Quantum Zeno Effect in Cavity QED: Experimental Proposal with Non Ideal Cavities and Detectors
aa r X i v : . [ qu a n t - ph ] O c t Quantum Zeno Effect in Cavity QED: Experimental Proposalwith Non Ideal Cavities and Detectors
R. Rossi Jr. and M. C. Nemes
Departamento de F´ısica, Instituto de Ciˆencias Exatas,Universidade Federal de Minas Gerais, C.P. 702,30161-970, Belo Horizonte, MG, Brazil
A. R. Bosco de Magalh˜aes
Departamento Acadˆemico de Ciˆencias B´asicas,Centro Federal de Educa¸c˜ao Tecnol´ogica de Minas Gerais,30510-000, Belo Horizonte, MG, Brazil
Abstract
We propose an experiment with two coupled microwave cavities and a “tunneling” photon ob-served by the passage of Rydberg atoms. We model the coupled cavities as in Ref. [1] and includedissipative effects as well as limited detection efficiency. We also consider realistic finite atom-fieldinteraction times and provide for a simple analytical expression for the photon “tunneling” proba-bility including all these effects. We show that for sufficiently small dissipation constants the effectcan be observed with current experimental facilities.
PACS numbers:Keywords: . INTRODUCTION The success of physical theories is intimately connected to its potentiality to describeexisting empirical data and to predict new, yet to be observed, phenomena [2]. Howeverthe interpretation of empirical data is not completely independent of the proposed theory.Therefore in natural sciences the measurement process plays a double role: it is at the sametime a testing tool of theories and also a physical process in itself, subjected to theoreticalanalysis. In the quantum domain theoretical descriptions of the measurement process are amatter of innumerous discussions.In 1932, in his famous treatise [3], J. von Neumann proposed a quantum measurementtheory, which became quickly well known. An initial premise of this theory is the postulatethat the measurement of a given observable always yields one of the eigenvalues of this ob-servable and, after the measurement, the system collapses to the corresponding eigenvector.This working hypothesis is known as “projection postulate” and is responsible for severalcounterintuitive aspects of the theory. It has led to the formulation of several paradoxes.The “Quantum Zeno Paradox” was presented in a mathematically rigorous fashion in 1977by B. Misra and E. C. Sudarshan [4]. In this formulation the authors show that a sequence ofprojective measurements on a system inhibits its time evolution. The paradoxical characterof this conclusion becomes explicit when one continuously observes the state of an unstableparticle. When the Quantum Zeno Effect (QZE) was first formulated, it has been associatedto two factors: an initially quadratic time decay and the projection postulate.In the 90ies, after the realization of the pioneer experiment [5] on the effect, whichshowed the interruption of the time evolution of a decaying system by means of continuousobservations, the QZE became the center of fervorous debates [6, 7]. The role attributedto the projection postulate was at the center of the discussions. New approaches have beenproposed [6, 8] and the strong association between the QZE and the projection postulatewas no longer a necessary ingredient. Nowadays the literature on the subject is vast andrange from experimental proposals to fundamental theoretical questions [9, 10, 11, 12].In the present contribution we will study the dynamics of the QZE in a (apparentlyfeasible [1, 13, 14]) experiment involving two coupled microwave cavities, one photon andRydberg atoms as probes. The novel aspects explored here are the effect of a lossy environ-ment and of limited efficiency detection on the visibility of the QZE.2n Section II, we describe the main elements of the proposed experiment and their inter-action. In Section III, the QZE is investigated in the situation where several atoms interactwith one cavity mode and next with ionization detectors. In Section IV, we show thatthese measurements of several atomic states are not essential for the QZE; the effects offinite atom-field interaction times and of field dissipation are also studied in this section. Insection V we draw the conclusions.
