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Franson et al. Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. B 209

Zeno logic gates using microcavities

James D. Franson and Todd B. Pittman

Physics Department, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA

Bryan C. Jacobs

Applied Physics Laboratory, Johns Hopkins University, Laurel, Maryland 20723, USA

Received May 24, 2006; revised July 21, 2006; accepted September 28, 2006;posted October 10, 2006 (Doc. ID 71352); published January 26, 2007

The linear optics approach to quantum computing has several potential advantages, but the logic operationsare probabilistic. We review the use of the quantum Zeno effect to suppress the intrinsic failure events in thesekinds of devices, which would produce deterministic logic operations without the need for ancilla photons orhigh-efficiency detectors. The potential advantages of implementing Zeno gates using microcavities and elec-tromagnetically induced transparency are discussed. © 2007 Optical Society of America

OCIS codes: 270.0270, 200.3760, 200.4660.

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. INTRODUCTIONhere has been considerable progress in the linear opticspproach to quantum computing,1,2 including the experi-ental demonstration of logic gates,3–6 small-scale

ircuits,7 quantum error correction,8 and cluster states.9

s first proposed by Knill et al.,1 probabilistic logic opera-ions can be performed using linear optical elements, an-illa photons, and feed-forward control based on measure-ents made on the ancilla. In principle, the intrinsic

ailure rate of these logic gates can be made arbitrarilymall by using a large number of ancilla, but that wouldequire a large increase in the required resources as wells the use of high-efficiency detectors. The use of clustertates9 appears to be a promising approach, since proba-ilistic logic operations can be used to set up the initialluster state, after which an arbitrary quantum calcula-ion can be performed without any intrinsic failurevents.

We recently proposed an alternative approach10 inhich the intrinsic failure events of linear optics logicates can be suppressed using the quantum Zeno effect11

ased on strong two-photon absorption. In addition to pro-iding an overview of the basic Zeno gate approach, thisaper discusses the use of microcavities to enhance thewo-photon absorption rate and to increase the perfor-ance of the logic gates. We will also discuss the use of aew technique for reducing the rate of single-photon scat-ering that is analogous to electromagnetically inducedransparency (EIT), except that it does not require these of strong laser beams.12

Since they rely on two-photon absorption, Zeno gatesre no longer linear optical devices. They can be viewed ashybrid approach that combines some aspects of a linear

ptics approach with the use of a nonlinear medium. Thisffers some potential advantages over other optical ap-roaches, such as cavity QED.13 For example, there is noeed to convert stationary qubits to flying qubits and backgain, and the quantum information remains in the form

0740-3224/07/020209-5/$15.00 © 2

f single photons at all times. In addition, the necessarywo-photon absorption can be achieved using an atomicapor instead of a single trapped atom or ion, which mayimplify the experimental apparatus.

. LINEAR OPTICS LOGIC GATESinear optics quantum computing was first discussed bynill et al.1 based on a set of nested interferometers. Atbout the same time, Koashi et al.14 proposed a similarpproach based on polarization-encoded qubits, and wentroduced a simplified CNOT gate shortly thereafter.15 Aeneric linear optics logic gate is illustrated in Fig. 1. Twoogical qubits in the form of single photons are mixed withn arbitrary number of ancilla photons in a “black box”ontaining only linear optical elements, such as beamplitters and phase shifters. The state of the ancilla iseasured after they leave the device, and the goal is to

esign the optical elements in such a way that the quan-um measurement process projects out the desired logicalutput state. This will be the case for some of the mea-urement results after single-qubit corrections have beenpplied (feed-forward control), while other measurementutcomes will correspond to output states that are knowno be incorrect and cannot be corrected. Incorrect resultsf that kind will be referred to as failure events, and theyccur with a probability that depends on the design of thepecific device.

As an example, the CNOT gate that we proposed15 ishown in Fig. 2. This is a relatively simple and stable de-ice in which the two input qubits are mixed with an en-angled pair of ancilla photons by using two polarizingeam splitters. The polarization states of the photons thatmerge into the two detectors are measured in a rotatedasis in order to avoid a determination of their logicalalue. It can be shown that this device will correctlymplement a CNOT logic operation provided that one andnly one photon is detected in each detector, which occursith a probability of 1 .

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007 Optical Society of America

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210 J. Opt. Soc. Am. B/Vol. 24, No. 2 /February 2007 Franson et al.

