Methods to improve single photon generation via photon-blockade in a photonic-crystal cavity with a strongly coupled quantum-dot are presented. With realistic system parameters, significant improvement in second-order-auto-correlation g^2 (0) (from 0.93 to 0.79) is achieved

Faraon, Andrei

Majumdar, Arka

Vučković, Jelena

English

Caltech Authors - Main

Engineering Anti-bunching via Photon Blockade in Photonic 
Crystal Cavity-Quantum Dot Systems 
Arka Majumdar, Andrei Faraon, Jelena Vuckovic                                                                                
E. L. Ginzton Laboratory, Stanford University, Stanford, CA-9430                                                                                                   
e-mail: arkam@stanford.edu 
Abstract: Methods to improve single photon generation via photon-blockade in a photonic-crystal 
cavity with a strongly coupled quantum-dot are presented. With realistic system parameters, significant 
improvement in second-order-auto-correlation g2 (0) (from 0.93 to 0.79) is achieved.                                                                  
©2009 Optical Society of America                                                                                                                           
OCIS codes: (050.5298) Photonic crystals; (270.5585) Quantum information and processing 
Growing interest in making scalable Quantum Information Processors (QIP) combined with modern semiconductor 
fabrication technology resulted in numerous proposals for making cavity QED system in semiconductors.  One of 
the major requirements of QIP is a non-classical light-source. We have previously demonstrated photon blockade in 
a photonic crystal – quantum dot (PhC-QD) system [1]; such an effect could be employed to make non-classical 
light sources on chip, but due to sub-optimal system parameters, only a limited blockade effect was achieved.  Here 
we analyze a possibility of achieving a stronger photon blockade, and thus of achieving a stronger antibunching 
upon transmitting a coherent light beam tuned to the first order dressed states through such a system, within the 
constraints of the state of the art technology. 
    If a QD is placed inside a cavity with two degenerate and orthogonal modes (which can be theoretically obtained 
in a cavity with C6 symmetry) and one of the modes is coherently driven by a laser, then the corresponding 
Hamiltonian (in rotating wave approximation with driving laser frequency) is given by: 
 
Here, a and b are the annihilation operators for the two cavity modes; Δc is the detuning between laser frequency and 
mode a , δ is the detuning between mode a and b (for a degenerate cavity δ = 0), Δa is detuning between driving 
laser and the QD resonance frequency; σ- ,σ+ are the lowering and raising operators for the QD; 2σz=[ σ-,σ+]; Ω is the 
Rabi frequency of the driving laser; go is the coupling strength between cavity modes and QD (same strength for 
both modes are assumed). We assume that in the experiment, light from only mode a (or mode b) can be collected as 
the two modes can be distinguished by polarization. By solving Master Equation with this Hamiltonian (1), and 
using damping parameters (cavity decay κ and dipole decay γ) from Ref. [1] (and assuming no detuning is present 
between cavity modes and the QD), g2(0) for mode a is obtained,  (Figure 1A, blue curve). It should be noted that as 
opposed to single cavity mode situation studied in Ref. [1] (Fig. 1A red curve), three antibunching points are 
obtained. If the cavity modes and QD are all resonant (frequency ωc), then the three energy levels correspond to  
cnm ω)2/1( ++  and )2(4)2/1(
2
0 ++±++ nmgnm cω , where m is the initial number of quanta in the 
mode a, n is the quanta in the mode b, and 1-quanta in the QD. Antibunching points B and C are due to ordinary 
Jaynes-Cummings ladder (formed by the last two energy levels) as explained by [2]. Antibunching point A (which is 
more antibunched relative to single mode case, with g2(0) = 0.79 compared to g2(0) = 0.93) is obtained when the 
laser frequency is tuned to cold-cavity frequency ωc. This happens as the successive levels at energy 
cnm ω)2/1( ++  are of opposite parity and hence transition between them is not allowed. For practical 
implementation, it is difficult to obtain simultaneous degeneracy and orthogonality in high quality PhC cavity modes 
and hence simulations are also done with non-degenerate and non-orthogonal cavity modes. To simulate non-
orthogonality of modes, a coupling term between two cavity modes is added in the Hamiltonian in Eq. 1 with 
coupling strength gab ( Coupling  term: )( ++ + abbagab ). Figure 1B shows the g
2(0) value at anti-bunching point 
A with different mode detuning δ and coupling gab values. The reduction of g2(0) with non-degeneracy of cavity 
mode is expected. For non-orthogonal modes, it is found that there is an optimum point. The mode b gets photon 
only through QD when modes a and b are orthogonal. Hence the mode b has more non-classical nature than mode a. 
When there is small leakage from a to b then the mode a becomes more anti-bunched due to light from mode b. But 
increasing leakage ultimately makes both the modes more classical.  
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    Another way to improve the antibunching, is to cascade two single mode PhC-QD systems but at the cost of 
reducing the efficiency of the source. As a non-classical light cannot be expressed in a finite Fock-state basis, hence 
the cascaded system is modeled as a single system (with coupling strength η) [3], and that the coupling is such that 
light travels in one direction (i.e., only forward traveling wave is present). Simulation results are shown in Figure 2. 
It should be noted that, the g2(0) is improved after cascading. Increasing η does not help in reduction of g2(0), but it 
increases the output intensity.  
   In conclusion, we analyzed two proposals for improving the anti-bunching of light by photon-blockade in a QD-
PhC cavity system as reported in [1]. If this elementary system can be improved (e.g., by pulse shaping of coherent 
light-source to increase coupling of photons to the cavity, or by increasing cavity Q-factor), then these two 
approaches will give even better performances than simulated here. 
  Financial support was provided by ONR Young Investigator Award, DARPA Young Faculty Award, SGF (T.I. 
Fellowship), Presidential Early Career Award for Engineers and Scientists (PECASE) and Army Research Office. 
 
                                                     (A)                                                                             (B) 
Figure 1: (a) g2(0) for three different situations. In all these cases g0 = κ = 16GHz, γ = κ/80, Ω = 50. Here it is assumed that cavity modes and the 
QD are resonant. The two modes are orthogonal and degenerate. (b) Variation of g2(0) at point A with detuning between cavity modes and gab. 
Same damping parameters and driving term are used. 
 
Figure 2: g2(0) with Δc for different coupling strength η. In both the cavities g0 = κ = 16GHz, γ = κ/80, Ω = 10. 
References: 
[1] A. Faraon et. al.; Coherent generation of nonclassical light on a chip via photon-induced tunneling and blockade; Nature Physics; (2008). 
[2] K. M. Birnbaum et. al.; Photon Blockade in an optical cavity with one trapped atom; Nature; vol. 436, 2005. 
[3] C.W. Gardiner, P. Zoller; Quantum Noise; Third Edition; Springer. 
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Engineering anti-bunching via photon blockade in photonic crystal cavity-quantum dot systems

Birnbaum et. al.; Photon Blockade in an optical cavity with one trapped atom;

Coherent generation of nonclassical light on a chip via photon-induced tunneling and blockade; Nature Physics;

Quantum Noise; Third Edition; Springer.

http://authors.library.caltech.edu/34124/1/05225646.pdf

Engineering anti-bunching via photon blockade in photonic crystal cavity-quantum dot systems

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