# Universal Dynamical Control of Open Quantum Systems

Authors: Gershon Kurizki
Journal: ISRN Optics
 (2013)

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## Abstract

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.


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## Body

## 1.  Introduction

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 [3]. 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 decoheringindependently, 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 ofnonlocal 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 [76], vibrational and collisional relaxation of trapped ions [1], and the relaxation of current-biased Josephson junctions [77]. Another source of decoherence in the same systems is proper dephasing or phase noise (PN) [78], 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, π-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 [93].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. Theseπ-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 [94]. 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 [89], continuous DD [88], concatenated DD [90], and optimal DD [95, 96]. DD has also been adapted to suppress other types of decoherence couplings such as internal state coupling [91] and heating [84].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 π-phase “bang-bang” (BB) is then found to be inadequate or nonoptimal compared to dynamic control based on the optimization of the universal formula [104].Our group has purported to substantially expand the arsenal of decay and decoherence control. We have presented auniversal 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 [103] 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 [99]: such modulation has been experimentally shown [106] 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 anarbitrary time-dependent field, which may cause the failure of the rotating-wave approximation [100]. It has also been extended to the analysis of multilevel systems, where quantum interference between the levels may either inhibit or accelerate the decay [107].Our general approach [99] 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) [114], wherein symmetrically degenerate many-qubit states, also known as “dark” or “trapping” states [78], 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 [71].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 isadvantageous 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 [138], or entanglement [139] 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 onoptically 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) [6]: 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 [9]. 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 [13]. 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.


### 1.4. Outline

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 obeyanalogous 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 Sections2 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 [123]. 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.


## 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 [76], vibrational and collisional relaxation of trapped ions [1], and the relaxation of current-biased Josephson junctions [77]. Another source of decoherence in the same systems is proper dephasing or phase noise (PN) [78], 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, π-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 [93].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. Theseπ-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 [94]. 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 [89], continuous DD [88], concatenated DD [90], and optimal DD [95, 96]. DD has also been adapted to suppress other types of decoherence couplings such as internal state coupling [91] and heating [84].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 π-phase “bang-bang” (BB) is then found to be inadequate or nonoptimal compared to dynamic control based on the optimization of the universal formula [104].Our group has purported to substantially expand the arsenal of decay and decoherence control. We have presented auniversal 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 [103] 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 [99]: such modulation has been experimentally shown [106] 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 anarbitrary time-dependent field, which may cause the failure of the rotating-wave approximation [100]. It has also been extended to the analysis of multilevel systems, where quantum interference between the levels may either inhibit or accelerate the decay [107].Our general approach [99] 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) [114], wherein symmetrically degenerate many-qubit states, also known as “dark” or “trapping” states [78], 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 [71].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 isadvantageous 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 [138], or entanglement [139] 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 onoptically 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) [6]: 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 [9]. 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 [13]. 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.


## 1.4. Outline

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 obeyanalogous 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 Sections2 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 [123]. 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

### 2.1. Framework

Consider the decay of a state|e〉 via its coupling to a bath, described by the orthonormal basis {|j〉}, which forms either a discrete or a continuous spectrum (or a mixture thereof). The total Hamiltonian is
(1)H=HS(t)+HB(t)+HI(t).
Here
(2)HS(t)=ħ(ωa+δa(t))|e〉〈e|
is the dynamically modulated Hamiltonian of the system, with ħωa being the energy of |e〉. The time-dependent frequency δa(t) can be attributed to the controllable dynamically imposed Stark shift, or to proper dephasing (uncontrolled, random fluctuation). The term
(3)HB(t)=ħ∑j‍(ωj+δf(t))|j〉〈j|
is the time-dependent Hamiltonian of the bath, with ħωj being the energies of |j〉. The time-dependent frequencies δf(t), like δa(t), may arise from proper dephasing or dynamical Stark shifts. Finally
(4)HI(t)=ϵ(t)μje|e〉〈j|+h.c.,
denotes the off-diagonal coupling of |e〉 with the continuum/bath, with ϵ~(t) being the dynamical modulation function and μje the system-bath coupling matrix elements.We write the wave function of the system as(5)|Ψ(t)〉=α(t)e-iωat-i∫0t‍δa(t′)dt′|e〉+∑j‍βj(t)e-iωjt-i∫0t‍δf(t′)dt′|j〉,
with the initial condition being
(6)|Ψ(0)〉=|e〉.
A one-level system |e〉 which can exchange its population with the bath states {|j〉} 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 |e〉 and |g〉, 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
(7)|e〉⟶|e〉|vac〉,|j〉⟶|g〉|j〉.


## 2.1. Framework

Consider the decay of a state|e〉 via its coupling to a bath, described by the orthonormal basis {|j〉}, which forms either a discrete or a continuous spectrum (or a mixture thereof). The total Hamiltonian is
(1)H=HS(t)+HB(t)+HI(t).
Here
(2)HS(t)=ħ(ωa+δa(t))|e〉〈e|
is the dynamically modulated Hamiltonian of the system, with ħωa being the energy of |e〉. The time-dependent frequency δa(t) can be attributed to the controllable dynamically imposed Stark shift, or to proper dephasing (uncontrolled, random fluctuation). The term
(3)HB(t)=ħ∑j‍(ωj+δf(t))|j〉〈j|
is the time-dependent Hamiltonian of the bath, with ħωj being the energies of |j〉. The time-dependent frequencies δf(t), like δa(t), may arise from proper dephasing or dynamical Stark shifts. Finally
(4)HI(t)=ϵ(t)μje|e〉〈j|+h.c.,
denotes the off-diagonal coupling of |e〉 with the continuum/bath, with ϵ~(t) being the dynamical modulation function and μje the system-bath coupling matrix elements.We write the wave function of the system as(5)|Ψ(t)〉=α(t)e-iωat-i∫0t‍δa(t′)dt′|e〉+∑j‍βj(t)e-iωjt-i∫0t‍δf(t′)dt′|j〉,
with the initial condition being
(6)|Ψ(0)〉=|e〉.
A one-level system |e〉 which can exchange its population with the bath states {|j〉} 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 |e〉 and |g〉, 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
(7)|e〉⟶|e〉|vac〉,|j〉⟶|g〉|j〉.


## 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|e〉,|g〉 and energy separation ħωa relaxing into a zero-temperature bath via off-diagonal (σx) coupling, Figure 1(a). The Hamiltonian is given by
(8)H=HS+HB+HI,HS=ħωa|e〉〈e|,HB=ħ∑j‍ωjaj†aj,HI=∑j‍μej|e,0〉〈g,1j|+H.c.,
the sum extending over all bath modes, where aj,aj† are the annihilation and creation operators of mode j, respectively,  with |{0}〉 and |1j〉 denoting the bath vacuum and jth-mode single excitation, respectively, and  μej being the corresponding transition matrix element and H.C. being Hermitian conjugate. We have also taken the rotating wave approximation (RWA). The general time-dependent state can be written as
(9)Ψ(t)=α~e(t)|e,0〉+∑j‍β~j(t)|g,1j〉.
The Schrödinger equation results in the following coupled equations [83]:
(10)∂tα~e(t)=-iωaα~e-iħ-1∑j‍μejβ~j,∂tβ~j(t)=-iωjβ~j-iħ-1μej*α~e.
One can go to the rotating frame, define αe(t)=α~e(t)eiωat, βj(t)=β~j(t)eiωjt, and get:
(11)βj(t)=-iħ-1∫0t‍dt′eiωjt′μej*e-iωat′αe(t′),(12)∂tαe=-∫0t‍dt′Φ(t-t′)eiωa(t-t′)αe(t′),
where
(13)Φ=ħ-2〈e,0|HIe-(iH0t/ħ)HI|e,0〉=ħ-2∑j‍|μej|2e-iωjt,
is the bath response/correlation function, expressible in terms of a sum over all transition matrix elements squared oscillating at the respective mode frequencies ωj.(a) Schematic drawing of a two-level system with off-diagonal coupling to a continuum or a bath. (b) Schematic drawing of a bath comprised of many harmonic oscillators with different frequencies, whose temporal dephasing after correlation timetc renders the system-bath interaction practically irreversible.
(a)
(b)It is the spread of oscillation frequenciesωj that causes the environment response to decohere after a (typically short) correlation timetc (Figure 1(b)). Hence, the Markovian assumption that the correlation function decays to 0 instantaneously, Φ(t)≈δ(t), is widely used: it is in particular the basis for the venerated Lindblad’s master equation describing decoherence [148]. It leads to exponential decay of αe at the Golden Rule (GR) rate [76, 78] as
(14)RGR=2π∑j‍|μej|2δ(ωa-ωj).We, however, are interested in the extremely non-Markovian time scales, much shorter thantc, 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 ofHS(t) and HI(t),
(15)HS(t)=ħ(ωa+δa(t))|e〉〈e|,HI(t)=∑j‍ϵ~(t)μej|e,0〉〈g,1j|+H.c.,ϵ(t)=ϵ~(t)ei∫0t‍dτδa(τ),
that result in the time-dependent modulation function ϵ(t), which has two components, namely, an amplitude modulation ϵ~(t) and phase modulation δa(t). The modulation function ϵ(t) is related to ϵ(t) in 3 via
(16)ϵ(t)=ϵ~(t)exp[-i∫0t‍δaf(t1)dt1],
with
(17)δaf(t)=δa(t)-δf(t).
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 ϵ(t) [149], (ii) coherent perturbations that describe phase modulations of the system-bath interactions [99, 124].For anyϵ(t), the exact equation (12) is then rewritten as
(18)∂tαe=-∫0t‍dt′Φ(t-t′)ϵ(t)ϵ*(t′)eiωa(t-t′)αe(t′).We now resort to thecrucial approximation that αe(t)varies slower than either ϵ(t) or Φ(t). This approximation is justifiable in the weak-coupling regime (to second order in HI), as discussed below. Under this approximation, (18) is transformed into a differential equation describing relaxation at a time-dependent rate as
(19)∂tαe=-q(t)αe,q(t)=∫0t‍dt′Φ(t-t′)ϵ(t)ϵ*(t′)eiωa(t-t′)≡r(t)2+iΔa(t),
where r(t) is the instantaneous time-dependent relaxation rate and Δa(t) is the Lamb shift due to the coupling to the bath. One can separate the spectral representation of Φ(t) into the real and imaginary parts
(20)Φ(ω)=∫0∞‍dtΦ(t)eiωt=πG(ω)-iχ(ω),
which satisfy the Kramers-Kronig relations
(21)χ(ω)=𝒫∫‍G(ω′)dω′ω′-ω,
with 𝒫 denoting the principal value. Henceforth, we shall concentrate on the relaxation rate, as it determines the excited state population,
(22)P(t)=|ϵ(t)|2=e-R(t)t,
where
(23)R(t)=1t∫0t‍dt′r(t)
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(24)ϵt(ω)=12π∫0t‍dt′ϵ(t′)eiωt′.The average time-dependent relaxation rate can be rewritten, by using the Fourier transforms ofΦ(t) and ϵ(t), in the following form:
(25)R(t)=2π∫-∞∞‍dωG(ω+ωa)Ft(ω),
where
(26)G(ω)=π-1Re∫0∞‍dteiωtΦ(t)
is the spectral-response function of the bath, and
(27)Ft(ω)=|ϵt(ω)|2t
is the finite-time spectral intensity of the (random or coherent) intervention/modulation function, where the 1/t factor comes about from the definition of the decoherence rate averaged over the (0,t) interval.The relaxation rateR(t) 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 G(ω), given to us by nature, and Ft(ω), 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 G(ω) and Ft(ω) 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 Ft(ω)) the broad spectral features of G(ω), 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

Ifϵ(t) corresponds to sufficiently slow rates of interruption/modulation νt, the spectrum of Ft(ω) is much narrower than the interval of change of G(ω) around ωa, the resonance frequency of the system. Then Ft(ω) can be replaced by δ(ω), so that the spectral width of G plays no role in determining R, and we may as well replace G(ω+ωa) 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
(28)R≈RGR=2π∫-∞∞‍dωδ(ω)G(ω+ωa)=2πG(ωa).
Namely, slow interventions do not affect the onset and rate of exponential decay.Frequency-domain representation of the dynamically controlled decoherence rate in various limits (Section7). (a) Golden Rule limit. (b) Quantum Zeno effect (QZE) limit. (c) Anti-Zeno effect (AZE) limit. Here, Ft(ω) and G(ω) are the modulation and bath spectra, respectively, ξ and ΓR are the interval of change and width of G(ω), respectively, and νt is the interruption rate.
(a)
(b)
(c)


