Creation of entanglement and implementation of …dguney/gate.pdfCreation of entanglement and implementation of quantum logic gate operations using a three-dimensional photonic crystal

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D. Ö. Güney and D. A. Meyer Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. B 283

Creation of entanglement and implementation ofquantum logic gate operations using a three-

dimensional photonic crystal single-mode cavity

Durdu Ö. Güney

Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive,La Jolla, California, 92093-0409

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla,California, 92093-0112

David A. Meyer

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla,California, 92093-0112

Received April 28, 2006; revised August 7, 2006; accepted August 27, 2006;posted September 1, 2006 (Doc. ID 70387); published January 26, 2007

We solve the Jaynes–Cummings Hamiltonian with time-dependent coupling parameters under the dipole androtating-wave approximations for a three-dimensional photonic crystal (PC) single-mode cavity with a suffi-ciently high-quality Q factor. We then exploit the results to show how to create a maximally entangled state oftwo atoms and how to implement several quantum logic gates: a dual-rail Hadamard gate, a dual-rail NOT gate,and a SWAP gate. The atoms in all of these operations are syncronized, which is not the case in previous studiesof PCs [J. Mod. Opt. 48, 1495 (2001); Eur. Phys. J. D 10, 285 (2000); Eur. Phys. J. D 18, 247 (2002)]. Ourmethod has the potential for extension to N-atom entanglement, universal quantum logic operations, and theimplementation of other useful, cavity QED-based quantum information processing tasks. © 2007 Optical So-ciety of America

OCIS codes: 020.5580, 200.4660, 220.4830, 230.5750, 270.5580.

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. INTRODUCTIONhe superiority of quantum computing over classical com-utation for problems with solutions based on the quan-um Fourier transform, as well as search and (quantum)imulation problems has attracted increasing attentionver the past decade. Despite promising developments inheory, however, progress in the physical realization ofuantum circuits, algorithms, and communication sys-ems has been extremely challenging to date. Majorodel physical systems include photons and nonlinear op-

ical media, cavity QED devices, ion traps, and nuclearagnetic resonance (NMR) with molecules, quantum

ots, superconducting gates, and spins inemiconductors.1

Quantum logic gates and quantum entanglement con-titute two building blocks, among others, of more sophis-icated quantum circuits and communication protocols.he latter also allows us to test basic postulates of quan-um mechanics.

In this work we propose an alternative method for cre-ting entanglement and implementing certain quantumogic gates based on single-mode photonic crystal (PC) mi-rocavities. Our scheme requires high-Q PC microcavitiesith a Q factor of approximately 108 where atoms can

reely propagate through the connected void regions. Weonsider 3D PCs for our purposes since they are more at-

0740-3224/07/020283-12/$15.00 © 2

ractive, compared to 2D PC slabs, for the realization ofuch high-Q microcavities. Although the fabrication ofhese high-Q 3D PC microcavities is currently challeng-ng, especially in the optical frequencies, mainly becausef the reduced size of the crystals at those wavelengths; inhe microwave or millimeter-wave regime, where our pro-osed PC cavity operates ��50 GHz�, the fabrication pro-ess is relatively easier thanks to the rather macroscopicature of the crystals at that scale. Thus simple machin-

ng or rapid prototyping methods can be exploited to buildhem.2 It is for this reason that the initial experiments onCs were performed in the microwave region. Subse-uently, much of the research concentrated on the infra-ed or the optical regime of the spectrum because of nu-erous applications that demanded telecommunicationsavelengths and/or miniaturization. In the less studiedut also less technically challenging microwave region the

factor is limited by the intrinsic loss of the material.3,4

sing an appropriate low-loss dielectric such as sapphire,or example, it has been predicted by Yablonovitch3 that a

factor of more than 109 should indeed be experimen-ally achievable. Furthermore, Qi et al.5 have predictedhat by using interference, imprint or x-ray lithographyombined with scanning-electron-beam lithography, theirayer-by-layer approach could allow the manufacturing ofarge area (several square centimeters) PCs that are low-

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284 J. Opt. Soc. Am. B/Vol. 24, No. 2 /February 2007 D. Ö. Güney and D. A. Meyer

ost, optical, or even visible spectrum versions of 3D PCavities similar to ours.

