quantum-nondemolition measurement of photon arrival using an atom-cavity system

Quantum-nondemolition measurement of photon arrival using an atom-cavity system

Kunihiro Kojima* and Akihisa Tomita†

Quantum Computation and Information Project, ERATO-SORST, JST, Miyukigaoka 34 Ibaraki 305-8501, Japan�Received 29 August 2006; revised manuscript received 25 December 2006; published 14 March 2007�

A simple and efficient quantum-nondemolition measurement �QND� scheme is proposed in which the arrivalof a signal photon is detected without affecting the qubit state. The proposed QND scheme functions even ifthe ancillary photon is replaced with weak light composed of vacuum and one-photon states. Although thedetection scheme is designed for entanglement sharing applications, it is also suitable for general purificationof a single-photon state.

DOI: 10.1103/PhysRevA.75.032320 PACS number�s�: 03.67.Hk, 32.80.�t, 42.50.�p

I. INTRODUCTION

Social needs for secure communications to prevent eaves-dropping, impersonation, falsification, and denial of servicehave increased dramatically in recent years accompanyingthe expansion of the service industry on the internet. Com-munication protocols based on quantum entanglement withnonlocal correlation have been the focus of intensive re-search as a possible means of providing secure communica-tions �1,2�. Such protocols often require prior sharing of en-tangled photons �3� or entangled atomic systems �4–8�among more than two nodes. However, there is a practicaldifficulty in sharing entangled photons between distantnodes, since the photons are usually transmitted over lossycommunication channels �2,9�. In such situations, photonsentangled at the input of the channel can readily become auseless mixture of vacuum and photons at the output. Toaddress this problem, it is necessary to purify the final mix-ture by removing the vacuum component �10,11�. Quantum-nondemolition �QND� measurements, by which the arrival ofa signal photon is detected without affecting the qubit state�encoded into the polarization mode�, may be very suitablefor this purpose �3,12�. In this study, a new scheme for QNDmeasurement that can be applied in cases involving the mix-ture of vacuum and photons is proposed.

Figure 1�a� shows the general process of QND detection.In this scheme, an ancillary photon prepared at the ancillaryinput Ain is transmitted and the photon is detected at thedetector D1 of the ancillary output Aout only when a signalphoton appears at the signal input Sin. The mixed state com-posed of vacuum and one-photon states at Sin is thus purifiedinto a one-photon state at the output port Sout after filteringby filter F, which allows the signal photon to pass throughthe filter only when the detector D1 detects the ancillaryphoton. When there is no photon at Sin, ancillary photonsappear at the reflection port Aref.

It is necessary to compose the corresponding realizationsfor the above scheme such that the qubit state of the signalphoton is unchanged. Figure 1�b� shows the proposed QNDscheme for the case that the qubit state is encoded into thepolarization mode of the signal photon. When the signal pho-

ton appears at Sin, the photon is transmitted or reflected at thepolarization beam splitter PBS1 depending on the polariza-tion mode. The polarization of the transmitted photon is or-thogonal to that of the reflected photon. After passingthrough PBS1, the wave packet of the signal photon is de-scribed by the superposition of the transmitted and reflectedwave-packet components. Each component then enters intothe QND with a single polarization mode �SQND�. The ele-ments SQND on each path are identical.

The process in SQND should be the same as that for QNDin Fig. 1�a� except in two aspects. To maintain coherencebetween the transmitted and reflected components of the sig-

*Electronic address: [email protected]†Electronic address: [email protected]

(a)

(b)

(c)

FIG. 1. �a� Schematic representation of QND measurement of asignal photon. �b� Proposed QND measurement scheme. �c� Pro-posed implementation of SQND for a single-mode polarized pho-ton. Sin/out: input-output channel of signal photon; Ain/out/ref: input-output-reflection channel of ancillary photon; F: filter; BS: halfbeam splitter; R: polarization rotator �changes the polarization ofthe input photon to the orthogonal polarization�; C: circulator; D:photodetector; PBS: polarization beam splitter.

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nal photon, the transmitted wave-packet components of theancillary photons at each ancillary output Aout should becombined at the half beam splitter BS1 before detection byD2 and D3 �Fig. 1�b�� in order to erase information on thepath taken by the signal photon. For the same purpose, theinput state at each ancillary input port Ain in Fig. 1�b� mustbe a superposition of vacuum and one-photon states. In par-ticular, when the input state at each Ain is a one-photon state,the wave-packet component at each ancillary reflection portAref must be detected by the same procedure as for the wave-packet component at each ancillary output port Aout. The sig-nal component at each output shown in Fig. 1�b� is recom-bined by PBS2 after or before detection for ancillaryphotons. The signal photon thus passes through the filter Fonly when an ancillary photon is detected at D2 or D3.

The functionality of SQND has been proposed and dem-onstrated based on a ��3� material for modulating the phaseof the ancillary photon only when a signal photon passesthrough the material �13,14�. The phase change is then de-tected by a single-photon self-interference measurement afterseparating the ancillary photon from the signal photon,which are orthogonally polarized, by PBS. However, phasemodulation by a single photon is very small �12�, and theinterference related to phase modulation will be suppresseddue to entanglement between the signal photon and the an-cillary photon, which changes the pulse shape of the ancil-lary photon. The self-interference effect is thus reduced, de-grading the efficiency of SQND. To avoid the use of weak��3� nonlinearities, a single-photon QND device composed oflinear optics and projective measurements has been pro-posed. However, such a scheme requires strict mode match-ing between the signal photon and the probe photon as one ofa maximally entangled photon pair �12,15�.