### 3.2. The Limit of Frequent Modulation

Frequent interruptions, intermittent with free evolution, are represented by a repetition of the free-evolution modulation spectrum(29)Ft=nτ(ω)=n|ϵτ(ω)|2t=n2πt|∫0τ‍dt′eiωt′|2,=n2πtτ2sinc2(ωτ2)=τ2πsinc2(ωτ2),
where τ=1/νt being the time-interval between consecutive interruptions. If ϵ(t) describes extremely frequent interruptions or measurements νt≫1/tc, Ft(ω) is much broader than G(ω+ωa). We may then pull Ft(ω) out of the integral, whereupon (25) yields
(30)R(t)≈τ∫-∞∞‍G(ω+ωa)dω.
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 ofFt(ω) is broader than the width of ξa (so that the Golden Rule is violated) but narrower than the width of G(ω) (so that the QZE does not hold), the overlap of Ft(ω) and G(ω)grows as the rate of interruptions, or modulations, increases. This brings about the increase of relaxation ratesR(t)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 ofFt(ω) and G(ω)do not overlap, that is, the resonant coupling is shifted from the maximum of G(ω). If, by contrast, the peaks of Ft(ω) and G(ω) do coincide, any rate of interruptions would result in QZE (Figure 2(b)). This can be understood by viewing Ft(ω) as an averaging kernel of G(ω) around ωa. If G(ωa) 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 functionϵ(t) 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,
(31)ϵ(t)=∑k‍ϵke-iωkt.
Here ωk (k=0,±1,…) are arbitrary discrete frequencies with the minimum spectral distance Ω. If ϵ(t) is periodic with the period Ω, then ωk=kΩ and ϵk become the Fourier components of ϵ(t). For a general quasiperiodic ϵ(t), one obtains
(32)|ϵt(ω)|2=ϵc2t∑k‍|λk|2S(ηk,t)+ϵc2∑k≠l‍λkλl*1+ei(ηk-ηl)t-eiηkt-e-iηlt2πηkηl.
Here
(33)ϵc2=∑k‍|ϵk|2
equals the average of |ϵ(t)|2 over a period of the order of 1/Ω, λk=ϵk/ϵc, and ηk=ω-ωk, whereas
(34)S(ηk,t)=2sin2(ηkt/2)πtηk2
is a bell-like function of ηk normalized to 1.For a sufficiently long time, the functionS(ηk,t) becomes narrower than the respective characteristic width ξ(ωa+ωk) of G(ω) around ωa+ωk, and one can set
(35)S(ηk,t)≈δ(ηk).Thus, when(36)t≫tc≡maxk{1ξ(ωa+ωk)},
where tc is the effective correlation (memory) time of the reservoir, (25) is reduced to
(37)R=2π∑k‍|λk|2G(ωa+ωk).
For the validity of (37), it is also necessary that
(38)ϵc2Rtc≪1.
This condition is well satisfied in the regime of interest, that is, weak coupling to essentially any reservoir, unless (for some harmonic k) ωa+ωk is extremely close to a sharp feature in G(ω), for example, a band edge [145], 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 ωk, with the weights |λk|2.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 P(ωa) (e.g., inhomogeneous or vibrational spectrum). We can then choose λk and ωk in (37) so as to minimize the decay convoluted with P(ωa). In what follows, various limits of (37) will be analyzed.


### 3.5. Coherent Phase Modulation (PM)

#### 3.5.1. Monochromatic Perturbation

Let(39)ϵ(t)=ϵ0e-iδaft.
Then
(40)R=2πG(ωa+Δ),
where Δ=δaf=const 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 R relative to RGR. It provides the maximal variation of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(ω) incurred by the width of Ft(ω): the modified R can even vanish, if the shifted frequency ωa+Δ is beyond the cut-off frequency of the coupling, where G(ω)=0. Conversely, the increase of R 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 amountϕ at times τ,2τ,…. Such modulation can be achieved by a train of n identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed frequency shifts δaf(t). Now
(41)ϵ(t)=ei[t/τ]ϕ,
where [⋯] is the integer part. One then obtains that
(42)ϵc=1,Fnτ(ω)=2sin2(ωτ/2)sin2[n(ϕ+ωτ)/2]πnτω2sin2[(ϕ+ωτ)/2].
The decay, according to (22), has then the form (at t=nτ)
(43)P(nτ)=exp[-R(nτ)nτ],
where R(nτ) is defined by (25).For sufficiently long times(44)ωk=2kπτ-ϕτ,|λk|2=4sin2(ϕ/2)(2kπ-ϕ)2.Forsmall phase shifts, ϕ≪1, the k=0 peak dominates,
(45)|λ0|2≈1-ϕ212,
whereas
(46)|λk|2≈ϕ24π2k2(k≠0).
In this case, one can retain only the k=0 term in (37) (unless G(ω) is changing very fast). Then the modulation acts as a constant shift
(47)Δ=-ϕτ.With the increase of|ϕ|, the difference between the k=0 and k=1 peak heights diminishes, vanishing for ϕ=±π. Then
(48)|λ0|2=|λ1|2=4π2,
that is, Ft(ω) for ϕ=±π contains two identical peaks symmetrically shifted in opposite directions (the other peaks |λk|2 decrease with k as (2k-1)-2, totaling 0.19).The above features allow one to adjust the modulation parameters for a given scenario to obtain an optimal decrease or increase ofR. The phase-modulation (PM) scheme with a small ϕ is preferable near the continuum edge, since it yields a spectral shift in the required direction (positive or negative). The adverse effect of k≠0 peaks in Ft(ω) then scales as ϕ2 and hence can be significantly reduced by decreasing |ϕ|. On the other hand, if ωa is near a symmetric peak of G(ω), R is reduced more effectively for ϕ≃π, as in [80, 81], since the main peaks of Ft(ω) at ω0 and ω1 then shift stronger with τ-1 than the peak at ω0=-ϕ/τ for ϕ≪1.


### 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-Q 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 τ1 and τ0-τ1, respectively, that is,
(49)ϵ(t)={1fornτ0<t<nτ0+τ1,0fornτ0+τ1<t<(n+1)τ0,
(n=0,1,…). Now [157]
(50)Fnτ0(ω)=2sin2(ωτ1/2)sin2(nωτ0/2)πnτ1ω2sin2(ωτ0/2),
so that (see (43))
(51)P(nτ0)=exp[-R(nτ0)nτ1],
where R(nτ0) is given by (25) and (50).This case is also covered by (37) and (38), where the parameters are now found to be
(52)ϵc2=τ1τ0,ωk=2kπτ0,|λk|2=τ1τ0sinc2(kπτ1τ0),
with
(53)sinc(x)=sinxx,sinc(0)=1.It is instructive to consider the limit whereinτ1≪τ0 and τ0 is much greater than the correlation time of the continuum; that is, G(ω) does not change significantly over the spectral intervals (2πk/τ0,2π(k+1)/τ0). In this case, one can approximate the sum (37) by the integral (25) with
(54)Ft(ω)≈(τ12π)sinc2(ωτ12),
characterized by the spectral broadening ~1/τ1. Then (25) for R reduces to that obtained when ideal projective measurements are performed at intervals τ1 [97]. Thus the AM scheme can imitate measurement-induced (dephasing) effects on quantum dynamics, if the interruption intervals τ0 exceed the correlation time of the continuum.The decay probabilityP(t), calculated for parameters similar to [106], completely coincides with that obtained for ideal impulsive measurements at intervals τ1 [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 forcurrent-biased Josephson junctions [77], the present theory has been extended to describe effects of current modulations on the rate of macroscopic quantum tunneling in Josephson junctions in [100].Projective measurements at an effective rateν, whether impulsive or continuous, usually result in a broadened (to a width ν) modulation function F(ω), without a shift of its center of gravity [97, 98, 101, 158, 159],
(55)Δ=∫dωωF(ω)≈0.
This feature was shown in [97] to be responsible for either the standard quantum Zeno effect whereby R scales as
(56)R∝1ν,
or the anti-Zeno effect whereby R grows with ν. In contrast, a weak and broadband chaotic field, such that
(57)|χa|I¯≪νB,
where I¯ is the mean intensity, νB is the bandwidth, and χa is the effective polarizability (electric or magnetic, depending on the system), would give rise to a Lorentzian dephasing function F(ω) with a substantial shift
(58)Δ=χaI¯.
This shift would have a much stronger effect on R than the QZE or AZE, which are associated with the rate ν, since
(59)ν~χa2I¯2νB≪|Δ|.


## 3.1. The Limit of Slow Modulation Rate

Ifϵ(t) corresponds to sufficiently slow rates of interruption/modulation νt, the spectrum of Ft(ω) is much narrower than the interval of change of G(ω) around ωa, the resonance frequency of the system. Then Ft(ω) can be replaced by δ(ω), so that the spectral width of G plays no role in determining R, and we may as well replace G(ω+ωa) 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
(28)R≈RGR=2π∫-∞∞‍dωδ(ω)G(ω+ωa)=2πG(ωa).
Namely, slow interventions do not affect the onset and rate of exponential decay.Frequency-domain representation of the dynamically controlled decoherence rate in various limits (Section7). (a) Golden Rule limit. (b) Quantum Zeno effect (QZE) limit. (c) Anti-Zeno effect (AZE) limit. Here, Ft(ω) and G(ω) are the modulation and bath spectra, respectively, ξ and ΓR are the interval of change and width of G(ω), respectively, and νt is the interruption rate.
(a)
(b)
(c)


## 3.2. The Limit of Frequent Modulation

Frequent interruptions, intermittent with free evolution, are represented by a repetition of the free-evolution modulation spectrum(29)Ft=nτ(ω)=n|ϵτ(ω)|2t=n2πt|∫0τ‍dt′eiωt′|2,=n2πtτ2sinc2(ωτ2)=τ2πsinc2(ωτ2),
where τ=1/νt being the time-interval between consecutive interruptions. If ϵ(t) describes extremely frequent interruptions or measurements νt≫1/tc, Ft(ω) is much broader than G(ω+ωa). We may then pull Ft(ω) out of the integral, whereupon (25) yields
(30)R(t)≈τ∫-∞∞‍G(ω+ωa)dω.
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 ofFt(ω) is broader than the width of ξa (so that the Golden Rule is violated) but narrower than the width of G(ω) (so that the QZE does not hold), the overlap of Ft(ω) and G(ω)grows as the rate of interruptions, or modulations, increases. This brings about the increase of relaxation ratesR(t)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 ofFt(ω) and G(ω)do not overlap, that is, the resonant coupling is shifted from the maximum of G(ω). If, by contrast, the peaks of Ft(ω) and G(ω) do coincide, any rate of interruptions would result in QZE (Figure 2(b)). This can be understood by viewing Ft(ω) as an averaging kernel of G(ω) around ωa. If G(ωa) 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 functionϵ(t) 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,
(31)ϵ(t)=∑k‍ϵke-iωkt.
Here ωk (k=0,±1,…) are arbitrary discrete frequencies with the minimum spectral distance Ω. If ϵ(t) is periodic with the period Ω, then ωk=kΩ and ϵk become the Fourier components of ϵ(t). For a general quasiperiodic ϵ(t), one obtains
(32)|ϵt(ω)|2=ϵc2t∑k‍|λk|2S(ηk,t)+ϵc2∑k≠l‍λkλl*1+ei(ηk-ηl)t-eiηkt-e-iηlt2πηkηl.
Here
(33)ϵc2=∑k‍|ϵk|2
equals the average of |ϵ(t)|2 over a period of the order of 1/Ω, λk=ϵk/ϵc, and ηk=ω-ωk, whereas
(34)S(ηk,t)=2sin2(ηkt/2)πtηk2
is a bell-like function of ηk normalized to 1.For a sufficiently long time, the functionS(ηk,t) becomes narrower than the respective characteristic width ξ(ωa+ωk) of G(ω) around ωa+ωk, and one can set
(35)S(ηk,t)≈δ(ηk).Thus, when(36)t≫tc≡maxk{1ξ(ωa+ωk)},
where tc is the effective correlation (memory) time of the reservoir, (25) is reduced to
(37)R=2π∑k‍|λk|2G(ωa+ωk).
For the validity of (37), it is also necessary that
(38)ϵc2Rtc≪1.
This condition is well satisfied in the regime of interest, that is, weak coupling to essentially any reservoir, unless (for some harmonic k) ωa+ωk is extremely close to a sharp feature in G(ω), for example, a band edge [145], 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 ωk, with the weights |λk|2.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 P(ωa) (e.g., inhomogeneous or vibrational spectrum). We can then choose λk and ωk in (37) so as to minimize the decay convoluted with P(ωa). In what follows, various limits of (37) will be analyzed.


