\documentclass[preprint]{revtex4} %\documentclass[prl,aps,multicolumn,showpacs]{revtex4} %\documentclass[prl,aps,twocolumn,preprintnumbers]{revtex4} \usepackage{epsfig,amssymb,amsfonts} \begin{document} \title{The world's smallest tunable interferometer and optical quantum computing} \date{\today} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \author{Dimitris G. \surname{Angelakis}$^{1}$}% \author{Marcelo F. \surname{Santos}$^{2}$}% \author{Vassilis \surname{Yannopapas}$^{3}$}% \author{Artur \surname{Ekert}$^{1}$}% \affiliation{$^{1}${\it Centre for Quantum Computation, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, U.K.} \\ {\it $^{2}$ Dept. de F\'{\i}sica, Universidade Federal de Minas Gerais, Belo Horizonte, 30161-970, MG, Brazil .}\\ {\it $^{3}$ Condensed Matter Physics Group, Blackett Laboratory, Imperial College, London, SW7 2BW, UK.}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} Highly efficient waveguides formed from linear chains of defects inside photonic crystals can provide the ideal environment for the manipulation of photonic pulses. With extremely high transmission rates, quantum superpositions of zero or one photon propagate through these waveguides virtually free of dissipation. Based on these waveguides, a fully tunable, micrometer size Mach-Zehnder Interferometer (MZI) can be realized. Furthermore efficient quantum logical manipulation of the photons can be implemented through the interaction with atoms doped in the structure. These atoms can be easily manipulated using external EM fields and provide the necessary tools to operate highly controllable one and two-qubit quantum gates on those propagating photons. \end{abstract} \maketitle \preprint{Version 20} % {\it NOTE FOR THE READER: This is a very provisional draft in progress. Specially % on the reference section very little has be done. Any comments welcome at %d.angelakis@damtp.cam.ac.uk.} \section{Introduction} {\it Universal quantum computation requires some basic elements such as decoherence free qubit manipulation, protected memory and information flow, and scalability. Photons have been proposed as possible qubits due to their resilience against decoherence, result of their usually non-interacting nature. However, this very same nature makes it difficult to manipulate photonic qubits. A new combination of propagation through photonic crystals waveguides, Mach-Zehnder interferometry and cavity quantum electrodynamics may present a complete set of tools for scalable optical quantum computation.} %It has been shown that defects waveguides in photonic crystals %provide a proper environment for photonic qubits propagation. With %extremely high transmission rates, quantum superpositions of zero %or one photon may propagate through these structures virtually %free of decoherence. However, qubit propagation alone does not %account for universal quantum computation. It turns out that built %in Mach-Zehnder interferometers allow for logical manipulation of %those qubits, while microcavity quantum electrodynamics inside %doped defects provide the remaining necessary tools to operate %highly controllable one and two-qubit quantum gates on those %propagating photons. \section{Part I-Tunable micro-MZI in a photonic crystal } \subsection{Introduction-Photonic crystals, defects and interferometry} A typical MZI consists of two arms of equal length, joined by two beam splitters \cite{rama}. In the simplest case, the introduction of a dielectric material inside one of the arms of the MZI induces a phase lag between the corresponding light paths which can be detected as oscillations in the intensity of the output ports after the second beam splitter. These oscillations are usually a harmonic function of the phase lag and result from the interference of the output light signals in the beam splitters. The MZI we have in mind is based on two-dimensional (2D) photonic crystals. The latter are artificial structures which consist of naturally occurring materials arranged so that the refractive index is periodic in two dimensions and homogeneous in the third \cite{joanno}. If the difference between the indices of refraction of the materials composing the photonic crystal is high enough, a photonic band gap can emerge, i.e. a forbidden frequency region in all in-plane directions and both polarizations. The size and periodicity of these structures are of the same order of magnitude as the wavelength of the light under investigation. A typical photonic crystal exhibiting gaps in optical frequencies should be periodic