Squeezed state generation in photonic crystal microcavities

M. G. Banaee; Jeff F. Young

doi:10.1364/OE.16.020908

1. Introduction

Deterministic and efficient nonclassical light sources play a central role in implementing future quantum information protocols [1, 2]. Two examples are single photon and squeezed state light sources. Optimum implementations of quantum cryptography [3] and quantum computation with linear optics [4] rely on single photon sources whereas quantum information processing with continuous variables mainly depends on squeezed state sources [5, 6, 7], which produce two strongly correlated (entangled) light beams. Semiconductor-based implementations of these nonclassical light sources are particularly attractive due to their small size, low power consumption and the possibility of integration with solid-state qubits [8, 9].

In the first protocol proposed for continuous variable teleportation [6], the entangled EPR (Einstein, Podolsky, and Rosen) beams were generated by combining two independent squeezed beams produced by a parametric down-conversion process in a bulk nonlinear crystal. The work described below investigates the feasibility of miniaturizing the squeezed state generation process by monolithically integrating a three dimensionally (3D)-confined photonic crystal microcavity with a single-mode ridge waveguide in a sub-micron thick slab of AlGaAs. This was motivated by quantitative measurements of second-order sum-frequency generation in an InP membrane-based microcavity [10], which suggested that the inverse process of parametric down-conversion in the microcavity could possibly be used as a practical, ultra-miniature source of squeezed light if it can be efficiently coupled to a single mode waveguide. Already there have been reported attempts to use one-dimensional (1D) photonic crystal slabs and 2D photonic crystal waveguides to generate squeezing [11] and entangled photon pairs [12, 13, 14], but as far as we know this is the first attempt to quantify the squeezing process in cubicwavelength scale, 3D photonic crystal microcavity structures.

Due to the nonclassical characteristics of squeezed light, a quantum mechanical model is needed to describe the down-conversion process in the cavity, and how it couples to various leakage channels. The overall aim of this work was to obtain a quantitative estimate for the (nonclassical) spectrum of radiation that propagates away from a 3D, wavelength-scale microcavity that supports a distinct localized electromagnetic mode with frequency at ω ₁, and which is pumped/excited by a classical source at 2ω ₁. We adapt a system-reservoir quantization formalism from Ref.[15], that allows us to rigorously relate the output channels’ quantum field variables to those of the input channels, and the leakage from the excited mode in the cavity (the so-called input-output relations [16, 17, 18, 19]).

In the following, an interaction Hamiltonian due to the second-order nonlinear coupling between photons of the fundamental cavity mode (degenerate down-conversion) is added to the system-reservoir Hamiltonian and all the appropriate approximations are explained explicitly. The crucial component of this model, that must be solved numerically, is the overlap integral describing the nonlinear interaction of the classical excitation field at 2ω ₁, with the quantized cavity mode. This is evaluated using a finite-difference time-domain (FDTD) solver [20] to find the relevant amplitudes for all the fields involved in the parametric down-conversion process. All of these calculations are done using a photonic crystal cavity coupled to a single-mode ridge waveguide following a design that was recently fabricated and shown to exhibit coupling efficiencies in excess of 55% [21].

2. Field quantization in an open optical cavity

The system of interest consists of a defect-state optical microcavity in a thin semiconductor membrane perforated by a regular 2D array of through-holes, Fig. 1(a). The light is confined in the cavity by Bragg reflection in the plane (due to the photonic crystal bandgap) and conventional total internal reflection in the vertical direction. For appropriately designed defect states and large enough surrounding photonic crystal material, the intrinsic loss of the cavity is due entirely to out-of-plane scattering [22, 23]. When a 1D single-mode waveguide is introduced in proximity to the microcavity, as in Fig. 1(b), it is possible to have the coupling of the cavity mode(s) to this waveguide channel dominate the total leakage. It was shown that preferential coupling to a single-mode output channel can account for up to 90% of the total leakage [24].

For modeling purposes, this system is conceptualized as consisting of a cavity field that can be driven by polarization induced in the microcavity by classical and/or quantum field sources incident via the continua channels (either from the top or bottom half space radiation modes, or other in-plane waveguide channel modes). Each continuum channel carries away radiation excited by the incident fields, possibly mediated by resonant scattering in the cavity. Using the system-reservoir formalism [15], the cavity is considered as a system with a discrete set of quantized modes (associated with normal modes of an isolated cavity), and the reservoir consists of a set of continuum field operators which model all the distinct channels that the cavity can leak into.

