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Volume 2013 (2013), Article ID 783865, 51 pages
Universal Dynamical Control of Open Quantum Systems
Weizmann Institute of Science, 76100 Rehovot, Israel
Received 25 March 2013; Accepted 24 April 2013
Academic Editors: M. D. Hoogerland, D. Kouznetsov, A. Miroshnichenko, and S. R. Restaino
Copyright © 2013 Gershon Kurizki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Due to increasing demands on speed and security of data processing, along with requirements on measurement precision in fundamental research, quantum phenomena are expected to play an increasing role in future technologies. Special attention must hence be paid to omnipresent decoherence effects, which hamper quantumness. Their consequence is always a deviation of the quantum state evolution (error) with respect to the expected unitary evolution if these effects are absent. In operational tasks such as the preparation, transformation, transmission, and detection of quantum states, these effects are detrimental and must be suppressed by strategies known as dynamical decoupling, or the more general dynamical control by modulation developed by us. The underlying dynamics must be Zeno-like, yielding suppressed coupling to the bath. There are, however, tasks which cannot be implemented by unitary evolution, in particular those involving a change of the system’s state entropy. Such tasks necessitate efficient coupling to a bath for their implementation. Examples include the use of measurements to cool (purify) a system, to equilibrate it, or to harvest and convert energy from the environment. If the underlying dynamics is anti-Zeno like, enhancement of this coupling to the bath will occur and thereby facilitate the task, as discovered by us. A general task may also require state and energy transfer, or entanglement of noninteracting parties via shared modes of the bath which call for maximizing the shared (two-partite) couplings with the bath, but suppressing the single-partite couplings. For such tasks, a more subtle interplay of Zeno and anti-Zeno dynamics may be optimal. We have therefore constructed a general framework for optimizing the way a system interacts with its environment to achieve a desired task. This optimization consists in adjusting a given “score” that quantifies the success of the task, such as the targeted fidelity, purity, entropy, entanglement, or energy by dynamical modification of the system-bath coupling spectrum on demand.
Due to the ongoing trends of device miniaturization, increasing demands on speed and security of data processing, along with requirements on measurement precision in fundamental research, quantum phenomena are expected to play an increasing role in future technologies. Special attention must hence be paid to omnipresent decoherence effects, which hamper quantumness [1–70]. These may have different physical origins, such as coupling of the system to an external environment (bath), noise in the classical fields controlling the system, or population leakage out of a relevant system subspace. Their consequence is always a deviation of the quantum state evolution (error) with respect to the expected unitary evolution if these effects are absent. In operational tasks such as the preparation, transformation, transmission, and detection of quantum states, these effects are detrimental and must be suppressed by dynamical control. The underlying dynamics must be Zeno-like yielding suppressed coupling to the bath.
Environmental effects generally hamper or completely destroy the “quantumness” of any complex device. Particularly fragile against environment effects is quantum entanglement (QE) in multipartite systems. This fragility may disable quantum information processing and other forthcoming quantum technologies: interferometry, metrology, and lithography. Commonly, the fragility of QE rapidly mounts with the number of entangled particles and the temperature of the environment (thermal “bath”). This QE fragility has been the standard resolution of the Schrödinger-cat paradox: the environment has been assumed to preclude macrosystem entanglement.
In-depth study of the mechanisms of decoherence and their prevention is therefore an essential prerequisite for applications involving quantum information processing or communications . The present paper aimed at furthering our understanding of these formidable issues. It is based on progress by our group, as well as others, towards a unified approach to the dynamical control of decoherence and disentanglement. This unified approach culminates in universal formulae allowing design of the required control fields.
Most theoretical and experimental methods that aimed at assessing and controlling (suppressing) decoherence of qubits (two-level systems that are the quantum mechanical counterparts of classical bits) have focused on one of two particular situations: (a) single qubits decohering independently, or (b) many qubits collectively perturbed by the same environment. Thus, quantum communication protocols based on entangled two-photon states have been studied under collective depolarization conditions, namely, identical random fluctuations of the polarization for both photons [71, 72]. Entangled qubits that reside at the same site or at equivalent sites of the system, for example, atoms in optical lattices, have likewise been assumed to undergo identical decoherence.
By contrast, more general problems of decay of nonlocal mutual entanglement of two or more small systems are less well understood. This decoherence process may occur on a time scale much shorter than the time for either body to undergo local decoherence, but much longer than the time each takes to become disentangled from its environment. The disentanglement of individual particles from their environment is dynamically controlled by interactions on non-Markovian time-scales, as discussed below. Their disentanglement from each other, however, may be purely Markovian [73–75], in which case the present non-Markovian approach to dynamical control/prevention is insufficient.
1.1. Dynamical Control of Single-Particle Decay and Decoherence on Non-Markovian Time Scales
Quantum-state decay to a continuum or changes in its population via coupling to a thermal bath is known as amplitude noise (AN). It characterizes decoherence processes in many quantum systems, for example, spontaneous emission of photons by excited atoms , vibrational and collisional relaxation of trapped ions , and the relaxation of current-biased Josephson junctions . Another source of decoherence in the same systems is proper dephasing or phase noise (PN) , which does not affect the populations of quantum states but randomizes their energies or phases.
For independently decohering qubits, a powerful approach for the suppression of decoherence appears to be the “dynamical decoupling” (DD) of the system from the bath [79–92]. The standard “bang-bang” DD, that is, -phase flips of the coupling via strong and sufficiently frequent resonant pulses driving the qubit [82–84], has been proposed for the suppression of proper dephasing .
This approach is based on the assumption that during these strong and short pulses there is no free evolution; that is, the coupling to the bath is intermittent with control fields. These -pulses hence serve as a complete phase reversal, meaning that the evolution after the pulse negates the deleterious effects of dephasing prior to the pulse, similar to spin-echo technique . However, some residual decoherence remains and increases with the interpulse time interval, and thus in order to combat decoherence effectively, the pulses should be very frequent. While standard DD has been developed for combating first-order dephasing, several extensions have been suggested to further optimize DD under proper dephasing, such as multipulse control , continuous DD , concatenated DD , and optimal DD [95, 96]. DD has also been adapted to suppress other types of decoherence couplings such as internal state coupling  and heating .