in the micrometer scale for example. An example of a 2D photonic crystal with an absolute band gap is an hexagonal lattice of air columns in GaAs or Si \cite{joanno}. One of the many ways to introduce a defect inside the above photonic crystal is to fill one of the air holes with a dielectric material which has a different refractive index from that of the material surrounding the air holes or to simply subtract a hole and replace it with the background material. A single defect introduces a bound state of the EM field within the photonic band gap. This defect can act as a high-Q cavity since the EM state is highly localized within the defect area and decays exponentially away from it in all directions. When many defects are brought together so as to form a linear chain, the states of neighboring defects interact with each other giving rise to a band of allowed modes within the absolute gap, allowing for light propagation solely along the direction of the chain. These chains are known as coupled-cavity waveguides (CCW) \cite{stef,yar1,oz1,yanno1}. Due to the weak coupling between the cavities light does not propagate freely along the CCW as in conventional dielectric waveguides but through a hopping mechanism between neighboring cavities. This mechanism allows for essentially lossless guiding, bending and splitting of light as well as for very small values of the group velocity \cite{oz2}. When an EM pulse is injected inside a CCW, it propagates almost undistorted with the same group velocity as long as the frequency width of the pulse is shorter than the width of the CCW band \cite{mook}. As can be seen from Fig.~\ref{pulse}, the intensity of the pulse is appreciable inside the defects of the CCW whilst vanishing in the space between them. The intensity of the field inside a CCW is higher than that of the input pulse in free space due to the reduced group velocity inside the pulse \cite{mook}. This spatial compression of the pulse is to our benefit since it facilitates the use of the quantum optical scheme we propose later in order to control the proposed photon qubit. \subsection{A tunable micrometer size, photonic crystal MZI} In the photonic crystal MZI presented here, each of the two arms is a CCW along which light is allowed to propagate. At the region of close proximity of the two arms/CCWs (one lattice constact apart) light can tunnel from one to the other enabling the splitting (or coupling) of light modes propagating in the arms. As can be seen from Fig.~\ref{fig1} our MZI has two input and two output ports. Assuming that only one of the two inputs, say input 1, is active, it can been shown \cite{marti1} that the power transfer function has the same form as that of a conventional MZI \cite{rama}. Namely, \begin{eqnarray} T_{11}=\sin^{2}(\Delta \phi) \nonumber \\ T_{12}=\cos^{2}(\Delta \phi) \label{eq:transfer} \end{eqnarray} where $T_{ij}$ represents the power transfer function from input $i$ to output $j$ and $\Delta \phi$ is the phase lag between the two output signals. Note that for the case of CCW-based MZIs \cite{marti1} as well as for other photonic crystal-based MZIs \cite{shih,borel}, no active control over the phase lag- between the two arms has been suggested. For different values of the phase lag, new structures with a different ratio of arms length needed to be constructed in each individual case. This could be avoided if a medium with tunable refractive index is placed in one of the arms. For example, one could use optically birefringent nematic liquid crystals infiltrated inside one or more defects \cite{BJ99}. In this work, we suggest to use quantum optical means to change the index of refraction of CCWs. This is possible by doping the defects in the arms of the interferometer with selections of atoms and controlling, with external classical fields, the quantum mechanical interaction between those doped atoms and the travelling photons. % %This phenomenon is known in CQED as frequency pulling and occurs %due to the dispersive interaction between the atom and the field. The controlling mechanism is based on the Stark shift of the electronic levels of the doped atoms. The particular choice of the externally applied field depends on the typical frequencies of the selected internal atomic states and the timescale of the required opearation; it can be a laser field for optical transitions or even a DC electric field for microwave transitions. The basic idea, commonly used in quantum optical experiments, is to use the energy shifts induced by the external field as switches