Fig. 1. 2D photonic crystal microcavities, a) an isolated cavity and b) adding a 1D waveguide channel to the cavity structure.

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Here we introduce the key elements of the system-reservoir Hamiltonian formalism, which is derived using the Feshbach projection technique [25, 26]. The exact eigenmodes of the vector potential for the multi-mode photonic crystal microcavity and several continuum channels that couple to the cavity, f m(r,ω), are expressed in terms of the discrete eigenmodes of the cavity, U _λ (r), and the continuum of modes, V _n(r,ω′), in the channels [15, 27]:

f_{m} (r, ω) = \sum_{λ} α_{λ}^{m} (ω) U_{λ} (r) + \sum_{n} \int d ω' β_{n}^{m} (ω, ω') V_{n} (r, ω') .

The discrete indexm labels an asymptotic solution in which an incoming wave through channel m is scattered by the structure into all other channels. Here U _λ (r) is a localized cavity mode with a discrete spectrum, and V _n(r,ω′) is a continuum eigenmode of channel n. These cavity and channel modes are orthogonal and they have nonzero values only in the appropriate regions (i.e. inside or outside of the [arbitrary] 3D boundary that delineates the cavity). The coefficients α^m _λ and β^m _n (ω,ω′) are the coupling coefficients between the cavity/channel modes and the exact eigenmode associated with excitation of the system through channel m.

The quantization of the fields may be achieved by expanding the vector potential in a complete set of mode functions and imposing canonical commutation relations for the expansion coefficients [15, 28]. By using the eigenmodes of the structure, Eq.(1), the Hamiltonian of the electromagnetic field in a linear medium becomes:

\hat{H} = \sum_{λ} \overset{̅}{h} ω_{λ} {\hat{a}}_{λ}^{†} {\hat{a}}_{λ} + \sum_{m} \int d ω \overset{̅}{h} ω {\hat{r}}_{m}^{†} (ω) {\hat{r}}_{m} (ω)

where â _λ and â ^† _λ are the creation and annihilation operators of the discrete cavity modes, whereas r̂_m and r̂^† _m are the corresponding operators of the continuum modes of channel m. In addition, W_λm and T_λm are the coupling coefficients between the cavity and channel operators at the interface that defines the cavity region [15]. Equation (2) exactly describes the electromagnetic degrees of freedom in a textured dielectric medium. Its utility, for structures designed to locally support discrete modes coupled preferentially to a set of waveguide continua channels, is that it represents a rigorous formulation in terms of heuristic localized and continua mode amplitudes, regardless of whether or not the last term can be approximated as a weak damping contribution to the localized mode amplitude(s). Although the following derivation will only utilize this weak coupling limit, Eq. (2) provides a much more general starting point for analysis of practical systems that may well not satisfy the associated approximations.

Equation (2) contains resonant (â ^† r̂, r̂^† â) and nonresonant (â r̂, â ^† r̂^†) terms. In the weak coupling limit, when the photonic crystal microcavity modes have high quality factors, a rotating-wave approximation can be used and the nonresonant terms in the Hamiltonian can be ignored.

3. Parametric down-conversion in an open cavity

To address the parametric down-conversion process inside a single-mode cavity with a non-zero second-order susceptibility, χ ⁽²⁾, the Hamiltonian in Eq. (2) has to include an additional term:

{\hat{H}}_{I} = \frac{2 ε_{0}}{3} \int d^{3} r \hat{E} (r, t) . {\overset{\leftrightarrow}{χ}}^{(2)} (r) : \hat{E} (r, t) \hat{E} (r, t),

where Ê(r,t), the total field in the cavity region, is the superposition of the fields associated with the cavity mode and the pump field. The quantized field due to the cavity mode can be written as

{\hat{E}}_{c} (r, t) = i \sqrt{\frac{\overset{̅}{h} ω_{1}}{2 ε_{0}} [{\hat{a}}_{1} (t) U_{1} (r) - {\hat{a}}_{1}^{†} (t) U_{1}^{*} (r)] .}