II. THE MODEL FOR AN EXPERIMENT
Let us consider two cavity modes coupled by a conducting wire (wave guide), as proposedin [1]. The Hamiltonian for the system is given by H AB = ~ ωa † a + ~ ωb † b + ~ g ( a † b + b † a ) , (1)where a † ( a ) and b † ( b ) are creation (annihilation) operators for modes M A and M B , ω theirfrequency and g a coupling constant [1]. The situation we shall consider concerning theelectromagnetic degree of freedom will always involve the following initial state ρ F (0) = | A , B ih A , B | = | , ih , | , where the bra (ket) | n, m i ( h n, m | ) refers to n excitations in mode M A and m excitations inmode M B . The evolution of this state according to (1) in a time interval T is given by ρ F ( T ) = | c ( T ) | | , ih , | + | c ( T ) | | , ih , | + ( c ( T ) c ∗ ( T ) | , ih , | + h.c. ) , (2)where c ( T ) = cos( gT ), c ( T ) = sin( gT ) and h.c. stands for Hermitian conjugate. Thus, dueto the coupling between the cavities, a photon initially in cavity A may be found at time T in cavity B with probability | c ( T ) | . At T = π/ g the photon has performed a completetransition from mode M A to mode M B : ρ C ( T ) = | , ih , | .In order to experimentally verify the occurrence of this transition, one can measure thenumber of photons in cavity B : if the value found is zero we know for sure that the transitiondid not occur. This may be realized by sending an effectively two level atom [15] in its loweststate through cavity B . The atom prepared in its lowest state works as a probe for the fieldstate. In order to realize this “two level atom” one uses a Rydberg atom whose relevanttransition may be tuned to the field quanta ~ ω . We denote by | e i ( | g i ) the higher (lower)3nergy atomic states. This tuning may be effected by using a quadratic Stark effect, as inRef. [16]. The control of the atom-field interaction time may be performed by this methodwith a precision of 1 µs . Since this time is small compared to the other relevant times inthe experiment, we will not consider imperfections in the atom-field interaction time. Theinteraction of the atom with the field mode in cavity B may be described by the Jaynes-Cummings model, which gives τ π = π/ Ω , where Ω is the vacuum Rabi frequency, for the π pulse time, the time in which one excitation moves from mode M B to the atom. If theatom-field coupling is much stronger than the coupling between modes M A and M B , τ π maybe disregarded [24], and we may write the density operator for the system composed by theatom an the field modes, after the atom-field interaction, as ρ AF ( T ) = | c ( T ) | | , , g ih , , g | + | c ( T ) | | , , e ih , , e | + ( c ( T ) c ∗ ( T ) | , , g ih , , e | + h.c. ) . (3)Since the atom-field state is maximally entangled, to measure the atomic level in an ion-ization detector is equivalent to measuring the number of photons in each cavity before theatom-field interaction. III. THE DETECTION PROCESS
In this section we will consider the measurement of the atomic state by ionization de-tectors D e and D g constructed in such a way as to ionize the atom in states | e i and | g i respectively. A. Perfect Detectors
If one has perfect detectors, each atom sent through cavity B will produce a click eitherin D e or D g . Thus the probability p , that a photon initially in mode M A did not reachcavity B is equal to the probability p clickD g of one click in detector D g : p , = p clickD g = | c ( T ) | . (4)If we send N atoms, one at each time t = iT /N ( i = 1 to N ), during the fixed timeinterval T = π/ g , we can in principle monitor the photon transition from mode M A tomode M B . The temporal evolution of the system under such conditions consists of N steps4omposed by a free evolution during a time interval τ A,B = T /N , followed by an atom-fieldinteraction.If in one of these steps we observe one click in D e , we must conclude that the photonwas found in cavity B . As may be seen in Eq. (3), after this click the field state becomes ρ F = | , ih , | , and all the subsequent atoms will be detected in | g i state.Let us now consider an experimental sequence where no clicks in D e are observed. Attime t = 0, the state of the atom-field system is given by ρ AF (0) = | , , g ih , , g | , (5)and during the period τ A,B = T /N the system evolves under the Hamiltonian (1): ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , g ih , , g | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , g | + h.c. ) . (6)At time τ A,B , the atom and the mode M B perform a π pulse (regarded as instantaneous),what leads to¯ ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , e ih , , e | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , e | + h.c. ) . (7)If we observe a click in D g , the state of the system ends up in ρ AF ( τ AB ) = | , , g ih , , g | = ρ AF (0) . (8)The probability of such a click in the first step is | c ( τ AB ) | , and in this case the systemis reprepared in state | , , g ih , , g | . If all atoms are detected in | g i state, the evolutionwill be composed by N identical steps to the one just described. Thus the probability of N clicks in D g is p ( N ) clickD g = (cid:0) | c ( τ AB ) | (cid:1) N , (9)which is equal to the probability p ( N )1 , that the photon is still in cavity A at time T , afterthe interaction between the field and the N atoms. If we consider the limit N → ∞ ,lim N →∞ p ( N )1 , = lim N →∞ p ( N ) clickD g = 1 . (10)Zeno effect becomes explicit: the continuous measuring of the number of photons in cavity B inhibits the transition of the photon from cavity A to cavity B .5 . Inefficient Detectors In order to take the limited efficiency of the detectors into account we need a model forthe detection process. In what follows we consider a schematic model for the atom-detectorinteraction [17]: H D = ~ ǫ g | g ih g | + ~ ǫ e | e ih e | + ~ Z dkǫ k | k ih k | + ~ v g Z dk ( | g ih k | + | k ih g | ) + ~ v e Z dk ( | e ih k | + | k ih e | ) , (11)where | e i and | g i represent the same atomic levels as in previous sections, and the set {| k i} concerns the continuum of atomic levels related to the ionization of the atom. We nextconsider several possibilities.