. THEORYn this paper we first explore the possibility of mutuallyntangling two Rb atoms by exploiting their interaction,ediated by a single defect mode confined to a 3D PC cav-

ty. We then describe various logic gates that can bemplemented by using the same interaction. We achievehis in three steps: analysis of a single-mode cavity (i) in aeneric 3D PC, (ii) in a specific 2D PC, and (iii) ultimatelyn a 3D version of the 2D PC of (ii). We assume that theefect frequency is resonant with the Rb atoms. Becausehe atoms are moving, the time-dependent HamiltonianJaynes–Cummings model) for this interaction in the di-ole and rotating wave approximations is

H�t� =��

2 �j

�zj + � ��†� + � �

j�Gj�t��+

j � + H.c.�, �1�

here the summation is over two atoms, A and B ,� is theesonant frequency, �z is the z component of the Paulipin operator, and �± are atomic raising and lowering op-rators. � and �† are photon destruction and constructionperators, respectively. The time-dependent coupling pa-ameters can be expressed as6

Gj�t� = �0fj�t�cos��j�, �2�

here �0 is the peak atomic Rabi frequency over the de-ect mode, and fj�t� is the spatial profile of the defect statebserved by the atom j at time t. �j is the angle betweenhe atomic dipole moment vector, �eg

j , for atom j and theode polarization at the atom location. For the rest of

his paper cos �A=1, and p=cos �B will be a parameterhat we adjust to effect the desired interactions betweentoms A and B.We ignore the resonant dipole–dipole interaction

RDDI), because the atoms have their transition frequen-ies close to the center of a wide photonic bandgap (PBG),nd the distance between them is always sufficientlyarge that the RDDI effect is not significant.7,8

Initially we prepare one of the atoms, A, in the excitedtate, and the cavity is left in its vacuum state, so

��0�� = �100�, �3�

here the tensor factors describe the states of atom A,tom B, and the cavity, successively.The PC should be designed to allow the atoms to go

hrough the defect. This can be achieved by injecting at-ms through the void regions of a PC with a defect state ofcceptor type. Since the spontaneous emission of a photonrom the atoms is suppressed in the periodic region of therystal, no significant interaction occurs outside the cav-ty. When the atoms enter the cavity, the interaction be-ween the atoms is enhanced by the single-mode cavity.his atom–photon–atom interaction allows us to designn entanglement process between atoms.Given the initial state, Eq. (3), the state of the system

t time t should be in the form

��t�� = a�t��100� + b�t��010� + �t��001� �4�

o satisfy probability and energy conservation. We willhow analytically that the amplitudes at time t can be ex-ressed in terms of the coupling parameters and the ve-ocities of the atoms.

We can write the Schrödinger equation for the time-volution operator in the form9

i ��

�tU�t,t0� = H�t�U�t,t0�. �5�

n the basis �100� , �010� , �001�, the matrix elements ofhe Hamiltonian, Eq. (1), in the rotating wave approxima-ion, are

H11 = H12 = H21 = H22 = H33 = 0, �6�

H13 = H31 = � GA�t�, �7�

H23 = H32 = � GB�t�. �8�

The Hamiltonian operators H�t� and H�t�� commute for�� t, if GB�t� is a constant multiple p of GA�t�. This con-ition can be satisfied easily by the appropriate orienta-ion of the atomic dielectric moment of the incoming at-ms with respect to the electric field in Eq. (2). Then theormal solution to Eq. (5) becomes

U�t,t0� = exp�−i

��

t0

t

d�H�� = exp�−i

�I� , �9�

here I is defined as the integral of the Hamiltonian op-rator. By expanding the exponential, Eq. (9), we obtain