Our group has proposed a universal strategy of approximate DD [97–103] for both decay and proper dephasing, by either pulsed or continuous wave (CW) modulation of the system-bath coupling. This strategy allows us to optimally tailor the strength and rate of the modulating pulses to the spectrum of the bath (or continuum) by means of a simple universal formula. In many cases, the standard -phase “bang-bang” (BB) is then found to be inadequate or nonoptimal compared to dynamic control based on the optimization of the universal formula .
Our group has purported to substantially expand the arsenal of decay and decoherence control. We have presented a universal form of the decay rate of unstable states into any reservoir (continuum), dynamically modified by perturbations with arbitrary time dependence, focusing on non-Markovian time-scales [97, 99, 100, 102, 105]. An analogous form has been obtained by us for the dynamically modified rate of proper dephasing [100, 101, 105]. Our unified, optimized approach reduces to the BB method in the particular case of proper dephasing or decay via coupling to spectrally symmetric (e.g., Lorentzian or Gaussian) noise baths with limited spectral width (see below). The type of phase modulation advocated for the suppression of coupling to phonon or photon baths with frequency cutoff  is, however, drastically different from the BB method. Other situations to which our approach applies, but not the BB method, include amplitude modulation of the coupling to the continuum, as in the case of decay from quasibound states of a periodically tilted washboard potential : such modulation has been experimentally shown  to give rise to either slowdown of the decay (Zeno-like behavior) or its speedup (anti-Zeno-like behavior), depending on the modulation rate.
The theory has been generalized by us to finite temperatures and to qubits driven by an arbitrary time-dependent field, which may cause the failure of the rotating-wave approximation . It has also been extended to the analysis of multilevel systems, where quantum interference between the levels may either inhibit or accelerate the decay .
Our general approach  to dynamical control of states coupled to an arbitrary “bath” or continuum has reaffirmed the intuitive anticipation that, in order to suppress their decay, we must modulate the system-bath coupling at a rate exceeding the spectral interval over which the coupling is significant. Yet our analysis can serve as a general recipe for optimized design of the modulation aimed at an effective use of the fields for decay and decoherence suppression or enhancement.
1.2. Control of Symmetry-Breaking Multipartite Decoherence
Control of multiqubit or, more generally, multipartite decoherence is of even greater interest, because it can help protect the entanglement of such systems, which is the cornerstone of many quantum information processing applications. However, it is very susceptible to decoherence, decays faster than single-qubit coherence, and can even completely disappear in finite time, an effect dubbed entanglement sudden death (ESD) [73, 74, 108–113]. Entanglement is effectively protected in the collective decoherence situation, by singling out decoherence-free subspaces (DFS) , wherein symmetrically degenerate many-qubit states, also known as “dark” or “trapping” states , are decoupled from the bath [87, 115–117].
Symmetry is a powerful means of protecting entangled quantum states against decoherence, since it allows the existence of a decoherence-free subspace or a decoherence-free subsystem [77, 78, 80–87, 102, 114–120]. In multipartite systems, this requires that all particles be perturbed by the same environment. In keeping with this requirement, quantum communication protocols based on entangled two-photon states have been studied under collective depolarization conditions, namely, identical random fluctuations of the polarization for both photons .
Entangled states of two or more particles, wherein each particle travels along a different channel or is stored at a different site in the system, may present more challenging problems insofar as combating and controlling decoherence effects are concerned: if their channels or sites are differently coupled to the environment, their entanglement is expected to be more fragile and harder to protect.
To address these fundamental challenges, we have developed a very general treatment. Our treatment does not assume the perturbations to be stroboscopic, that is, strong or fast enough, but rather to act concurrently with the particle-bath interactions. This treatment extends our earlier single-qubit universal strategy [97, 99, 100, 104, 121, 122] to multiple entangled systems (particles) which are either coupled to partly correlated (or uncorrelated) finite-temperature baths or undergo locally varying random dephasing [107, 123–126]. Furthermore, it applies to any difference between the couplings of individual particles to the environment. This difference may range from the large-difference limit of completely independent couplings, which can be treated by the single-particle dynamical control of decoherence via modulation of the system-bath coupling, to the opposite zero-difference limit of completely identical couplings, allowing for multiparticle collective behavior and decoherence-free variables [86, 87, 115–117, 127–130]. The general treatment presented here is valid anywhere between these two limits and allows us to pose and answer the key question: under what conditions, if any, is local control by modulation, addressing each particle individually, preferable to global control, which does not discriminate between the particles?
We show that in the realistic scenario, where the particles are differently coupled to the bath, it is advantageous to locally control each particle by individual modulation, even if such modulation is suboptimal for suppressing the decoherence of a single particle. This local modulation allows synchronizing the phase-relation between the different modulations and eliminates the cross coupling between the different systems. As a result, it allows us to preserve the multipartite entanglement and reduces the multipartite decoherence problem to the single particle decoherence problem. We show the advantages of local modulation, over global modulation (i.e., identical modulation for all systems and levels), as regards the preservation of arbitrary initial states, preservation of entanglement, and the intriguing possibility of entanglement increase compared to its initial value.
The experimental realization of a universal quantum computer is widely recognized to be difficult due to decoherence effects, particularly dephasing [1, 131–133], whose deleterious effects on entanglement of qubits via two-qubit gates [134–136] are crucial. To help overcome this problem, we put forth a universal dynamical control approach to the dephasing problem during all the stages of quantum computations [125, 137], namely, (i) storage, wherein the quantum information is preserved in between gate operations, (ii) single-qubit gates, wherein individual qubits are manipulated, without changing their mutual entanglement, and (iii) two-qubit gates, that introduce controlled entanglement. We show that in terms of reducing the effects of dephasing, it is advantageous to concurrently and specifically control all the qubits of the system, whether they undergo quantum gate operations or not. Our approach consists in specifically tailoring each dynamical quantum gate, with the aim of suppressing the dephasing, thereby greatly increasing the gate fidelity. In the course of two-qubit entangling gates, we show that cross dephasing can be completely eliminated by introducing additional control fields. Most significantly, we show that one can increase the gate duration, while simultaneously reducing the effects of dephasing, resulting in a total increase in gate fidelity. This is at odds with the conventional approaches, whereby one tries to either reduce the gate duration, or increase the coherence time.
A general task may also require state and energy transfer , or entanglement  of noninteracting parties via shared modes of the bath [123, 140] which call for maximizing the shared (two-partite) couplings with the bath, but suppressing the single-partite couplings.