1. Only Detector D g is Present This case corresponds to the Hamiltonian (11) with v e = 0. The atom-field system startsin the state ρ AF (0) = | , , g ih , , g | (12)and evolves to ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , g ih , , g | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , g | + h.c. ) . (13)Now the system performs a π Rabi pulse, regarded as instantaneous, ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , e ih , , e | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , e | + h.c. ) . (14)Next the atom interacts with D g during a time interval τ g , ρ AF ( τ A,B + τ g ) = | c ( τ AB ) | | , ih , | (cid:18)Z dµ h ψ gµ | g i e − iǫ gµ τ g | ψ gµ i (cid:19) (cid:18)Z dµ h g | ψ gµ i e iǫ gµ τ g h ψ gµ | (cid:19) + ( c ( τ AB ) c ∗ ( τ AB ) | , i (cid:18)Z dµ h ψ gµ | g i e − iǫ gµ τ g | ψ gµ i (cid:19) h , , e | + h.c. )+ | c ( τ AB ) | | , , e ih , , e | , where {| ψ gµ i} and ǫ gµ correspond to the set of eigenvectors and eigenvalues of H D with v e = 0.This atom-detector interaction time will be considered to have the same order magnitude of6he π Rabi pulse time, and will be disregarded: τ A,B + τ g ≃ τ A,B [25]. A click in D g meansthe atom was ionized, i.e. , its state is described by the set {| k i} ; hence the probability ofsuch a click is given by p clickD g = Z dkT r {| k ih k | ρ AF ( τ A,B + τ g ) } (15)= | c ( τ AB ) | p g , (16)where p g is the efficiency of the detector D g : p g = Z dk (cid:12)(cid:12)(cid:12)(cid:12)Z dµ h ψ µ | g i|h k | ψ µ i e − iǫ µ τ g (cid:12)(cid:12)(cid:12)(cid:12) . (17)If one observes a click in D g , the state of the cavity field collapses to ρ A ( τ A,B ) = | , ih , | , returning to its initial state; thus the probability that D g clicks for the N atoms is P ( N ) clickD g = (cid:0) | c ( τ AB ) | p g (cid:1) N . (18)Of course in the limit p g = 1 one recovers the result of the previous section:lim N →∞ P ( N ) clickD g = 1 . The effect of having an inefficient measurement, i.e. , having a detection efficiency p g < N consecutive clicks in D g as a function of N for different values of p g . In this case the limit N → ∞ yields lim N →∞ P ( N ) clickD g = 0 . (19)This does not mean that Zeno effect is not present. Given the detector’s inefficiency onecan not associate the effect to the statistics of D g clicks: no click in D g does not necessarilymean that the photon in fact decayed from cavity A to B . The intrinsic detection inefficiencylimits the experimental visibility of the Zeno effect in the present experimental scheme.