The QND measurement scheme is implemented in thepresent study by a simple and efficient method involving atwo-sided atom-cavity system consisting of two identicalmirrors and an atom coupled with a single mode of a cavity.Figure 1�c� shows a schematic of the proposed implementa-tion for SQND. The polarizations of the signal and ancillaryphotons are orthogonal, and the two photons are combinedby the polarization beam splitter PBS3 and directed to thetwo-sided atom-cavity system. The solid circle in Fig. 1�c�represents the intracavity atom, which interacts with photonsvia the cavity mirrors. The ancillary photons are consideredto be totally reflected by the atom cavity when there is nosignal photon at the input. This is realizable as suggested bythe experiments of Turchette and co-workers �16,17�. Whena signal photon arrives at the input, reflection of ancillaryphotons at the atom cavity is suppressed by saturation of theatomic transition due to the absorption of signal photons.Detection of the ancillary photon at D2 and D3 thus impliesthe arrival of the signal photon.

In the scheme shown in Fig. 1�c�, the circulator C1 playsthe role of separating the signal photon reflected at the atomcavity from the photon at the input port. Likewise, C2 sepa-rates the ancillary photon reflected at the atom cavity fromthe photon at the ancillary input port Ain. PBS4 separates thesignal photon transmitted at the atom cavity from the trans-mitted ancillary photon. The signal photon thus appears atoutput port 1 or 2, where the wave packet of the signal pho-

ton is described by the superposition of the wave-packetcomponents of output ports 1 and 2. These components arethen combined by the half beam splitter �BS2� and the signalwave-packet appears at uutput when perfect mode-matchingat BS2 is achieved. For the unachievable case, the wave-packet component of the signal photon leaks out on the op-posite side of the output at BS2. The leaked component willstill be useful for QND as shown in Fig. 1�b� if the leakedcomponents on each path are combined by PBS and anothersignal output port is prepared.

The performance of the proposed QND scheme �Fig.1�b�� is characterized in terms of efficiency and successprobability, where the efficiency is defined as the probabilitythat the signal photon appears at output 1 or 2 �Fig. 1�c��when the ancillary photon is detected, while the successprobability is defined as the probability that the ancillaryphoton is detected when the signal photon appears at outputport 1 or 2. To estimate these quantities, the responses of thetwo-sided atom-cavity system for one- and two-photon inputwas analyzed considering a range of pulse durations for theinput photons. Useful conditions for the QND are also exam-ined. The pulse shape of the output signal photon after thetime-resolved detection of the ancillary photon is analyzedqualitatively, since information on the pulse shape is impor-tant for processing the signal photons with other photons. Itis found to be possible to increase the efficiency by up to100% by increasing the pulse duration of the ancillary pho-ton. However, the success probability is decreased simulta-neously to 0% in a trade-off relationship. The efficiency ismaintained even if the ancillary photon is replaced withweak light described by the superposition of vacuum andone-photon states, although the success probability is re-duced in such a case. These results suggest that the modematching between the input light and the cavity mode is lesscritical in the proposed scheme �18� than in the QND pro-posals based on interferometry �12,13,15�.

The remainder of this paper is organized as follows. InSec. II, the Hamiltonian for the two-sided atom-cavity sys-tem is presented and the corresponding model is introduced.The output state for the atom-cavity system is then derivedfor one-photon pulse input �Sec. III�, and the output state fortwo-photon input is obtained using the results for one-photoninput �Sec. IV�. In Sec. V, the performance of the proposedQND is examined quantitatively and the coherence of theoutput signal photons is analyzed qualitatively for the casethat the influence of the dephasing in the QND process isnegligible. The implementation of the proposed QND is thendiscussed in Sec. VI.

II. THEORETICAL MODEL

To analyze the responses of a two-sided cavity, in which asingle two-level system couples with the single mode of thecavity, for one- and two-photon pulse input, a model of spa-tiotemporal propagation to and from the two-sided atom-cavity system is necessary. The proposed model is illustratedin Fig. 2. The cavity couples with the left-side field mode FLand the right-side field mode FR via the two mirrors, whichhave transmittance T and T� �T=T��. In the figure, g and e

KUNIHIRO KOJIMA AND AKIHISA TOMITA PHYSICAL REVIEW A 75, 032320 �2007�

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denote the ground and excited states of the single two-levelatom. It is assumed that only one longitudinal and transversalmode is allowed in the cavity. The vertical arrow on the leftside of the cavity �FL� represents the radiative input �rL

�0� and output �rL�0� fields at the cavity, where rL corre-sponds to the spatial coordinate. The vertical arrow on theright side �FR� similarly represents the input �rR�0� andoutput �rR�0� fields. The two-way arrows at the origin onthe left and right arrows represent coupling of the atom-cavity system with the radiative fields FL and FR.

The total Hamiltonian for this model is as follows:

H = �i=L,R

�HFi+ HintFi

� + Hintac �1�

with

HFi= �

−�

�

dk �ckibFi

† �k�bFi�k� ,

HintFi= �

−�

�

dki��c�

��bFi

† �k�a − a†bFi�k�� ,

Hintac = �g�a†− + −†a� ,

where −= �ge�, and a and bFi�k� are the annihilation opera-

tors for the single mode of the cavity and the radiative fieldFi �i=L ,R�, respectively. The single mode of the cavity isresonantly coupled with the atomic system and all the Hamil-tonians have been formulated in a rotating frame defined bythe transition frequency of the atomic system 0. The wavevector is likewise defined in the rotating frame, that is, k isdefined relative to the resonant wave vector 0 /c. The factor�c� /� is the coupling constant between the single mode ofthe cavity and the radiative field, where � is the cavity decayrate due solely to the coupling of the cavity mode with theradiative field Fi. The factor g is the coupling constant be-tween the cavity mode and the two-level system.