It is therefore desirable to have a general framework for optimizing the way a system interacts with its environment to achieve a desired task. This optimization consists in adjusting a given “score” that quantifies the success of the task, such as the targeted fidelity, purity, entropy, entanglement, or energy by dynamical modification of the system-bath coupling spectrum on demand. The goal of this work is to develop such a framework.
1.3. Dynamical Protection from Spontaneous Emission
Schemes of quantum information processing that are based on optically manipulated atoms face the challenge of protecting the quantum states of the system from decoherence, or fidelity loss, due to atomic spontaneous emission (SE) [1, 141, 142]. SE becomes the dominant source of decoherence at low temperatures, as nonradiative (phonon) relaxation becomes weak [4, 5]. SE suppression cannot be achieved by frequent modulations or perturbations of the decaying state, because of the extremely broad spectrum of the radiative continuum (“bath”) [76, 97]. A promising means of protection from SE is to embed the atoms in photonic crystals (three-dimensionally periodic dielectrics) that possess spectrally wide, omnidirectional photonic bandgaps (PBGs) : atomic SE would then be blocked at frequencies within the PBG [6–8]. Thus far, studies of coherent optical processes in a PBG have assumed fixed values of the atomic transition frequency . However, in order to operate quantum logic gates, based on pairwise entanglement of atoms by field-induced dipole-dipole interactions [10, 143, 144], one should be able to switch the interaction on and off, most conveniently by AC Stark-shifts of the transition frequency of one atom relative to the other, thereby changing its detuning from the PBG edge. The question then arises: should such frequency shifts be performed adiabatically, in order to minimize the decoherence and maximize the quantum-gate fidelity? The answer is expected to be affirmative, based on the existing treatments of adiabatic entanglement and protection from decoherence [11, 12, 129] and on the tendency of nonadiabatic evolution to spoil fidelity and promote transitions to the continuum . Surprisingly, our analysis (Section 6) demonstrates that only an appropriately phased sequence of “sudden” (strongly nonadiabatic) changes of the detuning from the PBG edge may yield higher fidelity of qubit and quantum gate operations than their adiabatic counterparts. This unconventional nonadiabatic protection from decoherence is valid for qubits that are strongly coupled to the continuum edge [14, 145], as opposed to the weak coupling approach in Sections 2–5.
In this paper we develop, step by step, the framework for universal dynamical control by modulating fields of multilevel systems or qubits, aimed at suppressing or preventing their noise, decoherence, or relaxation in the presence of a thermal bath. Its crux is the general master equation (ME) of a multilevel, multipartite system, weakly coupled to an arbitrary bath and subject to arbitrary temporal driving or modulation. The present ME, derived by the technique [146, 147], is more general than the ones obtained previously in that it does not invoke the rotating wave approximation and therefore applies at arbitrarily short times or for arbitrarily fast modulations.
Remarkably, when our general ME is applied to either AN or PN, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided that the corresponding density-matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically starting from the ubiquitous spin-boson Hamiltonian.
In Sections 2 and 3, we present a universal formula for the control of single-qubit zero-temperature relaxation and discuss several limits of this formula. In Sections 4 and 5, we extend this formula to multipartite or multilevel systems. In Section 6 dynamical control in the strong coupling regime is considered. In Section 7, the treatment is extended to the control of finite-temperature relaxation and decoherence and culminates in single-particle Bloch equations with dynamically modified decoherence rates that essentially obey the universal formula of Section 3. We then discuss in Section 7.4 the possible modulation arsenal for either AN or PN control. In Section 8, we discuss the extensions of the universal control formula to entangled multipartite systems. The formalism is applicable in a natural and straightforward manner to such systems . It allows us to focus on the ability of symmetries to overcome multipartite decoherence [87, 114–117]. In Section 8, we discuss the implementations of the universal formula to multipartite quantum computation. Section 9 discusses some general aspects of multipartite dynamical control. We develop a general optimization strategy for performing a chosen unitary or nonunitary task on an open quantum system. The goal is to design a controlled time-dependent system Hamiltonian by variationally minimizing or maximizing a chosen function of the system state, which quantifies the task success (score), such as fidelity, purity, or entanglement. If the time dependence of the system Hamiltonian is fast enough to be comparable to or shorter than the response time of the bath, then the resulting non-Markovian dynamics is shown to optimize the chosen task score to second order in the coupling to the bath. This strategy can not only protect a desired unitary system evolution from bath-induced decoherence but also take advantage of the system-bath coupling so as to realize a desired nonunitary effect on the system. Section 10 summarizes our conclusions whereby this universal control can effectively protect complex systems from a variety of decoherence sources.
2. Modulation-Affected Control of Decay into Continua and Zero-Temperature Baths: Weak-Coupling Theory
Consider the decay of a state via its coupling to a bath, described by the orthonormal basis , which forms either a discrete or a continuous spectrum (or a mixture thereof). The total Hamiltonian is Here is the dynamically modulated Hamiltonian of the system, with being the energy of . The time-dependent frequency can be attributed to the controllable dynamically imposed Stark shift, or to proper dephasing (uncontrolled, random fluctuation). The term is the time-dependent Hamiltonian of the bath, with being the energies of . The time-dependent frequencies , like , may arise from proper dephasing or dynamical Stark shifts. Finally denotes the off-diagonal coupling of with the continuum/bath, with being the dynamical modulation function and the system-bath coupling matrix elements.
We write the wave function of the system as with the initial condition being A one-level system which can exchange its population with the bath states represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states and , respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the rotating-wave approximation (RWA), which is alleviated in Section 7, the present formalism applies to a relaxing qubit, under the substitutions
3. Single-Qubit Zero-Temperature Relaxation
To gain insight into the requirements of decoherence control, consider first the simplest case of a qubit with states and energy separation relaxing into a zero-temperature bath via off-diagonal () coupling, Figure 1(a). The Hamiltonian is given by the sum extending over all bath modes, where are the annihilation and creation operators of mode , respectively, with and denoting the bath vacuum and th-mode single excitation, respectively, and being the corresponding transition matrix element and . being Hermitian conjugate. We have also taken the rotating wave approximation (RWA). The general time-dependent state can be written as The Schrödinger equation results in the following coupled equations : One can go to the rotating frame, define , , and get: where is the bath response/correlation function, expressible in terms of a sum over all transition matrix elements squared oscillating at the respective mode frequencies .