2. Only Detector D e is Present Another possibility of investigating the limited detection efficiency in the same experimen-tal scheme consists in having only detector D e present. This corresponds to the Hamiltonian(11) with v g = 0. 7 IG. 1: Probability of consecutive clicks in D g as a function of N , for T = π g and different valuesof p g : p g = 1 (dashed), p g = 0 , p g = 0 , Note that in this case one click in D e projects the cavity state to | , ih , | ; thus, in orderto observe the effect we must study sequences of events which do not give rise to any clickin D e . In the first step of such a sequence the initial atom-field state is given by ρ AF (0) = | , , g ih , , g | , (20)which evolves to ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , g ih , , g | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , g | + h.c. ) , (21)and next performs an instantaneous π Rabi pulse, leading to the state¯ ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , e ih , , e | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , e | + h.c. ) . (22)In the sequence the atom interacts with the detector according to Eq. (11) with v g = 0,leading to the state ρ AF ( τ A,B ) = | c ( τ AB ) | | , ih , | (cid:18)Z dµ h ψ eµ | e i e − iǫ eµ τ e | ψ eµ i (cid:19) (cid:18)Z dµ h e | ψ eµ i e iǫ eµ τ e h ψ eµ | (cid:19) + | c ( τ AB ) | | , , g ih , , g | + ( c c ∗ | , i (cid:18)Z dµ h ψ eµ | e i e − iǫ eµ τ e | ψ eµ i (cid:19) h , , g | + h.c. ) , {| ψ eµ i} and ǫ eµ correspond to the set of eigenvectors and eigenvalues of H D with v g = 0. τ e is the atom-detector interaction time which will be neglected as in the previous section.If no click in D e is observed, the state of the cavity field ends up in ρ F ( τ A,B ) = | c ( τ A,B ) | | , ih , | + | c ( τ AB ) | (1 − p e ) | , ih , || c ( τ AB ) | + | c ( τ AB ) | (1 − p e ) . This statistical mixture is the initial state of the next step, whose final state can becalculated in an analogous way as above, giving ρ F (2 τ A,B ) = ( | c ( τ AB ) | ) | , ih , | + | c ( τ AB ) | (1 − p e )(1 + | c ( τ AB ) | ) | , , ih , | ( | c ( τ AB ) | ) + | c ( τ AB ) | (1 − p e )(1 + | c ( τ AB ) | ) . (23)All subsequent steps will present distinct final states, but always statistical mixtures of | , ih , | and | , ih , | . Since the part related to | , ih , | does not vary with time, onlythe part concerning | , ih , | will be responsible for changes in the state, which will be thesame in every step, and may be expressed as | , ih , | −→ | c ( τ AB ) | | , ih , | + | c ( τ AB ) | (1 − p e ) | , ih , | . (24)Consequently, it is easy to obtain the state of the fields after i no clicks in D e : ρ ( i ) F ( iτ A,B ) = ( | c ( τ AB ) | ) i | , ih , | + | c ( τ AB ) | (1 − p e )( P ik =1 | c ( τ AB ) | k − ) | , ih , | ( | c ( τ AB ) | ) i + | c ( τ AB ) | (1 − p e )( P ik =1 | c ( τ AB ) | k − ) . (25)The probability of no click in D e in the i -th step may be calculated as p ( i )˜ nclickD e = Z dkT r n ( | g ih g | + | e ih e | ) ρ ( i ) F,A o (26)Where ρ ( i ) F,A is the state of the system at the N-th step immediately before the interac-tion between atom and detector. This state operator can be calculated from ρ ( i − F . Theprobability of N consecutive no clicks in D e may be computed as the product P ( N )˜ nclickD e = N Y i =1 p ( i )˜ nclickD e = ( | c ( τ AB ) | ) N + | c ( τ AB ) | (1 − p e )( N X k =1 | c ( τ AB ) | k − ) , (27)where p e , the efficiency of the detector, is given by p e = Z dk (cid:12)(cid:12)(cid:12)(cid:12)Z dµ h ψ µ | e i|h k | ψ µ i e − iǫ µ τ e (cid:12)(cid:12)(cid:12)(cid:12) . In the limit N → ∞ , 9 IG. 2: Probability of consecutive no-clicks in D e as a function of N , for T = π g and differentvalues of p e : p e = 1 (dashed), p e = 0 , p e = 0 , lim N →∞ P ( N )˜ nclickD e = 1 . In Fig.2 we show the probability of N consecutive no-clicks in D e for different values of p e . For p e = 1 the curve is the same as the one for p g = 1, since no clicks in a perfect D e isequivalent to clicks in a perfect D g . For inefficient detectors, the probability of N consecutiveno-clicks must be larger than this probability for perfect detectors. This is illustrated inFig.2, where the curves representing smaller p e tend to reach the asymptotic value 1 fasteras N → ∞ . Note that for inefficient detectors no-click in D e does not necessarily mean thatthe photon is for sure in cavity A : the monitoring of the photon transition is not perfect.However, the asymptotic behavior of P ( N )˜ nclickD e , tending to 1 for any value of p e , is mostcertainly a consequence of the Zeno effect. IV. NO INTERMEDIATE MEASUREMENTS