U�t,t0� = 1 + �− i

��I +

1

2!�− i

��2

I2 + ¯ + � 1

n!��− i

��n

In

+ ¯ . �10�

ultiplying Eq. (10) by the initial state �100� from theight and comparing it with Eq. (4) gives

a�t� = 1 + GA2 �

n=1�− 1�n

1

2!�GA

2 + GB2 �n−1, �11�

b�t� = GAGB�n=1

�− 1�n1

2n!�GA

2 + GB2 �n−1, �12�

�t� = iGA�n=1

�− 1�n1

�2n − 1�!�GA

2 + GB2 �n−1, �13�

here we have defined

Gj =�t0

t

Gj��d�. �14�

ecognizing the Taylor series for sine and cosine allowss to rewrite Eqs. (11)–(13) as

a�t� = 1 +GA

2

GA2 + GB

2 �cos�GA2 + GB

2 �1/2 − 1�, �15�

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b�t� =GAGB

GA2 + GB

2 �cos�GA2 + GB

2 �1/2 − 1�, �16�

�t� = − iGA

�GA2 + GB

2 �1/2sin�GA

2 + GB2 �1/2. �17�

or a single initially excited atom, A, passing across theavity we can set GB=0 in Eqs. (15)–(17), which gives theame result as Eq. (15) of Ref. 10. The exact solution forhe time-independent problem of N identical two level at-ms with a resonant single-mode quantized field given inef. 11 could be helpful to generalize our results to the-atom case, see for example Ref. 7.Absent a rigorous calculation of the defect mode in the

D PC, which we have postponed until Section 4, let usrst assume a generic spatial profile for the mode, whichscillates and decays exponentially. Thus fj�t� in Eq. (2)an be expressed as6

fi = exp�− �Vjt − L�/Rdef�cos��

l�Vjt − L� , �18�

here Vj ,L ,Rdef, and l are the velocity of atom j, the totalath length of the atoms, the defect radius, and the latticeonstant of the PC, respectively.

We chose the velocities of the atoms to be the same,j=V, with 150 m/s V 650 m/s, a typical velocity

ange appropriate for both experiments and our calcula-ions. Setting the atoms to have the same velocity has twommediate advantages: First, it makes the Hamiltoniansommute at different times, since the coupling param-ters of the atoms differ by only a constant factor [see Eq.2)], and thus greatly simplifies the analysis. Second, itynchronizes the atoms, providing a cyclical readout thatould also be synchronized with the cycle time of a quan-um computer.12 These features were not present in pre-ious studies.6,7,13

It was shown by Hagley et al.14 in 1997 that atomic ve-ocity resolution could be as small as 0.4 m/s in experi-

ents, which means that our proposed entangler andogic gates are rather robust and insensitive to experi-

ental velocity fluctuations. In fact, with our design theelocity resolution of the atomic pairs can be further re-

ig. 1. (Color online) Probability amplitudes as a function of thnitial state is �100� (or �010�). (c) Surface b�V ,p�, if the system is

axed up to slightly more than ±1 m/s, increasing the ef-ciency of the system at the expense of less than 1% de-iation in the output probability amplitudes.

Using Eqs. (15)–(18) we can express the asymptoticrobability amplitudes, a�t� and b�t�, as functions of Vnd p. Figure 1(a) illustrates a�V ,p�, when the initialtate is �100�. The result for b�V ,p� (or a�V ,p�), if the ini-ial state is �100� (or �010�), is displayed in Fig. 1(b). Notehat Fig. 1(b) illustrates two probability amplitudes si-ultaneously. This is a consequence of the symmetry inq. (16). To compute these surfaces we used thesymptotic (constant) values of the Gj, which describe theccumulated atom–cavity coupling during the interactionime [see Eq. (14)]. Once this interaction ceases to occurs the atoms leave the cavity, those asymptotic values ofhe Gj are reached, and thus must be used in Eqs.15)–(17).

