quantum electrodynamics

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1 Folie: 1 Theory for strongly coupled quantum dot cavity quantum electrodynamics Alexander Carmele

2 OUTLINE Folie: 2 I: Introduction and Motivation 1.) Atom quantum optics and advantages of semiconductor nanostructures 2.) Cluster expansion approach in the single-photon regime II: Mathematical induction method 1.) General set of equations of motion 2.) Examples (1+2): LO-phonon cavity feeding and induced antibunching III: Photon-probability cluster expansion (PPCE) 1.) Photon probability expansion and modified Hartree-Fock factorization rule 2.) Examples (3+4): Electrically-driven single photon emitter IV: Conclusions

electron-light The simplest fully quantized model of interest (J.H.

3 Jaynes-Cummings model Folie: 3 Atom cavity-qed, solved by the Jaynes-Cummings model 1 : Isolated two-level system (no losses) One-electron assumption One interaction: electron-light The simplest fully quantized model of interest (J.H. Eberly) analytically solvable, e.g. ha y ca c i(t) = cos 2 (M p N + 1 t) 1 E. Jaynes and F. Cummings, Proc.IEEE 51, 89 (1963)

3, 253 (2007) Typical realization based on: (i) Trapped atoms (ii) Atom

4 Atom-photon interfaces: experimental realizations Folie: 4 Single-photon server with just one atom, Hijlkema et al., Nat. Phys. 3, 253 (2007) Typical realization based on: (i) Trapped atoms (ii) Atom beam Entanglement source for quantum repeaters, Chen et al., Phys.Rev.Lett. 99, (2007)

frequencies, Cavity-system ultra-small, Electrical pumping, Semiconductor QD cavity-qed: Theoretical

5 Semiconductor nanostructures for device fabrication Folie: 5 Advantageous semiconductor QD properties for future technological applications in microcavity systems: Fixed position, tailorable coupling strengths and frequencies, Cavity-system ultra-small, Electrical pumping, Semiconductor QD cavity-qed: Theoretical simulations for device optimization are desirable BUT: more interactions and additional loss mechanisms need to be considered

6 Hierarchy problem and cluster expansion Folie: 6 1.) more interactions (electron-phonon, electron-electron) 2.) number of carriers may not be fixed ha y va c c y i i¹h@ t hoi = h[h; O]i ha y ca c c y ci For non-markovian description, equation of motion approach Hierarchy problem ha y va c c y c y ci Typical approach: Factorization / truncation scheme via cluster expansion

7 Example and breakdown of cluster expansion Folie: 7 Assuming Fock photons, no coherent contributions: Vacuum Rabi oscillation (N=0) are not approximated good enough!!

8 Cluster expansion in weak correlated dynamics Folie: 8 Assuming Fock photons, no coherent contributions: But: Rabi oscillations with N=50 are well approximated!!

9 Limit of cluster expansion approach Folie: 9 Weak coupling regime: (e.g. resonance fluorescence, biexciton cascade) Strong coupling regime (dynamics weakly correlated ): (e.g. laser, superradiance effects) Strong coupling regime (dynamics strongly correlated ): (e.g. single photon emitter)

Technological application of single photons Folie: 10 Since

improved measurement (ii) quantum cryptography (iii) quantum

simulations become necessary (i): mathematical induction method

10 Technological application of single photons Folie: 10 Since single-photon of interest for technological application, (i) improved measurement (ii) quantum cryptography (iii) quantum information processing (entanglement) New approaches for device simulations become necessary (i): mathematical induction method (atom-like QDs) (ii): photon-probability cluster expansion (electrical pumping)

11 Folie: 11 (i) mathematical induction method

12 Sanvitto et al., Appl.Phys.Lett. 86 (2005) Theory of strongly coupled quantum dot cavity-qed Marquez et al., Appl.Phys.Lett. 78 (2001) Semiconductor QD cavity-qed Hamiltonian Folie: 12 QD, assumed as a 2-level system with one electron, interacts with the cavity photons, bulk LO-phonons, a classical pump field

13 Solving LO-phonon QD cavity-qed without factorization Folie: 13 Using product rule for operators: and generalized commutation relations: for every possible combination of phonon, photon, and electron operators: and their dynamics, e.g.