In the experimental set ups discussed in the previous sections the photon transition wasmonitored by N probe atoms and a macroscopic signal was generated. We were interestedin the probability of occurrence of selected sequences, namely, N consecutive clicks in D g or10 consecutive no-clicks in D e , which would be associated to the permanence of the photonin cavity A . Obviously, a complete correlation can not be achieved due to the inefficiencyof the detectors.Pascazio and Namiki propose in Ref. [8] a dynamical approach to QZE and show theessential role of the generalized spectral decomposition . They propose that QZE occurs evenin the absence of intermediate measures, what explains Itano results in [7]. For the systemcomposed by two coupled cavity modes, the generalized spectral decomposition is broughtabout by the interaction between the two level probe atom and the cavity B mode. As wewill see, the classical signals generated by the ionization detectors in each step (intermediatemeasures) are not necessary for inhibiting the photon transition and, accordingly with theapproach in [8], are not essential for the characterization of the QZE.The idea now is to send atoms through cavity B , also in T /N intervals, and not tomeasure the outcome of the atom-cavity interaction each time. After N such interactionsone atom is sent through cavity A and measured by a detector D e .As in the previous schemes, the first step of the evolution starts with the atom-fields stategiven by ρ AF (0) = | , , g ih , , g | , (28)which evolves to ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , g ih , , g | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , g | + h.c. ) , (29)and then to ρ AF ( τ A,B ) = | c ( τ AB ) | | , , g ih , , g | + | c ( τ AB ) | | , , e ih , , e | +( c ( τ AB ) c ∗ ( τ AB ) | , , g ih , , e | + h.c. ) . (30)Since this atom is not measured, the field state must be represented in the end of the stepby ρ F ( τ AB ) = T r A { ρ AF ( τ A,B ) } (31)= | c ( τ AB ) | | , ih , | + | c ( τ AB ) | | , ih , | , (32)where T r A is the trace over the variables of the atom, and accounts for the lack of informationabout the atomic state. 11n order to calculate the final state of the following steps, we must observe that only thepart of ρ A related to | , ih , | changes with time, in a way that may be described by | , ih , | −→ | c ( τ AB ) | | , ih , | + | c ( τ AB ) | | , ih , | . (33)Thus, it is easy to see that the state operator for the fields in the cavities, after the interactionof M B with N atoms, can be written as ρ F ( T ) = ( | c ( τ AB ) | ) N | , ih , | + | c ( τ AB ) | N X k =1 ( | c ( τ AB ) | ) k − | , ih , | . (34)The probability that the photon transition from cavity A to cavity B has not occurred is p ( N )1 , = ( | c ( τ AB ) | ) N , (35)and, in the limit N → ∞ , lim N →∞ p ( N )1 , = 1 . (36)This, according to the dynamical approach in [8], characterizes Zeno effect. The measure-ment of this probability can be done by using one probe atom prepared in | g i state and sentthrough cavity A immediately after the interaction of M B with the N -th atom. If this probeatom and mode M A perform a π Rabi pulse, the atom-fields state will be given by ρ AF ( T ) = ( | c ( τ AB ) | ) N | , , e ih , , e | + | c ( τ AB ) | N X k =1 ( | c ( τ AB ) | ) k − | , , g ih , , g | , (37)and measuring the energy level of the atom with an ionization detector tell us about thefield state. The inefficiency of the detector enters just as a multiplicative factor in the data. A. Finite Interaction Times and Lossy Cavities
