June 1, 2007

Fig. 1: Schematic plot for a two-dimensional array of
photonic crystal microcavities doped with substitutional
donor/acceptor impurities. |

Quantum many-body systems, such as strongly correlated electrons, are generally difficult to understand due to the lack of appropriate theoretical tools. A brute-force matrix diagonalization method is limited by the exponentially growing Hilbert space dimension with the number of particles. A quantum Monte-Carlo simulation method often suffers from the so-called sign problem. An analytical mean-field method provides good approximate solutions for three-dimensional systems but limited applications for two-dimensional systems. An interesting alternative is to construct a quantum simulator that implements a model Hamiltonian with controllable parameters [1]. One such example was demonstrated using Bose-condensed cold atoms in an optical lattice potential. The superfluid to Mott-insulator QPT predicted by the Bose-Hubbard model was observed by changing the ratio of on-site repulsive interaction to hopping matrix element U/t [2]. This experiment opened a door for simulating complex many-body systems with more controllable artificial systems [3,4].

Simulating the Bose-Hubbard model using photons has recently attracted an intense interest [5,6,7]. Even though photons do not interact with each other in free space, by confining light inside a small cavity with an active medium, the nonlinear photon-photon interaction can be effectively introduced. A related concept is a photon blockade: an optical analog of single electron Coulomb blockade effect [8,9]. These schemes require an extremely high-Q cavity even though EIT in a four-level atomic ensemble or single-atom cavity QED in the strong coupling regime are employed.

Here we show that an optical quantum simulator for
such strongly correlated photonic systems can be constructed using a
simpler approach. Our scheme consists of a two-dimensional array of
coupled photonic crystal microcavities [10] doped with substitutional
donor/acceptor impurities [11,12]. A schematic plot of the system is
shown in Fig. 1. The photons hop from site to site via optical
evanescent field coupling and interact with each other through the
nonlinearity induced by the many-exciton cavity QED effect. Setting bulk
doping density to 10^{14} cm^{-3}, cavity photon-bound
exciton frequency detuning to hundreds of GHz and cavity Q factor to
10^{5}, the QPT from photonic superfluid to Mott-insulator
should be observed. The proposed scheme combines the large oscillator
strength and small inhomogeneous linewidth of donor/acceptor-bound
excitons embedded in bulk semiconductor matrix [11], and the recent
advancement in photonic crystal microcavities with high cavity Q factor
and small mode volume [13]. In particular, we will show that when using
a blue detuning, the increase of exciton fraction of the lower branch of
the cavity polaritons decreases the required cavity Q factor.

We start our analysis by considering the optical evanescent field coupling between adjacent microcavities. The tunneling part of the Hamiltonian is given as

where <*ij*> indicates that only the nearest
neighbor coupling is considered, t is the tunneling energy determined by
the overlapping of the nearest neighbor cavity fields, and a_{i}
is the annihilation operator of the ith site cavity mode. To
quantitatively estimate the condition of QPT, we perform a mean field
analysis by applying the decoupling approximation [7,14] i.e. let
a_{i}^{†}a_{j} = <
a_{i}^{†}> a_{j} +
a_{i}^{†} < a_{j}> - <
a_{i}^{†}> < a_{j}> and define a
real-valued superfluid parameter ψ = < a_{i}>. The
Hamiltonian can then be rewritten as

where *z* is
the number of nearest neighbors. Next we consider the free and
interacting part of the total Hamiltonian

ω_{e}, ω_{p} and g are
the bound exciton transition frequency, cavity photon resonance
frequency and exciton-photon coupling constant. N is the number of
impurities per cavity. L_{z} is the collective angular momentum
operator in the z-direction and L_{±} are the collective
creation/annihilation operators. The single site eigenenergy spectrum
considering only the free and interacting Hamiltonian is sketched in
Fig. 2. In general, the number of eigenstates for each excitation
manifold n is equal to n+1 if n≤N and equal to N+1 if n>N. The
ground state (lower branch of the cavity polaritons for n=1 excitation
manifold) interaction energy U can then be identified and is shown in
Fig. 2. Notice that in the small detuning limit, U approaches zero if N
is much larger than n. In this case, the system behaves linearly because
the collective angular momentum operator satisfies a bosonic commutation
relation [L_{+}, L_{-}] ~ N [15].

Equipped with these equations, we are now in position to evaluate the QPT condition. We consider only a single site Hamiltonian

where μ is the chemical potential in grand canonical ensemble. Given t and μ, one can calculate the eigenenergies by diagonalizing the effective Hamiltonian using bare states as a complete set of basis. The ground state energy can be found by minimizing the lowest eigenenergy with adjusting the superfluid parameter. The accuracy of such calculation depends on how many state vectors are used to span the Hilbert space. The convergence of the eigenenergies is usually a good indication that a large enough basis set is considered.

In Fig. 3, we plot the superfluid parameter as a
function of t and μ given N = 8 and Δ = ω_{p} -
ω_{e} = 0. Unlike the single-atom cavity QED systems
[6,7], the Mott lobe sizes in the chemical potential direction are
relatively unchanged for low filling factors. The nature of such a
photonic QPT is neither purely fermionic nor bosonic, but shares similar
features with the Bose-Hubbard model: localization of one additional
photon per cavity upon entering the next Mott-insulator regime.