The response of the two-sided atom cavity depends on therelative magnitude of the cavity decay rate � with respect to

the coupling constant g. The bad-cavity regime characterizedby ��g �19,20� is assumed, as described below.

III. ONE-PHOTON PROCESSES

In the following calculations, it is assumed that the two-level system is in the ground state before the arrival of theinput one-photon pulse from the left side of the cavity. Thestate of the field-atom-cavity system for one-photon pro-cesses can be expanded on the basis of the wave-numbereigenstates �kL and �kR of the radiative fields, the excitedstate of the two-level system �E, and the cavity one-photonstate �C. The state �kL denotes a state with the atom in theground state g, the cavity mode a, and all modes of the “Rfield” kR in the vacuum state, and one mode of the “L field”kL in the first excited state, with the remaining states beingthe vacuum state, i.e., �kL= �g ,0a ,1kL

,0kR. Likewise, �kR

= �g ,0a ,0kL,1kR

, �C= �g ,1a ,0kL,0kR

, and �E= �e ,0a ,0kL

,0kR. The quantum state for the one-photon pro-

cess can then be written as

��t� = �E;t��E + ��C;t��C +� dkL��kL;t��kL

+� dkR��kR;t��kR . �2�

On these bases, the Hamiltonian given by Eq. �1� can beexpressed as

H1ph = �ckL + �ckR + i��c�

��

−�

�

dkL��kLC� − �CkL��

+ i��c�

��

−�

�

dkR��kRC� − �CkR�� + �g��CE�

+ �EC�� ,

where

kj = �−�

�

dkj�kjkj� . �3�

The equations for the temporal evolution of the probabilityamplitudes �E ; t�, ��C ; t�, ��kL ; t�, and ��kR ; t� can thus beobtained from the Schrödinger equation i�d /dt��t�= H��t� using Eqs. �2� and �3� as follows:

d

dt �E;t� = − ig��C;t� , �4�

d

dt��C;t� = − ig �E;t� −�c�

�� dkL��kL;t�

−�c�

�� dkR��kR;t� , �5�

d

dt��kL;t� = − ikLc��kL;t� +�c�

��C;t� , �6�

�

�

�

�

�

�

�

�

��

�

�

��

�

�

��

�

�

��

� �

�

� � �

�

�

� �

FIG. 2. Schematic representation of cavity geometry.

QUANTUM-NONDEMOLITION MEASUREMENT OF PHOTON… PHYSICAL REVIEW A 75, 032320 �2007�

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d

dt��kR;t� = − ikRc��kR;t� +�c�

��C;t� . �7�

The evolutions ��kL ; t� and ��kR ; t� can be obtained by inte-grating Eqs. �6� and �7�:

��kL;t� = e−ikLc�t−ti��kL;ti� +�c�

��

ti

t

dt�e−ikLc�t−t��C;t�� ,

�8�

��kR;t� = e−ikRc�t−ti��kR;ti� +�c�

��

ti

t

dt�e−ikRc�t−t��C;t�� ,

�9�

where ti is the initial time of the evolution. To describe theevolution in real space, the results of the integrations of Eqs.

�8� and �9� are subjected to Fourier transformation using

� j�rj;t� � �1

�2��

−�

�

dkjeikjrj��kj;t� for rj � 0

−1

�2��

−�

�

dkjeikjrj��kj;t� for rj � 0,

where j = L,R . �10�

As the incoming field rj �0 is discontinuously connected tothe outgoing field rj �0 via the mirror of the cavity, whichchanges the phase of the incoming field by �, the phase ofthe incoming field amplitude is different from that of theoutgoing amplitude by �.

The real-space representation of the temporal evolutionon the field FL is then given by

�L�rL;t� = ��L„rL − c�t − ti�;ti… for rL � 0

− �L„rL − c�t − ti�;ti… for c�t − ti� � rL

− �L„rL − c�t − ti�;ti… −�2�

c��C;t −

rL

c� for 0 � rL � c�t − ti� . �11�

The first case corresponds to the single-photon amplitude propagating on the incoming field rL�0, the second case corre-sponds to reflection of the single-photon amplitude by the left mirror of the cavity and then propagation on the outgoing fieldrL�0, and the third case consists of two parts; the component reflected by the left mirror, and the amplitude of a single photonre-emitted into the outgoing field rL�0 after absorption by the cavity.

Likewise, the real-space representation of the temporal evolution on the field FR reads as

�R�rR;t� = ��R„rR − c�t − ti�;ti… for rR � 0

− �R„rR − c�t − ti�;ti… for c�t − ti� � rR

− �R„rR − c�t − ti�;ti… −�2�

c��C;t −

rR

c� for 0 � rR � c�t − ti� . �12�

The temporal evolution of the cavity one-photon amplitudecan be obtained by integrating Eq. �5� and using the Fouriertransform �10�:

��C;t� = − ig�ti

t

dt�e−2��t−t�� E;t�� + e−2��t−ti��C;ti�

− �2�c�ti

t

dt�e−2��t−t��L„− c�t� − ti�;ti…

− �2�c�ti

t

dt�e−2��t−t��R„− c�t� − ti�;ti… . �13�

Since, in the present analysis, the atom-cavity system is in

the ground state before the one-photon input pulse propagat-ing on the field FL arrives at the system, the cavity-stateamplitude ��C ; ti� and the excited-state amplitude �E ; ti� atthe initial time are zero, and the field amplitude �L�rL ; ti� iszero for the region rL�0. Moreover, it is assumed that thestate of the field FR is initially the vacuum state, that is, thefield amplitude �R�rR ; ti� is zero. Under these assumptions,Eqs. �12� and �13� can be reduced as follows:

