Control of non-Markovian decay and decoherence by measurements and interference

Novel methods are discussed for the state control of atoms coupled to multi-mode reservoirs with non-Markovian spectra: 1) Excitation decay control : we point out that the quantum Zeno effect, i.e., inhibition of spontaneous decay by frequent measurements, is observable in open cavities and waveguides using a sequence of evolution- interrupting pulses or randomly-modulated CW fields. 2) Location-dependent interference of decay channels - nonadiabatic (resonant) control : We show that the control of populations and coherences of two metastable states is feasible via resonant single-photon absorption to an intermediate state, by controlled spontaneous emission in a cavity. 3) Decoherence control by conditionally interfering parallel evolutions: We demonstrate that an arbitrary internal atomic state can be completely protected from decoherence by interference of its interactions with the reservoir over many different time interals in parallel . Such interference is conditional upon the detection of appropriate atomic-momentum observables. Realization in cavities is suggested. The rich arsenal of control methods described above can improve the performance of single-atom devices. It can also advance the state-of-the-art of quantum information encoding and processing.


Introduction
It is our purpose here to demonstrate that e ective control of atomic states is possible in the presence of spontaneous relaxation into non-Markovian reservoirs, characterized by non-smooth spectra.Cavities, waveguides and periodic dielectric structures are prime examples of environments that give rise to such reservoirs for electromagnetic (EM) eld modes, whereas condensed media or multi-ion traps are their analogs for phonon modes.In cavities, one may employ the following arsenal of means for the control of radiative decay: (i) mode density control, which allows the suppression or enhancement of the decay rate in selected spectral ranges; (ii) location control, which allows the adjustment of the strength and phase of coupling to a standing-wave eld mode of the cavity, by varying the position (relative to eld nodes) of atoms or molecules deposited on a thin lm; and (iii) measurement control, namely, non-selective or conditional measurements (CMs) of an auxiliary variable that is correlated to the internal state of the atom.
In what follows, we shall demonstrate how this arsenal can be implemented in several methods we have developed for quantum state control in cavities: (a) population decay control by frequent or continuous measurements, realizing the quantum Zeno e ect 1 ; (b) complete control of state preparation by location-dependent interference of decay channels 2 ; and (c) control of decoherence by CM-induced interference of evolution histories 3 .The aim will be to clarify the essentials of these methods, by means of computer movies and other graphical illustrations, and nally compare their advantages and disadvantages.

Quantum Zeno e ect: control of excitation decay
The "watchdog" or quantum Zeno e ect (QZE) is a spectacular manifestation of the in uence of measurements on the evolution of a quantum system.The original QZE prediction has been that irreversible decay of an excited state into a reservoir can be inhibited 4 , by repeated interruption of the system-reservoir coupling, which is associated with measurements 5;6 .However, this prediction has not been experimentally veri ed as yet!Instead, the interruption of Rabi oscillations and analogous forms of nearlyreversible evolution has been at the focus of interest 7{14 .
We have recently demonstrated 15 that the inhibition of nearly-exponential excitedstate decay by the QZE in two-level atoms, in the spirit of the original suggestion 4 , is amenable to experimental veri cation in resonators.Although this task has been widely believed to be very di cult, we have shown, by means of our uni ed theory of spontaneous emission into reservoirs with arbitrary mode-density spectra 16 , that several realizable con gurations based on two-level emitters in cavities or in waveguides are in fact adequate for QZE observation.We have now developed a more comprehensive view of the possibilities of excited-state decay by QZE 1 .Here we wish to demonstrate, by means of computer movies and illustrations, that QZE is indeed achievable by repeated or continuous measurements of the excited state, in various non-Markovian reservoirs.