It is the spread of oscillation frequencies that causes the environment response to decohere after a (typically short) correlation time (Figure 1(b)). Hence, the Markovian assumption that the correlation function decays to instantaneously, , is widely used: it is in particular the basis for the venerated Lindblad’s master equation describing decoherence . It leads to exponential decay of at the Golden Rule (GR) rate [76, 78] as
We, however, are interested in the extremely non-Markovian time scales, much shorter than , on which all bath modes excitations oscillate in unison and the system-bath exchange is fully reversible. How does one probe, or, better still, maintain the system in a state corresponding to such time scales?
To this end, we assume modulations of and , that result in the time-dependent modulation function , which has two components, namely, an amplitude modulation and phase modulation . The modulation function is related to in 3 via with This modulation may pertain to any intervention in the system-bath dynamics: (i) measurements that effectively interrupt and completely dephase the evolution, describable by stochastic , (ii) coherent perturbations that describe phase modulations of the system-bath interactions [99, 124].
For any , the exact equation (12) is then rewritten as
We now resort to the crucial approximation that varies slower than either or . This approximation is justifiable in the weak-coupling regime (to second order in ), as discussed below. Under this approximation, (18) is transformed into a differential equation describing relaxation at a time-dependent rate as where is the instantaneous time-dependent relaxation rate and is the Lamb shift due to the coupling to the bath. One can separate the spectral representation of into the real and imaginary parts which satisfy the Kramers-Kronig relations with denoting the principal value. Henceforth, we shall concentrate on the relaxation rate, as it determines the excited state population, where is the average relaxation rate.
It is advantageous to consider the frequency domain, as it gives more insight into the mechanisms of decoherence. For this purpose, we define the finite-time Fourier transform of the modulation function as
The average time-dependent relaxation rate can be rewritten, by using the Fourier transforms of and , in the following form: where is the spectral-response function of the bath, and is the finite-time spectral intensity of the (random or coherent) intervention/modulation function, where the factor comes about from the definition of the decoherence rate averaged over the interval.
The relaxation rate described by (25)–(27) embodies our universal recipe for dynamically controlled relaxation [99, 124], which has the following merits: (a) it holds for any bath and any type of interventions, that is, coherent modulations and incoherent interruptions/measurements alike; (b) it shows that in order to suppress relaxation we need to minimize the spectral overlap of , given to us by nature, and , which we may design to some extent; (c) most importantly, it shows that in the short-time domain, only broad (coarse-grained) spectral features of and are important. The latter implies that, in contrast to the claim that correlations of the system with each individual bath mode must be accounted for, if we are to preserve coherence in the system, we actually only need to characterize and suppress (by means of ) the broad spectral features of , the bath response function. The universality of (25)–(27) will be elucidated in what follows, by focusing on several limits.
3.1. The Limit of Slow Modulation Rate
If corresponds to sufficiently slow rates of interruption/modulation , the spectrum of is much narrower than the interval of change of around , the resonance frequency of the system. Then can be replaced by , so that the spectral width of plays no role in determining , and we may as well replace by a spectrally finite, flat (white-noise) reservoir; that is, we may take the Markovian limit. The result is that (25) coincides with the Golden Rule (GR) rate, (14) (Figure 2(a)) as Namely, slow interventions do not affect the onset and rate of exponential decay.
3.2. The Limit of Frequent Modulation
Frequent interruptions, intermittent with free evolution, are represented by a repetition of the free-evolution modulation spectrum where being the time-interval between consecutive interruptions. If describes extremely frequent interruptions or measurements , is much broader than . We may then pull out of the integral, whereupon (25) yields This limit is that of the quantum Zeno effect (QZE), namely, the suppression of relaxation as the interval between interruptions decreases [150–152]. In this limit, the system-bath exchange is reversible and the system coherence is fully maintained (Figure 2(b)). Namely, the essence of the QZE is that sufficiently rapid interventions prevent the excitation escape to the continuum, by reversing the exchange with the bath.
3.3. Intermediate Modulation Rate
In the intermediate time-scale of interventions, where the width of is broader than the width of (so that the Golden Rule is violated) but narrower than the width of (so that the QZE does not hold), the overlap of and grows as the rate of interruptions, or modulations, increases. This brings about the increase of relaxation rates with the rate of interruptions, marking the anti-Zeno effect (AZE) [85, 102, 153] (Figure 2(c)). On such time-scales, more frequent interventions (in particular, interrupting measurements) enhance the departure of the evolution from reversibility. Namely, the essence of the AZE is that if you do not intervene in time to prevent the excitation escape to the continuum, then any intervention only drives the system further from its initial state.
We note that the AZE can only come about when the peaks of and do not overlap, that is, the resonant coupling is shifted from the maximum of . If, by contrast, the peaks of and do coincide, any rate of interruptions would result in QZE (Figure 2(b)). This can be understood by viewing as an averaging kernel of around . If is the maximum of the spectrum, any averaging can only be lower than this maximum, which is the Golden Rule decay rate. Hence, any rate of interruptions can only decrease the decay rate with respect to the Golden Rule rate, that is, cause the QZE.
3.4. Quasiperiodic Amplitude and Phase Modulation (APM)
The modulation function can be either random or regular (coherent) in time, as detailed below. Consider first the most general coherent amplitude and phase modulation (APM) of the quasiperiodic form, Here () are arbitrary discrete frequencies with the minimum spectral distance . If is periodic with the period , then and become the Fourier components of . For a general quasiperiodic , one obtains Here equals the average of over a period of the order of , , and , whereas is a bell-like function of normalized to 1.
For a sufficiently long time, the function becomes narrower than the respective characteristic width of around , and one can set
Thus, when where is the effective correlation (memory) time of the reservoir, (25) is reduced to For the validity of (37), it is also necessary that This condition is well satisfied in the regime of interest, that is, weak coupling to essentially any reservoir, unless (for some harmonic ) is extremely close to a sharp feature in , for example, a band edge , a case covered by Section 6. Otherwise, the long-time limit of the general decay rate (25) under the APM is a sum of the GR rates, corresponding to the resonant frequencies shifted by , with the weights .
Formula (37) provides a simple general recipe for manipulating the decay rate by APM. Its powerful generality allows for the optimized control of decay, not only for a single level but also for a band characterized by a spectral distribution (e.g., inhomogeneous or vibrational spectrum). We can then choose and in (37) so as to minimize the decay convoluted with . In what follows, various limits of (37) will be analyzed.