��C;t� = − ig�ti

t

dt�e−2��t−t�� E;t��

− �2�c�ti

t

dt�e−2��t−t��L„− c�t� − ti�;ti… �14�


032320-4

�R�rR;t� = �0 for rR � 0 or c�t − ti� � rR

−�2�

c��C;t −

rFR

c� for 0 � rR � c�t − ti� . �15�

It is assumed above that the atom-cavity system is in thebad-cavity regime characterized by ��g. The cavity one-photon amplitude given by Eq. �14� can then be approxi-mated as

��C;t� � − ig

2� �E;t�

− �2�c�ti

t

dt�e−2��t−t��L„− c�t� − ti�;ti… . �16�

The excited-state amplitude �E ; t� can be obtained by inte-grating Eq. �4� and then substituting Eq. �16�, affording

�E;t� � ig�2�c�ti

t

dt�e−��/2��t−t��

��ti

t�dt�e−2��t�−t��L„− c�t� − ti�;ti…

where � = g2/� . �17�

The temporal evolution of the excited-state amplitude is ex-plicitly dominated by the atomic dipole relaxation character-ized by the rate � and is implicitly and effectively restricted

by the cavity decay characterized by the rate � and the inputpulse duration. The reduction of that amplitude due to thecavity decay prevents efficient interaction between the inputphoton and the two-level system, which is the starting pointof efficient nonlinear two-photon interaction. It is thereforeassumed that the pulse duration of the input one-photon ismuch larger than the cavity decay time 1/�. The excited-state amplitude �17� can then be approximated as

�E;t� � i�c�

2�

ti

t

dt�e−��/2��t−t��L„− c�t� − ti�;ti…

for t − ti � 1/� . �18�

Likewise, the cavity one-photon amplitude given by Eq. �16�can be approximated with Eq. �18� as

��C;t� ��

2g�c�

2�

ti

t

dt�e−��/2��t−t��L„− c�t� − ti�;ti…

−� c

2��L„− c�t − ti�;ti… for t − ti � 1/� . �19�

The effective field amplitude for �L�rL ; t� can be obtainedby substituting Eq. �19� into Eq. �11�, giving

�L�rL;t� � ��L„rL − c�t − ti�;ti… = 0 for c�t − ti� � rL

�L„rL − c�t − ti�;ti… for rL � 0

−�

2�

ti

t

dt�e−��/2��t−rL/c−t��L„− c�t� − ti�;ti… for 0 � rL � c�t − ti� . �20�

Here, the first case is equal to zero, since the field amplitude �L�rL ; ti� is assumed to be initially equal to zero for rL�0. Thesecond case corresponds to the incoming field amplitude at the atom-cavity system, and the third case corresponds to the fieldamplitude of the photon re-emitted by the intracavity atomic system.

The effective-field amplitude for �R�rR ; t� can similarly be obtained as

�R�rR;t� � ��R„rR − c�t − ti�;ti… = 0 for rR � 0 or c�t − ti� � rR

−�

2�

ti

t

dt�e−��/2��t−rR/c−t��L„− c�t� − ti�;ti… + �L„rR − c�t − ti�;ti… for 0 � rR � c�t − ti� . �21�

Here, the first case is equal to zero according to the initialcondition, and the second case represents the interferencebetween the field amplitude of the transmitted photon with-out absorption by the atomic system and the field amplitude

of the photon re-emitted by the atomic system.To investigate the outgoing amplitudes �L�rL�0; t�;

�R�rR�0; t� for an arbitrary incoming amplitude under theabove-mentioned initial conditions, it is convenient to repre-


032320-5

sent the outgoing amplitudes as a matrix element of the evo-lution operator, as follows.

�L�rL;t� = rL��t� = �−�

�

drL�u1ph�L� �rL,rL� ;t − ti� � �L�rL� ;ti� ,

�22�

�R�rR;t� = rR��t� = �−�

�

drL�u1ph�R� �rR,rL� ;t − ti� � �L�rL� ;ti� .

�23�

Here, u1ph�L� �rL ,rL� ; t− ti� and u1ph

�R� �rR ,rL� ; t− ti� are the matrix el-

ements of the evolution operator e−�i/��H1ph, representing thetransition probability amplitude from the state �rL� at time tito the state �rL or �rR at time t, where �rj� 1

�2��−�

� dkje−ikjrj�kj for j=L ,R. These matrix elements can

be obtained approximately by comparing the results of Eqs.�20�–�23�, as follows:

u1ph�L� �rL,rL� ;t − ti� � uabs

�L��rL,rL� ;t − ti� �24�

u1ph�R� �rR,rL� ;t − ti� � utrans

�R� �rR,rL� ;t − ti� + uabs�R��rR,rL� ;t − ti�

�25�

with

utrans�R� �rR,rL� ;t − ti� = �„rR − c�t − ti� − rL�…

and

uabs�j� �rj,rL� ;t − ti� = � −

�

2ce−��/2c��c�t−ti�+rL�−rj�

for 0 � rj � c�t − ti� + rL� and rL� � 0

0 for rj � c�t − ti� + rL� or rL� � 0.

�26�

for j = L,R .