Figure 2(a) illustrates a slice from the surface in Fig.(a) where the velocity of the atoms is V=433 m/s. Notehat we obtain the maximally entangled state

�10� ��10� + �01�

�2, �19�

p to an overall phase, −1, when the velocity of the atoms,=433 m/s, and the initial state is �100�. Similarly if weeep the velocities of the atoms the same but set the atomto be excited initially (i.e., the initial state is �010�), we

btain the slice in Fig. 2(b) and thus the maximally en-angled state

�01� ��10� − �01�

�2, �20�

p to the same overall phase factor as in Eq. (19). [In Eqs.19) and (20) we omit the cavity state in the kets, since itactors out and does not contribute to the logic operationsith which we are concerned. When this is done in the fol-

owing, we use the same notation.]Figure 3 shows the coupling parameters calculated in

he reference frame of the atoms as a function of time,j�t�, with the chosen values of �0 ,L ,Rdef, and l. The solid

urve and dashed curve correspond to atoms A and B, re-pectively. The total interaction time is less than 20 ns.

city and p. Surfaces (a) a�V ,p� and (b) b�V ,p� [or a�V ,p�] if thelly prepared in the �010� state.

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The probabilities �a�t��2 , �b�t��2, and ��t��2 from Eqs.15)–(17) are graphed in Fig. 4. Figure 4(a) shows the evo-ution of the probabilities when the initial state is �100�.ote that the cavity is disentangled from the atoms, ande end up with the final state, Eq. (19). On the otherand, if the initial state is �010�, the time evolution of therobabilities is illustrated in Fig. 4(b). Note that the finaltate in this setting becomes Eq. (20), since the cavity isgain disentangled. From Eqs. (19) and (20) it is clearhat the quantum system we have described not only en-angles the atoms but also operates as a dual-rail Had-mard gate,1 up to an overall phase.Using the same quantum system one can also design a

ual-rail NOT gate under certain conditions. If we set VAVB=565 m/s, for example, we obtain the following logi-al transformations, up to an unimportant global phaseactor, which define a dual-rail NOT gate (see Fig. 5):

�10� � �01�, �21�

�01� � �10�. �22�

ig. 2. (Color online) Slices from each surface in Fig. 1 (a) Prob-bility amplitudes a�V ,p�—solid curve—and b�V ,p�—dashedurve—with V=433 m/s. The entangled state, Eq. (19), is ob-ained at p=0.414. (b) Probability amplitudes a�V ,p�—dashedurve—and b�V ,p�—solid curve—with the same velocity, V433 m/s. The entangled state, Eq. (20), is observed at the samealue, p=0.414.

Furthermore, using the Hamiltonian, Eq. (1), it can behown that a �00� initial state gains only a deterministichase factor of −1 in the interaction picture. Once the con-itions, Eqs. (21) and (22), are satisfied, a �11� initial state

ig. 3. (Color online) Coupling parameters in the referencerame of moving atoms with velocities V=433 m/s at p=0.414.

0=11�109 Hz, �=2.4�1015 Hz, l=1.6�c� , L=10l, Rdef= l.6 Atom

experiences the coupling parameter shown with the solid curvend atom B the one shown with the dashed curve.

ig. 4. (Color online) Time evolution of the probabilities show-ng the final entanglement when (a) the initial state is �100� andb) �010�.

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s transformed into itself up to the same global phase ashe states in Eqs. (21) and (22). Thus including these asossible initial states, our dual-rail NOT gate also operatess a SWAP gate up to the relative phase of the �00� state. Aimilar analysis shows that a dual-rail Z gate is also pos-ible for certain parameter choices.