The problems related to the inefficiency of the ionization detectors, which imposed im-portant limitations for the observation of Zeno effect in the proposals of Sec. III, have beenovercome by the experimental proposal of the present section. However there are otherlimitations if a realistic experiment is to be performed. Firstly the cavity is not perfectand dissipation/decoherence will also affect the visibility of the effect. And secondly theinteraction time is finite. We consider all these effects in the present section.Fig.3 sketches the time evolution, divided in N steps, each one composed by two parts:no atom is present and the cavities are coupled (clear zones), the atom interacts with mode12 IG. 3: Sketch of the total time of one experimental sequence. M B during a π Rabi pulse (dark zones). Each clear zone corresponds to the time interval τ AB = T /N , where T is, as in previous sections, the time during which a photon passesfrom cavity A to cavity B if no atom is present: T = π/ g . Since our goal here is to studythe inhibition (due to intermediate interactions) of such a photon transition, the cavitieswill be uncoupled during the atom-field interactions, in order to keep the total interactiontime between modes M A and M B fixed in T [26]. For the Rubydium atoms used in theexperiment [18], the π Rabi pulse time is τ π ≃ − s , and the increase in the number ofprobe atoms, N , may turn the total time of atom-field interactions, N τ π , quantitativelyimportant. In order to take this time into account, we must consider T ′ = T + N τ π as the total time of one experimental sequence.Let us start by modeling the environment as a large set of harmonic oscillators linearlycoupled to the system of interest (modes M A and M B ) [19]. This model has been used tocalculate the time evolution of two microwave modes constructed in a single cavity, andthe theoretical results showed good agreement with experimental ones [20]. In Ref. [21] itis shown that, for identical cavities and zero temperature, the model leads to the masterequation ddt ρ F ( t ) = k (cid:0) aρ F ( t ) a † − ρ F ( t ) a † a − a † aρ F ( t ) (cid:1) − iω (cid:2) a † a, ρ F ( t ) (cid:3) (38)+ k (cid:0) bρ F ( t ) b † − ρ F ( t ) b † b − b † bρ F ( t ) (cid:1) − iω (cid:2) b † b, ρ F ( t ) (cid:3) − ig (cid:2) b † a + a † b, ρ F ( t ) (cid:3) , where ω is the frequency of the modes of interest, g is their coupling constant and k givesthe decay rate of the cavities; cross decay rates and shifts in ω and g , which tend to be small[23], were disregarded. Using this master equation, we calculate the time evolution of the13tate ρ F (0) = | A , B ih A , B | = | , ih , | (39)as ρ F ( t ) = ( f ( t ) | , i + l ( t ) | , i )( h.c. ) + (1 − | f ( t ) | − | l ( t ) | ) | , ih , | , (40)where f ( t ) = exp [ − ( k + iω ) t ] cosh [ − igt ] , (41) l ( t ) = exp [ − ( k + iω ) t ] sinh [ − igt ] . The probability of finding the photon in cavity A , in this case, is given by | f ( t ) | = e − kt cos ( gt ) . (42)If the field state has evolved from t = 0 to t = τ AB in the manner described above, and attime t = τ AB an atom prepared in | g i state begins its interaction with mode M B , the stateof the whole system will be given by ρ AF ( τ AB ) = ( f ( τ AB ) | , , g i + l ( τ AB ) | , , g i )( h.c. )+(1 −| f ( τ AB ) | −| l ( τ AB ) | ) | , , g ih , , g | . (43)During the atom-field interaction, the field modes evolve independently, since they are un-coupled. The evolution of state (43) is described by the master equation ddt ρ AF ( t ) = k (cid:0) aρ AF ( t ) a † − ρ AF ( t ) a † a − a † aρ AF ( t ) (cid:1) + iω (cid:2) a † a, ρ AF ( t ) (cid:3) (44)+ k (cid:0) bρ AF ( t ) b † − ρ AF ( t ) b † b − b † bρ AF ( t ) (cid:1) − i Ω b † σ − + bσ + , ρ AF ( t )] , where Ω is vacuum Rabi frequency, and σ − = σ † + = | g ih e | . The first line of Eq. (44)describes the dissipation of mode M A ; the second line describes the interaction of the atomwith mode M B according to the dissipative Jaynes-Cummings model [23]. In previous cal-culations, τ π was the time spent by an atom to absorb the excitation of mode M B . Here, τ π plays an analogous role, and will be defined as τ π = 1 p Ω − k arccos (cid:18) k − Ω Ω (cid:19) . (45)This time, which depends not only on the vacuum Rabi frequency, but also on the dissipationconstant, is the time for a complete transfer of the excitation of mode M B , to the atom or14o the environment. This definition coincides with the previous one if no dissipation isconsidered ( k = 0). Using master equation (44) to describe the evolution of the system from t = τ AB to t = τ AB + τ π , we get ρ AF ( τ AB + τ π ) = | f ( τ AB ) | e − kτ π | , , g ih , , g | + | l ( τ AB ) | e − kτ π | , , e ih , , e | (46)+ (1 − | f ( τ AB ) | e − kτ π − | l ( τ AB ) | e − kτ π ) | , , g ih , , g | . The state of the fields after the interaction with the first atom is obtained by taking thetrace over the atomic variables: ρ F ( τ AB + τ π ) = T r A { ρ AF ( τ AB + τ π ) } = | f ( τ AB ) | e − kτ π | , ih , | + (1 − | f ( τ AB ) | e − kτ π ) | , ih , | . Observing that the part of the density operator associated to | , ih , | does not changewith time, it is easy to calculate the probability to find the photon in cavity A after theinteraction with N atoms: p ( N )1 , = ( | f ( τ AB ) | e − kτ π ) N (47)= e − k ( T + Nτ π ) (cid:18) cos (cid:18) gT N (cid:19)(cid:19) N . This equation explicitates the effect of N intermediate interactions over two kinds of tem-poral dependencies. The term (cid:0) cos (cid:0) gT N (cid:1)(cid:1) N represents no transition of the photon fromcavity A to cavity B . It grows when N increases, tending to 1 when N → ∞ . The term e − k ( T + Nτ π ) , related to the probability that the photon has not decayed to the environment,decreases to zero when N → ∞ . Of course this decrease is due to the enhancement of thetotal time in which the field is exposed to the environment, not being related to any kindof anti-Zeno effect. Since the dynamics of dissipation is exponential, it is not affected byintermediate measures. The role played by the finite interaction time τ π is also explicitatedand will become quantitatively important as N → ∞ .In order to observe the dependence of p ( N )1 , on N , an atom prepared in | g i state is sentinto cavity A just after the interaction of the N -th atom with mode M B . The atom thenperforms a π Rabi pulse, and passes through a D e detector. If the efficiency of D e is p e , theprobability of a click will be given by p ( N ) D e click = p e e − k ( T + Nτ π ) (cid:18) cos (cid:18) gT N (cid:19)(cid:19) N . (48)15 IG. 4: Probability of one click in D e as a function of N , for T = π g , Ω = 10 s − , p e = 1, g = 10 s − and different values of k : k = 10 s − (continuous) and k = 10 s − (dashed). This is the empirical quantity to be measured in the present proposal.There will be no problems associated to the efficiency p e , since it enters just as a multi-plicative factor that does not depend on N . However, the term e − k ( T + Nτ π ) depends on N ,and may prevent the observation of Zeno effect if the decay constant k is not small enough.In Fig.4, we may observe the competition between the tendencies of p ( N ) D e click when N grows:the increasing one, due to Zeno effect, and the decreasing one, due to dissipation. In thecontinuous curve k = 10 s − , corresponding to the cavities used in several experiments [16].In this case it would be very difficult to observe Zeno effect, since dissipation dominateseven for small values of N . For the dashed curve k = 10 s − ; this value corresponds to thecavity described in [13, 14], and turns the observation of Zeno effect possible.16 . CONCLUSION We consider some realistic aspects related to the observation of the QZE in Cavity QED.They are: the effect of a lossy environment, of limited detection efficiency and finite atom-field interaction time. The calculations are fully analytical and the experiment is apparentlyfeasible [13, 14]. Our main result is the equation for the probability of a no-click detec-tion as a function of the number of incoming atoms, the cavities dissipation constants, theprobability of a click in detector D e and a finite atom-field interaction time τ π , p ( N ) D e click = p e e − k ( T + Nτ π ) (cid:18) cos (cid:18) gT N (cid:19)(cid:19) N . (49)This is the main result of the present contribution. It explicitates, within the context ofthe present model the role played on the visibility of the QZE by a realistic apparatus andrealistic detectors. We hope this result may encourage the experimental realization of thepresent proposal. [1] J. M. Raimond, M. Brune, and S. Haroche, Phys. Rev. Lett. , 1964 (1997).[2] B. C. Van Fraassen, The Scientific Image , Clarendon Press, Oxford, 1980.[3] J. von Neumann,
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