The component utrans is the transition component for thetransmitted photon without absorption by the atomic system,while uabs is the transition component for the photon re-emitted by the atomic system.

IV. TWO-PHOTON PROCESSES

The interaction between two photons in the atom-cavitysystem is treated as follows. It is assumed that the two pho-tons, Photon1 and Photon2, are distinguishable by the polar-ization mode in this case. The atomic system described bythe theoretical model can thus be implemented as a V-typethree-level system. In the V-type system, there are two ex-cited states, ��1 and ��2, with orthogonal polarizations butsharing the same ground state �g �see Fig. 3�. The transitionto the excited state ��1�2� is caused by Photon1 �2�. Accord-ing to this representation, the total Hamiltonian is given by

H = �i=L,R;j=1,2

�HFi

�j� + HintFi

�j� � + Hintac�j�

with

HFi

�j� = �−�

�

dk �ckibFij

† �k�bFij�k�

HintFi

�j� = �−�

�

dki��c�

��bFij

† �k�aj − aj†bFij

�k��

Hintac�j� = �g�aj

†−�j� + −

†�j�aj� , �27�

where −�j�= �g� j�, and aj and bFij

�k� are the annihilationoperators for the jth mode of the cavity and the radiativefield Fij �i=L ,R and j=1,2�, respectively. The other condi-tions are the same as in the theoretical model.

To extend the response function for the one-photon inputgiven by Eqs. �24� and �25� to the two-photon case, the statedescription for one-photon processes presented above mustbe extended to two-photon processes to afford the totalHamiltonian given by Eq. �27� for the distinguishable two-photon input involving Photon1 and Photon2. The states fortwo-photon processes obtained by extending the state de-scription for one-photon processes are as follows:

�C1 � �C2, �E1 � �C2

�C1 � �E2, �ki1 � �ki�2

�C1 � �ki2, �ki1 � �C2

�E1 � �ki2, �ki1 � �E2, �E1 � �E2 . �28�

Here, i , i�=L ,R. The state �kLj denotes a state with the atomin the ground state gj, the cavity mode aj, and all modes ofthe“Rj field” kRj in the vacuum state, and one mode of the “Ljfield” kLj in the first excited state, with the remaining statesbeing the vacuum state, i.e., �kLj= �gj ,0aj

,1kLj,0kRj

. Like-wise, �kRj= �gj ,0aj

,0kLj,1kRj

, �Cj= �gj ,1aj,0kLj

,0kRj, and

�Ej= �ej ,0aj,0kLj

,0kRj for j=1,2.

Denoting the states �g, ��1, and ��2 of the V-type systemby �g1 ,g2, �e1 ,g2, and �g1 ,e2, the interaction Hamiltonian

Hintac�j� given in Eq. �27� can be rewritten for the interaction

with two distinguishable photons �Photon1 and Photon2� as

� � � � � � � � � � � � � �

� �

�

� � �

�

�

� � �

FIG. 3. V-type three-level system.


032320-6

�j=1,2

Hintac�j� = �g �

i=L,R��C1E1� + �E1C1��

� ��C2C2� + �−�

�

dki2�ki2ki2��+ ��C1C1� + �

−�

�

dki1�ki1ki1�� C2E2�

+ �E2C2�� . �29�

See the Appendix for the derivation of Eq. �29�. To facilitateformulation of the matrix element of temporal evolution fortwo-photon processes, the Hamiltonian given by Eq. �29� isfurther divided into a linear term and a nonlinear term asfollows:

Hintac = Hintaclin + Hintac

Nonlin,

where

Hintaclin = �g��C1E1� + �E1C1�� I1ph

�2� + I1ph�1�

� ��C2E2�

+ �E2C2�� 30�

HintacNonlin = − �g��C1E1� + �E1C1�� E2E2� + �E1E1�

� ��C2E2� + �E2C2��

with

I1ph�j� = �

i=L,R��

−�

�

dkij�kijkij� + �CjCj� + �EjEj�� .

�31�

The linear term given by Eq. �30� describes the dynamics ofthe two photons, Photon1 and Photon2, which are absorbedand emitted independently. The linear Hamiltonian includestransitions to the state �E1 ,E2, where both photons are ab-sorbed by the atomic system. This transition is impossible ina two-level system �a V-type three-level system is consideredhere�. The nonlinear term given by Eq. �31� suppresses tran-sitions to the state �E1 ,E2.

The total Hamiltonian given by Eq. �27� for two-photonprocesses is thus given by

H2ph = Hlin + HNonlin. �32�

Hlin = H1ph�1�

� I1ph�2� + I1ph

�1�� H1ph

�2� , �33�

HNonlin = − �Hintac�1�

� �E2E2� + �E1E1� � Hintac�2� � , �34�

where

H1ph�j� = Hintac

�j� + �i=L,R

�ck�ij� + Hintfc�ij� + �EjEj�

with

k�ij� = �−�

�

dkijkij�kijkij� ,

Hintfc�ij� = i��c�

��

−�

�

dkij��kijCj� − �Cjkij�� ,

and

Hintac�j� = �g��CjEj� + �EjCj�� ,

where the indices i=L ,R distinguish the left-side field of thetwo-sided cavity from the right-side field, and the indices j=1,2 distinguish the two photons and the two excited states.