## 3.5. Coherent Phase Modulation (PM)

### 3.5.1. Monochromatic Perturbation

Let(39)ϵ(t)=ϵ0e-iδaft.
Then
(40)R=2πG(ωa+Δ),
where Δ=δaf=const 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 R relative to RGR. It provides the maximal variation of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(ω) incurred by the width of Ft(ω): the modified R can even vanish, if the shifted frequency ωa+Δ is beyond the cut-off frequency of the coupling, where G(ω)=0. Conversely, the increase of R 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 amountϕ at times τ,2τ,…. Such modulation can be achieved by a train of n identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed frequency shifts δaf(t). Now
(41)ϵ(t)=ei[t/τ]ϕ,
where [⋯] is the integer part. One then obtains that
(42)ϵc=1,Fnτ(ω)=2sin2(ωτ/2)sin2[n(ϕ+ωτ)/2]πnτω2sin2[(ϕ+ωτ)/2].
The decay, according to (22), has then the form (at t=nτ)
(43)P(nτ)=exp[-R(nτ)nτ],
where R(nτ) is defined by (25).For sufficiently long times(44)ωk=2kπτ-ϕτ,|λk|2=4sin2(ϕ/2)(2kπ-ϕ)2.Forsmall phase shifts, ϕ≪1, the k=0 peak dominates,
(45)|λ0|2≈1-ϕ212,
whereas
(46)|λk|2≈ϕ24π2k2(k≠0).
In this case, one can retain only the k=0 term in (37) (unless G(ω) is changing very fast). Then the modulation acts as a constant shift
(47)Δ=-ϕτ.With the increase of|ϕ|, the difference between the k=0 and k=1 peak heights diminishes, vanishing for ϕ=±π. Then
(48)|λ0|2=|λ1|2=4π2,
that is, Ft(ω) for ϕ=±π contains two identical peaks symmetrically shifted in opposite directions (the other peaks |λk|2 decrease with k as (2k-1)-2, totaling 0.19).The above features allow one to adjust the modulation parameters for a given scenario to obtain an optimal decrease or increase ofR. The phase-modulation (PM) scheme with a small ϕ is preferable near the continuum edge, since it yields a spectral shift in the required direction (positive or negative). The adverse effect of k≠0 peaks in Ft(ω) then scales as ϕ2 and hence can be significantly reduced by decreasing |ϕ|. On the other hand, if ωa is near a symmetric peak of G(ω), R is reduced more effectively for ϕ≃π, as in [80, 81], since the main peaks of Ft(ω) at ω0 and ω1 then shift stronger with τ-1 than the peak at ω0=-ϕ/τ for ϕ≪1.


## 3.5.1. Monochromatic Perturbation

Let(39)ϵ(t)=ϵ0e-iδaft.
Then
(40)R=2πG(ωa+Δ),
where Δ=δaf=const 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 R relative to RGR. It provides the maximal variation of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(ω) incurred by the width of Ft(ω): the modified R can even vanish, if the shifted frequency ωa+Δ is beyond the cut-off frequency of the coupling, where G(ω)=0. Conversely, the increase of R 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 amountϕ at times τ,2τ,…. Such modulation can be achieved by a train of n identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed frequency shifts δaf(t). Now
(41)ϵ(t)=ei[t/τ]ϕ,
where [⋯] is the integer part. One then obtains that
(42)ϵc=1,Fnτ(ω)=2sin2(ωτ/2)sin2[n(ϕ+ωτ)/2]πnτω2sin2[(ϕ+ωτ)/2].
The decay, according to (22), has then the form (at t=nτ)
(43)P(nτ)=exp[-R(nτ)nτ],
where R(nτ) is defined by (25).For sufficiently long times(44)ωk=2kπτ-ϕτ,|λk|2=4sin2(ϕ/2)(2kπ-ϕ)2.Forsmall phase shifts, ϕ≪1, the k=0 peak dominates,
(45)|λ0|2≈1-ϕ212,
whereas
(46)|λk|2≈ϕ24π2k2(k≠0).
In this case, one can retain only the k=0 term in (37) (unless G(ω) is changing very fast). Then the modulation acts as a constant shift
(47)Δ=-ϕτ.With the increase of|ϕ|, the difference between the k=0 and k=1 peak heights diminishes, vanishing for ϕ=±π. Then
(48)|λ0|2=|λ1|2=4π2,
that is, Ft(ω) for ϕ=±π contains two identical peaks symmetrically shifted in opposite directions (the other peaks |λk|2 decrease with k as (2k-1)-2, totaling 0.19).The above features allow one to adjust the modulation parameters for a given scenario to obtain an optimal decrease or increase ofR. The phase-modulation (PM) scheme with a small ϕ is preferable near the continuum edge, since it yields a spectral shift in the required direction (positive or negative). The adverse effect of k≠0 peaks in Ft(ω) then scales as ϕ2 and hence can be significantly reduced by decreasing |ϕ|. On the other hand, if ωa is near a symmetric peak of G(ω), R is reduced more effectively for ϕ≃π, as in [80, 81], since the main peaks of Ft(ω) at ω0 and ω1 then shift stronger with τ-1 than the peak at ω0=-ϕ/τ for ϕ≪1.


## 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-Q 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 τ1 and τ0-τ1, respectively, that is,
(49)ϵ(t)={1fornτ0<t<nτ0+τ1,0fornτ0+τ1<t<(n+1)τ0,
(n=0,1,…). Now [157]
(50)Fnτ0(ω)=2sin2(ωτ1/2)sin2(nωτ0/2)πnτ1ω2sin2(ωτ0/2),
so that (see (43))
(51)P(nτ0)=exp[-R(nτ0)nτ1],
where R(nτ0) is given by (25) and (50).This case is also covered by (37) and (38), where the parameters are now found to be
(52)ϵc2=τ1τ0,ωk=2kπτ0,|λk|2=τ1τ0sinc2(kπτ1τ0),
with
(53)sinc(x)=sinxx,sinc(0)=1.It is instructive to consider the limit whereinτ1≪τ0 and τ0 is much greater than the correlation time of the continuum; that is, G(ω) does not change significantly over the spectral intervals (2πk/τ0,2π(k+1)/τ0). In this case, one can approximate the sum (37) by the integral (25) with
(54)Ft(ω)≈(τ12π)sinc2(ωτ12),
characterized by the spectral broadening ~1/τ1. Then (25) for R reduces to that obtained when ideal projective measurements are performed at intervals τ1 [97]. Thus the AM scheme can imitate measurement-induced (dephasing) effects on quantum dynamics, if the interruption intervals τ0 exceed the correlation time of the continuum.The decay probabilityP(t), calculated for parameters similar to [106], completely coincides with that obtained for ideal impulsive measurements at intervals τ1 [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 forcurrent-biased Josephson junctions [77], the present theory has been extended to describe effects of current modulations on the rate of macroscopic quantum tunneling in Josephson junctions in [100].Projective measurements at an effective rateν, whether impulsive or continuous, usually result in a broadened (to a width ν) modulation function F(ω), without a shift of its center of gravity [97, 98, 101, 158, 159],
(55)Δ=∫dωωF(ω)≈0.
This feature was shown in [97] to be responsible for either the standard quantum Zeno effect whereby R scales as
(56)R∝1ν,
or the anti-Zeno effect whereby R grows with ν. In contrast, a weak and broadband chaotic field, such that
(57)|χa|I¯≪νB,
where I¯ is the mean intensity, νB is the bandwidth, and χa is the effective polarizability (electric or magnetic, depending on the system), would give rise to a Lorentzian dephasing function F(ω) with a substantial shift
(58)Δ=χaI¯.
This shift would have a much stronger effect on R than the QZE or AZE, which are associated with the rate ν, since
(59)ν~χa2I¯2νB≪|Δ|.


## 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,ωa→ωa+δr(t), where δr(t) is a stochastic variable with zero mean, 〈δ(t)〉=0 and 〈δ(t)δ(t′)〉=ΦP(t-t′) is the second moment. For multipartite systems, where each qubit can undergo different proper dephasing, δj(t), one has an additional second moment for the cross dephasing, 〈δj(t)δj′(t′)〉=Φjj′P(t-t′). A general treatment of multipartite systems undergoing this type of proper dephasing is given in [107]. 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(60)|B1,2〉=12eiωat(|e〉1|g〉2±|g〉1|e〉2),|B3,4〉=12(ei2ωat|e〉1|e〉2±|g〉1|g〉2).
For an initial Bell-state ρ¯l(0)=|Bl〉〈Bl|, where l=1,…,4, one can then obtain the fidelity, Fl(t)=〈Bl|ρ¯l(t)|Bl〉, as
(61)Fl(t)=cos(ϕ±(t))Re[eiϕ±(t)(1-12∑jj′‍Jjj′,lP(t))],
where
(62)ϕj(t)=2∫0t‍dτV0,j(τ),(63)Jjj′,lP(t)=2π∫-∞∞‍dωGjj′P(ω)Ft,jj′,l(ω),(64)Gjj′P(ω)=∫-∞∞‍dtΦjj′P(t)eiωt,(65)Ft,jj,l(ω)=|ϵt,j(ω)|2,(66)Ft,jj′,3(ω)=-Ft,jj′,1(ω)=ϵt,j*(ω)ϵt,j′*(ω),(67)Ft,jj′,4(ω)=-Ft,jj′,2(ω)=ϵt,j(ω)ϵt,j′*(ω),
where V0,j(t) is the amplitude of the resonant field applied on qubit j, ϕ±(t)=(ϕ1(t)±ϕ2(t))/2, and the ϕ+ corresponds to k=1,3 and ϕ- to k=2,4. 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,Jjj′,l(t)=J(t). This results in complete preservation of the singlet only, that is, F2(t)=1, for all t, 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 Ft,jj′,l(ω)=0 for all  j≠j′. This requirement can be important for quantum communication schemes.Figure 3
Cross decoherence as a function of local modulation. Here two qubits are modulated by continuous resonant fields, with amplitudesΩ1,2. The cross decoherence decays as the two qubits’ modulations become increasingly different. The bath parameters are ΦT(t)=e-t/tc, where tc=0.5 is the correlation time, and Ω1=3.


## 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,ωa→ωa+δr(t), where δr(t) is a stochastic variable with zero mean, 〈δ(t)〉=0 and 〈δ(t)δ(t′)〉=ΦP(t-t′) is the second moment. For multipartite systems, where each qubit can undergo different proper dephasing, δj(t), one has an additional second moment for the cross dephasing, 〈δj(t)δj′(t′)〉=Φjj′P(t-t′). A general treatment of multipartite systems undergoing this type of proper dephasing is given in [107]. 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(60)|B1,2〉=12eiωat(|e〉1|g〉2±|g〉1|e〉2),|B3,4〉=12(ei2ωat|e〉1|e〉2±|g〉1|g〉2).
For an initial Bell-state ρ¯l(0)=|Bl〉〈Bl|, where l=1,…,4, one can then obtain the fidelity, Fl(t)=〈Bl|ρ¯l(t)|Bl〉, as
(61)Fl(t)=cos(ϕ±(t))Re[eiϕ±(t)(1-12∑jj′‍Jjj′,lP(t))],
where
(62)ϕj(t)=2∫0t‍dτV0,j(τ),(63)Jjj′,lP(t)=2π∫-∞∞‍dωGjj′P(ω)Ft,jj′,l(ω),(64)Gjj′P(ω)=∫-∞∞‍dtΦjj′P(t)eiωt,(65)Ft,jj,l(ω)=|ϵt,j(ω)|2,(66)Ft,jj′,3(ω)=-Ft,jj′,1(ω)=ϵt,j*(ω)ϵt,j′*(ω),(67)Ft,jj′,4(ω)=-Ft,jj′,2(ω)=ϵt,j(ω)ϵt,j′*(ω),
where V0,j(t) is the amplitude of the resonant field applied on qubit j, ϕ±(t)=(ϕ1(t)±ϕ2(t))/2, and the ϕ+ corresponds to k=1,3 and ϕ- to k=2,4. 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,Jjj′,l(t)=J(t). This results in complete preservation of the singlet only, that is, F2(t)=1, for all t, 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 Ft,jj′,l(ω)=0 for all  j≠j′. This requirement can be important for quantum communication schemes.Figure 3
Cross decoherence as a function of local modulation. Here two qubits are modulated by continuous resonant fields, with amplitudesΩ1,2. The cross decoherence decays as the two qubits’ modulations become increasingly different. The bath parameters are ΦT(t)=e-t/tc, where tc=0.5 is the correlation time, and Ω1=3.