. TWO-DIMENSIONAL PHOTONICRYSTALS

n the preceding analysis we have assumed the genericorm, Eq. (18), for the spatial profile of the defect mode toemonstrate that, in principle, PC microcavities can besed as entanglers, and more specifically, as certain logicates. In the following we apply these ideas to 2D and fi-ally real 3D PC microcavity designs, to show that imple-entations of these quantum devices are indeed possible

n these photonic systems. As the authors of Ref. 7 ob-erved, however, “A rigorous calculation of the electro-agnetic field in the presence of a defect in a 3D photonic

rystal can be a difficult task.” In the following we ad-ress this task systematically.First, we consider a 2D photonic crystal design with a

riangular lattice of dielectric rods with a dielectric con-tant of 12 (i.e., silicon) in Fig. 6. The radius of the rods is.175l, where l is the lattice constant. The symmetry isroken in the center by introducing a defect with reduced

ig. 5. (Color online) Time evolution of the probabilities thateads to a dual-rail NOT (Pauli �x) logic operation when the initialtate is (a) �100� and (b) �010�.

od radius of 0.071l to form the microcavity of our quan-um system. We assume that the atoms A and B travellong the vertical dashed lines shown in Fig. 6(a), al-hough any two of the obvious paths shown by dashedines (or any void regions with line of sight to the cavity)ork as well.We should note that the 2D cavity is designed only for

reliminary purposes. Since it assumes infinite height, itannot be intended for actual implementation of our en-angler or logic operations. However, the real 3D struc-ure to be fabricated, described in Section 4, is sufficientlylose to the 2D cavity designed in this section that theode profiles along the 1D atomic trajectories in the 2D

avity overlap to a great extent with those of the 3D cav-ty. Regardless of the lack of vertical confinement in thisreliminary 2D design, designing it first reduces the ex-ensive computational resources needed in more sophisti-ated 3D logic designs.

To find the probability amplitudes for the atoms as aunction of time while travelling through the crystal, weeed to calculate the coupling parameters in Eq. (14) andubstitute them in Eqs. (15)–(17). Once we obtain therobability amplitudes, we can demonstrate entangle-ent creation and design logic gates as before. We alsoote that the quality factor of the cavity should be highnough that the interaction time (i.e., the logic operationime) is much less than the photon lifetime in the cavity.elow we show that typical logic operations take 50 �s for

he Hadamard gate and 30 �s for the NOT or the SWAP gateFigs. 8 and 12, 9 and 13, respectively). Thus a qualityactor of 108 should be sufficient for reliable gate opera-ions in real 3D PC microcavities. Since our preliminaryD PC cavity is not realistic, however, only an in-plane Qactor of the same value is taken into consideration toatch the mode profiles along the atomic trajectories in

oth 2D and 3D. To achieve this matching the Q factor ofhe cavity could be increased exponentially with addi-ional periods of rods15 in Fig. 6(a). We observe, however,hat the spatial profile of the mode in Fig. 6(b) does nothange significantly (and hence degrade the gate) after aertain number of periods. Thus the exact number of pe-iods required for a quality factor of 108 is not essential toemonstrate our main goal in this paper, namely, that

ig. 6. (Color online) (a) Single-mode microcavity in a 2D pho-onic crystal with a triangular lattice (see the text for details). (b)orresponding electric field spatial profile for the TM mode al-

owed in the cavity.

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uch logic operations and entanglement creation are in-eed possible in these PC structures.In our generic profile, Eq. (18), the coupling parameter

s assumed to be real. For generality, however, we mustllow it to be a complex parameter. Thus the interactionart of the Hamiltonian, Eq. (1), for a single-atom cavitynteraction can be written as16–19

HI = � �g�r��†�− + ��+� �23�

y incorporating a complex-coupling parameter g�r� intoq. (1).In a PC we can express the atom-field coupling

arameter16,20 at the position of atom j as

g�rj� = g0�rj�cos��j�, �24�

here g0 and �rj� are defined as

g0 =�eg

��

2�0�mVmode�1/2

, �25�

�rj� � E�rj�/�E�rm��. �26�

m denotes the position in the dielectric where ��r� �E�r��2s maximum, and �m is defined as the dielectric constantt that point. The cavity mode volume, Vmode, is given by