As the temporal evolution described by Hlin is composedof the evolution of a single photon, the corresponding matrixelement of the temporal-evolution operator can be expressedas the product of the individual single-photon matrix ele-ments given by Eqs. �24� and �25�, i.e.,

u2phlin�jk��rj1,rk2;rL1� ,rL2� ;t − ti� = u1ph

�j� �rj1;rL1� ;t − ti�

�u1ph�k� �rk2;rL2� ;t − ti�

for j,k = L,R . �35�

The components of re-emission from the state �E1 ,E2 canbe effectively described by

uabs�j� �rj1,rL1� ;t − ti�uabs

�k� �rk2,rL2� ;t − ti�

for 0 � rj1,rk2 � c�t − ti� + Min�rL1� ,rL2� � �36�

using Eq. �26�. These re-emission components refer to theprocess in which the two photons are absorbed and remittedsimultaneously by an atom. However, the nonlinear term�34� eliminates these components �19�. The total matrix ele-ment for two-photon processes can thus be described by

u2ph�jk��rj1,rk2;rL1� ,rL2� ;t − ti�

= u2phlin�jk��rj1,rk2;rL1� ,rL2� ;t − ti�

+ u2phNonlin�jk��rj1,rk2;rL1� ,rL2� ;t − ti� ,

where

u2phNonlin�jk��rj1,rk2;rL1� ,rL2� ;t − ti� � − �Eq. �36�� . �37�

The output wave function on the left-side and right-side out-put fields can then be expressed as

� jk�rj1,rk2;t� = �−�

�

drL1� drL2� u2ph�jk��rj1,rk2;rL1� ,rL2� ;t − ti�

� �LL�rL1� ,rL2� ;ti� for j,k = L,R . �38�

The output wave function describes the far-field state of thephotons after interaction with the atom-cavity system. Ingeneral, a two-photon wave function propagating in freespace is given by � jk�rj1 ,rk2 ; t�=� jk�rj1−ct ,rk2−ct�. The re-sults of Eq. �37� and �38� can therefore be simplified bytransformation to a moving coordinate system, i.e.,

rj1 − ct = xj1,

rj2 − ct = xj2,

rL1� − cti = xL1� ,


032320-7

rL2� − cti = xL2�

for j,k = L,R .

In this coordinate system, the output wave function in theoutgoing far field is expressed as

�out�jk��xj1,xk2� � �

−�

�

dxL1� dxL2� � u2ph�jk��xj1,xk2;xL1� ,xL2� �

�in�LL��xL1� ,xL2� � for j,k = L,R , �39�

where u2ph�jk��xj1 ,xk2 ;xL1� ,xL2� � is given by

u2ph�jk��xj1,xk2;xL1� ,xL2� � = u2ph

lin�jk��xj1,xk2;xL1� ,xL2� �

+ u2phNonlin�jk��xj1,xk2;xL1� ,xL2� � ,

where

u2phlin�jk��xj1,xk2;xL1� ,xL2� � = u1ph

�j� �xj1;xL1� �u1ph�k� �xk2;xL2� �

with

u1ph�L� �xL,xL�� = uabs

�L��xL,xL�� ,

u1ph�R� �xR,xL�� = utrans

�R� �xR,xL�� + uabs�R��xR,xL�� ,

utrans�j� �xj;xL�� = ��xL� − xj� ,

uabs�j� �xj;xL�� = −

�

2ce−��/2c��xL�−xj� for xj � xL� ,

and

u2phNonlin�jk��xj1,xk2;xL1� ,xL2� �

= − uabs�j� �xj1,xL1� �uabs

�k� �xk2,xL2� �

for 0 � xj1,xk2 � Min�xL1� ,xL2� � .

V. PERFORMANCE OF QND MEASUREMENT

The performance of the proposed QND �Fig. 1� is evalu-ated using Eq. �39�. The pulsed mode of the one- and two-photon input is assumed to be a Gaussian mode �in�xL�=� 2

d��exp�−2xL

2 /d2�, where d is the input pulse duration. Theone- and two-photon pulsed state can thus be described by

��in1ph =� dxL�in�xL��xL �40�

and

��in2ph = ��in1 � ��in2 =� dxL1dxL2�in�xL1��in�xL2�

��xL1;xL2 . �41�

The corresponding output states can be formulated using Eq.�39� as

��out1ph = �

j=L,R� dxL�out

�j� �xj��xj , �42�

��out2ph = �

j,k=L,R� dxj1dxk2�out

�jk��xj1,xk2��xj1;xk2 , �43�

where

�out�j� �xj� =� dxj�u1ph

�j� �xj;xL��in�xL�� ,

�out�jk��xj1,xk2� =� dxL1� dxL2� ,

�u2ph�jk��xj1,xk2;xL1� ,xL2� ��in�xL1� ��in�xL2� � .

The transmittance and reflectance of the atom-cavity systemfor one- and two-photon input can be characterized by thedetection probabilities on the left and right sides of the cav-ity, as given by

P1ph�j ;d� =� dxj��out�j� �xj��2, �44�

P2ph�j1,k2;d� =� dxj1dxk2��out�jk��xj1,xk2��2 for j,k = L,R .

�45�

As mentioned in the Introduction, the performance of theQND can be characterized in terms of efficiency and successprobability. The efficiency is defined as the probability thatthe signal photon appears at the output when the detector D1or D2 �Fig. 1�b�� detects an ancillary photon. Note that theprobability associated with detection of the arrival of thesignal photon is different from the probability related to de-tection of the ancillary photon: the former is given byP2ph�R1,L2;d�+ P2ph�R1,R2;d�, while the latter is given byadding P1ph�R ;d� to the former. The efficiency is thus givenby the former divided by the latter, i.e.,

EQND�d� =P2ph�R1,R2;d� + P2ph�R1,L2;d�

P1ph�R;d� + P2ph�R1,R2;d� + P2ph�R1,L2;d�.