to tune the interaction between the atoms and the propagating photons. Assume a photon is inserted in the structure from the left, and starts hopping its way to the right-Fig. 2. The atomic transition of the doped atoms $\omega_{gh}$, is initially far off resonant with those of the hopping photon $\omega_p$ and the defects $\omega_d$, while the latter two are in resonance with each other: $\omega_{gh}>>\omega_p=\omega_d$. The photon hops freely and unchanged from one defect to the other and the CCW supports the propagating mode(Fig. \ref{MZI}). As soon as the photon reaches the arms area the switching field is applied in one of these arms(this could be up to the full length of the structure), bringing the atomic frequencies of these dopants close but still of resonance to that of the hopping photon. This generates what is commonly known as a near-resonant dispersive interaction. Here the detuning $\Delta$, between an atomic transition frequency $\omega_{gh}$ and the frequency $\omega_{p}$ of the incoming light field, is smaller than each of those frequencies individually while, at the same time, it remains much bigger than the coupling constant $\Omega$ between the atom and the light field: $\omega_{gh},\omega_{p} >>\Delta=\omega_{gh}-\omega_p>>\Omega$. In this case the combined system eigenstates undergoe the well known AC-stark shift and acquire a phase proportional to $(\Omega^2/\Delta)T$, where $T$ is the tunable interaction time. This translates to the amplitude of the photon being in one of the CCWs to experience a phase shift in respect to the other one and thus solves the tunability problem. We note the simplicity of the shifting mechanism which originates from the fact that knowledge for the exact location of the photon is not essential. The photon will interact only with the atoms in the specific defect that it happened to reside in at the time of the field shift. Due to the physics of the hopping mechanism (tunneling) and the strong localization of the photon within only one defect each time, the photon spends most of its life in defects rather than between them. This allows for the scheme to work efficiently as the switching of the external field always ``catches" the photon inside some defect rather in between them(see analytical discussion at the end on relevant time scales and coupling strengths) Once the required shift is applied, the photon needs to be ``released" to continue its forward propagation towards the second beamsplitter. This is done as follows: First note that the application of the switching field inducing the phase shift causes the resonance frequencies $\omega_{d}$ of the neighboring empty defects to change, a phenomenon known in Cavity QED as frequency pulling\footnote {This also contributes to the prevention of leakage during the phase shift operation}. This shift of the defects' frequencies means they don't support the photon mode anymore and they actually reflect(the CCW channel has closed). Selectively turning off the external field to the right of the defect where the phase shift was induced (supposing left to right propagation) opens half of the channel and forces the photon to leak to the right and continue propagating in the proper direction (Fig. \ref{MZI}). A rough estimate of the photon flying time inside the structure suffices in this case to estimate where the external field should be turned off. %The easiest way would be to time %the shifting operation when the photon is roughly half way %through the device. In this case the releasing would occur %by switching off the field in the right half. Photon flying times %can be estimated by dividing the length of the device with the group %velocity. Typical values of these are $100\mu m$ and $10^{-4}c$ %which gives crossing times of the order of ns. We note again the applicability of the above mechanism, characteristic of the ``global" addressing of the external field through the full extend of the structure. This idea will also be the main tool for the implementation of non-local quantum gates between pairs of travelling photons as discussed next.