General set of equations of motion Folie: 14 For example, in the case of LO-phonon assisted vacuum Rabi oscillations ( = 0 ) : E00 11 E20 00 E10 00 E00 00 E01 00 E02 00 T 10 20 T 10 10 T 10 00 T 10

14 General set of equations of motion Folie: 14 For example, in the case of LO-phonon assisted vacuum Rabi oscillations ( = 0 ) : E00 11 E20 00 E10 00 E00 00 E01 00 E02 00 T T T T T P ps 00 = hcyp c s i Phonon interaction: Photon interaction: P P E T P numerically solvable up to arbitrary accuracy, reproducing analytical solutions of the IBM and JCM

15 LO-phonon QD cavity-qed: additional anti-crossings Folie: 15 Probing with δ-pulse at 300K:

16 Modified Rabi frequency due to phonon cavity feeding Folie: 16 Modified Rabi frequency at the Stokes-position (2): (temperature dependence weak) 3K 300K Effective oscillator strength (0 K) via Huang-Rhys factor:

17 Quantum beating at resonance position Folie: 17 Quantum beating at resonance position (1): Impact on intensity-intensity correlation function: FOCK STATE: initially one fock photon in the cavity and an excited QD

18 Quantum beating at resonance position Folie: 18 Quantum beating at resonance position (1) : Impact on intensity-intensity correlation function: THERMAL STATE: for a cavity field, prepared initially in the thermal state

LO-phonon induced thermal anti-bunching Slide: 19 Impact on intensity-intensity correlation function mean value: 1.1 0.

19 LO-phonon induced thermal anti-bunching Slide: 19 Impact on intensity-intensity correlation function mean value: Thermal cavity field is transformed into a non-classical field via LO-phonon interaction 1 Semiconductor environment enforces non-classical light features 1 PRL 104, (2010)

20 Applications of the mathematical induction method Slide: 20 Systems, for which the induction method is successfully applied: 1. QD as a two-level system with one-electron (photon statistics) 1 2. QD as a three-level system (quantum coherence) 2 3. QD as a four-level system with two electrons (biexciton cascade) 3 1 PRL 104, (2010) 2 PSSB, accepted (2010) 3 PRB 81, (2010)

21 Slide: 21 (ii) photon-probability cluster expansion

QD cavity-qed in the presence of a carrier reservoir Slide: 22 In the presence of a carrier reservoir, the number of carriers inside the QD is not fixed.

22 QD cavity-qed in the presence of a carrier reservoir Slide: 22 In the presence of a carrier reservoir, the number of carriers inside the QD is not fixed. The one-electron assumption is not valid anymore: Typically, a Hartree Fock factorization is applied: But in case of strongly correlated electron-photon dynamics:

23 Photon-probability cluster expansion Slide: 23 A factorization approach for strongly correlated systems, the photon probability cluster expansion (PPCE) is introduced. 1 Expansion of observables: Introducing a modified Hartree Fock approximation 1 : 1 PRL 103, (2009)

24 Modified Rabi oscillation amplitude Slide: 24 Equation of motion changed due to many-particle contribution 1 : The amplitude of the Rabi flops depends on the number of electrons and holes in the QD. 1 PSSB 247, 809 (2010)

Simulation of an electrically driven single photon source Slide: 25 Inclusion of phenomenological pumping terms: @ t fnj e pump = Se in (p n fn) e Se out @ t fnj h pump = Sh

25 Simulation of an electrically driven single photon source Slide: 25 Inclusion of phenomenological pumping t fnj e pump = Se in (p n fn) e Se t fnj h pump = Sh in(p n fn) h Sh out f e n f h n microscopical derived with corresponding interaction Hamiltonian Pump mechanism and losses balance into a stationary state 1 1 RRL 4, 289 (2010)

26 Theory of strongly coupled quantum dot cavity-qed Parameter studies of single QD laser Slide: 26 Microscopic model reveals: Why are QD very promising single-photon sources? 1 Comparison with one-electron approximation (OEA): OEA: f e sp = f e CR: f e sp = P n f e n f h n p n Due to the WL, enhanced Pauli-blocking in the QD states occurs Advantageous for single-photon emission in a wide pump interval 1 Semic.Sci.Technology, accepted (2010)

Conclusion Slide: 27 Microscopic theory helps to reveal advantageous quantum optical properties of semiconductor QD, for example: LO-phonon cavity feeding

27 Conclusion Slide: 27 Microscopic theory helps to reveal advantageous quantum optical properties of semiconductor QD, for example: LO-phonon cavity feeding (phonon induced strong coupling) LO-phonon induced anti-bunching of a thermal cavity field Many-particle induced enhanced Pauli-blocking enforces single-photon emission

28 Folie: 28 Theory for strongly coupled quantum dot cavity quantum electrodynamics

29 Thermal antibunching with LO-phonon life time Folie: 29

30 PPCE: general canonical statistical operator Folie: 30

Theory of quantum dot cavity-qed

03.01.2011 Slide: 1 Theory of quantum dot cavity-qed -- LO-phonon induced cavity feeding and antibunching of thermal radiation -- Alexander Carmele, Julia Kabuss, Marten Richter, Andreas Knorr, and Weng