�46�

This equation is obtained using Eqs. �44� and �45�. Likewise,the success probability is defined as the probability that thedetector D2 or D3 detects an ancillary photon when the sig-nal photon appears at output 1 or 2 �Fig. 1�c��, as given by

Psuc�d� = P2ph�R1,R2;d� + P2ph�R1,L2;d� . �47�

This equation holds when the state at the ancillary input portAin on each path �Fig. 1�b�� is a single photon state. Thiscondition gives the upper limit of the success probability.

Figure 4 shows the efficiency EQND�d� and success prob-ability Psuc�d� for the proposed scheme. The symmetric caseis that in which the pulse durations of the ancillary and signalphotons are identical, while the asymmetric case is that inwhich the pulse duration of the ancillary photon is fixed at40/�, corresponding to an ancillary photon transmittance of0.49%. The efficiency can be increased to 100% by increas-ing the pulse duration of the ancillary photon. However, thesuccess probability is simultaneously decreased to 0%. Con-


032320-8

sidering this trade-off relationship, the efficiency at a successprobability of 10% is 94.3%. A success probability of 8% isobtained for pulse durations of 40/� in the symmetric case.In the asymmetric case, efficiency of 95.5% was obtainedwith a success probability of 10% for pulse durations of12.5/� �signal� and 40/� �ancillary�. The asymmetric casethus relaxes the requirements for the signal pulse in theQND. For example, efficiency of greater than 90% is achiev-able over a wide range of signal photon pulse durationswhile maintaining a success probability of around 10%. Fur-thermore, the jitter of the signal photon is larger than thepulse duration of the signal photon yet smaller than the pulseduration of the ancillary photon. The asymmetric case there-fore allows the arrival time and pulse shape of the signalpulse to be determined by time-resolved photodetection ofthe ancillary photon in conjunction with the output functionEq. �39�. Information on the pulse shape of the output signalphoton will become important when the signal photon is pro-cessed with another photon.

When the pulse durations of the signal and ancillary pho-tons are much longer than the radiative relaxation time 1/�,the output wave functions associated with the detection ofthe ancillary photons, �out

�RL��xR1 ,xL2� and �out�RR��xR1 ,xR2�,

can be approximated by the nonlinear component

�−�� dx1�dx2�u2ph

Nonlin�x1 ,x2 ;x1� ,x2��in�x1��in�x2��. For simplicity,if the pulse shapes of the signal and ancillary photons areassumed to be rectangular and the pulse duration of the sig-nal photon d2 is set much shorter than that of the ancillaryphoton d1, the nonlinear component can be expressed as−�1/d2�e−��/2c��x1−x2�. This function indicates that the pulseshape of the signal photon is not dependent on the detectiontiming of the ancillary photon. For a Gaussian pulse shape,the nonlinear component at x1=x2 is described by a Gaussianfunction. However, the conclusion remains the same as in therectangular case.

VI. DISCUSSION

One of the promising candidates for experimental realiza-tion of the two-sided atom-cavity is a single quantum dotexciton system coupled with the cavity mode of a photoniccrystal �21�. Such a system provides design capabilities fortemporal stability and reproductivity of the dipole couplingwith the cavity mode. For a transition frequency and oscilla-tor strength of the exciton of �0=2.35�105 GHz �0.97 eV�and f =100, the mode volume of a two-dimensional photoniccrystal is Vm=0.02 �m3, with a coupling constant g of 0�132 GHz. As the bad-cavity regime is typically ��4g, theresultant radiative relaxation rate � can take values of 0�33 GHz. If the decoherence time of the exciton byphonons is approximately 1 ns �22�, the pulse duration of theancillary photon should be less than 500 ps in order to avoiddecoherence by phonons. Under this condition, the maximalefficiency for the QND is 86% and the success probability is20% in the symmetric case. As there is little difference be-tween the symmetric and asymmetric cases in terms of effi-ciency and success probability, the corresponding values forthe asymmetric case should be similar. Note that the effi-ciency of 86% is a maximum, since the temporal evolutionof a single exciton dipole under interaction with phonons,driven by weak coherent light with pulse duration of 500 psor more is unknown. This temporal evolution should there-fore be investigated experimentally as part of future research.It will also be necessary to conduct detailed theoreticalanalyses of the decoherence time by phonons beyond theindependent boson model �20,23� in order to discuss effi-ciencies of greater than 90%.

VII. CONCLUSION

A QND measurement scheme involving a two-sidedatom-cavity system for the detection of photon arrival inentanglement sharing was proposed. The efficiency and suc-cess probability of the scheme were estimated by analyzingthe responses of the two-sided atom-cavity system for one-and two-photon input over a range of input pulse duration.The conditions for improved QND performance were alsoexamined. Efficiency of up to 100% was found to be possibleby increasing the pulse duration of the ancillary photon, al-though the success probability is simultaneously reduced to0% in a trade-off relationship. For a success probability of10%, with relaxation of the requirements for the signal pulsein the QND, efficiency of 95.5% was obtained. In the case of

(a)

(b)

FIG. 4. �a� Efficiency and �b� success probability of QND mea-surement. Solid and broken lines denote symmetric and asymmetriccases of pulse duration. The rate � is the dipole relaxation ratedescribed by g2 /�, where g and � represent the coupling constant ofthe atom-cavity system and the cavity decay rate, respectively.