%[{\it More research needed on current literature re MZIs. %Elaborate a bit more on Eq. 1. Describe possible uses in optical %circuitry, integrated photonics. Add appropriate references}] \section{Part II-Efficient optical quantum computing} \subsection{Mechanism for strong nonlinear interaction} %Before elaborating on the explicit form using the CCWs to %implement quantum The qubits to be manipulated are encoded in the dual rail representation. Each logical qubit consists of two modes of light carrying a single photon between them. The first logical qubit will be characterized by $|0_{L} \rangle_{1}= |10 \rangle_{1}$ where the photon travels in the lower mode of the pair and $|1_{L} \rangle_{1}= |10 \rangle_{1}$ when it travels on the upper one. Similarly the corresponding states for qubit 2 will be $|0_{L} \rangle_{2}= |10 \rangle_{2}$, $|1_{L} \rangle_{2}= |01 \rangle_{2}$ In our device each of the above physical photon modes will be realized by a CCW. The beamsplitting operation is perfomed by bringing the corresponding modes together in a beamsplitter (beam-coupler) fashion as described in the previous section. The single qubit operation can be easily implemented using a beamsplitter and a phase gate. {\it describe in detail single qubit gate here?} %For a single qubit rotation of the form $\psi_{in}$ In order to achieve universal quantum computation, one also needs to operate the controlled phase gate. We show that three-level-atoms doped in the arms of the photonic crystal MZI allow for the efficient implementation of strong nonlinear phase-shifts allowing for the efficient implementation of any non-local quantum gate. Before elaborating on the dynamics of the atom-photon interaction, let us recall that the sought action of our non-linear device on two photons should be the following: $|00 \rangle \rightarrow |00 \rangle$, $|01 \rangle \rightarrow |01 \rangle$, $|10 \rangle \rightarrow |10 \rangle$, $|11 \rangle \rightarrow -|11 \rangle$. Here 0 and 1 correspond to the photon occupation numbers for each input mode. In the specific case considered here, the states before the arrows are the entries in the MZI device while the states after the arrows are its outcomes. Note now that the evolution of two identical input photons in our MZI when no phase shift is applied is as follows: $|00 \rangle \rightarrow |00 \rangle \rightarrow |00 \rangle$, $|01 \rangle \rightarrow \frac{|01 \rangle - |10 \rangle}{\sqrt{2}} \rightarrow |01 \rangle$, $|10 \rangle \rightarrow \frac{|01 \rangle + |10 \rangle}{\sqrt{2}} \rightarrow |10 \rangle$, $|11 \rangle \rightarrow \frac{|20 \rangle - |02 \rangle}{\sqrt{2}} \rightarrow |11 \rangle$, where the first and second arrows denote the transformations in the first and second beam splitters of the MZI respectively. Assume now that there exists a nonlinear medium that when inserted in the arms of our MZI does not affect the vacuum and 1-photon states, but rotates the two-photon state $|2 \rangle$ to $-|2 \rangle$. Consequently the states $|00 \rangle$, $|01 \rangle$ and $|10 \rangle$ do not change, while the states $|20 \rangle$ and $|02 \rangle$ both acquire the same phase $\pi$. It's straightforward to see that this exactly implements the controlled-phase gate. Consider a scheme similar to the one we described above for the case of the simple phase shift, where groups of three-level atoms (levels $g$, $h$ and $e$) are placed inside the defects forming the arms of the MZI and pairs of photons are inserted from the left. The electronic levels form a cascade with similar transition frequencies $\omega_{gh} \sim \omega_{he}$ and they couple linearly to the hopping photons through electro-dipole interactions, as shown in Fig.\ref{MZI}. In the rotating wave approximation, the dynamics of this system is described by the following Hamiltonian ($\hbar=1$) \begin{eqnarray} H_0=\omega_{0}a^\dagger a + \omega_{g} \sigma_{gg} + \omega_{e} \sigma_{ee} + \omega_{h} \sigma_{hh} +\nonumber\\ g_1(\sigma_{gh} a^\dagger + \sigma_{hg} a) + g_2(\sigma_{he} a^\dagger + \sigma_{eh} a) \end{eqnarray} which, written in the interaction picture, reads \begin{eqnarray} H_{int}=g_1 \sigma_{gh} a^\dagger e^{i\delta t} + g_2 \sigma_{he} a^\dagger e^{-i \delta t} + h.c. \end{eqnarray} $\delta=\omega_0-(\omega_h-\omega_g) = (\omega_e-\omega_h)-\omega_0$. $a^\dagger$, $a$ are the creation and annihilation operators for a photon trapped in a defect and $\sigma_{ij}$ are the corresponding atom operators with $i,j=g,h,e$. $g_{1},g_{2}$ are the corresponding coupling constants for the two transitions. %These could either be through coupling each one of them %to extra atomic levels like Zeeman level - radio frequency, %excited levels - optical frequency, or they can be shifted %together by means of static electric fields. The photons are injected to the MZI from the left and when they are through the first beam splitter, the external field is applied to the whole extent of the MZI, shifting the atomic levels in all defects so that $|\delta_1|=|\delta_2|>>g_1,g_2$. This translates in having the individual single photons transitions being equally detuned from the corresponding atomic ones, whereas the two photon $|g2 \rangle \rightarrow |e0 \rangle$ one is resonant(Fig. \ref{MZI}). In this case the dynamics of this system can be efficiently described by the following effective Hamiltonian~\cite{twophotons}: \begin{equation} H_{eff}=-\frac{g_1^2}{\delta}\sigma_{gg} (a^\dagger a) + \frac{g_2^2}{\delta}(\sigma_{ee} a a^\dagger) + \frac{g_1 g_2}{\delta}(\sigma_{ge} {a^\dagger} ^2 + \sigma_{eg} a^2) \label{eq:eff} \end{equation} Due to the second-order nature of this interaction, there are energy corrections to both atomic states $e$ and $g$ that depend on the energy of the photons. The intensity of the applied external field can be chosen in order correct this and tune the atoms back to resonance with the two photon transition $|g \rangle |2 \rangle \rightarrow |e \rangle |0 \rangle$. This translates into adding the extra amount of $2\frac{g_1^2}{\delta} - \frac{g_2^2}{\delta}$ to the energy of most excited level. Assume, now, that the atom is initially in level $|g \rangle$. If the two photons are in resonance with the $|g \rangle \rightarrow |e \rangle$ transition, it is straightforward to see that the following table for the evolution of the joint atom-field state after a time T is true\cite{BRH87} \begin{eqnarray} &&|g \rangle |00 \rangle \rightarrow |g\rangle |00 \rangle, \\ &&|g \rangle |01 \rangle \rightarrow e^{-i \phi_1}|g \rangle |01 \rangle, \\ &&|g \rangle |10 \rangle \rightarrow e^{-i \phi_1}|g \rangle |10 \rangle,\\ &&|g \rangle |20 \rangle \rightarrow e^{2 i \phi_1} [\cos{k T}|g \rangle |20 \rangle + \sin{k T} |e \rangle |00 \rangle],\\ &&|g \rangle |02 \rangle \rightarrow e^{2 i \phi_1} [\cos{k T}|g \rangle |02 \rangle + \sin{k T} |e \rangle |00 \rangle] \end{eqnarray} where $k=\frac{(g_1 g_2 \sqrt{2})}{\delta}$, and $\phi_1 = \frac{(g_1)^2 T}{\delta}$. Notice that the effective Rabi frequency of this transition is typically from a two-photon, second order process. When $kT = \pi$, the two-photon interaction completes a full Rabi oscillation, acquiring a total phase $\phi_T = \pi + 2 \phi_1$, where $\phi_1 = \frac{g_1 \pi}{g_2 \sqrt{2}}$. If the ratio between coupling constants is $g_1/g_2 = 2 \sqrt{2}$, then $\phi_1 = 2 \pi$ which means that the two-photon state acquires a minus sign while the remaining states are brought back to their originals. Under these conditions, the time-evolution table showed above reproduces exactly the nonlinear $\pi$ phase shift required. If $g_1 \neq g_2 2 \sqrt{2}$, then small corrections in the energy of state $|g \rangle$ are enough to assure the $2 \pi$ dephasing of the one-photon states and to implement the desired quantum gate. %?????????????????? For the experimental realization, we only need to find atoms having two similar frequency transitions which couple to the field according to the above ratio of coupling strengths. Note here that obviously the control phase gate can be also achieved for any ratio $g_{1}/g_{2}=2\sqrt{2}k/k^{\prime}$ where $k$ and $k^{\prime}$ are integers but this is the simplest one. We stress the fact that again the initial and final shifting, the rotation and the releasing to the proper direction can be performed in the same global fashion as the simple MZI phase shift mechanism described above. For the non-local phase shift, it is actually even simpler than the former as the external fields act in both arms of the MZI in the same way. \subsection{Experimental feasibility and range of parameters} The range of experimental parameters for the feasibility of above scheme is discussed next. The important quantities are the relevant timescales of the operations described above, compared to various decoherence times for the photonic qubits. In our case there are almost zero leaking losses from the CCW, the dielectric is highly non absorptive and as always, photons don't interact with each other at all except when we want them to. The only source of decoherence is from the decay of the defect cavity modes during the interaction with the doped atoms. This needs to be compared with the typical coupling constants and the switching times of the external fields. % To evaluate the coupling strength and the defect decay time we % need following quantities: % The modal volume $V_{mode}$ % of a defect cavity and the critical atom and photon numbers %$m_0=(\frac{\gamma_t}{2g})^2$, $N_0= \frac{2 \kappa %%\gamma_t}{g^2}$. %The cavity decay rate $\kappa = \frac{\omega_d}{4 \pi Q}$ (Q is %the quality factor and $\omega_d$ the resonance frequency), %$\gamma_t$ is the atomic dipole decay rate and % g is the coupling constant %($g_0 = %%\gamma_t \sqrt {\frac{V_0}{V_{mode}}}$ is the Rabi coupling %%constant). Typical values of the above quantities for the defect-doped atom couplings can be found in Ref.