Vmode =�� r��E�r��2dr

�m�E�rm��2. �27�

Using the block iterative plane-wave expansionethod21 we found the normalized frequency of the cavityode shown in Fig. 6(b) to be 0.3733 c / l. By setting l2.202 mm, we tune the resonant wavelength to 5.9 mm.t this wavelength, �eg for the Rb atom is 2�10−26 C·m.6

ince we observe that the energy density is concentratedn the center of the cavity, � in Eqs. (25) and (27) be-

ig. 7. (Color online) Normalized coupling parameters (i.e., div374 m/s at p=0.414, where g0 is found to be 2.765 MHz. Thetrengths for atoms A and B, respectively.

m

omes 12. On the other hand, cos��j� in Eq. (24) can beafely assumed to be constant for each atom because ouravity mode is a TM mode, and its electric-field polariza-ion direction can always be assumed to have a constantngle with the atomic dipole moment vector, �eg

j. We canompute the coupling parameters as functions of time inhe reference frame of the moving atoms, as shown in Fig.. The tails of the coupling parameter functions do notontribute significantly to the results due to exponentialecay away from the cavity.With these settings Fig. 8 demonstrates an entangler

hat operates as a dual-rail Hadamard gate. If the initialtate is �10� (i.e., atom A is excited), we end up with state10� [see Eq. (19) above and Fig. 8(a)]. However, if atom

is initially excited (i.e., the initial state is �01�), we ob-ain the state �01�, up to an unimportant global phaseactor of −1 [see Eq. (20) above and Fig. 8(b)] as the out-ut of the logic gate.To get the system to act as a dual-rail NOT gate, we

imply set the velocities of the atoms to be VA=VB490 m/s. The evolution of the probability amplitudes for

he atoms is shown in Fig. 9. When the excitation is ini-ially at atom A (or at atom B), it is transferred to atom

(or atom A). Note that we also get an unimportanthase factor of −1 in the output. Furthermore as ex-lained above, the input states �00� (or �11�) are not trans-ormed into different states, and only �00� gains a differ-nt phase factor of +1. If this phase problem with the �00�tate were solved, the system could also be exploited as aWAP gate.

. THREE-DIMENSIONAL PHOTONICRYSTALSlthough the implementation of logic gates in 2D PCs

ooks promising, in reality we need 3D devices. The 2Dnalysis is useful, however, for reducing the substantialomputation required for analyzing more realistic 3D

y g0) in the reference frame of moving atoms with velocities Vcurve and the gray curve correspond to normalized coupling

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tructures, because there is great similarity between theodes allowed in these 2D crystals and in their carefully

hosen 3D counterparts (see Fig. 10), which we describeext.To engineer logic operations in 3D PCs, we employ the

tructure5,22,23 shown in Fig. 10(a). The 3D PC we havehosen has various advantages over others24–26: emula-ion of 2D properties in 3D,5,23 polarization of the modes,nd simplified design and simulation. It consists of alter-ating layers of a triangular lattice of air holes and a tri-ngular lattice of dielectric rods, where the centers of theoles are stacked along the [111] direction of the face-entered cubic (fcc) lattice. The parameters for the crystalre given in the caption of Fig. 10(a). Using the block it-rative plane-wave expansion method,21 we calculate thathe structure exhibits a 3D bandgap of over 20%, acrosshe frequency range of 0.507c / l–0.623c / l.

As in the 2D geometry, we introduce a defect by reduc-ng the radius of a rod inside the crystal as shown in Fig.

ig. 8. (Color online) Probability amplitudes for entangled atomsee Fig. 6(a)] when (a) atom A is initially in the excited state andhe probability amplitudes for the states �100� , �010� and �001�, r

0(b). Using the supercell method we can compute thathis cavity supports only a single mode with a frequencyf 0.539c / l. Setting l=3.18 mm tunes the cavity mode tohe atomic transition wavelength of 5.9 mm. Thus we canesign a 3D single-mode cavity for our system to operates the desired quantum logic gates.The spatial profile for the electric field of the mode is

hown in Figs. 10(c) and 10(d). Note that it has a profileimilar to its 2D counterpart in Fig. 6(b), where the en-rgy density is also maximized in the center of the cavity.t is this similarity that simplifies our design and analysisor the 3D case.