3.5. Coherent Phase Modulation (PM)
3.5.1. Monochromatic Perturbation
Let Then where is a frequency shift, induced by the ac Stark effect (in the case, e.g., of atoms) or by the Zeeman effect (in the case of spins). In principle, such a shift may drastically enhance or suppress relative to . It provides the maximal variation of achievable by an external perturbation, since it does not involve any averaging (smoothing) of incurred by the width of : the modified can even vanish, if the shifted frequency is beyond the cut-off frequency of the coupling, where . Conversely, the increase of due to a shift can be much greater than that achievable by repeated measurements, that is, the anti-Zeno effect [97, 98, 101, 102]. In practice, however, ac Stark shifts are usually small for (cw) monochromatic perturbations, whence pulsed perturbations should often be used.
3.5.2. Impulsive Phase Modulation
Let the phase of the modulation function periodically jump by an amount at times . Such modulation can be achieved by a train of identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed frequency shifts . Now where is the integer part. One then obtains that The decay, according to (22), has then the form (at ) where is defined by (25).
For sufficiently long times
For small phase shifts, , the peak dominates, whereas In this case, one can retain only the term in (37) (unless is changing very fast). Then the modulation acts as a constant shift
With the increase of , the difference between the and peak heights diminishes, vanishing for . Then that is, for contains two identical peaks symmetrically shifted in opposite directions (the other peaks decrease with as , totaling 0.19).
The above features allow one to adjust the modulation parameters for a given scenario to obtain an optimal decrease or increase of . The phase-modulation (PM) scheme with a small is preferable near the continuum edge, since it yields a spectral shift in the required direction (positive or negative). The adverse effect of peaks in then scales as and hence can be significantly reduced by decreasing . On the other hand, if is near a symmetric peak of , is reduced more effectively for , as in [80, 81], since the main peaks of at and then shift stronger with than the peak at for .
3.6. Amplitude Modulation (AM)
Amplitude modulation (AM) of the coupling arises, for example, for radiative-decay modulation due to atomic motion through a high- cavity or a photonic crystal [154, 155] or for atomic tunneling in optical lattices with time-varying lattice acceleration [106, 156]. Let the coupling be turned on and off periodically, for the time and , respectively, that is, (). Now  so that (see (43)) where is given by (25) and (50).
It is instructive to consider the limit wherein and is much greater than the correlation time of the continuum; that is, does not change significantly over the spectral intervals . In this case, one can approximate the sum (37) by the integral (25) with characterized by the spectral broadening ~1. Then (25) for reduces to that obtained when ideal projective measurements are performed at intervals . Thus the AM scheme can imitate measurement-induced (dephasing) effects on quantum dynamics, if the interruption intervals exceed the correlation time of the continuum.
The decay probability , calculated for parameters similar to , completely coincides with that obtained for ideal impulsive measurements at intervals [97, 98, 101] and demonstrates either the quantum Zeno effect (QZE) or the anti-Zeno effect (AZE) behavior, depending on the rate of modulation.
Since the Hamiltonian for atoms in accelerated optical lattices is similar to the Legett Hamiltonian for current-biased Josephson junctions , the present theory has been extended to describe effects of current modulations on the rate of macroscopic quantum tunneling in Josephson junctions in .
Projective measurements at an effective rate , whether impulsive or continuous, usually result in a broadened (to a width ) modulation function , without a shift of its center of gravity [97, 98, 101, 158, 159], This feature was shown in  to be responsible for either the standard quantum Zeno effect whereby scales as or the anti-Zeno effect whereby grows with . In contrast, a weak and broadband chaotic field, such that where is the mean intensity, is the bandwidth, and is the effective polarizability (electric or magnetic, depending on the system), would give rise to a Lorentzian dephasing function with a substantial shift This shift would have a much stronger effect on than the QZE or AZE, which are associated with the rate , since
4. Multipartite Decay Control
4.1. Multipartite PN Control by Resonant Modulation
One can describe phase noise, or proper dephasing, by a stochastic fluctuation of the excited-state energy, , where is a stochastic variable with zero mean, and is the second moment. For multipartite systems, where each qubit can undergo different proper dephasing, , one has an additional second moment for the cross dephasing, . A general treatment of multipartite systems undergoing this type of proper dephasing is given in . Here we give the main results for the case of two qubits.
Let us take two TLS, or qubits, which are initially prepared in a Bell state. We wish to obtain the conditions that will preserve it. In order to do that, we change to the Bell basis, which is given by For an initial Bell-state , where , one can then obtain the fidelity, , as where where is the amplitude of the resonant field applied on qubit , , and the corresponds to and to . Expressions (61)–(67) provide our recipe for minimizing the Bell-state fidelity losses. They hold for any dephasing time-correlations and arbitrary modulation.
One can choose between two modulation schemes, depending on our goals. When one wishes to preserve and initial quantum state, one can equate the modified dephasing and cross dephasing rates of all qubits, . This results in complete preservation of the singlet only, that is, , for all , but reduces the fidelity of the triplet state. On the other hand, if one wishes to equate the fidelity for all initial states, one can eliminate the cross dephasing terms, by applying different modulations to each qubit (Figure 3), causing for all . This requirement can be important for quantum communication schemes.
5. Dynamical Control of Zero-Temperature Decay in Multilevel Systems
5.1. General Formalism
Here we discuss in detail a model for dynamical decay modifications in a multilevel system. The system with energies , , is coupled to a zero-temperature bath of harmonic oscillators with frequencies . Using the factorized coupling defined in Section 2.1, the corresponding Hamiltonian is found to be as in 1, where where now each level has a different modulation and a different coupling to the bath and denotes a gate operation.
The system evolution is divided into two phases, one of storage without gate operations and a gate operation of finite duration
The full wave function is given by Similarly to what was said in Section 2.1, one can consider two types of situations. The above equations (68)–(72) were written for an -level system which can exchange its population with the reservoir. In addition, one can consider an -level system, where transitions are possible between any level and a lower level , the reservoir consisting of quantum systems, as described in Section 2.1. The theory in Section 5 holds for both situations, with the minor difference that one should substitute as in (70) and (72) and perform a similar substitution in (76) below.