~\onlinecite{scherer,Ye04}. Specifically for the hexagonal case of air holes in a dielectric where defects are formed by the absence of a hole as described in section IV of Ref.~\onlinecite{scherer}, quality factors of few thousand can be achieved. The cavity decay rate in this case is roughly $10^{10}$Hz corresponding to a photon lifetime of the order of $T_{def}=1ns$. Both the MZI phase shift operation and the two photon nonlinear phase shift should occur faster than that. The coupling constant $g$ for the individual atom-photon coupling for say the D2 atomic transition (852nm) of a doped atom of $^{133}Cs$ in each micrometer defect is equal to $3* 10^9Hz$. For the MZI phase shift, the maximum induced phase is $\sqrt Ng^2/\Delta*T_{def}$. If $\Delta\sim 10^{12}$Hz, a collection of 10000 atoms results in a required operation time of 0.01ns, which is more than sufficient to induce any phase between 0 and $\pi$ before the photon leaks out. For the two photon nonlinear phase shift, the two photon Rabi frequency goes as $Ng_1g_2/\Delta$. As $g_{1}$ is very close to $g_{2}$ and both of the order of $3*10^{9}Hz$, with the same typical value of $\Delta$ we get similar operation time of 0.1ns. We see that in both cases we are a few orders of magnitude faster than the defect cavity decay time. We note that these can be improved even more by just adding more atoms in the defects and making the coupling stronger. As far as the switching times of the gates, the photon device crossing time for a group velocity of the order of $10^{-4}c$ is of the order ns. All these mean that the switching of the external fields need to be operated in ns or 10 ps regime which is within the limits of current electronics. The other technological requirement would be to efficiently couple the input photons from the incoming fiber to the crystal which can also be met using techniques studied in Ref. ... \section{Conclusions} In this work we suggested to use highly efficient waveguides formed from linear chains of defects inside 2D photonic crystals to built a micrometer size, tunable interferometer. We described the possible quantum optical means for inducing phase shifts and thus adding tunability to the integrable, micrometer size structure. We showed that the loss free propagation of photonic qubits in this device, combined with the possibility of inducing strong nonlinear phase shifts through the interaction with doped atoms in the structure, can provide for the efficient implementation of any quantum gate. 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The spheres denote the positions of the defects of the CCW.} \label{pulse} \end{figure} \begin{figure} %\includegraphics[width=0.45\textwidth]{MZI_1b.eps} \centering \epsfxsize=17 cm \epsfbox{MZI5.eps} \caption{The proposed MZI is integrated in a 2D, micrometer size photonic crystal. Pairs of photons are inserted from the left side and start propagating (hopping from defect to defect) towards the right. The blue defect regions in the arms betweeen the beamsplitters are doped with atoms providing for efficient implementation of both the MZI phase shift and the non-linear two photon phase shift. The green and red boxes stand for the external shifting gates producing the fields used for the operations. The green colour corresponds to the ``field on, nonlinear phase inducing" operation(where all four gates are on), whereas the half red half green one (the one shown here) to the ``afterwards release" and ending of the operation(see text){\it[ Thesingle qubit lasers shown are there from old version, pls ignore.]}. } \label{MZI} \end{figure} %\newpage \begin{figure} \centering \epsfxsize=5 cm \epsfbox{atom1.eps} \caption{The atomic level structure of the doped atoms. Each single photon is equally detuned from the corresponding atomic transition whereas the two photon one is on resonance. A simple efficient gating mechanism for the incident photons can be implemented using ideas similar to those known in the micromaser literature.} \label{fig3} \end{figure} %\begin{figure} %\centering \epsfxsize=18 cm \epsfbox{mzi.eps} \caption{Another %possible side view of the device suggested, currently under %construction} \label{fig3} %\end{figure} \end{document}