QZE by frequent impulsive measurements
Consider an initially excited two-level atom coupled to an arbitrary density-of-modes (DOM) spectrum (!) of the electromagnetic eld in the vacuum state.At time its evolution is interrupted by a short optical pulse, which serves as an impulsive quantum measurement 7{14 .Its role is to break the evolution coherence, by transferring the populations of the excited state jei to an auxiliary state which then decays back to jei incoherently.
The atomic response, i.e., the emission rate into this reservoir at frequency !, which is jg(!)j 2 (!), ~g(!) being the eld-atom coupling energy, can be divided into (1) Here G s (!) stands for the sharply-varying (nearly-singular) part of the DOM distribution, associated with narrow cavity-mode lines or with the frequency cuto in waveguides or photonic band edges.The complementary part G b (!) stands for the broad portion of the DOM distribution (the "background" modes), which always coincides with the free-space DOM (!) ! 2 at frequencies well above the sharp spectral features.In an open structure (see below), G b (!) represents the atom coupling to the uncon ned free-space modes.
We cast the excited-state amplitude in the form e ( )e ?i!a , where ! a is the atomic resonance frequency.Restricting ourselves to su ciently short interruption intervals such that e ( ) ' 1, yet long enough to allow the rotating wave approximation, we obtain e ( ) ' 1 ?Z 0 dt( ?t) s (t)e i t ?b =2; (2) where s (t) = Z 1 0 d!G s (!)e ?i(!?!s)t : ( = !a ?!s is the detuning of the atomic resonance from the peak !s of G s (!), and is the e ective rate of spontaneous emission into the background modes.
To rst order in the atom-eld interaction, the excited state probability after n interruptions (measurements), W(t = n ) = j e ( )j 2n , can be written as W(t = n ) 2Re e ( ) ? 1] n e ?t ; (4) where = 2Re 1 ?e ( )]= : (5) In most structures b is comparable to the free-space decay rate f and gives rise to an exponential decay factor in the excited state probability, regardless of how short is, i.e., = s + b ; (6) where s is the contribution to from the sharply-varying modes.
Thus the background-DOM e ect cannot be modi ed by QZE.Only the sharplyvarying DOM contribution s allows for QZE, provided that s = (2= )Re Z 0 dt( ?t) s (t)e i t (7)   decreases with for su ciently short .This essentially means that the correlation (or memory) time of the eld reservoir is longer (or, equivalently, s (t) falls o slower) than the chosen interruption interval .

.1 Application to a Lorentzian line
The simplest application of the above analysis is to the case of a two-level atom coupled to a near-resonant Lorentzian line centered at ! s , characterizing a high-Q cavity mode 15 .In this case, G s (!) = g 2 s ?s =f ? 2 s +(!?! s ) 2 ]g, where g s is the resonant coupling strength and ?s is the linewidth (Fig. 1).In the short-time approximation, taking into account that the Fourier transform of the Lorentzian G s (!) is s (t) = g 2 s e ??st , Eq. ( 2) yields (without the background-modes contribution) e ( ) 1 ?g 2 s ?s ?i + e (i ??s) ? 1 ?s ?i : The QZE condition is then (? s + j j) ?1 ; g ?1 s : On resonance, when = 0, Eq. (8) yields = s + b ; s = g 2 s : (10)   Only the s term decreases with , indicating the QZE inhibition of the nearly-exponential decay into the Lorentzian eld reservoir as !0. Since ?s has dropped out of Eq. (10), the decay rate is the same for both strong-coupling (g s > ?s ) and weak-coupling (g s ?s ) regimes.Physically, this comes about since for g ?1 s the energy uncertainty of the emitted photon is too large to distinguish between reversible and irreversible evolutions.
The evolution inhibition, however, has rather di erent meaning for the two regimes.In the weak-coupling regime, where, in the absence of the external control, the excited-state population decays nearly exponentially at the rate g 2 s =? s + b (at = 0), one can speak about the inhibition of irreversible decay, in the spirit of the original QZE prediction 4 .By contrast, in the strong-coupling regime in the absence of interruptions (measurements), the excited-state population undergoes damped Rabi oscillations at the frequency 2g s .In this case, the QZE slows down the evolution during the rst Rabi half-cycle (0 t =2g ?1 s ), the evolution on the whole becoming irreversible.A possible realization of this scheme is as follows.Within an open cavity the atoms repeatedly interact with a pump laser, which is resonant with the jei !jui transition frequency.The resulting jei !jgi uorescence rate is collected and monitored as a function of the pulse repetition rate 1= .Each short, intense pump pulse of duration t p and Rabi frequency p is followed by spontaneous decay from jui back to jei, at a rate u , so as to destroy the coherence of the system evolution, on the one hand, and reshu e the entire population from jei to jui and back, on the other hand (Fig. 2).The demand that the interval between measurements signi cantly exceed the measurement time, u if the \measurements" are performed with pulses: p t p = ; t p ?1 u .This calls for choosing a jui !jei transition with a much shorter radiative lifetime than that of jei !jgi.
Movie A, describing the QZE for a Lorentz line on resonance ( = 0), has been programmed for feasible cavity parameters: (1 ?f) f , where R is the geometric-mean re ectivity of the two mirrors, f is the fractional solid angle (normalized to 4 ) subtended by the confocal cavity, and L is the cavity length.It shows, that the population of jei decays nearly-exponentially well within interruption intervals , but when those intervals become too short, there is signi cant inhibition of the decay.Movie B shows the e ect of the detuning = !a ?!s on the decay: The decay now becomes oscillatory.The interruptions now enhance the decay, the degree of enhancement depends on the phase between interruptions.