The pump field, which is taken to be a classical coherent state at 2ω ₀ (assuming ω ₀≃ω ₁, where ω ₁ is the cavity mode frequency) is written as:

E_{p} (r, t) = - i A_{p} (t) [U_{p} (r) e^{- 2 i ω_{0} t} - U_{p}^{*} (r) e^{+ 2 i ω_{0} t}],

where A_p and U _p(r) are the envelope function and the normalized spatial distribution of the pump field in the cavity region respectively. By substituting the total field Ê(r,t)=Ê_c(r,t)+E _p(r,t) in the Hamiltonian of Eq.(3) and only considering the resonant terms we have

{\hat{H}}_{I} = i \overset{̅}{h} [g {\hat{a}}_{1}^{†} (t) {\hat{a}}_{1}^{†} (t) e^{- 2 i ω_{0} t} - g^{*} {\hat{a}}_{1} (t) {\hat{a}}_{1} (t) e^{2 i ω_{0} t}],

where

g = \frac{ω_{1}}{3} A_{p} (t) \int d^{3} r [U_{1} . {\overset{\leftrightarrow}{χ}}^{(2)} : U_{1} U_{p} + U_{1} . {\overset{\leftrightarrow}{χ}}^{(2)} : U_{p} U_{1} + U_{p} . {\overset{\leftrightarrow}{χ}}^{(2)} : U_{1} U_{1}] .

Here we suppose the spatial distribution of the fields in Eq.(7) are real valued functions and therefore g is a real number. By adding the Hamiltonian of the nonlinear interaction, Eq.(6), to the original system-reservoir Hamiltonian (of a single-mode cavity), Eq.(2), and Fourier transforming the Heisenberg equation of motion for the cavity mode we get:

{\tilde{a}}_{1} (Ω) = \frac{2 g \sqrt{2 γ_{1}} {\tilde{r}}_{1}^{in †} (- Ω) + 2 g \sum_{m = 2} \sqrt{2 γ_{m}} {\tilde{r}}_{m}^{in †} (- Ω)}{[Γ - i (Δ + Ω)] [Γ + i (Δ - Ω)] - 4 g^{2}}

where we used a rotating frame in which, $a_{1} (t) = {\tilde{a}}_{1} (t) e^{- i ω_{0} t}$ , and Δ=ω ₀-ω ₁ is the offset of the cavity mode with respect to the half of the pump frequency. Here γ_m=π|W _1m|² (m=1, …) represents damping of the cavity mode due to the coupling into channel m and Γ=Σ_m γ_m accounts for the total loss of the cavity mode.

We made two assumptions to derive the above equation of motion, Eq.(8). First we supposed that due to the weak coupling of the cavity mode to the reservoir, its eigenfrequency is not affected by the reservoir. The second assumption is the Markov approximation (reservoir correlation times are assumed negligibly short compared to the characteristic time scale associated with the cavity mode dynamics [17]), in which we assume that the coupling coefficients γ_m(ω) are independent of frequency.

In writing Eq. (8), we separated the channels that a microcavity mode can communicate with into two parts. Channel 1 is a 1D single-mode waveguide and r̃_m (m=2, …) are the rest of the reservoir operators which the cavity modes can couple with, Fig. (2).

Fig. 2. The model cavity that communicates with several output channels.

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To measure the degree of squeezing in the output channel 1, we have to evaluate the following correlation functions [17, 28]:

S_{X} (Ω) = < X^{out} (Ω), X^{out} (- Ω) > - 1,

S_{Y} (Ω) = < Y^{out} (Ω), Y^{out} (- Ω) > - 1,

where S_X (Ω) and S_Y (Ω) are called the spectrum of squeezing for the X and Y field quadratures respectively, with

X^{out} (Ω) = {\tilde{r}}_{1}^{out} (Ω) + {\tilde{r}}_{1}^{out †} (- Ω),

The usual derivation of S_X and S_Y [16, 17] for a single continuum channel combines Eq.(8) with the input-output relation [15, 16]