The CNOT gate of Fig. 2 will fail if two photons emergerom the device in the same optical path, such as two pho-ons entering the top detector. This is the only failureode of this device, and it would operate properly all of

he time if the probability of two photons emerging in theame path could be suppressed in some way.

. ZENO GATESn the quantum Zeno effect,11 a randomly occurring eventan be suppressed by frequent measurements to deter-ine whether it has occurred. The basic idea is to use this

ffect to suppress the emission of two photons into theame output path of our CNOT gate, which would elimi-ate the intrinsic failure events and produce a determin-

stic logic gate.The origin of the Zeno effect is illustrated in Fig. 3. For

ufficiently short amounts of time, the probability ampli-

ig. 1. Generic logic gate using linear optical elements. The twonput qubits are mixed with a number of ancilla photons by usinginear optical elements. Corrections to the output may be appliedased on measurements made on the ancilla.

ig. 2. Controlled-NOT logic gate implemented using two polar-zing beam splitters, an entangled pair of ancilla, and two detec-ors. This gate will produce the correct logical output whenever aingle photon is found in both detectors, which occurs with arobability of 1

4 .

ude for a failure event will increase linearly, as illus-rated in Fig. 3(a), while the probability itself increasesuadratically. If a measurement is made after a shortmount of time to determine whether an error has oc-urred, it will be found with high probability that no erroras occurred, as illustrated in Fig. 3(b). The quantumtate of the system will then collapse into the originaltate with no error, and a sequence of such measurementsill continually “reset” the system and prevent an error

rom occurring.For the Zeno effect to work, the probability amplitude

or an error must increase linearly in time, which is nothe case for the beam splitters in the CNOT gate of Fig. 2.o solve this problem, we can replace the beam splittersith coupled optical fibers as illustrated in Fig. 4. Here

he cores of two optical fibers are brought sufficientlylose to each other that their evanescent fields will gradu-lly couple a photon in one fiber core into the other fiber.evices of this kind are commercially available in the

orm of 50–50 couplers, for example.The probability that two photons will be coupled into

he same fiber core can be suppressed, at least in prin-iple, by making frequent measurements to determinehether that is the case. It will be assumed that the mea-

urement process has no effect if only a single photon isresent in each fiber. The total probability PF of a failurevent in which two photons are found in the same fiberfter N such measurements is given10 by

PF = 1 − cos2N��/2N�. �1�

he dots in Fig. 5 show the failure probability as a func-ion of the number of measurements made during theime that the photons propagated through the coupled setf fibers. It can be seen that a sequence of measurementsf this kind can suppress the probability that two photonsill emerge in the same optical fiber.

ig. 3. Basic origin of the quantum Zeno effect in which fre-uent measurements to determine whether an error has oc-urred will continuously collapse the state vector of the systemack into the initial state corresponding to no errors.

ig. 4. Zeno logic gate implemented using two coupled opticalbers and strong two-photon absorption due to atoms in the fiberores or their evanescent fields. The length of the coupled fiberss chosen in such a way that a single photon in either input portill be completely coupled into the other fiber (a SWAP opera-

ion). When two photons are input at the same time, the Zeno ef-ect will prevent two photons from being in the same fiber core,hich results in a controlled phase gate in addition to the SWAP.

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Franson et al. Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. B 211

As a practical matter, equivalent results can be ob-ained if atoms that can absorb two photons, but not one,re placed in the optical fiber cores or in their evanescentelds. Roughly speaking, the atoms “watch” for the pres-nce of two photons in the same fiber core, which shouldnhibit that from occurring. The solid curve in Fig. 5hows the results of a density-matrix calculation10 inhich the probability of a failure event (in which two pho-

ons emerge from the same path or one or more photonsre absorbed) as a function of the strength of the two-hoton absorption rate (in arbitrary units). It can be seenhat no actual measurements are required and thattrong two-photon absorption is sufficient for the imple-entation of Zeno logic gates.Although coupled optical fibers combined with two-

hoton absorption could be used to implement the CNOT

ate of Fig. 2, the coupled-fiber device of Fig. 4 can imple-ent a universal two-qubit logic gate by itself. Here the

ength of the device is chosen in such a way that a singlehoton entering one path will be transferred completelyo the other path, which is equivalent to a Rabi oscillationn a two-level atom. As in a conventional Rabi oscillation,uch a process is accompanied by a phase shift of � /2. Ifhe system were linear and two photons were incident,ne in each path, a phase shift of � would occur. But if twohotons are incident, the Zeno effect eliminates any cou-ling between the two fiber cores and no phase shift oc-urs; this is equivalent to a nonlinear phase shift of �ompared with the linear response. As a result, the devicef Fig. 4 can implement a SWAP� operation that is de-ned (aside from a constant overall phase factor) in thewo-qubit computational basis as