In our designs, although the maximum field intensity isoncentrated in high-dielectric regions, the resultanttom–cavity interaction is sufficient to achieve our goals.avities maximizing the field intensity in low-index re-ions, where atoms can strongly interact with the mode,ould lead to higher performance. However, from the de-ign and implementation perspective we find the former

ed by a dual-rail Hadamard operation in the 2D photonic crystalen atom B is initially in the excited state. a�t�, b�t�, and �t� are

vely.

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ore appropriate. Some of the drawbacks of the latter cane stated as follows: If we remove the dielectric com-letely in the center of the defect to maximize the field in-ensity in air, then the defect frequency would be close tohe band edge, which would thus strengthen the other-ise negligible RDDI mechanism.7 If we use a holey lat-

ice, on the other hand, emulating the 2D PBG inside aD crystal and simultaneously allowing the free passagef atoms, without interfering with the dielectric backbonef the 3D crystal, would complicate the design consider-bly. Nevertheless, a further optimization of the proposedtructure could be possible.

We can quantify23 the TM polarization of the mode in alane as

P =� d2r�Ez��;r��2

� d2r�E��;r��2. �28�

e compute that P is almost 0.99 in the defect planehown in Fig. 10(b) for the spatial profile exhibited in

ig. 9. (Color online) Probability amplitudes for atoms A andnitially (a) only atom A is excited and (b) only atom B is excited

igs. 10(c) and 10(d). Thus it is safe to assume a TMolarized mode in the system Hamiltonian because thisoesn’t affect the probability amplitudes significantlyn the microwave regime for the parameters we have cho-en.

The coupling parameters as functions of time in the ref-rence frame of the atoms are shown in Fig. 11, where theelocities of the atoms are both set to V=353 m/s, with=0.414.In Fig. 12 we demonstrate the 3D version of our 2D

ual-rail Hadamard gate, which also acts as an atomic en-angler. Just as in the 2D case, if the initial excitation isn atom A, the resulting state is �10� [see Eq. (19) above]s shown in Fig. 12(a), while if it is on atom B the outputtate is �01� [see Eq. (20) above] as displayed in Fig.2(b), up to an unimportant global phase of −1. The inter-ction between atoms A and B is mediated by the photo-ic qubit when the parameters are set correctly.We set the velocity of both atoms to VA=VB=459 m/s to

btain a dual-rail NOT gate in the 3D photonic crystal, upo an unimportant global phase factor, −1. The probabilitymplitude evolution of the atoms is displayed in Fig. 13.

er the dual-rail NOT operation in the 2D photonic crystal when
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FawTmsdi

F=a


The excitation of the excited atom is transferred to theround state atom with the help of the photonic qubitllowed in the designed cavity. Note that because ofhe symmetry of the problem, the b�t�s in Figs. 5(a), 9(a),nd 13(a) are indeed approximately the same as the a�t�sf Figs. 5(b), 9(b), and 13(b), respectively. Furthermore forhe same reason as in the 2D case, analyzed in the previ-

ig. 10. (Color online) (a) Top view of the 3D photonic crystal wir holes and a triangular lattice of dielectric rods (for details oithin either a hole or rod layer is �1/�2�l, where l is the fcc latthe thicknesses of a hole layer and a rod layer are taken to beaterial of dielectric constant 12. (b) Horizontal cross section of th

imulations we assumed the path shown by the dashed horizontown to 0.050l to hold a single mode in the cavity. The (c) real pn the cavity at a particular instant in time with the frequency o

ig. 11. (Color online) Coupling parameters in the reference0.414, where g0 is found to be 2.899 MHz. The black curve and tnd B, respectively.