In order to find the solution, one has to diagonalize the system hamiltonian by introducing a matrix that rotates the amplitudes as such that, by defining , one gets where are the eigenvalues of the new rotated system. Thus the transformed wave function becomes Using these rotated state amplitudes, a procedure similar to that used for one level, one finds that they obey the following integrodifferential equations, assuming slowly varying as Here, the and matrices are given by with and being the modulation and reservoir-response matrices, respectively, given by where During the storage phase, one has , and , and during the gate-operation phase, , , and .
The solution to (77) is of the form
To simplify the analysis, one can define the fluence and the modulation spectral matrices as The relevant imaginary parts of the spectral response of the reservoir can be expressed, analogously to (20) and (21), by the Kramers-Kronig relations
Defining we shall now represent in different regimes (phases).
(i) As a reference, it is important to consider the decoherence effects with no modulations at all, that is, . In this case, one obtains a diagonal decoherence matrix This means that interference of decaying levels and cancels out in the long time limit, and the decoherence is without cross relaxation.
(ii) During the storage phase, (84) results in One can easily see that for the off-diagonal terms, a simple separation into decay rates and energy shifts is inapplicable in this formulation.
6. The Strong-Coupling Regime: Decay Control Near Continuum Edge by Nonadiabatic Interference
The analysis expounded thus far has been based on a perturbative treatment of the system-bath coupling. Here, we address the regime of strong system-bath coupling, as in the case of a resonance frequency very near to the continuum edge, a situation that may be encountered in atomic excitation near the ionization energy, vibrational excitation frequency in a solid near the Debye cutoff, or an atomic excitation in a photonic crystal near a photonic bandgap. In the strong-coupling regime, it is advantageous to work in the combined basis of the system (qubit) and field (bath) states that incorporate the system-bath interaction. Dynamical control of the decay can then be analysed by exact solution of the Schrödinger equation in this basis. Analytical expressions are obtainable for alternating static evolutions with different parameters (e.g., resonant frequency), the dynamical control resulting from their interference. Specifically, we shall consider optical manipulations of atoms embedded in photonic crystals with atomic transition frequencies near a photonic bandgap (PBG), that is, near the edge of the photonic mode continuum, where the qubit is strongly coupled to the continuum, and spontaneous emission (SE) is only partially blocked, because an initially excited atom then evolves into a superposition of decaying and stable states, the stable state representing photon-atom binding [14, 145]. In what follows we shall demonstrate the ability of appropriately alternating sudden changes of the detuning to augment the interference of the emitted and back-scattered photon amplitudes, thereby increasing the probability amplitude of the stable (photon-atom bound) state. As a result, phase-gate operations affected by dipole-dipole interactions can be performed with higher fidelity than in the case of adiabatic frequency change.
6.1. Hamiltonian and Equations of Motion
We consider a two-level atom with excited and ground states and coupled to the field of a discrete (or defect) mode and to the photonic band structure (PBS) in a photonic crystal. The hamiltonian of the system in the rotating-wave approximation assumes the form  Here, is the energy of the atomic transition frequency, and are, respectively, the creation and annihilation operators of the field mode at frequency , is the mode density of the PBS, and and are the coupling rates to the atomic dipole of a mode from the continuum and the discrete mode, respectively.
Let us first consider the initial state obtained by absorbing a photon from the discrete mode as where is the vacuum state of the field. Then the evolution of the wavefunction has the general form where we have denoted by and the single-photon state of the relevant modes. The Schrödinger equation then leads to the set of coupled differential equations This evolution reflects the interplay between the off-resonant Rabi oscillations of and , at the driving rate , and the partly inhibited oscillatory decay from to via coupling to the continuum . This decay depends on the detuning of from the continuum edge at (the upper cutoff of the PBG). For a spectrally steep edge (see below), we are in the regime of strong coupling to the mode continuum (as in a high-Q cavity ) which allows for the existence of an oscillatory, nondecaying, component of , associated with a photon-atom bound state [7, 145].
6.2. Periodic Sudden Changes of the Detuning
Let us now introduce abrupt changes of , that is, of the detuning from the upper cutoff, , of the PBG (by fast AC-Stark modulations as discussed below), at intervals . In the sudden-change approximation for , the amplitudes of the excited state, the discrete mode and the continuum still evolve according to (91), except that from to the atomic transition frequency is , that is, the detuning , while for , we have , that is, . This dynamics leads to the relation Here, and are solutions of (91) with a static (fixed) atomic transition frequency, or . However, the initial condition at the instant of the frequency change from to is no longer the excited state (89) but the superposition In other words, the dynamics is equivalent to two successive static evolutions, the second one starting from initial conditions .
Using the Laplace transform of the system (91) with the initial condition (93), it is possible to express the dynamic amplitude of the excited state after the sudden change as where we have used the initial conditions and the solution of (91) for the initial condition (89).
There is an advantageous feature to the sudden change: since the time dependence of in (92) arises from the static amplitudes , , and at the shifted time , a consequence of the sudden change is to revive the excited-state population oscillations, which tend to disappear at long times in the static case. Hence, by applying several successive sudden changes, we should be able to maintain large-amplitude oscillations of the coherence between and . The scenario leading to the largest amplitude consists in periodic shifts of the energy detuning from to . When the initial detuning is large and we first reduce it to before it increases to , the dynamic population and the coherence, thanks to the revival of oscillations, are periodically larger than the static ones. This remarkable result occurs unexpectedly: it implies that successive abrupt changes can reverse the decay to the continuum, even though they cannot be associated with the Zeno effect: they occur at intervals much longer than the correlation (Zeno) time of the radiative continuum, which is utterly negligible ( s) , or even longer than the static-oscillation half period. The fact that this happens only for the rather “counter-intuitive” ordering of detuning values (from large to small then back again) is a manifestation of interference between successive static evolutions: their relative phases determine the beating between the emitted and reabsorbed (back-scattered) photon amplitudes and thereby the oscillation of .
Let us now consider the initial superposition and a nonnegligible coupling constant . In this case, the periodic dynamic population of the excited state also strongly exceeds the static one. Most importantly, the instantaneous dynamic fidelity is periodically enhanced as compared to the static one, as demonstrated numerically.