## 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 energiesħωn, 1≤n≤N, is coupled to a zero-temperature bath of harmonic oscillators with frequencies ωj. Using the factorized coupling defined in Section 2.1, the corresponding Hamiltonian is found to be as in 1, where
(68)HS(t)=ħ∑n‍ωn|n〉〈n|+∑nn′‍Onn′(t)|n〉〈n′|,(69)HB(t)=ħ∑j‍ωj|j〉〈j|,(70)HI(t)=∑n,j‍ϵn(t)μjn|n〉〈j|+h.c.,
where now each level has a different modulation ϵn(t) and a different coupling to the bathμjn and Onn′(t) 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(71)Onn′(t)={gnn′,t′≤t≤t′+τ,0,elsewhere.The full wave function is given by(72)|Ψ(t)〉=∑n‍αn(t)e-iωnt|n〉+∑j‍βj(t)e-iωjt|j〉.
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 N-level system which can exchange its population with the reservoir. In addition, one can consider an (N+1)-level system, where transitions are possible between any level 〈n| and a lower level 〈g|, 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
(73)|n〉⟶|n〉|vac〉,|j〉⟶|g〉|j〉
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 amplitudesα(t)={αn(t)} as
(74)α(t)=Θ(t)α~(t),
such that, by defining Ωnn′(t)=ωnδnn′+Onn′(t), one gets
(75)Ω~nn′(t)=(Θ-1(t)ΩΘ(t))nn′=ω~n(t)δnn′,
where ωn(t) are the eigenvalues of the new rotated system. Thus the transformed wave function becomes
(76)|Ψ(t)〉=∑n~‍α~n(t)e-i∫0t‍ω~n(τ)dτ|n~〉+∑j‍βj(t)e-iωjt|j〉.
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 αn(t) as
(77)α~˙(t)=-[f(t)+q(t)]α~.
Here, the f(t) and q(t) matrices are given by
(78)fnn′′(t)=itω~˙n(t)δnn′′++∑n′‍[Θ-1(t)]nn′Θ˙n′n′′(t)ei∫0t‍[ω~n(τ)-ω~n′′(τ)]dτ,qnn′′(t)=∑n′‍∫0t‍dt′Knn′(t,t′)Φnn′(t-t′)××ei∫0t‍ω~n(τ)dτ-i∫0t′‍ω~n′′(τ)dτΘn′n′′(t′),
with K(t,t′) and Φ(t) being the modulation and reservoir-response matrices, respectively, given by
(79)Knn′(t,t′)=ϵn*(t)ϵn′(t′),Φnn′(t)=∫‍dωGnn′(ω)e-iωt,
where
(80)Gnn′(ω)=ħ-2∑j‍μnj*μn′jδ(ω-ωj).
During the storage phase, one has Θnn′(t)=δnn′, ω~n(t)=ωn and g~nn′′(t)=δnn′′, and during the gate-operation phase, Θnn′(t)=Θnn′, ω~n(t)=ω~n, and g~nn′′(t)=gnn′′.The solution to (77) is of the form
(81)α~(t)=T+e-∫0t‍[f(t′)+q(t′)]dt′α~(0).To simplify the analysis, one can define the fluence and the modulation spectral matrices as(82)Qnn′(t)=∫0t‍dτϵn*(τ)ϵn′(τ),Fnn′,t(ω)=ϵt,n*(ω-ωn)ϵt,n′(ω-ωn′)Qnn′(t).
The relevant imaginary parts of the spectral response of the reservoir can be expressed, analogously to (20) and (21), by the Kramers-Kronig relations
(83)Φ~nn′(ω)=πGnn′(ω)-iχnn′(ω),χnn′(ω)=𝒫∫‍Gnn′(ω′)dω′ω′-ω.Defining(84)J(t)=∫0t‍dt′q(t′),
we shall now represent Jnn′(t) in different regimes (phases).(i) As a reference, it is important to consider the decoherence effects with no modulations at all, that is,Fnn′,t(ω)=δ(ω-ωn)δ(ωn-ωn′). In this case, one obtains a diagonal decoherence matrix
(85)Jnn′(t)=tδnn′Φ~nn′(ωn).
This means that interference of decaying levels n and n′ cancels out in the long time limit, and the decoherence is without cross relaxation.(ii) During the storage phase, (84) results in
(86)Jnn′(t)=Qnn′(t)∫-∞∞‍dωFnn′,t(ω)Φ~nn′(ω).
One can easily see that for the off-diagonal terms, a simple separation into decay rates and energy shifts is inapplicable in this formulation.(iii) During gate operations, (84) assumes the form
(87)Jnn′(t)=∑n′′‍Qnn′′(t)Θn′′n′∫-∞∞‍dωFnn′′,t(ω)Φ~nn′′(ω).
In a more compact and enlightening form, one can rewrite this equation as Jgate=Jstorage×Θ, where Jstorage is given in (86).


## 5.1. General Formalism

Here we discuss in detail a model for dynamical decay modifications in a multilevel system. The system with energiesħωn, 1≤n≤N, is coupled to a zero-temperature bath of harmonic oscillators with frequencies ωj. Using the factorized coupling defined in Section 2.1, the corresponding Hamiltonian is found to be as in 1, where
(68)HS(t)=ħ∑n‍ωn|n〉〈n|+∑nn′‍Onn′(t)|n〉〈n′|,(69)HB(t)=ħ∑j‍ωj|j〉〈j|,(70)HI(t)=∑n,j‍ϵn(t)μjn|n〉〈j|+h.c.,
where now each level has a different modulation ϵn(t) and a different coupling to the bathμjn and Onn′(t) 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(71)Onn′(t)={gnn′,t′≤t≤t′+τ,0,elsewhere.The full wave function is given by(72)|Ψ(t)〉=∑n‍αn(t)e-iωnt|n〉+∑j‍βj(t)e-iωjt|j〉.
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 N-level system which can exchange its population with the reservoir. In addition, one can consider an (N+1)-level system, where transitions are possible between any level 〈n| and a lower level 〈g|, 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
(73)|n〉⟶|n〉|vac〉,|j〉⟶|g〉|j〉
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 amplitudesα(t)={αn(t)} as
(74)α(t)=Θ(t)α~(t),
such that, by defining Ωnn′(t)=ωnδnn′+Onn′(t), one gets
(75)Ω~nn′(t)=(Θ-1(t)ΩΘ(t))nn′=ω~n(t)δnn′,
where ωn(t) are the eigenvalues of the new rotated system. Thus the transformed wave function becomes
(76)|Ψ(t)〉=∑n~‍α~n(t)e-i∫0t‍ω~n(τ)dτ|n~〉+∑j‍βj(t)e-iωjt|j〉.
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 αn(t) as
(77)α~˙(t)=-[f(t)+q(t)]α~.
Here, the f(t) and q(t) matrices are given by
(78)fnn′′(t)=itω~˙n(t)δnn′′++∑n′‍[Θ-1(t)]nn′Θ˙n′n′′(t)ei∫0t‍[ω~n(τ)-ω~n′′(τ)]dτ,qnn′′(t)=∑n′‍∫0t‍dt′Knn′(t,t′)Φnn′(t-t′)××ei∫0t‍ω~n(τ)dτ-i∫0t′‍ω~n′′(τ)dτΘn′n′′(t′),
with K(t,t′) and Φ(t) being the modulation and reservoir-response matrices, respectively, given by
(79)Knn′(t,t′)=ϵn*(t)ϵn′(t′),Φnn′(t)=∫‍dωGnn′(ω)e-iωt,
where
(80)Gnn′(ω)=ħ-2∑j‍μnj*μn′jδ(ω-ωj).
During the storage phase, one has Θnn′(t)=δnn′, ω~n(t)=ωn and g~nn′′(t)=δnn′′, and during the gate-operation phase, Θnn′(t)=Θnn′, ω~n(t)=ω~n, and g~nn′′(t)=gnn′′.The solution to (77) is of the form
(81)α~(t)=T+e-∫0t‍[f(t′)+q(t′)]dt′α~(0).To simplify the analysis, one can define the fluence and the modulation spectral matrices as(82)Qnn′(t)=∫0t‍dτϵn*(τ)ϵn′(τ),Fnn′,t(ω)=ϵt,n*(ω-ωn)ϵt,n′(ω-ωn′)Qnn′(t).
The relevant imaginary parts of the spectral response of the reservoir can be expressed, analogously to (20) and (21), by the Kramers-Kronig relations
(83)Φ~nn′(ω)=πGnn′(ω)-iχnn′(ω),χnn′(ω)=𝒫∫‍Gnn′(ω′)dω′ω′-ω.Defining(84)J(t)=∫0t‍dt′q(t′),
we shall now represent Jnn′(t) in different regimes (phases).(i) As a reference, it is important to consider the decoherence effects with no modulations at all, that is,Fnn′,t(ω)=δ(ω-ωn)δ(ωn-ωn′). In this case, one obtains a diagonal decoherence matrix
(85)Jnn′(t)=tδnn′Φ~nn′(ωn).
This means that interference of decaying levels n and n′ cancels out in the long time limit, and the decoherence is without cross relaxation.(ii) During the storage phase, (84) results in
(86)Jnn′(t)=Qnn′(t)∫-∞∞‍dωFnn′,t(ω)Φ~nn′(ω).
One can easily see that for the off-diagonal terms, a simple separation into decay rates and energy shifts is inapplicable in this formulation.(iii) During gate operations, (84) assumes the form
(87)Jnn′(t)=∑n′′‍Qnn′′(t)Θn′′n′∫-∞∞‍dωFnn′′,t(ω)Φ~nn′′(ω).
In a more compact and enlightening form, one can rewrite this equation as Jgate=Jstorage×Θ, where Jstorage is given in (86).


## 6. The Strong-Coupling Regime: Decay Control Near Continuum Edge by Nonadiabatic Interference

The analysis expounded thus far has been based on aperturbative 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|e〉 and |g〉 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 [145]
(88)H=ħωat|e〉〈e|+ħ∫0+∞‍ωaω†aωρ(ω)dω+ħ(κd⋆ad†|g〉〈e|+h.c.)+ħ∫0+∞‍[κ⋆(ω)aω†|g〉〈e|+h.c.]ρ(ω)dω.
Here, ħωat is the energy of the atomic transition frequency, aω† and aω are, respectively, the creation and annihilation operators of the field mode at frequency ω, ρ(ω) is the mode density of the PBS, and  κ(ω) and κd 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(89)|Ψ(0)〉=|e,{0ω}〉,
where |{0ω}〉 is the vacuum state of the field. Then the evolution of the wavefunction |Ψ(t)〉 has the general form
(90)|Ψ(t)〉=α(t)|e,{0ω}〉+βd(t)|g,1d〉+∫0+∞‍βω(t)|g,1ω〉ρ(ω)dω,
where we have denoted by |1ω〉 and |1d〉 the single-photon state of the relevant modes. The Schrödinger equation iΨ˙(t)=HΨ(t) then leads to the set of coupled differential equations
(91)α˙(t)=-iωatα(t)-iκdβd(t)-i∫0+∞‍κ(ω)βω(t)ρ(ω)dω,β˙d(t)=-iωdβd(t)-iκd⋆α(t),β˙ω(t)=-iωβω(t)-iκ⋆(ω)α(t).
This evolution reflects the interplay between the off-resonant Rabi oscillations of |e,{0ω}〉 and |g,1d〉, at the driving rate κd, and the partly inhibited oscillatory decay from |e,{0ω}〉 to |g,{1ω}〉 via coupling to the continuum ρ(ω). This decay depends on the detuning of ωat from the continuum edge at ωU (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 [8]) which allows for the existence of an oscillatory, nondecaying, component of α(t), associated with a photon-atom bound state [7, 145].