SWAP� � �1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 − 1� . �2�

his is equivalent to an ordinary SWAP operation com-ined with a controlled phase gate, which is a universalwo-qubit operation that can be combined with single-

ig. 5. Reduction in the intrinsic error (failure) rate of a linearptics logic gate due to the quantum Zeno effect. The dots corre-pond to the error probability in the device of Fig. 4 as a functionf the number of measurements performed. The solid curvehows the equivalent error probability as a function of the rate ofwo-photon absorption (arbitrary units) (from Ref. 10).

ubit operations to produce a CNOT gate or any other op-ration.

The main challenge in implementing Zeno gates of thisind will be to achieve a sufficiently large amount of two-hoton absorption while keeping the single-photon losseso a small level. The rate of single-photon loss due to scat-ering by the atoms will be comparable to the two-photonbsorption rate if the diameter of the fiber cores is equalo the wavelength of the photons and if the atoms are ran-omly distributed10; that would be unacceptable for logicperations. The single-photon scattering rate can be re-uced substantially below that level if the atoms have aearly uniform density, such as in a crystal structure,ince a uniform medium does not scatter light. Otherethods for reducing the single-photon scattering rate

ompared with the two-photon absorption rate are dis-ussed in the next two sections.

. USE OF MICROCAVITIEShe rate of two-photon absorption can be enhanced com-ared with the rate of single-photon scattering by confin-ng the photons to a resonant cavity with a small modeolume. The electric field associated with a single photons inversely proportional to the square root of the modeolume and can reach values of the order of 104 volts/mn typical cavities.10 The rate of two-photon scattering isroportional to the fourth power of the field, while theate of single-photon scattering is proportional to the sec-nd power of the field, so that the ratio of two-photon ab-orption to single-photon scattering is inversely propor-ional to the mode volume.

One potential implementation of a Zeno gate usingesonant cavities is illustrated in Fig. 6. Here the inputubits q1 and q2 (single photons) are carried in twoaveguides that are coupled to two ring resonators R1nd R2. The two resonators are weakly coupled to eachther in a manner that is analogous to the coupling be-ween the two fiber cores in Fig. 4. The couplings are ad-usted so that a single photon incident in one waveguideill be completely coupled into the other waveguide

hrough the two resonators (which is analogous to a Rabiscillation). In the presence of strong two-photon absorp-ion, this will implement a SWAP� operation as before.

The geometry of Fig. 6 has several advantages over theoupled optical fibers of Fig. 4. First of all, the overall sizef the device is reduced, which would be especially impor-ant if an atomic vapor were used for the two-photon ab-

ig. 6. Implementation of a Zeno logic gate �SWAP�� using twoave guides coupled to two ring resonators, R1 and R2, with two-hoton absorbing atoms in the evanescent fields of the resona-ors. This device is equivalent to the coupled optical fibers of Fig., except that the rate of two-photon absorption is enhanced byhe small mode volume of the resonators.

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212 J. Opt. Soc. Am. B/Vol. 24, No. 2 /February 2007 Franson et al.

orption. In addition, the reduced mode volume enhanceshe rate of two-photon absorption compared with theingle-photon absorption rate. As was mentioned above,eno gates of this kind do not require the trapping ofingle atoms, as is the case in cavity QED experiments.

The performance of the Zeno gate could be improved ifariable couplings were added between the wave guidesnd the ring resonators, as illustrated in Fig. 7. The ideas to vary the coupling in such a way that an incidentingle-photon wave packet could be coupled into one ofhe resonators and stored for some period of time (as lim-ted by the quality factor Q of the cavity). This would al-ow more time for the SWAP� operation to be performed ifhe rate of two-photon absorption is limited, and it wouldnsure that both photons were in either R1 or R2 at theame time. The variable coupling could consist of amaller ring resonator constructed from a material whosendex of refraction could be externally controlled, for ex-mple. This would allow the resonant frequency of themaller ring to be adjusted to control the effective cou-ling between the wave guides and the larger resonators.