In order to use these results for quantum logic gates, let us consider the example of the dipole-dipole induced control-phase gate, which consists in shifting the phase of the target-qubit excited state by via interaction with the control qubit [10, 143, 144]. The phase shift must be accumulated gradually, to preserve the coherence of the system. We have found that ten or twenty sudden shifts of or , respectively, alternating with appropriate detuning changes, can keep the fidelity high, with little decoherence. The system begins to evolve following the “counter-intuitive” detuning sequence discussed above (not to be confused with the adiabatic STIRAP method [11, 12, 129]). As soon as two sudden changes of the detuning have been performed, the conditional phase shift of or takes place and the process is further repeated. The total gate operation is completed within the time interval of maximum fidelity. The fidelity of the system relative to its initial state during the realization of a control phase gate, with alternating detunings, is perhaps our most impressive finding. We find that the fidelity is increased using the “counterintuitive” sequence of detunings (solid line) as compared to the static (fixed) choice of maximal detuning (long-dashed line), or compared to the dynamically enhanced fidelity obtained without gate operations (dot-dashed line).
6.3. Comparison with the Weak-Coupling Regime
We have compared the results of this method, which allows for possibly strong coupling of with the continuum edge, with those of the universal formula of Section 2 (25), which expresses the decay rate of by the convolution of the modulation spectrum and the PBS coupling spectrum. We find good agreement with this formula only in the regime of weak coupling to the PBG edge, when the dimensionless detuning parameter , as expected from the limitations of the theory in Section 2.
6.4. Experimental Scenario
The following experimental scenario may be envisioned for demonstrating the proposed effect: pairs of qubits are realizable by two species of active rare-earth dopants [17, 18] or quantum dots in a photonic crystal. The transition frequency of one species is initially detuned by from the PBG edge with coupling constant and by ~3 MHz from the resonance of the other species. This is abruptly modulated by nonresonant laser pulses which exert ~3 MHz AC Stark shifts. Between successive shifts, the qubits are near resonant with their neighbours and therefore become dipole-dipole coupled, thus affecting the high-fidelity phase-control gate operation [10, 143, 144]. The required pulse rate is , much lower than the pulse rate stipulated under similar conditions by previously proposed strategies [81, 99, 118].
7. Finite-Temperature Relaxation and Decoherence Control
So far we have treated the case of an empty (zero-temperature) bath. In order to account for finite-temperature situations, where the bath state is close to a thermal (Gibbs) state, we resort to a master equation (ME) for any dynamically controlled reduced density matrix of the system [100, 124] that we have derived using the Nakajima-Zwanzig formalism [70, 146, 147, 160]. This ME becomes manageable and transparent under the following assumptions. (i) The weak-coupling limit of the system-bath interaction prevails, corresponding to the neglect of terms. This is equivalent to the Born approximation, whereby the back effect of the system on the bath and their resulting entanglement are ignored. (ii) The system and the bath states are initially factorisable. (iii) The initial mean value of vanishes.
We present the general form of the Nakajima-Zwanzig formalism and resort to the aforementioned assumptions only when necessary. Hence, the formalism may seem cumbersome, yet it can be simplified greatly if the assumptions are made from the outset (see ).
7.1. Explicit Equations for Factorisable Interaction Hamiltonians
We now wish to write the ME explicitly for time-dependent Hamiltonians of the following form : where and are the system and bath Hamiltonians, respectively, and , the interaction Hamiltonian, is the product of operators and which act on the system and bath, respectively.
Finally, defining the correlation function for the bath, we obtain the ME for in the Born approximation as
We focus on two regimes: a two-level system coupled to either an amplitude- or phase-noise (AN or PN) thermal bath. The bath Hamiltonian (in either regime) will be explicitly taken to consist of harmonic oscillators and be linearly coupled to the system Here are the annihilation and creation operators of mode , respectively, and is the coupling amplitude to mode .
7.1.1. Amplitude-Noise Regime
We first consider the AN regime of a two-level system coupled to a thermal bath. We will use off-resonant dynamic modulations, resulting in AC-Stark shifts. The Hamiltonians then assume the following form: where is the dynamical AC-Stark shifts, is the time-dependent modulation of the interaction strength, and the Pauli matrix .
7.1.2. Phase-Noise Regime
Next, we consider the PN regime of a two-level system coupled to a thermal bath via operator. To combat it, we will use near-resonant fields with time-varying amplitude as our control. The Hamiltonians then assume the following forms: where is the time-dependent resonant field, with real envelope , is the time-dependent modulation of the interaction strength, and .
Since we are interested in dephasing, phases due to the (unperturbed) energy difference between the levels are immaterial.
7.2. Universal Master Equation
To derive a universal ME for both amplitude- and phase-noise scenarios, we move to the interaction picture and rotate to the appropriate diagonalizing basis, where the appropriate basis for the AN case of (100) is while for the PN case of (102) the basis is
In this rotated and tilted frame, where is the phase-modulation due to the time-dependent control in the system Hamiltonian.
Allowance for arbitrary time-dependent intervention in the system and interaction dynamics , , respectively, yields the following universal ME for a dynamically controlled decohering system [100, 124]: Here is the modulated interaction operator, where denotes the rotated and tilted frame, and . The modulation function is given by for both AN and PN. It is important to note that is a function of (not of ): this convolutionless form of the ME is fully non-Markovian to second order in , as proven exactly in .
7.3. Universal Modified Bloch Equations
The resulting modified Bloch equations, in the appropriate diagonalizing basis (see (104) for AN and (105) for PN), are given by The time-dependent relaxation rates are real, and the only difference between them is the complex conjugate of the combined modulation function, . They can be very different for a complex correlation function.
One can derive the corresponding time-averaged relaxation rates of the upper and lower states as
For both AN (see (100)) and PN (see (102)), where is the zero-temperature bath spectrum, and are the frequency-dependent density of bath modes and the transition matrix element, respectively, is the temperature-dependent bath mode population, and is the inverse temperature. Also, is the Heaviside function, that is, the zero-temperature bath spectrum is defined only for positive frequencies . Hence, the first right-hand side of (114) is nonzero for positive frequencies and the second right-hand side is nonzero for negative frequencies.
For either AN or PN, we may control the decoherence by either off-resonant or near-resonant modulations, respectively. The modulation spectrum has the same form for both (see Section 7.4) as where the modulation function is given in (107) and (109). The time-dependent modulation phase factor is obtained for AN in the form of an AC-Stark shift, time-integrated over where is the Rabi frequency of the control field and is the detuning. The corresponding phase factor for PN is the integral of the Rabi frequency , that is, the pulse area of the resonant control field, (107) (Figure 4).