### 6.2. Periodic Sudden Changes of the Detuning

Let us now introduce abrupt changes ofωat,  that is, of the detuning Δat=ωU-ωat from the upper cutoff, ωU, of the PBG (by fast AC-Stark modulations as discussed below), at intervals τ. In the sudden-change approximation for ωat, the amplitudes (αdyn(t),βddyn,{βωdyn(t)}) of the excited state, the discrete mode and the continuum still evolve according to (91), except that from t=0 to t=τ the atomic transition frequency is ωat=ωA, that is, the detuning Δat=ωU-ωA=ΔA, while for t>τ, we have ωat=ωB, that is, Δat=ΔB. This dynamics leads to the relation
(92)αdyn(t)=αA(t),βddyn(t)=βd,A(t),βωdyn(t)=βω,A(t),(t≤τ);αdyn(t)=αB(s)(t),βddyn(t)=βd,B(s)(t),βωdyn(t)=βω,B(s)(t),(t>τ).
Here, (αA(t),βd,A(t),{βω,A(t)}) and (αB(s)(t),βd,B(s)(t),{βω,B(s)(t)}) are solutions of (91) with a static (fixed) atomic transition frequency, ωA or ωB. However, the initial condition at the instant t=τ of the frequency change from ΔA to ΔB is no longer the excited state (89) but the superposition
(93)|Ψ(τ)〉=αA(τ)|e,{0ω}〉+βd,A(τ)|g,1d〉+∫0+∞‍βω,A(τ)|g,1ω〉ρ(ω)dω.
In other words, the dynamics is equivalent to two successive static evolutions, the second one starting from initial conditions (αA(τ),βd,A(τ),{βω,A(τ)}).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
(94)αdyn(t)=αA(τ)αB(t-τ)+βd,A(τ)βd,B(t-τ)+∫0+∞‍βω,A(τ)βω,B(t-τ)ρ(ω)dω,(t>τ),
where we have used the initial conditions (αA(τ),βd,A(τ),{βω,A(τ)}) and the solution (αB(t),βd,B(t),{βω,B(t)}) of (91) for the initial condition (89).There is an advantageous feature to the sudden change: since the time dependence ofαdyn(t) in (92) arises from the static amplitudes αB, βd,B, and βω,B at the shifted time t-τ, 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 |e〉 and |g〉. The scenario leading to the largest amplitude consists in periodic shifts of the energy detuning from ΔA to ΔB. When the initial detuning ΔA is large and we first reduce it to ΔB before it increases to ΔA, the dynamic population and the|e〉-|g〉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 (10-18 s) [97], 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 αdyn(t).Let us now consider the initial superposition(95)|Ψ(0)〉=α(0)|e,{0ω}〉+βd(0)|g,1ωd〉
and a nonnegligible coupling constant κd. In this case, the periodic dynamic population of the excited state also strongly exceeds the static one. Most importantly, the instantaneous dynamic fidelity|〈Ψ(0)|Ψ(t)〉|2 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 byπ 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 π/10 or π/20, 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 π/10 or π/20 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 |〈Ψ(0)|Ψ(t)〉|2 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 forpossibly strong coupling of |e〉 with the continuum edge, with those of the universal formula of Section 2 (25), which expresses the decay rate of α(t) 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 Δat/γc>5, 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 Δat~1MHz from the PBG edge with coupling constant γc~10MHz and by ~3 MHz from the resonance of the other species. This Δat is abruptly modulated by fsec 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 γc/10~1MHz, much lower than the pulse rate stipulated under similar conditions by previously proposed strategies [81, 99, 118].


## 6.1. Hamiltonian and Equations of Motion

We consider a two-level atom with excited and ground states|e〉 and |g〉 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 [145]
(88)H=ħωat|e〉〈e|+ħ∫0+∞‍ωaω†aωρ(ω)dω+ħ(κd⋆ad†|g〉〈e|+h.c.)+ħ∫0+∞‍[κ⋆(ω)aω†|g〉〈e|+h.c.]ρ(ω)dω.
Here, ħωat is the energy of the atomic transition frequency, aω† and aω are, respectively, the creation and annihilation operators of the field mode at frequency ω, ρ(ω) is the mode density of the PBS, and  κ(ω) and κd 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(89)|Ψ(0)〉=|e,{0ω}〉,
where |{0ω}〉 is the vacuum state of the field. Then the evolution of the wavefunction |Ψ(t)〉 has the general form
(90)|Ψ(t)〉=α(t)|e,{0ω}〉+βd(t)|g,1d〉+∫0+∞‍βω(t)|g,1ω〉ρ(ω)dω,
where we have denoted by |1ω〉 and |1d〉 the single-photon state of the relevant modes. The Schrödinger equation iΨ˙(t)=HΨ(t) then leads to the set of coupled differential equations
(91)α˙(t)=-iωatα(t)-iκdβd(t)-i∫0+∞‍κ(ω)βω(t)ρ(ω)dω,β˙d(t)=-iωdβd(t)-iκd⋆α(t),β˙ω(t)=-iωβω(t)-iκ⋆(ω)α(t).
This evolution reflects the interplay between the off-resonant Rabi oscillations of |e,{0ω}〉 and |g,1d〉, at the driving rate κd, and the partly inhibited oscillatory decay from |e,{0ω}〉 to |g,{1ω}〉 via coupling to the continuum ρ(ω). This decay depends on the detuning of ωat from the continuum edge at ωU (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 [8]) which allows for the existence of an oscillatory, nondecaying, component of α(t), associated with a photon-atom bound state [7, 145].


## 6.2. Periodic Sudden Changes of the Detuning

Let us now introduce abrupt changes ofωat,  that is, of the detuning Δat=ωU-ωat from the upper cutoff, ωU, of the PBG (by fast AC-Stark modulations as discussed below), at intervals τ. In the sudden-change approximation for ωat, the amplitudes (αdyn(t),βddyn,{βωdyn(t)}) of the excited state, the discrete mode and the continuum still evolve according to (91), except that from t=0 to t=τ the atomic transition frequency is ωat=ωA, that is, the detuning Δat=ωU-ωA=ΔA, while for t>τ, we have ωat=ωB, that is, Δat=ΔB. This dynamics leads to the relation
(92)αdyn(t)=αA(t),βddyn(t)=βd,A(t),βωdyn(t)=βω,A(t),(t≤τ);αdyn(t)=αB(s)(t),βddyn(t)=βd,B(s)(t),βωdyn(t)=βω,B(s)(t),(t>τ).
Here, (αA(t),βd,A(t),{βω,A(t)}) and (αB(s)(t),βd,B(s)(t),{βω,B(s)(t)}) are solutions of (91) with a static (fixed) atomic transition frequency, ωA or ωB. However, the initial condition at the instant t=τ of the frequency change from ΔA to ΔB is no longer the excited state (89) but the superposition
(93)|Ψ(τ)〉=αA(τ)|e,{0ω}〉+βd,A(τ)|g,1d〉+∫0+∞‍βω,A(τ)|g,1ω〉ρ(ω)dω.
In other words, the dynamics is equivalent to two successive static evolutions, the second one starting from initial conditions (αA(τ),βd,A(τ),{βω,A(τ)}).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
(94)αdyn(t)=αA(τ)αB(t-τ)+βd,A(τ)βd,B(t-τ)+∫0+∞‍βω,A(τ)βω,B(t-τ)ρ(ω)dω,(t>τ),
where we have used the initial conditions (αA(τ),βd,A(τ),{βω,A(τ)}) and the solution (αB(t),βd,B(t),{βω,B(t)}) of (91) for the initial condition (89).There is an advantageous feature to the sudden change: since the time dependence ofαdyn(t) in (92) arises from the static amplitudes αB, βd,B, and βω,B at the shifted time t-τ, 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 |e〉 and |g〉. The scenario leading to the largest amplitude consists in periodic shifts of the energy detuning from ΔA to ΔB. When the initial detuning ΔA is large and we first reduce it to ΔB before it increases to ΔA, the dynamic population and the|e〉-|g〉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 (10-18 s) [97], 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 αdyn(t).Let us now consider the initial superposition(95)|Ψ(0)〉=α(0)|e,{0ω}〉+βd(0)|g,1ωd〉
and a nonnegligible coupling constant κd. In this case, the periodic dynamic population of the excited state also strongly exceeds the static one. Most importantly, the instantaneous dynamic fidelity|〈Ψ(0)|Ψ(t)〉|2 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 byπ 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 π/10 or π/20, 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 π/10 or π/20 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 |〈Ψ(0)|Ψ(t)〉|2 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 forpossibly strong coupling of |e〉 with the continuum edge, with those of the universal formula of Section 2 (25), which expresses the decay rate of α(t) 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 Δat/γc>5, 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 Δat~1MHz from the PBG edge with coupling constant γc~10MHz and by ~3 MHz from the resonance of the other species. This Δat is abruptly modulated by fsec 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 γc/10~1MHz, 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 isclose 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 HI prevails, corresponding to the neglect of O(HI3) 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 HI 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 [70]).


### 7.1. Explicit Equations for Factorisable Interaction Hamiltonians

We now wish to write the ME explicitly for time-dependent Hamiltonians of the following form [100]:
(96)H(t)=HS(t)+HB+HI(t),HI(t)=S(t)B,
where HS and HB are the system and bath Hamiltonians, respectively, and HI, the interaction Hamiltonian, is the product of operators S and B which act on the system and bath, respectively.Finally, defining the correlation function for the bath,(97)ΦT(t)=〈BB~(t)〉B,
we obtain the ME for ρ in the Born approximation as
(98)ρ˙(t)=-i[HS,ρ(t)]+∫0t‍dτ{ΦT(t-τ)[S~(t,τ)ρ(t),S(t)]+H.c.}.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(99)HB=∑λ‍ωλaλ†aλ,B=∑λ‍(κλaλ+κλ*aλ†).
Here aλ,aλ† are the annihilation and creation operators of mode λ, respectively, and κλ is the coupling amplitude to mode λ.


#### 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:(100)HS(t)=(ωa+δa(t))|e〉〈e|,(101)S(t)=ϵ~(t)σx,
where δa(t) is the dynamical AC-Stark shifts, ϵ~(t) is the time-dependent modulation of the interaction strength, and the Pauli matrix σx=|e〉〈g|+|g〉〈e|.


#### 7.1.2. Phase-Noise Regime

Next, we consider the PN regime of a two-level system coupled to a thermal bath viaσz operator. To combat it, we will use near-resonant fields with time-varying amplitude as our control. The Hamiltonians then assume the following forms:
(102)HS(t)=ωa|e〉〈e|+V(t)σx,(103)S(t)=ϵ~(t)σz,
where V(t)=Ω(t)e-iωat+c.c is the time-dependent resonant field, with real envelope Ω(t), ϵ~(t) is the time-dependent modulation of the interaction strength, and  σz=|e〉〈e|-|g〉〈g|.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
(104)|↑,↓〉=|e,g〉,
while for the PN case of (102) the basis is
(105)|↑,↓〉=12(e-iωat|e〉±|g〉).In this rotated and tilted frame,(106)US(t,τ)=US(t,0)US†(τ,0),US(t,0)=e-iϕ(t)|↑〉〈↑|+eiϕ(t)|↓〉〈↓|,
where
(107)ϕ(t)={ωat+∫0t‍dt′δa(t′)forAN,∫0t‍dt′Ω(t′)2forPN
is the phase-modulation due to the time-dependent control in the system Hamiltonian.Allowance forarbitrary time-dependent intervention in the system and interaction dynamics HS(t), HI(t), respectively, yields the following universal ME for a dynamically controlled decohering system [100, 124]:
(108)ρ~˙(t)=∫0t‍dt′{Φ(t-t′)[S~(t′)ρ~(t),S~(t)]+H.c.}.
Here S~(t)=ϵ(t)σ~x is the modulated interaction operator, where ~ denotes the rotated and tilted frame, and σ~x=|↑〉〈↓|+|↓〉〈↑|. The modulation function is given by
(109)ϵ(t)=ϵ~(t)eiϕ(t)
for both AN and PN. It is important to note that ρ~ is a function of   tonly (not oft′): this convolutionless form of the ME is fully non-Markovian to second order in HI, as proven exactly in [124].