032320-9

signal photons with pulse duration of 12.5/� and ancillaryphotons with pulse duration of 40/�, the obtained successprobability was 10%. The success probability can be in-creased to 100% by returning the signal photon to the inputwhen no ancillary photon is detected on the right side of thecavity. Decoherence on the signal wave packet after detec-tion of the ancillary photon is suppressed by choosing signaland ancillary photon pulse durations much larger than theradiative relaxation time of the atom cavity, in which casethe pulse shape of the signal photon is approximated by thewave function ��x�=�� ·e−��/2c��x�, where � is a normaliza-tion factor.

The proposed QND scheme functions correctly even if theancillary photon is replaced with weak light described by thesuperposition of vacuum and one-photon states. Thereforethe proposal is applicable not only for entanglement sharingbut also for general purification of a single-photon state. Re-alization of the proposal scheme is therefore expected todrive substantial progress in photon manipulation technol-ogy.

ACKNOWLEDGMENTS

K.K. thanks NEC researchers M. Shirane and A. Kiriharafor giving valuable information about the current state on thestudy of photonic crystal cavity and sharing entangledatomic systems.

APPENDIX: DERIVATION OF EQ. (29)

In the temporal evolution under the total Hamiltoniangiven by Eq. �27�, the initial number of energy quanta isalways preserved. For example, the number of energy quantafor a two-photon input is two, and this number is alwayspreserved even upon interaction with the atom-cavity sys-tem. The possible states in the interaction of the V-typethree-level system with two distinguishable photons, Pho-ton1 and Photon2 are thus

�g � �F1�l� � �F2

�l��

��1 � �F1�0� � �F2

�l��

��2 � �F1�l� � �F2

�0� for l,l� = 1,2,3,

where

�Fm�0� � �0am

,0kLm,0kRm

,

�Fm�1� � �1am

,0kLm,0kRm

,

�Fm�2� � �0am

,1kLm,0kRm

and

�Fm�3� � �0am

,0kLm,1kRm

.

The state �0am,1kLm

,0kRm denotes a state in which the cavity

mode am and all modes of the Rm field kRm are in the vacuumstate, and one mode of the Lm field kLm is in the first excitedstate, with the remaining states being the vacuum state. Thestate �0am

,0kLm,0kRm

denotes a state in which the cavitymode am and all modes of the Rm field kRm and the Lm fieldkLm are in the vacuum state. The same holds for�1am

,0kLm,0kRm

and �0am,0kLm

,1kRm.

On the truncated Hilbert space composed of these states,the matrix representation of the operators a1

†−�1� and a2

†−�2�

are given by

a1†−

�1� = �g�1� � �F1�1�F1

�0��

� ��F2�1�F2

�1�� + �−�

�

dkL2�F2�2�F2

�2��

+ �−�

�

dkR2�F2�3�F2

�3�� , �A1�

a2†−

�2� = �g�2� � ��F1�1�F1

�1�� + �−�

�

dkL1�F1�2�F1

�2��

+ �−�

�

dkR1�F1�3�F1

�3�� F2�1�F2

�0�� . �A2�

To express the above operators as those acting on the Hilbertspace spanned by the state descriptions given by Eq. �28�, westart from expressing the quantum state of the V-type atomicsystem as the quantum state of the two two-level atomicsystems where the Hilbert space is spanned by the basis��g1 ,g2 , �e1 ,g2 , �g1 ,e2 , �e1 ,e2�. The V-type atomic system,where the excited state is ��1 or ��2 or the superposition ofthese states, does not emit two photons simultaneously. Onthe other hand, the two two-level atomic systems in thedouble excited state ��e1 ,e2� emit two photons simulta-neously. This difference implies that the quantum state of theV-type system should be expressed as a quantum state in thesubspace spanned by the basis except for the double excitedstate �e1 ,e2 of the two two-level atomic systems. Theground state of the V-type atomic system �g corresponds tothe ground state of the two two-level atomic systems �g1 ,g2.From the viewpoint of single-photon resonant transition pro-cesses, the allowed transition to and from either of the twoexcited states ��1 and ��2 corresponds to the transition toand from either of the two excited states �e1 ,g2 and �g1 ,e2.Using these correspondences, the operator �g�1� in Eq. �A1�is expressed as the operator �g1 ,g2e1 ,g2�= �g1e1� � �g2�g2� on the Hilbert space of the two two-level atomic sys-tems. Likewise, The operator �g�2� in Eq. �A2� is expressedas the operator �g1g1� � �g2e2�. The operators given byEqs. �A1� and �A2� are then expressed by substituting theseexpressions as


032320-10

a1†−

�1� = �g1e1� � �F1�1�F1

�0�� g2g2�

� ��F2�1�F2

�1�� + �−�

�

dkL2�F2�2�F2

�2��

+ �−�

�

dkR2�F2�3�F2

�3�� , �A3�

a2†−

�2� = �g1g1� � ��F1�1�F1

�1�� + �−�

�

dkL1�F1�2�F1

�2��

+ �−�

�

dkR1�F1�3�F1

�3�� g2e2� � �F2�1�F2

�0�� .

�A4�

These operators can be expressed on the state descriptions�kLj�kRj, �Cj, and �Ej for j=1,2 as

a1†−

�1� = �C1E1� � ��C2C2� + �i=L,R

�−�

�

dki2�ki2ki2�� ,

�A5�

a2†−

�2� = ��C1C1� + �i=L,R

�−�

�

dki1�ki1ki1�� .

� �C2E2� . �A6�

The matrix representation of the interaction Hamiltonian

� j=1,2Hintac�j� given by Eq. �29� is thus obtained by the above

equations.

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quantum-nondemolition measurement of photon arrival using an atom-cavity system

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