us section, our 3D dual-rail NOT gate also operates as aWAP gate up to some deterministic phase. That is,

�00� � − �00�, �29�

�01� � �10�, �30�

lattice. It consists of alternating layers of a triangular lattice oftructure see Refs. 5, 22, and 23). The nearest-neighbor spacingstant. The hole and rod radii are 0.293l and 0.124l, respectively.l and 0.354l, respectively. Silicon is assumed as the high-indextal. Dashed lines show the obvious paths atoms can travel. In our. A defect is introduced by reducing the radius of the middle rod

(d) imaginary part of the electric field of the TM mode allowedc / l. The imaginary part is one-half a period later.

of the moving atoms with both velocities V=353 m/s, and py curve correspond to normalized coupling strengths for atoms A

ith fccf the sice con

0.225e crys

al lineart andf 0.539

framehe gra

atltsv

5Pag

gumiIatcistHpfimate

F(p


�10� � �01�, �31�

�11� � �11�. �32�

Note also that the values of the velocities of the atoms,nd g0, for the 2D logic gates are within 10% of those inhe 3D gates. This is a consequence of the fact that theocalized modes allowed in 2D and 3D cavities have morehan 90% overlap in their spatial profiles.23 Thus our re-ults are also consistent with Ref. 23 and justify first in-estigating the 2D case.

. CONCLUSIONShotonic bandgap materials could be especially promisings robust quantum circuit boards for the delicate nexteneration quantum computing and networking technolo-

ig. 12. (Color online) Probability amplitudes for the entangledsee Fig. 10) when (a) atom A is initially in the excited state androbability amplitudes for the states �100� , �010� and �001�, respe

ies. The high-quality factor and extremely low-mode vol-me achieved successfully in microcavities have alreadyade PCs an especially attractive paradigm for quantum

nformation processing experiments in cavity QED.6,7,27

n our paper we have extended this paradigm by solvingnalytically the Jaynes–Cummings Hamiltonian underhe dipole and rotating-wave approximations for two syn-hronized two-level atoms moving in a PC and by apply-ng the solution to produce the two maximally entangledtates in Eqs. (19) and (20). We have also demonstratedhe design of quantum logic gates, including dual-railadamard and NOT gates and SWAP gate operations. Ourroposed system is quite tolerant to calculation and/orabrication errors, in the sense that most errors remain-ng after the design can be corrected by simply experi-

entally adjusting the velocity or the angle between thetomic moment vector of atom B and the mode polariza-ion. Our technique can be generalized to N-atomntanglement,7 and it also has potential for universal

under dual-rail Hadamard operation in the 3D photonic crystalom B is initially in the excited state. a�t�, b�t�, and �t� are the.

atoms(b) at

ctively

qccWdorc

ATAADPa0(

da

R

Fi


uantum logic gates, atom–photon entanglement pro-esses, as well as the implementation of various, usefulavity QED-based quantum information processing tasks.e should also mention the methodological result that

ue to the emulation of 2D PC cavity modes in 3D PCs,5,23

ne can design the more sophisticated circuit first in 2D toeduce the difficulty of the 3D computations, where typi-ally much more computational power is needed.

CKNOWLEDGMENTShis work was supported in part by the National Securitygency (NSA) and Advanced Research and Developmentctivity (ARDA) under Army Research Office (ARO) grantAAD19-01-1-0520, by the Defense Advanced Researchroject Agency (DARPA) Quantum Information Sciencend Technology (QuIST) program under contract F49620-2-C-0010, and by the National Science FoundationNSF) under grant ECS-0202087.

ig. 13. (Color online) Probability amplitudes for atoms A andnitially (a) only atom A is excited and (b) only atom B is excited

D. Ö. Güney may be reached by e-mail [email protected]. D. A. Meyer may be reached by e-mailt [email protected].

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Creation of entanglement and implementation of …dguney/gate.pdfCreation of entanglement and implementation of quantum logic gate operations using a three-dimensional photonic crystal