{\tilde{r}}_{1}^{out} (Ω) - {\tilde{r}}_{1}^{in} (Ω) = - \sqrt{2 γ_{1}} {\tilde{a}}_{1} (Ω),

and uses the fact that for a reservoir in the vacuum state <rⁱⁿ ₁(Ω)rⁱⁿ ^† ₁(Ω)>=1, and <rⁱⁿ ₁ (Ω)rⁱⁿ ₁ (Ω)>=<rin ^† ₁(Ω)rⁱⁿ ₁(Ω)>=<rⁱⁿ ^† ₁(Ω)r_in ^† ₁(Ω)>=0. The result is

S_{X} (Ω) = \frac{8 g γ_{1}}{{(γ_{1} - 2 g)}^{2} + Ω^{2}},

and

S_{Y} (Ω) = \frac{- 8 g γ_{1}}{{(γ_{1} + 2 g)}^{2} + Ω^{2}} .

By using Eq.(8) with the generalized input-output relation,

{\tilde{r}}_{m}^{out} (Ω) - {\tilde{r}}_{m}^{in} (Ω) = - \sqrt{2 γ_{1}} {\tilde{a}}_{1} (Ω),

and the generalized expectation values for reservoir vacuum fields (<rⁱⁿ_n(Ω)rⁱⁿ ^† _m(Ω)>=δ_nm and <rⁱⁿ_n(Ω)rⁱⁿ_m(Ω)>=<rⁱⁿ ^† _n(Ω)rⁱⁿ ^† _m(Ω)>=<rⁱⁿ ^† _n(Ω)rⁱⁿ_m(Ω)>=0), one gets

S_{X} (Ω) = \frac{8 g γ_{1}}{{[(γ_{1} + γ_{x}) - 2 g]}^{2} + Ω^{2}},

and

S_{Y} (Ω) = \frac{- 8 g γ_{1}}{{[(γ_{1} + γ_{x}) + 2 g]}^{2} + Ω^{2}},

where

γ_{x} = \sum_{m = 2} \sqrt{γ_{m}},

Before we show the numerical calculation of the spectrum of squeezing in a parametric down conversion process generated in a photonic crystal microcavity, it is useful to compare the squeezing spectra for the single and multiple channel cases in the limit of zero detuning. Figure 3(a) shows the spectrum of squeezing for the Y quadrature for two different cases, when there is only one output channel, the waveguide channel (solid line), and when loss through channels 1 and all other channels are equal, for the threshold situation when 2g=γ ₁+γ_x. When the nonlinear cavity is coupled only to a single-mode channel, S _Y=-1 at threshold when Ω=0. In this situation there is a perfect correlation between the Y field quadratures of the degenerate photons at the waveguide output [31, 32, 33]. By plotting Eq. (17) versus g for Ω=0, Fig. 3(b), in the single channel case (solid), and the case where there is equal coupling to the waveguide and all other channels (dashed), we see that the degree of squeezing is always less in the latter case, and at threshold the maximum amount of squeezing is

S_{Y} (Ω = 0) = - \frac{γ_{1}}{γ_{1} + γ_{x}},

rather than unity. In another words the squeezing degree is equal to the coupling efficiency into the single-mode waveguide channel.

Fig. 3. a) Spectrum at threshold of squeezing for the Y quadrature in a degenerate down-conversion process. b) Squeezing versus g factor at Ω=0. The solid lines are for the case when the cavity couples to a single channel (γ_x=0) and the dashed lines are when γ_x=γ ₁.

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4. Numerical estimation of the squeezing spectrum

Crucial steps in quantitatively estimating the degree of squeezing expected for a realistic cavity/waveguide system, excited by a realistic pump source involve obtaining numerical estimates of g (the nonlinear coupling coefficient in Eq.(7)), and γ ₁ and γ_x (cavity mode losses through channel 1 and all other channels). To estimate g for a particular microcavity and host material, one needs to determine the tensor χ ^⃡(2), and the 3D field profiles in the cavity region corresponding to the localized microcavity mode and the pump field.