. ELECTROMAGNETICALLY INDUCEDRANSPARENCYhe rate of single-photon scattering can be further re-uced by using EIT. In conventional EIT,16,17 the resonantcattering of photons by a two-level atom can be elimi-ated by applying a strong laser beam tuned to a differentransition. The laser beam splits the initial excited atomictate into two dressed states with slightly different fre-uencies, and the scattering amplitudes from these twotates are equal and opposite. This results in a dramaticecrease in the rate of single-photon absorption on theriginal transition.

One potential problem in using conventional EIT inuantum Zeno gates is the difficulty in separating aingle photon from an intense laser beam, even if the pho-on and laser beam have different frequencies. A similareduction in the single-photon scattering can bebtained,12 however, without any laser beams if the pho-ons are tuned between two of the resonant modes of anptical cavity, as illustrated in Fig. 8. Here one or morehotons with energy �� are incident in a waveguide that

ig. 7. Zeno logic gate implemented as in Fig. 6 but with vari-ble couplings included between the wave guides and the reso-ators to improve the performance.

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s strongly coupled to a ring resonator. The resonator inurn is coupled to a large number of two-photon absorbingtoms. (The strong coupling between the resonator andhe cavity will reduce the Q of the cavity, which allows aignificant response even between two of the resonantodes.)Under these conditions, there is destructive interfer-

nce between the scattering amplitudes from resonatortates lR and mR, since they correspond to equal and op-osite detunings. A second-order perturbation theoryalculation12 of the single-photon scattering rate R1 giveshe result

R1 = 2NA� �1

�12 + ���1�2�� M1MW

� − ��0/2+

M1MW

� + ��0/2�2

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ere � is the difference in energy between the incidenthotons and the average energy of the two resonatortates, and �±��0 /2 is the detuning in the two interme-iate states. The number of atoms in a mode volume haseen denoted by NA, �1 is the linewidth of the first excitedtomic state and �1 is its detuning, and M1 and MW arehe matrix elements associated with the absorption of ahoton by the atom and the resonator, respectively. For=0, the detunings are equal and opposite and the totalcattering amplitude vanishes, just as is the case in EIT.

The calculated12 two-photon absorption rate andingle-photon scattering rates for this system are shownn Fig. 9. When the incident photons are tuned between

ig. 8. Optical transparency using the interference between twoodes of a resonant cavity. One or more photons in a waveguide

re strongly coupled into a ring resonator, which in turn isoupled into a large number of two-photon absorbing atoms.uantum interference eliminates single-photon scattering when

he incident photons are tuned between two resonator levels, al-hough strong two-photon absorption can still occur.

ig. 9. Numerical results (solid curve) showing a reduction inhe single-photon scattering rate due to quantum interference inhe device of Fig. 8. The dotted curve shows the two-photon ab-orption rate (both in arbitrary units) (from Ref. 12).

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Franson et al. Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. B 213

he two resonant modes ��=0�, there is no single-photoncattering, as expected. On the other hand, the two-hoton absorption has a strong peak at the same locationrovided that the upper level of the atomic transition isesonant with the energy of two photons. Numerical esti-ates for a realistic set of parameters suggest that the

ingle-photon scattering rate can be 4 or 5 orders of mag-itude less than the two-photon absorption rate underhese conditions, which may allow the operation of Zenoates with relatively low loss.

. SUMMARYinear optics is a promising approach for quantum com-uting, but the logic operations have an intrinsic failureate, which can be overcome by using cluster states andther techniques. Here we have reviewed an alternativepproach10 in which the quantum Zeno effect is used touppress the failure events and produce nearly determin-stic logic operations. This reduces the overhead in themount of resources that are required, and it also elimi-ates the need for high-efficiency detectors. The Zeno ef-

ect can be implemented using strong two-photon absorp-ion to suppress the emission of two photons into theame output port, which is the only failure mode of ourNOT gate.One of the challenges in implementing Zeno gates is to

roduce sufficiently strong two-photon absorption with ainimal rate of single-photon scattering. The single-

hoton scattering can be reduced compared with the two-hoton absorption by using a medium with a uniform den-ity, microcavities with small mode volume, and a neworm of EIT that does not require any strong lasereams.12 It may be possible to implement high-erformance Zeno logic gates by using a combination ofhese techniques.

CKNOWLEDGMENTShis work was funded by the Army Research Office andhe Disruptive Technology Office under grant W911NF-5-0397.Corresponding author J. D. Franson can be reached by

-mail at jfranson@umbc.edu.

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