### 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
(110)ρ˙↑↑=-ρ˙↓↓=-r↑(t)ρ↑↑+r↓(t)ρ↓↓,(111)r↑(↓)=2Re∫0t‍dt′Φ(t-t′)K±(t,t′),(112)K+(t,t′)=K-*(t,t′)=ϵ(t)ϵ*(t′).
The time-dependent relaxation rates are real, and the only difference between them is the complex conjugate of the combined modulation function, K±(t,t′). 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(113)R↑=∫-∞∞‍dωG(+ω)Ft(ω),R↓=∫-∞∞‍dωG(-ω)Ft(ω).For both AN (see (100)) and PN (see (102)),
(114)G(ω)=G0(ω)(nT(ω)+1)+G0(-ω)nT(-ω),(115)G0(ω)=ρ(ω)|μeg|2(ω)Θ(ω),
where G0(ω) is the zero-temperature bath spectrum, ρ(ω) and μeg(ω) are the frequency-dependent density of bath modes and the e↔g transition matrix element, respectively, nT(ω)=(eβω-1)-1 is the temperature-dependent bath mode population, and β=ħ/kBT is the inverse temperature. Also, Θ(ω) is the Heaviside function, that is, the zero-temperature bath spectrum is defined only for positive frequencies G0(ω<0)=0. 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 thesame form for both (see Section 7.4) as
(116)Ft(ω)=1t|∫0t‍dt′eiωt′ϵ(t′)|2,
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
(117)δa(t)=∫0t‍dt′|Ω(t′)|2Δ(t′),
where Ω(t) is the Rabi frequency of the control field and Δ(t′) is the detuning. The corresponding phase factor for PN is the integral of the Rabi frequency Ω(t), that is, the pulse area of the resonant control field, (107) (Figure 4).Schematic drawing of system and bath. (a) Amplitude noise (AN) (red) combatted by AC-Stark shift modulation (green). (b) Phase noise (PN) (red) combatted by resonant-field modulation (green).
(a)
(b)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 ratesRe(g)(t) rather than the level shifts. The average rate of the |e〉→|g〉 transition Re(t) and its |g〉→|e〉 counterpart Rg(t) are given by
(118)Re(g)(t)=2π∫-∞∞‍dωFt(ω)GT(±ω).
Here the upper (lower) sign corresponds to the subscript e(g), and
(119)GT(ω)=(2π)-1∫-∞∞‍ΦT(t)eiωtdt
can be shown [161] to be nonnegative, with GT(-ω)=e-βωGT(ω), and vanishes for ω<0 at T=0: G0(ω)=0(ω<0). For the oscillator bath, one finds that
(120)GT(ω)=[n(ω)+1]G0(ω)+n(-ω)G0(-ω),
where G0(ω)=∑λ‍|κλ|2δ(ω-ωλ) and n(ω)=(eβω-1)-1 is the average number of quanta in the oscillator (bath mode) with frequency ω.We apply (118) to the case of coherent modulation of quasiperiodic form, (see (31)). Without a limitation of the generality, we can assume that ∑k‍|ϵk|2=1. We then find, using (118), that the rates Re(g)(t) tend to the long-time limits
(121)Re(g)=2π∫-∞∞‍dωF(ω)GT(±ω),
where
(122)F(ω)=limt→∞Ft(ω)=∑k‍|ϵk|2δ(ω-ωa-ωk),
or
(123)Re(g)=2π∑k‍|ϵk|2GT(±(ωa+ωk)).
Equation (121) shows that Re(Rg)is given by the overlap of the modulation spectrumF(ω)with the bath-CF spectrumGT(ω)[GT(-ω)]. The limits (123) are approached when Ωt≫1 and t≫tc≡maxk{1/ξ(±(ωa+ωk)}. Here tc is the bath memory (correlation) time, defined as the inverse of ξ(ω), the spectral interval over which GT(ω) changes around the relevant frequencies.Had we used the standard dipolar RWA hamiltonian in the case of an oscillator bath, dropping the antiresonant terms inHI(t), we would have arrived at the transition rates
(124)Re(g)RWA=2π∫0∞‍dωF(ω)GT(±ω),
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 F(ω)≃0 at ω<0, being peaked near ωa. However, whenever the suppression of Re(g) requires modulation at a rate comparable to ωa, the RWA is inadequate. For instance, (120) and (124) imply that, at T=0, the rate RgRWA vanishes identically, irrespective of F(ω), in contrast to the true upward-transition rate Rg in (121), which may be comparable to Re 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 Re(g) 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|e〉to be shifted below|g〉.The validity of the (decohering) qubit model in the presence of modulation at a rate≳ωa is now elucidated: it requires that Re(g)jt≪1, Re(g)j being the effective transition rate from level e(g) to any other level j, and, in particular, Re(g)t≪1. If  Re(g) 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 fluctuationsδd(t) are typically characterized by a (single) correlation time td, with ensemble mean δ-d=0. When the field V(t) is used only for gate operations, we assume that it does not affect proper dephasing. The ensemble average over δd(t) results in R(t)→R(t)+Rd(t) with the dephasing rate
(125)Rd(t)=∫0t‍dt′Φd(t′),Φd(t)=δd(t)δd(0)¯.
The dephasing CF Φd(t) is the counterpart of the bath CF ΦT(t).Att≫td, the decoherence rate and shift approach their asymptotic values
(126)Rd=limt→∞Rd(t),Δd=limt→∞Δd(t).
For the validity of (127), it is necessary that
(127)Rd,|Δd|≪1td.
We assume the secular approximation, which holds if
(128)V0(t)≫Rd,|Δd|.By analogy with (118), one can obtain that
(129)Rd(t)=π∫-∞∞‍dωFt(ω)Gd(ω),
where Ft(ω) is given by (117) with
(130)ϵ(t)=exp[-i∫0t‍V(t′)dt′],(131)Gd(ω)=1π∫0∞‍Φd(t)cosωtdt.
As follows from (131), Gd(ω) is a symmetric function,
(132)Gd(-ω)=Gd(ω).Theproper dephasing rate associated with
(133)Φd(t)=Ae-t/td
is
(134)Rd=Atd.
In the presence of a constantV0 [cw V(t)], it is modified into
(135)Rd=AtdV02td2+1.
For a sufficiently strong field, the dephasing rate Rd can be suppressed by the factor 1/(V0td)2≪1. 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 τ-periodic π-pulses [2, 82, 84] is an analog of the above “parity kicks.” Using the analog of  (121), such pulses can be shown to suppress Rd approximately according to (135) with V0=π/τ. This BB method requires pulsed fields with Rabi frequencies ≫1/τ, that is, much stronger fields than the cw field in (135). Using td~10-7 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(136)ϵ(t)=∑k‍ϵke-iνkt,
where νk(k=0,±1,…) are arbitrary discrete frequencies such that
(137)|νk-νk′|≥Ω∀k≠k′,
where Ω is the minimal spectral interval.One can define the long-time limit of the quasi-periodic modulation, when(138)Ωt≫1,t≫tc,
where tc is the bath-memory (correlation) time, defined as the inverse of the largest spectral interval over which G(ω) and G(-ω) change appreciably near the relevant frequencies ωa+νk. In this limit, the average decay rate is given by (Figure 5(a)) as
(139)R↑=2π∑k‍|λk|2G(ωa+νk),(140)λk=ϵkϵc,ϵc=∑k‍|ϵk|2.Spectral representation of the bath coupling,G(ω), and the modulation, Ft(ω). (a) General quasi-periodic modulation, with peaks at ωk. (b) On-off modulation, with 1/τ1 repetition rate for τ1≪τ0. (c) Impulsive phase modulation, (π-pulses), ϕ=π. (d) Monochromatic modulation, or impulsive phase modulation, with small phase shifts, ϕ≪1, and 1/τ repetition rate.
(a)
(b)
(c)
(d)


#### 7.4.1. Phase Modulation (PM) of the Coupling

Monochromatic Perturbation.  Let
(141)ϵ(t)=ϵ0e-iΔt.
Then
(142)R↑=2πG(ωa+Δ),
where Δ=const. 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 R relative to the Golden Rule decay rate, that is, the decay rate without any perturbation as
(143)RGR=2πG(ωa).Equation (40) provides the maximal change of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(ω) incurred by the width of Ft(ω): the modified R can even vanish, if the shifted frequency ωa+Δ is beyond the cutoff frequency of the coupling, where G(ω)=0 (Figure 5(d)). This would accomplish the goal of dynamical decoupling [81–87, 118, 162]. Conversely, the increase of R 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 νk 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 ofπ-phases; that is, each pulse has a total area of π, whose effects are similar to time-reversal or the spin-echo technique [94]. Thus, the free evolution propagator after the control π-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 amountϕ at times τ,2τ,…. Such modulation can be achieved by a train of identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed AC Stark shifts of ωa. When ϕ=π, this modulation corresponds to dynamical-decoupling (DD) pulses.For sufficiently long times (see (138)), one can use (139), with
(144)νk=2kπτ-ϕτ,|λk|2=4sin2(ϕ/2)(2kπ-ϕ)2.Forsmall phase shifts, ϕ≪1, the k=0 peak dominates,
(145)|λ0|2≈1-ϕ212,
whereas
(146)|λk|2≈ϕ24π2k2(k≠0).
In this case, one can retain only the k=0 term in (139), unless G(ω) is changing very fast with frequency. Then the modulation acts as a constant shift (Figure 5(d)) as
(147)Δ=-ϕτ.As|ϕ| increases, the difference between the k=0 and k=1 peak heights diminishes, vanishing for ϕ=±π. Then
(148)|λ0|2=|λ1|2=4π2,
that is, Ft(ω) for ϕ=±π contains two identical peaks symmetrically shifted in opposite directions (Figure 5(c)) (the other peaks |λk|2 decrease with k as (2k-1)-2, totaling 0.19).The foregoing features allow one to adjust the modulation parameters for a given scenario to obtain anoptimal decrease or increase of R. Thus, the phase-modulation (PM) scheme with a small ϕ is preferable near a continuum edge (Figure 5(d)), since it yields a spectral shift in the required direction (positive or negative). The adverse effect of k≠0 peaks in Ft(ω) then scales as ϕ2 and hence can be significantly reduced by decreasing |ϕ|. On the other hand, if ωa is near a symmetric peak of G(ω), R is reduced more effectively for ϕ≃π, as in [80, 81], since the main peaks of Ft(ω) at ω0 and ω1 then shift stronger with τ-1 than the peak at ω0=-ϕ/τ for ϕ≪1.