Here we consider a “3-missing hole” cavity [34] made in an Al _0.3Ga_0.7As membrane (the refractive index, n=3.23 around 1.5 µm). The concentration of Al in Al _0.3Ga_0.7As is chosen such that its bandgap [35] is at 1.798 ev (or 689 nm), so that the pump beam (at ~720 nm, twice the cavity mode frequency) does not generate any free carriers. Free carrier generation in the cavity region changes the refraction index of the medium and induces undesired losses due to free carrier absorption (Any loss in the waveguide-cavity structure introduces uncorrelated fluctuations, which degrade the squeezing degree [16]. The overall loss can be modeled as an extra channel that the light can couple into). Free carriers can still be generated by two photon absorption of the pump, but that is a weaker process and can’t be easily avoided using conventional III–V semiconductors.

The value of the g coefficient, Eq.(7), depends on the χ ^ˡ(2) symmetry of the host photonic crystal material. We used a coordinate system in which the microcavity and incident pump field polarization are in the x-y plane. Al_0.3Ga_0.7As belongs to the 4̄3m symmetry group and if the wafer used to fabricate the microcavity is oriented in the [001] direction, then χ ^ˡ(2) has the form

χ_{[001]}^{(2)} (- ω_{1}; 2 ω_{0}, - ω_{1}) = (\begin{matrix} χ_{xxx}^{(2)} & χ_{xxy}^{(2)} & χ_{xxz}^{(2)} & χ_{xyx}^{(2)} & χ_{xyy}^{(2)} & χ_{xyz}^{(2)} & χ_{xzx}^{(2)} & χ_{xzy}^{(2)} & χ_{xzz}^{(2)} \\ χ_{yxx}^{(2)} & χ_{yxy}^{(2)} & χ_{yxz}^{(2)} & χ_{yyx}^{(2)} & χ_{yyy}^{(2)} & χ_{yyz}^{(2)} & χ_{yzx}^{(2)} & χ_{yzy}^{(2)} & χ_{yzz}^{(2)} \\ χ_{zxx}^{(2)} & χ_{zxy}^{(2)} & χ_{zxz}^{(2)} & χ_{zyx}^{(2)} & χ_{zyy}^{(2)} & χ_{zyz}^{(2)} & χ_{zzx}^{(2)} & χ_{zzy}^{(2)} & χ_{zzz}^{(2)} \end{matrix})

where β=216 pm/V for ω ₁~1.5 µm [36]. The sparse nature of this tensor restricts the possible orientations of the photonic crystal with respect to the electronic axes of the host crystal. If a transverse pump field is incident normal to a [001] oriented crystal, only a z-oriented second order polarization can be generated, which does not couple to the microcavity modes that derive from TE-polarized slab modes, and are therefore polarized in-plane. To reduce these restrictions, we consider wafers grown along [111] direction. The χ ^ˡ(2) tensor for wafers oriented along [111] is of the form:

χ_{[111]}^{(2)} (- ω_{1}; 2 ω_{0}, - ω_{1}) = β (\begin{matrix} \frac{3}{\sqrt{3}} & 0 & \frac{\sqrt{2} - 2}{\sqrt{3}} & 0 & - \frac{1}{\sqrt{3}} & 0 & \frac{\sqrt{2} - 2}{\sqrt{3}} & 0 & \frac{1 - 2 \sqrt{2}}{\sqrt{3}} \\ 0 & - \frac{1}{\sqrt{3}} & 0 & - \frac{1}{\sqrt{3}} & 0 & - \frac{\sqrt{2}}{\sqrt{3}} & 0 & - \frac{\sqrt{2}}{\sqrt{3}} & 0 \\ \frac{\sqrt{2} - 2}{\sqrt{3}} & 0 & \frac{1 - 2 \sqrt{2}}{\sqrt{3}} & 0 & - \frac{\sqrt{2}}{\sqrt{3}} & 0 & \frac{1 - 2 \sqrt{2}}{\sqrt{3}} & 0 & \frac{3 \sqrt{2}}{\sqrt{3}} \end{matrix})

The nonlinear coupling coefficient also depends on the spatial distribution of the cavity mode and the pump laser, which can be determined using an FDTD simulation [20], as described in the next section.

5. Degenerate parametric down-conversion

We consider a 3-missing hole cavity in an hexagonal 2D photonic crystal structure to estimate the squeezed spectrum in the case of degenerate parametric down-conversion involving a single cavity mode pumped at twice its natural frequency. The structure is shown schematically in Fig. 4(a), it has a lattice constant of a=420 nm, an air hole radius r=0.29a, and the semiconductor membrane thickness is h=200 nm. Three holes next to the cavity are shifted to increase the cavity Q factor [37]. The shift at the A, B, and C locations are 0.2a, 0.025a, and 0.2a respectively. The cavity is tilted with respect to the waveguide axis, Fig. 4(b), to increase its coupling efficiency to the waveguide [24].