.2 Application to mode distributions with cuto
We can extend the above analysis to DOM distributions characterized by a cuto frequency, as in a waveguide, a photonic band edge or a phonon reservoir (with Debye cuto ).A speci c model for the spectral response of a DOM distribution with a cuto ω+Γ s s G s ω s ω where !s is the cuto (or band-edge) frequency, ?s is the cuto -smoothing parameter, C is the strength of the coupling of the atomic dipole to this reservoir, and () is the Heaviside step function (Fig. 5).Upon computing the Fourier transform of Eq. ( 11), we nd from Eqs. (3), (7) that the QZE condition is minf??1 s ; j j ?1 ; C ?2=3 g: (12)   Under this condition, Eq. ( 7) As seen in our movie C, the QZE is now less pronounced as compared to the Lorentzian-line case (movie A).This case is realizable for an active dipole layer embedded in a dielectric waveguide, using a level scheme similar to that of Fig. 2. The same evolution obtains for a state relaxing vibrationally into a phonon reservoir, if the vibrational transition is near the Debye cuto of the reservoir.

QZE by continuous measurement: noisy-eld dephasing
Instead of disrupting the coherence of the evolution by a sequence of "impulsive" measurements, i.e., short -pulses, we can achieve this goal by noisy-eld dephasing of e (t): Random ac-Stark shifts by an o -resonant intensity-uctuating eld result in the replacement of Eq. ( 7) by (Fig. 7) Here the sharply varying spectral response G s ( + ! a ) Eq. ( 1)] replaces the Fourier transform of s (t) in Eq. ( 7), whereas F( ) is the Fourier transform of the relaxation function of the coherence element eg (t).For the common dephasing model corresponding to exponential decay of eg (t), F( ) is a Lorentzian spectrum whose width is h ! 2 i c , the product of the mean-square Stark shift and the noisy-eld correlation  time.The QZE condition is that this width be larger than the width of G s (!) (Fig. 7).The advantage of this realization is that it does not depend on u , and is realizable for any atomic transition.Its importance for molecules is even greater: if we start with a single vibrational level of jei, no additional levels will be populated by this process.
The random ac-Stark shifts described above cause both shifting and broadening of the spectral transition.If we wish to avoid the shifting altogether, we may employ random phase modulation of a driving eld that is nearly resonant with the jei $ jui transition.If the spectral width of this phase modulation, ?, is much larger than all the other rates speci ed below, then we can show that s 2g 2 s ?= 2 ; (15)   where is the driving-eld Rabi frequency.When g s ?, s is strongly suppressed compared to its zero-eld value, and QZE is at its best!
Figure 8: (a) Scheme of location-dependent interference of decay channels in a cavity.Vertical cyan line denotes a thin lm, f in(out) denote incoming and outgoing photon, r(l) and 1(2) subscripts denote photon propagation to right (left) near ! 1 -green (! 2 -red).(b) Level scheme for atoms in lm.