Hence, upon making the appropriate substitutions, the Bloch equations (110) have the same universal form for either AN or PN. An arbitrary combination of AN and PN requires a more detailed treatment, yet the universal form is maintained.
7.3.1. Dynamically Modified Decay Rates
Since we are interested here in dynamical control of relaxation, we shall concentrate on the transition rates rather than the level shifts. The average rate of the transition and its counterpart are given by Here the upper (lower) sign corresponds to the subscript , and can be shown  to be nonnegative, with , and vanishes for at : . For the oscillator bath, one finds that where and is the average number of quanta in the oscillator (bath mode) with frequency .
We apply (118) to the case of coherent modulation of quasiperiodic form, (see (31)). Without a limitation of the generality, we can assume that . We then find, using (118), that the rates tend to the long-time limits where or Equation (121) shows that is given by the overlap of the modulation spectrum with the bath-CF spectrum . The limits (123) are approached when and . Here is the bath memory (correlation) time, defined as the inverse of , the spectral interval over which changes around the relevant frequencies.
Had we used the standard dipolar RWA hamiltonian in the case of an oscillator bath, dropping the antiresonant terms in , we would have arrived at the transition rates wherein the integration is performed from 0 to , rather than from to , as in (121). This means that the RWA transition rates hold for a slow modulation, when at , being peaked near . However, whenever the suppression of requires modulation at a rate comparable to , the RWA is inadequate. For instance, (120) and (124) imply that, at , the rate vanishes identically, irrespective of , in contrast to the true upward-transition rate in (121), which may be comparable to for ultrafast modulation. The difference between the RWA and non-RWA decay rates stems from the fact that the RWA implies that a downward (upward) transition is accompanied by emission (absorption) of a bath quantum, whereas the non-RWA (negative-frequency) contribution to in (121) allows for just the opposite: downward (upward) transitions that are accompanied by absorption (emission). The latter processes are possible since the modulation may cause level to be shifted below .
The validity of the (decohering) qubit model in the presence of modulation at a rate is now elucidated: it requires that , being the effective transition rate from level to any other level , and, in particular, . If are strongly suppressed by the modulation, the TLS model holds for long times.
7.3.2. Dynamically Modified Proper Dephasing
We turn now to proper dephasing when it dominates over decay. The random frequency fluctuations are typically characterized by a (single) correlation time , with ensemble mean . When the field is used only for gate operations, we assume that it does not affect proper dephasing. The ensemble average over results in with the dephasing rate The dephasing CF is the counterpart of the bath CF .
At , the decoherence rate and shift approach their asymptotic values For the validity of (127), it is necessary that We assume the secular approximation, which holds if
The proper dephasing rate associated with is In the presence of a constant [cw ], it is modified into For a sufficiently strong field, the dephasing rate can be suppressed by the factor . This suppression reflects the ability of strong, near-resonant Rabi splitting to shift the system out of the randomly fluctuating bandwidth, or average its effects. Quantum gate operations may be performed by slight modulations of the control field, which can flip the qubit without affecting proper dephasing. By comparison, the “bang-bang” (BB) method involving -periodic -pulses [2, 82, 84] is an analog of the above “parity kicks.” Using the analog of (121), such pulses can be shown to suppress approximately according to (135) with . This BB method requires pulsed fields with Rabi frequencies , that is, much stronger fields than the cw field in (135). Using s, cw Rabi frequencies exceeding 1 MHz achieve a significant dephasing suppression.
7.4. Modulation Arsenal
Any modulation with quasi-discrete, finite spectrum is deemed quasiperiodic, implying that it can be expanded as where are arbitrary discrete frequencies such that where is the minimal spectral interval.
One can define the long-time limit of the quasi-periodic modulation, when where is the bath-memory (correlation) time, defined as the inverse of the largest spectral interval over which and change appreciably near the relevant frequencies . In this limit, the average decay rate is given by (Figure 5(a)) as
7.4.1. Phase Modulation (PM) of the Coupling
Monochromatic Perturbation. Let Then where is a frequency shift, induced by the AC Stark effect (in the case of atoms) or by the Zeeman effect (in the case of spins). In principle, such a shift may drastically enhance or suppress relative to the Golden Rule decay rate, that is, the decay rate without any perturbation as
Equation (40) provides the maximal change of achievable by an external perturbation, since it does not involve any averaging (smoothing) of incurred by the width of : the modified can even vanish, if the shifted frequency is beyond the cutoff frequency of the coupling, where (Figure 5(d)). This would accomplish the goal of dynamical decoupling [81–87, 118, 162]. Conversely, the increase of due to a shift can be much greater than that achievable by repeated measurements, that is, the anti-Zeno effect [97, 98, 101, 102]. In practice, however, AC Stark shifts are usually small for (cw) monochromatic perturbations, whence pulsed perturbations should often be used, resulting in multiple shifts, as per (139).
Dynamical Decoupling. Dynamical decoupling (DD) is one of the best known approaches to combat decoherence, especially dephasing [79–92, 95, 96]. A full description of this approach is beyond the scope of this work, but we present its most essential aspects and how it can be incorporated into the general framework described above.
7.4.2. Standard DD
DD is based on the notion that the phase-modulation control fields are short and strong enough such that the free evolution can be neglected during these pulses. Hence, the propagator can be decomposed into the free propagator, followed by the control-field propagator, free propagator, and so forth. The control fields used result in the periodic accumulation of -phases; that is, each pulse has a total area of , whose effects are similar to time-reversal or the spin-echo technique . Thus, the free evolution propagator after the control -pulse negates the effects of the free evolution propagator prior to the control fields, up to first order of the noise in the Magnus expansion.
While the formalism of dynamical decoupling is quite different from the formalism presented here, it can be easily incorporated into the general framework of universal dynamical decoherence control by introducing impulsive phase modulation. Let the phase of the modulation function periodically jump by an amount at times . Such modulation can be achieved by a train of identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed AC Stark shifts of . When , this modulation corresponds to dynamical-decoupling (DD) pulses.
For small phase shifts, , the peak dominates, whereas In this case, one can retain only the term in (139), unless is changing very fast with frequency. Then the modulation acts as a constant shift (Figure 5(d)) as
As increases, the difference between the and peak heights diminishes, vanishing for . Then that is, for