To understand the general behavior of the QPT
condition, we plot the required t for the system to reach the QPT as a
function of N and Δ. This is shown in Fig. 4a. The numerical value
of ω_{e}/2π is chosen as 365.8 THz that corresponds to
820 nm Si donor-bound exciton emission wavelength [11]. g is estimated
as 33.2 GHz by calculating the bound exciton oscillator strength using
the experimentally measured 1 ns lifetime [11], and a cavity mode volume
equal to (820/3.25)^{3} nm^{3}. (3.25 is the refractive
index of GaAs.) As previously explained, the system behaves linearly if
N is sufficiently large and hence t decreases as N increases. In
addition, by having a blue detuning i.e. &Delta > 0, the ground state is
of more exciton fraction. Therefore, t is enhanced because the optical
evanescent field coupling has to be stronger in order to maintain the
QPT condition.

The leakage of the cavity photons imposes an important constraint on the required cavity Q factor. If the polariton decay rate in the Mott-insulator regime is faster than the polariton tunneling rate, the system never reaches an equilibrium state described by Hamiltonian (5). Assuming that the ground state decay rate is slower than the tunneling rate, we find the required cavity Q factor to be

Fig. 4: The required (a) tunneling energy and (b)
cavity Q factor for the system to enter the Mott-insulator
regime with one photon per cavity as a function of N and
Δ. |

where |c_{e}^{2} and
|c_{p}|^{2} are the exciton and photon fraction of the
ground state, t is the required tunneling energy for the system to reach
the QPT, τ_{e} is the bound exciton spontaneous emission
lifetime, and F is the Purcell factor due to the inhibition of
spontaneous emission in a photonic crystal results from the reduced
local optical density of states. Notice that the polariton tunneling
rate is well described by |c_{p}|^{2}t. We plot the
required Q value in Fig. 4b as a function of N and Δ where F = 0.2
is used [16,17]. In general, Q about 10^{5} to 10^{6} is
needed to reach the OPT point. However, with a blue detuning, the cavity
Q factor is reduced. This way, in principle, cavity Q factor can be
relaxed by choosing a sufficiently large blue detuning. For example, Q
is slightly smaller than 10^{5} when Δ = 12 g given N =
9.

Observing the QPT from a photonic superfluid to Mott-insulator requires the ability to control t and U. The frequency detuning can be used as such parameter. Varying a temperature can modulate the bound exciton transition frequency while keeping the cavity photon resonance frequency constant. Alternatively, injecting a molecular gas can modulate the cavity photon resonance frequency while keeping the bound exciton transition frequency constant. Signature of the QPT can be experimentally identified by observing the far field radiation just like in the optical lattice experiments [2]. Another way to identify the QPT is to inject photons by end-firing the membrane layer and measure the transmission from the whole structure. High and low transmissions correspond to the superfluid and the Mott-insulator regimes, respectively.

In conclusion, we propose an experimental scheme to observe the photonic QPT. Our scheme is based on two recent experimental breakthroughs, highly homogeneous substitutional donor/acceptor impurities in semiconductor, and photonic crystal microcavities with high cavity Q factor and small mode volume. The QPT condition is robust against the impurity number fluctuation, and only a moderate cavity Q factor is required when using a large blue detuning. Finally, we point out that due to the flexibility of designing microcavity array topology, systems such as extended Bose-Hubbard model [18] or one-dimension Tonks-Girardeau gas [19] could also be simulated by our scheme.

© 2007 Y.C. Neil Na. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] R. P. Feynman, Int. J. Theor. Phys. **21**,
467 (1982).

[2] M. Greiner et al., Nature **415**, 39
(2002).

[3] M. Köhl et al., Phys. Rev. Lett. **94**,
080403 (2005).

[4] D. Jaksch and P. Zoller, Ann. Phys. **315**, 52
(2005).

[5] M. J. Hartmann, F. G. S. L. Brandão and M.
B. Plenio, Nature Phys. **2**, 849 (2006).

[6] D. G. Angelakis, M. F. Santos and S. Bose, Preprint at http://arxiv.org/abs/quant-ph/0606159 (2006).

[7] A. D. Greentree et al., Nature Phys. **2**, 856
(2006).

[8] A. Imamoglu, H. Schmidt, G. Woods and M. Deutsch,
Phys. Rev. Lett. **79**, 1467 (1997).

[9] K. M. Birnbaum et al., Nature **436**, 87
(2005).

[10] H. Altug and J. Vučković,
Appl. Phys. Lett. **84**, 161 (2004).

[11] C. J. Hwang, Phys. Rev. B **8**, 646
(1973).

[12] K. C. Fu et al., Phys. Rev. Lett. **95**,
187405 (2005).

[13] T. Yoshie et al., Nature **432**, 200
(2004).

[14] K. Sheshadri, H. R. Krishnamurthy, R. Pandit and
T. V. Ramakrishnan, Europhys. Lett. **22**, 257 (1993).

[15] Y. Yamamoto and A. Imamoglu, *Mesoscopic
Quantum Optics* (Wiley, 1999).

[16] M. Fujita et al., Science **308**, 1296
(2005).

[17] D. Englund et al., Phys. Rev. Lett. **95**,
013904 (2005).

[18] V. W. Scarola and S. Das Sarma, Phys. Rev. Lett.
**95** 033003 (2005).

[19] E. H. Lieb and W. Liniger, Phys. Rev.
**130**, 1605 (1963).