State control by location-dependent interference of decay channels
The conventional approaches to state control in a multilevel system are based on interference of absorbed or emitted radiation in two or more channels (transitions), which allows for coherences between eld-dressed states of the system 17{25 .In these approaches, spontaneous emission has an adverse e ect on interference and its suppression is highly desirable.The adiabatic population transfer method (STIRAP), for example, achieves this goal by preventing the intermediate decaying state from being populated 23 .Here, by contrast, we outline a state-control method 2 wherein spontaneous emission from the intermediate state is, in fact, bene cial, since it exhibits interference between competing channels (transitions), owing to the boundary conditions on the cavity.We envisage a thin lm of atoms or molecules located in a certain position relative to the cavity mirrors.The coupling to the cavity eld in the "bad cavity" regime e ectively amounts to having a one-dimensional (1D) mode continuum, corresponding to spontaneous emission predominantly into one or several Lorentz-broadened modes along the cavity axis.Let us assume that one of the cavity mirrors admits a single photon, a spectrally narrow wavepacket around ! 1 .This input condition can be realized by unidirectional mirror transparency near ! 1 , or by a measurement verifying that the incoming photon has not been re ected by this mirror.Let us consider the atoms on the thin lm to be in the -con guration, such that ! 1 is resonant with the j1i $ j3i transition and ! 2 is resonant with the j2i $ j3i transition (Fig. 8).
Suppose that the task at hand is population transfer from j1i to j2i, by means of resonant absorption of the ! 1 -photon together with spontaneous emission, which is concentrated in the bands around ! 1 and ! 2 .The Wigner-Weisskopf solution for this system, at times much longer than the spontaneous lifetime in the cavity ?1 c , can be obtained upon taking account of the boundary conditions at the mirrors and neglecting the retardation of the photon (due to its travel between the mirrors) relative to the atomic response 2 .This solution satis es the requirement for complete (100%) population transfer from j1i to j2i, provided that R 1 e i 1 (1 + R 2 e i 2 )(1 + r 2 e i 2 ) Here g 1(2) are the vacuum eld-dipole couplings (vacuum Rabi frequencies) for the j1i $ j3i and j2i $ j3i transitions, respectively; R 1(2) and r 1(2) are the ! 1 -or !2re ectivities of the left-hand and right-hand mirrors, respectively, whereas 1(2) ( 1(2) ) are the corresponding phase delays accumulated by re ection and round-trip travel to and from the left-hand (right-hand) mirror.Note that r 1 , 1 do not appear in this the evolution of an internal atomic state that is coupled to many nearly discrete modes in a cavity.
If the atom is initially in a superposition of its two internal states, and the eld in the vacuum state, j (i) A+F i = (Bjei + Cjgi)j0i, then the atomic interaction with M eld modes during a time t results in an entangled state with certain h(t) and f m (t) dictated by the unitary evolution operator U(t) of the system.The atom-eld entanglement in (18) translates, upon tracing over the eld degrees of freedom, into decoherence of the reduced atomic density matrix.In order to eliminate the decoherence in the atomic state we suggest to superpose many evolution paths of the atom-eld system di ering in the duration of the atom-eld interaction 3 .This can be achieved by conditional measurement (CM) of \ancilla" atomic variables which determine the eld-atom interaction time, e.g., the atomic momentum.By passing the atom through an appropriately designed di raction grating, before it enters the cavity, we obtain at the exit from the cavity a combined ( eldatom) state of the form j (f) A+F i = X j j jp j iU(t j )j (i) A+F i: (19)   Here the superposed intracavity A ? F evolution operators U(t j ) are functions of the evolution time, t j = mL=p ?j , which is the corresponding cavity traversal time determined by the momentumprojections p ?j perpendicular to the cavity axis (Fig. 10), while j are the corresponding di raction amplitudes.The CM projects this state onto an appropriate superposition of momentum states.This can be done by passing the atom through a second grating after the cavity whose transmission function is centered at the same momenta jp j i, but with di erent amplitudes ~ j .The CM amounts to detecting the atom in the direction corresponding to jp l i.The system obtained by such a CM is then given by j (f) A+F i = X j j U(t j )j (i) A+F i (20)   with j = P ?1=2 j l?j , where P is the success probability of the momentum state detection.Tracing over the eld degrees of freedom, we nd that in order for the nal atomic state to be identical with the initial state, namely (f) A = (i) A , it is necessary and su cient that the following (M + 2) conditions be satis ed by the set of j 's and t j 's parameters: Hence, provided that the number of control (momentum-transmission) parameters is su ciently large, it should be possible in principle to satisfy them simultaneously and thus to ensure that any internal atomic state remains unspoilt in spite of the atom's exposure to many interaction channels.Furthermore, additional control parameters can be used to optimize the CM success probability P. In Fig. (11) we show that the entanglement caused by di erent evolutions U(t j ), and the ensuing tracing over M = 20 eld modes yield t j -dependent decoherence of the initial atomic state Bjei + Cjgi.By contrast, when this tracing follows the interference of M + 2 evolution operations U(t j ) with the rightly chosen j , it yields an atomic state that is identical to the initial one, i.e., completely protected from decoherence.Considerable deviations from the prescribed conditions on j and t j , i.e., a \failed" measurement, may result in worse decoherence than that associated with each t j .Hence the need for accurate assignment of the control parameters and for optimization of the CM success probability, which is achievable by means of extra control parameters.