#### 7.4.3. Optimal DD

One of the recent extensions of standard DD is the optimal DD, introduced by Uhrig [95]. While the standard DD applies an equidistant sequence of π-pulses, optimal DD finds the temporal spacing of the pulses such that dephasing is minimized. In [95], the optimal temporal spacing was found to satisfy a simple analytic equation and the resulting modulation power spectrum for nπ-pulses during time t can be approximated by
(149)Ft(ω)≈2t(n+1)2Jn+12(ωt/2)(ωt/2)2,
where J is the Bessel function. This optimal DD sequence was derived by making the first n derivatives of Ft(ω) vanish. This requires the modulation spectrum to have higher frequency components, by minimizing the lower frequency peaks observed in the standard DD. However, it is important to note that this optimization is done irrespective of the coupling spectrum G(ω). Hence, this modulation is optimal only if there is weaker coupling to higher frequencies of the bath, which can be easily seen in (121) by noting that the coupling spectrum G(ω) must have lower values at higher frequencies for the optimal DD to be effective. By contrast, we analyze below an optimal modulation based on our universal formula [104] that takes the coupling spectrum into account.Other Extensions. We considered here only two forms of dynamical decoupling that can be easily incorporated into the general framework. However, DD has many extensions that go beyond what has been discussed above. An important extension is the universal DD sequence [119] that combats all forms of coupling to a bath, namely, both amplitude and phase noises. The general formalism presented above has not yet addressed this issue, but attempts are currently being made to generalize the treatment.Another important extension is known as concatenated DD [90] that treats increasingly higher order corrections of the noise, where each concatenation level of the control pulses reduces the previous level’s induced errors. This powerful protocol cannot be easily incorporated into our formalism since it goes beyond the second-order approximation used in our derivation of the universal formula.In general, two major differences make our universal formula approach advantageous compared to the DD approach. The first difference is that we relax the DD assumption that the control fields must be either very short or very strong. In our formalism, the control fields are consideredconcurrently with the coupling to the bath, hence allowing a much wider variety of pulse sequences, ranging from continuous modulation all the way to DD sequences. The second difference relates to the consideration of the bath spectrum. Dynamical decoupling suggests using the same pulse sequence (be it periodic, optimized, or concatenated), no matter what the shape of the bath spectrum. By contrast, our formalism explicitly considers the bath spectrum and allows optimal tailoring of the modulation to a given bath spectrum (see below).


### 7.5. Optimal Decoherence Control of a Qubit

The aforementioned arsenal of the modulation schemes is not general and neither it nor DD [82, 90, 95, 119, 163, 164] have shown optimality with respect to the decoherence suppression, for any given coupling spectrum. Here we apply variational principles to our universal-formula dynamical control of decoherence in order to find the optimal modulation for any given decoherence process. We first derive an equation for the optimal, energy-constrained control by modulation that minimizes decoherence, for any given bath-coupling spectrum and then numerically solve this equation and compare the optimal modulation to energy-constrained dynamical-decoupling pulses.The objective is to minimize the average decoherence rate,R(t), given a bath-coupling spectrum, G(ω), by finding the optimal phase modulation, ϵ(t)=eiϕ(t) under an energy constraint, respectively, given by
(150)R(t)=2π∫-∞∞‍dωG(ω)Ft(ω),G(ω)=(2π)-1Re∫-∞∞‍dtΦ(t)eiωt,Ft(ω)=|ϵt(ω)|2t,ϵt(ω)=12π∫0t‍dt1eiϕ(t1)eiωt1,∫0T‍dt|ϕ˙(t)|2=E.


#### 7.5.1. Calculus of Variations

#### 7.5.2. Derivation of the Euler-Lagrange Equation

The boundary conditions for the accumulated phase areϕ(0)=ϕ˙(0)=0, which results in a smooth solution and accounts for turning the control field on at t=0. Eliminating λ, we find that the optimal control field shape is the solution to the following equation:
(151)ϕ¨(t)=-EZ[t,ϕ(t)]∫0T‍dt1|∫0t1‍dt2Z[t2,ϕ(t2)]|2,
where λ is the Lagrange multiplier, and
(152)Z[t,ϕ(t)]=1T∫0T‍dt1Φ~(|t-t1|)0000000×sin[ϕ(t)-ϕ(t1)+Δ(t-t1)].We may compare the optimal dephasing rate to the one obtained by the popular periodic DD control (bang bang) procedure. But to make the comparison meaningful, we impose the same energy constraint. Finite-duration periodic DD against pure dephasing is the “bang bang” application ofnπ-pulses and is given in our setting by
(153)Ω(t)={πνjτ≤t<jτ+ν,j=0,…,n-1,0otherwise,
where ν<τ is the width of each pulse and τ is the interval between pulses. The energy constraint E and the total modulation duration T=nτ+ν are related via n=νE/π2. In the frequency domain, the spectral modulation intensity can be described by a series of peaks, where the two main peaks are at ±π/τ. Thus, the peaks are shifted in proportion to the energy invested in the modulation.(a) Single-Peak Resonant Dephasing Spectrum. This simple dephasing spectrum describes a common scenario where Φ(t)=e-t/tcγ/tc, where γ is the long-time dephasing rate [R(t→∞)=2πγ] and tc is the noise correlation time. Figure 6(a) shows R(T), normalized to the bare (unmodulated) dephasing rate, as a function of the energy constraint. As expected, the more energy is available for modulation, the lower is the dephasing rate. For low energies, the optimal modulation significantly outperforms DD, while at higher energies this difference disappears. These results can be understood from Figure 7(a), by noticing that the two central DD peaks have significant overlap with G(ω) at the low energy value shown. As E is increased at fixed T, the DD peaks move farther apart and have less overlap with G(ω), leading to improved performance. Applying the linearized EL equation with DD as initial guess yields only mild improvements (not shown). The explanation for the superior performance of the optimal modulation is also evident from Figure 7(a): since higher frequencies have lower coupling strength in this case, the optimal control “reshapes” so as to maximize its weight in the high-frequency range, to the extent permitted by the energy constraint. The modulation can be well approximated by Ω(t)=a[1+e-t/T(t/T-1)], where a is determined by the energy constraint, which fits the results in Figure 6(a).Average modified final decoherence rateR(T), normalized with respect to the unmodulated rate, as a function of energy constraint. DD—dash, magenta. Optimal modulation—solid, blue. Insets: optimal modulation Ω(t) for different energy constraints. (a) Single-peak resonant dephasing spectrum (inset: E=20). (b) Single-peak off-resonant spectrum (inset: E=50). (c) 1/f spectrum (inset: E=30). (d) Multipeaked spectrum (inset: E=30).
(a)
(b)
(c)
(d)Dephasing spectrumG(ω) (solid, red), optimal (dot-dash, green), and DD (dash, magenta) modulation spectra FT(ω), in arbitrary units (a.u.). Same parameters as in the insets of Figure 6.
(a)
(b)
(c)
(d)(b) Single-Peak Off-Resonant Dephasing Spectrum. This dephasing spectrum describes a variation on the aforementioned scenario, where the spectral peak is shifted (Δ≠0 in Φ(t)=Φ~(t)eiΔt), for example, coupling to a nonresonant bath. With no other constraints, the optimal modulation is trivially similar to the one of the resonant spectrum, with a shifted energy-constraint Enon-res=Eres+Δ. To demonstrate how this formalism can accommodate additional constraints, we impose a positivity constraint, ϕ˙(t)≥0 (positive field amplitude) and obtain nontrivial behavior of R(T) as a function of the energy constraint—see Figure 6(b). Here we used the linearized EL equation with the DD modulation (153) as an initial guess. For both the DD and optimal modulations, we observe an initial increase in the dephasing rate as a function of energy, followed by a decrease. For DD, this can be interpreted as a manifestation of the initial anti-Zeno effect and the subsequent quantum Zeno effect [100, 165]. Because of the positivity constraint, modulations with low energy (both DD and optimal) couple to more resonant modes of the bath (Figure 7(b)) and thus do worse than the unmodulated case. The DD modulation is optimal for small energy constraints, and hence the decoherence rates of DD and our optimal solution coincide. This is because the DD peaks do not overlap with the off-resonant spectral peak. However, as the positive-frequency main DD peak (Figure 7(b)) nears the off-resonant spectral peak, with increased energy, the optimal modulation diverges from the DD modulation and “reshapes” itself so as to couple to higher modes of the bath. In the time domain (Figure 6(b) inset), this is seen as a smoothing of the abrupt DD modulation. At even higher energy constraints, there is once more no improvement by the optimal modulation over DD, yet there is an improvement over the unmodulated case. Over the entire range of E, the optimal modulation results in a much flatter R(T) than DD, which is an indicator of its robustness. While DD is strongly influenced by the off-resonant peak, the optimal modulation exploits the energy available to find the minimal overlap with the dephasing spectrum.(c)
1
/
f
Dephasing Spectrum. The ubiquitous 1/f dephasing spectrum that describes a variety of experiments—for example, charge noise in superconducting qubits [166]—is given in our notation by G(ω)∝1/ω, with cutoffs ωmin and ωmax. Figure 6(c) shows that, as expected, the more the energy available for modulation, the lower the dephasing rate. Notice that due to the cut-off ωmin, a low-energy modulation does worse than the unmodulated case. Furthermore, since, as in case (a), higher frequencies now have lower coupling strength, the optimal control “reshapes” so as to have as high a weight in the high frequency range as the energy constraint allows (Figure 7(c)). This is expressed in the time domain (Figure 6(c) inset) as the initial increase in the modulation strength (t<50). The later decrease in modulation strength can be attributed to the lower cutoff, where the optimal modulation benefits from lower frequencies, that is, lower modulation amplitudes. Upon comparing the 1/f case to the Lorentzian spectrum, Figure 6(a), we observe a similar optimal initial chirped modulation in the time domain.Despite the differences in the long-time behavior (due to the lower cutoff in the1/f case), these two examples allow us to generalize to any dephasing spectrum with a monotonically decreasing system-bath coupling strength as a function of frequency. The optimal modulation for such spectra will be an energy-constrained chirped modulation, with modifications due to other spectral characteristics, for example, cutoffs.(d) Multipeaked Dephasing Spectrum. This describes the most general scenario, where there can be several resonances and noise correlation times. Figure 6(d) shows R(T) as a function of the energy constraint. Once again, because DD does not account for the dephasing spectrum, its performance is much worse than our (universal-formula) optimal modulation, whose “reshaping” results in monotonically improving performance: the peaks of our optimal modulation are predominantly anticorrelated with the peaks of G(ω).


## 7.1. Explicit Equations for Factorisable Interaction Hamiltonians

We now wish to write the ME explicitly for time-dependent Hamiltonians of the following form [100]:
(96)H(t)=HS(t)+HB+HI(t),HI(t)=S(t)B,
where HS and HB are the system and bath Hamiltonians, respectively, and HI, the interaction Hamiltonian, is the product of operators S and B which act on the system and bath, respectively.Finally, defining the correlation function for the bath,(97)ΦT(t)=〈BB~(t)〉B,
we obtain the ME for ρ in the Born approximation as
(98)ρ˙(t)=-i[HS,ρ(t)]+∫0t‍dτ{ΦT(t-τ)[S~(t,τ)ρ(t),S(t)]+H.c.}.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(99)HB=∑λ‍ωλaλ†aλ,B=∑λ‍(κλaλ+κλ*aλ†).
Here aλ,aλ† are the annihilation and creation operators of mode λ, respectively, and κλ is the coupling amplitude to mode λ.


### 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:(100)HS(t)=(ωa+δa(t))|e〉〈e|,(101)S(t)=ϵ~(t)σx,
where δa(t) is the dynamical AC-Stark shifts, ϵ~(t) is the time-dependent modulation of the interaction strength, and the Pauli matrix σx=|e〉〈g|+|g〉〈e|.


### 7.1.2. Phase-Noise Regime

Next, we consider the PN regime of a two-level system coupled to a thermal bath viaσz operator. To combat it, we will use near-resonant fields with time-varying amplitude as our control. The Hamiltonians then assume the following forms:
(102)HS(t)=ωa|e〉〈e|+V(t)σx,(103)S(t)=ϵ~(t)σz,
where V(t)=Ω(t)e-iωat+c.c is the time-dependent resonant field, with real envelope Ω(t), ϵ~(t) is the time-dependent modulation of the interaction strength, and  σz=|e〉〈e|-|g〉〈g|.Since we are interested in dephasing, phases due to the (unperturbed) energy difference between the levels are immaterial.


## 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:(100)HS(t)=(ωa+δa(t))|e〉〈e|,(101)S(t)=ϵ~(t)σx,
where δa(t) is the dynamical AC-Stark shifts, ϵ~(t) is the time-dependent modulation of the interaction strength, and the Pauli matrix σx=|e〉〈g|+|g〉〈e|.