The cavity shows a fundamental mode at f ₁=207.98 THz (1.44 µm). The spatial distribution of the cavity mode and the pump beam at 2f ₁=415.96 THz (~721 nm), with the same polarization as the cavity mode and focussed to a spot size of 2.5 µm, are shown in Fig. (5). The quality factor of both an isolated cavity and one connected to the single-mode ridge waveguide as in Fig. 4 (b), are found to be Q_i=79420, and Q_T=20940. By using

\frac{1}{Q_{T}} = \frac{1}{Q_{i}} + \frac{1}{Q_{wg}},

we can extract the quality factor of the cavity mode due to its coupling with the waveguide, which is Q_wg=28440. The values of the cavity losses for the model calculation are thus γ1

γ_{1} = ω_{1} ⁄ Q_{wg} = 46 GHz, γ_{x} = ω_{1} ⁄ Q_{i} = 16 GHz .

The design and properties of the ridge-waveguide and its interface with the PhC waveguide is described in Ref. [21]. If a CW excitation beam with an average power of 10 mW is assumed, Eq. (7) yields

g_{[111]} = 0.839 GHz .

By using 500 ps pulses (the linewidth of the cavity mode is ~16 larger than the pump laser linewidth in this case.) with 80 MHz repetition rate and 10mWaverage power, the g coefficients for the [111] direction increases to:

g_{[111]} = 4.199 GHz .

Figure (6) shows the calculated spectrum of squeezing of the Y quadrature for the [111] growth direction for the 500 ps pump laser case. The squeezing is ~30% below the shot noise level. The threshold of the degenerate down-conversion process for this structure is at 2_g=γ ₁+γ_x=62 GHz, therefore, with the above chosen pump beam parameters we are well below the threshold. The optimum squeezing degree for the structure at threshold is ~70%.

Fig. 4. a) Shift of the holes next to the cavity in order to increase its Q and also its coupling efficiency to the waveguide, b) the cavity is tilted with respect to the waveguide to boost their coupling efficiencies.

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Fig. 5. Intensity profile of a) X component and b) Y component of the electric field associated with the cavity in Fig.(4), and c) total intensity of the pump beam in the vicinity of the cavity. Center of the cavity is located at (x=0,y=0).

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Fig. 6. The spectrum of squeezing for the Y quadrature of the sample in Fig. (4) for a crystal oriented along the [111] direction pumped with 500 ps pulses of 80 MHz repetition rate and 10 mW average power.

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In summary, a 30% squeezing of the light coupled into a single-mode ridge waveguide at λ=1.44µm is predicted, when that waveguide is terminated in an L3 planar photonic crystal microcavity that is excited from the top half-space by a classical source at λ=0.72µm of 10 mW average power with 500 ps pulse duration and an 80 MHz repetition rate. The calculation is based on a degenerate down-conversion formalism for a cavity coupled to multiple continua, with the linear and nonlinear coupling parameters obtained using FDTD simulations of a realistic Al_0.3Ga_0.7As membrane structure. In order to maximize the coupling efficiency between the cavity and the single-mode channel, the cavity was tilted 60 degrees with respect to the waveguide axis on a [111] oriented AlGaAs crystal. The Q of the cavity without the waveguide was 80000. This amount of squeezing is within a factor of 2 of the maximum possible at the threshold of the process, given the coupling efficiency to the single channel output of 70%. The calculation was based on using a bulk AlGaAs alloy membrane with no attempt made to resonantly enhance χ ⁽²⁾ using quantum wells or quantum dot resonances.

Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Canadian Institute for Advanced Research, the Canadian Foundation for Innovation, and the British Columbia Knowledge Development Fund.

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Squeezed state generation in photonic crystal microcavities

Abstract

1. Introduction

2. Field quantization in an open optical cavity

3. Parametric down-conversion in an open cavity

4. Numerical estimation of the squeezing spectrum

5. Degenerate parametric down-conversion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (30)

Optics Express