Conclusions
The present outline and graphical illustration of several methods for combating population decay and state decoherence in cavities underscores their essential traits and allows a comparison of their merits and limitations: a.The quantum Zeno e ect (QZE): Our uni ed analysis of two-level system coupling to eld reservoirs has revealed the general optimal conditions for observing the QZE in various structures (cavities, waveguides, phonon reservoirs, and photonic band structures).We note that the wavefunction collapse notion is not involved here, since the measurement is explicitly described as an act of coherence-breaking.This analysis also clari es that QZE cannot combat the background-modes contribution to exponential decay.The best way to prevent this contribution is by an AC Stark shift of the resonance well into a photonic band gap.However, if the resonant transition has a considerable width, caused by vibrational-state multiplicity or by inhomogeneous broadening, such that this width exceeds the band-gap width or the AC Stark shift, then we should resort to the QZE strategies outlined above.This method achieves the inhibition of population decay at the expense of increasing decoherence.21)-( 23)) we can recreate eg = 1 at any desired time t if the required CM is successful.b.State control by location-dependent interference of decay channels: This method is remarkable in that, rather than try to avoid spontaneous emission by its inhibition (in cavities or in photonic bandgap 16 or by adiabatic o -resonance transfer 23 ), it guides this decay into desired channels, and thereby can populate any desired superposition of two (or more) atomic (molecular) states and, correspondingly, of radiation frequencies in the photon wavepacket emerging from the cavity.The limitations of this method are: (i) It is appropriate only for thin lms, whose location is controllable to within a fraction of a wavelength.(ii) As in the case of QZE, it requires the spontaneous emission rates into the axial cavity modes c to be much larger than the corresponding rates into the uncon ned \background" modes b .The decay probability of the atom into uncon ned modes, b during the time needed for the creation of the superposition state and the photon wavepacket transformation is the error probability of this state control method.The method is e ective only if 1= c 1= b .c. Conditionally interfering parallel evolutions (CIPE): The main merit of this method is that it allows us to control the evolution of M 1 coupled degrees of freedom by essentially as many ancilla, which de nitely makes it a \low cost" control method.In particular, its applicability to arbitrary atomic superposition states implies that we can use this method as part of the quantum computing/processing chain of operations.By optimizing the CM, we can achieve high success probability, or con dence in the method.Nevertheless, the probabilistic nature of this method remains a limitation.Furthermore, the method does not apply to decay into mode continua, but only to coupling with discrete (albeit large) number of modes.

Figure 10 :
Figure 10: Scheme of CM preparation of superposed parallel eld-atom evolutions in a cavity.Red arrows denote atomic momenta.Dash-dotted blue lines denote di raction gratings.

Figure 11 :
Figure 11: Coherence term eg (t) of the atomic reduced density matrix as a function of evolution time t, showing decoherence of the atomic state with time.By the CIPE method (Eqs.(21)-(23))we can recreate eg = 1 at any desired time t if the required CM is successful.