## 7.1.2. Phase-Noise Regime

Next, we consider the PN regime of a two-level system coupled to a thermal bath viaσz operator. To combat it, we will use near-resonant fields with time-varying amplitude as our control. The Hamiltonians then assume the following forms:
(102)HS(t)=ωa|e〉〈e|+V(t)σx,(103)S(t)=ϵ~(t)σz,
where V(t)=Ω(t)e-iωat+c.c is the time-dependent resonant field, with real envelope Ω(t), ϵ~(t) is the time-dependent modulation of the interaction strength, and  σz=|e〉〈e|-|g〉〈g|.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
(104)|↑,↓〉=|e,g〉,
while for the PN case of (102) the basis is
(105)|↑,↓〉=12(e-iωat|e〉±|g〉).In this rotated and tilted frame,(106)US(t,τ)=US(t,0)US†(τ,0),US(t,0)=e-iϕ(t)|↑〉〈↑|+eiϕ(t)|↓〉〈↓|,
where
(107)ϕ(t)={ωat+∫0t‍dt′δa(t′)forAN,∫0t‍dt′Ω(t′)2forPN
is the phase-modulation due to the time-dependent control in the system Hamiltonian.Allowance forarbitrary time-dependent intervention in the system and interaction dynamics HS(t), HI(t), respectively, yields the following universal ME for a dynamically controlled decohering system [100, 124]:
(108)ρ~˙(t)=∫0t‍dt′{Φ(t-t′)[S~(t′)ρ~(t),S~(t)]+H.c.}.
Here S~(t)=ϵ(t)σ~x is the modulated interaction operator, where ~ denotes the rotated and tilted frame, and σ~x=|↑〉〈↓|+|↓〉〈↑|. The modulation function is given by
(109)ϵ(t)=ϵ~(t)eiϕ(t)
for both AN and PN. It is important to note that ρ~ is a function of   tonly (not oft′): this convolutionless form of the ME is fully non-Markovian to second order in HI, as proven exactly in [124].


## 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
(110)ρ˙↑↑=-ρ˙↓↓=-r↑(t)ρ↑↑+r↓(t)ρ↓↓,(111)r↑(↓)=2Re∫0t‍dt′Φ(t-t′)K±(t,t′),(112)K+(t,t′)=K-*(t,t′)=ϵ(t)ϵ*(t′).
The time-dependent relaxation rates are real, and the only difference between them is the complex conjugate of the combined modulation function, K±(t,t′). 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(113)R↑=∫-∞∞‍dωG(+ω)Ft(ω),R↓=∫-∞∞‍dωG(-ω)Ft(ω).For both AN (see (100)) and PN (see (102)),
(114)G(ω)=G0(ω)(nT(ω)+1)+G0(-ω)nT(-ω),(115)G0(ω)=ρ(ω)|μeg|2(ω)Θ(ω),
where G0(ω) is the zero-temperature bath spectrum, ρ(ω) and μeg(ω) are the frequency-dependent density of bath modes and the e↔g transition matrix element, respectively, nT(ω)=(eβω-1)-1 is the temperature-dependent bath mode population, and β=ħ/kBT is the inverse temperature. Also, Θ(ω) is the Heaviside function, that is, the zero-temperature bath spectrum is defined only for positive frequencies G0(ω<0)=0. 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 thesame form for both (see Section 7.4) as
(116)Ft(ω)=1t|∫0t‍dt′eiωt′ϵ(t′)|2,
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
(117)δa(t)=∫0t‍dt′|Ω(t′)|2Δ(t′),
where Ω(t) is the Rabi frequency of the control field and Δ(t′) is the detuning. The corresponding phase factor for PN is the integral of the Rabi frequency Ω(t), that is, the pulse area of the resonant control field, (107) (Figure 4).Schematic drawing of system and bath. (a) Amplitude noise (AN) (red) combatted by AC-Stark shift modulation (green). (b) Phase noise (PN) (red) combatted by resonant-field modulation (green).
(a)
(b)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 ratesRe(g)(t) rather than the level shifts. The average rate of the |e〉→|g〉 transition Re(t) and its |g〉→|e〉 counterpart Rg(t) are given by
(118)Re(g)(t)=2π∫-∞∞‍dωFt(ω)GT(±ω).
Here the upper (lower) sign corresponds to the subscript e(g), and
(119)GT(ω)=(2π)-1∫-∞∞‍ΦT(t)eiωtdt
can be shown [161] to be nonnegative, with GT(-ω)=e-βωGT(ω), and vanishes for ω<0 at T=0: G0(ω)=0(ω<0). For the oscillator bath, one finds that
(120)GT(ω)=[n(ω)+1]G0(ω)+n(-ω)G0(-ω),
where G0(ω)=∑λ‍|κλ|2δ(ω-ωλ) and n(ω)=(eβω-1)-1 is the average number of quanta in the oscillator (bath mode) with frequency ω.We apply (118) to the case of coherent modulation of quasiperiodic form, (see (31)). Without a limitation of the generality, we can assume that ∑k‍|ϵk|2=1. We then find, using (118), that the rates Re(g)(t) tend to the long-time limits
(121)Re(g)=2π∫-∞∞‍dωF(ω)GT(±ω),
where
(122)F(ω)=limt→∞Ft(ω)=∑k‍|ϵk|2δ(ω-ωa-ωk),
or
(123)Re(g)=2π∑k‍|ϵk|2GT(±(ωa+ωk)).
Equation (121) shows that Re(Rg)is given by the overlap of the modulation spectrumF(ω)with the bath-CF spectrumGT(ω)[GT(-ω)]. The limits (123) are approached when Ωt≫1 and t≫tc≡maxk{1/ξ(±(ωa+ωk)}. Here tc is the bath memory (correlation) time, defined as the inverse of ξ(ω), the spectral interval over which GT(ω) changes around the relevant frequencies.Had we used the standard dipolar RWA hamiltonian in the case of an oscillator bath, dropping the antiresonant terms inHI(t), we would have arrived at the transition rates
(124)Re(g)RWA=2π∫0∞‍dωF(ω)GT(±ω),
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 F(ω)≃0 at ω<0, being peaked near ωa. However, whenever the suppression of Re(g) requires modulation at a rate comparable to ωa, the RWA is inadequate. For instance, (120) and (124) imply that, at T=0, the rate RgRWA vanishes identically, irrespective of F(ω), in contrast to the true upward-transition rate Rg in (121), which may be comparable to Re 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 Re(g) 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|e〉to be shifted below|g〉.The validity of the (decohering) qubit model in the presence of modulation at a rate≳ωa is now elucidated: it requires that Re(g)jt≪1, Re(g)j being the effective transition rate from level e(g) to any other level j, and, in particular, Re(g)t≪1. If  Re(g) 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 fluctuationsδd(t) are typically characterized by a (single) correlation time td, with ensemble mean δ-d=0. When the field V(t) is used only for gate operations, we assume that it does not affect proper dephasing. The ensemble average over δd(t) results in R(t)→R(t)+Rd(t) with the dephasing rate
(125)Rd(t)=∫0t‍dt′Φd(t′),Φd(t)=δd(t)δd(0)¯.
The dephasing CF Φd(t) is the counterpart of the bath CF ΦT(t).Att≫td, the decoherence rate and shift approach their asymptotic values
(126)Rd=limt→∞Rd(t),Δd=limt→∞Δd(t).
For the validity of (127), it is necessary that
(127)Rd,|Δd|≪1td.
We assume the secular approximation, which holds if
(128)V0(t)≫Rd,|Δd|.By analogy with (118), one can obtain that
(129)Rd(t)=π∫-∞∞‍dωFt(ω)Gd(ω),
where Ft(ω) is given by (117) with
(130)ϵ(t)=exp[-i∫0t‍V(t′)dt′],(131)Gd(ω)=1π∫0∞‍Φd(t)cosωtdt.
As follows from (131), Gd(ω) is a symmetric function,
(132)Gd(-ω)=Gd(ω).Theproper dephasing rate associated with
(133)Φd(t)=Ae-t/td
is
(134)Rd=Atd.
In the presence of a constantV0 [cw V(t)], it is modified into
(135)Rd=AtdV02td2+1.
For a sufficiently strong field, the dephasing rate Rd can be suppressed by the factor 1/(V0td)2≪1. 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 τ-periodic π-pulses [2, 82, 84] is an analog of the above “parity kicks.” Using the analog of  (121), such pulses can be shown to suppress Rd approximately according to (135) with V0=π/τ. This BB method requires pulsed fields with Rabi frequencies ≫1/τ, that is, much stronger fields than the cw field in (135). Using td~10-7 s, cw Rabi frequencies exceeding 1 MHz achieve a significant dephasing suppression.


## 7.3.1. Dynamically Modified Decay Rates

Since we are interested here in dynamical control of relaxation, we shall concentrate on the transition ratesRe(g)(t) rather than the level shifts. The average rate of the |e〉→|g〉 transition Re(t) and its |g〉→|e〉 counterpart Rg(t) are given by
(118)Re(g)(t)=2π∫-∞∞‍dωFt(ω)GT(±ω).
Here the upper (lower) sign corresponds to the subscript e(g), and
(119)GT(ω)=(2π)-1∫-∞∞‍ΦT(t)eiωtdt
can be shown [161] to be nonnegative, with GT(-ω)=e-βωGT(ω), and vanishes for ω<0 at T=0: G0(ω)=0(ω<0). For the oscillator bath, one finds that
(120)GT(ω)=[n(ω)+1]G0(ω)+n(-ω)G0(-ω),
where G0(ω)=∑λ‍|κλ|2δ(ω-ωλ) and n(ω)=(eβω-1)-1 is the average number of quanta in the oscillator (bath mode) with frequency ω.We apply (118) to the case of coherent modulation of quasiperiodic form, (see (31)). Without a limitation of the generality, we can assume that ∑k‍|ϵk|2=1. We then find, using (118), that the rates Re(g)(t) tend to the long-time limits
(121)Re(g)=2π∫-∞∞‍dωF(ω)GT(±ω),
where
(122)F(ω)=limt→∞Ft(ω)=∑k‍|ϵk|2δ(ω-ωa-ωk),
or
(123)Re(g)=2π∑k‍|ϵk|2GT(±(ωa+ωk)).
Equation (121) shows that Re(Rg)is given by the overlap of the modulation spectrumF(ω)with the bath-CF spectrumGT(ω)[GT(-ω)]. The limits (123) are approached when Ωt≫1 and t≫tc≡maxk{1/ξ(±(ωa+ωk)}. Here tc is the bath memory (correlation) time, defined as the inverse of ξ(ω), the spectral interval over which GT(ω) changes around the relevant frequencies.Had we used the standard dipolar RWA hamiltonian in the case of an oscillator bath, dropping the antiresonant terms inHI(t), we would have arrived at the transition rates
(124)Re(g)RWA=2π∫0∞‍dωF(ω)GT(±ω),
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 F(ω)≃0 at ω<0, being peaked near ωa. However, whenever the suppression of Re(g) requires modulation at a rate comparable to ωa, the RWA is inadequate. For instance, (120) and (124) imply that, at T=0, the rate RgRWA vanishes identically, irrespective of F(ω), in contrast to the true upward-transition rate Rg in (121), which may be comparable to Re 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 Re(g) 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|e〉to be shifted below|g〉.The validity of the (decohering) qubit model in the presence of modulation at a rate≳ωa is now elucidated: it requires that Re(g)jt≪1, Re(g)j being the effective transition rate from level e(g) to any other level j, and, in particular, Re(g)t≪1. If  Re(g) 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 fluctuationsδd(t) are typically characterized by a (single) correlation time td, with ensemble mean δ-d=0. When the field V(t) is used only for gate operations, we assume that it does not affect proper dephasing. The ensemble average over δd(t) results in R(t)→R(t)+Rd(t) with the dephasing rate
(125)Rd(t)=∫0t‍dt′Φd(t′),Φd(t)=δd(t)δd(0)¯.
The dephasing CF Φd(t) is the counterpart of the bath CF ΦT(t).Att≫td, the decoherence rate and shift approach their asymptotic values
(126)Rd=limt→∞Rd(t),Δd=limt→∞Δd(t).
For the validity of (127), it is necessary that
(127)Rd,|Δd|≪1td.
We assume the secular approximation, which holds if
(128)V0(t)≫Rd,|Δd|.By analogy with (118), one can obtain that
(129)Rd(t)=π∫-∞∞‍dωFt(ω)Gd(ω),
where Ft(ω) is given by (117) with
(130)ϵ(t)=exp[-i∫0t‍V(t′)dt′],(131)Gd(ω)=1π∫0∞‍Φd(t)cosωtdt.
As follows from (131), Gd(ω) is a symmetric function,
(132)Gd(-ω)=Gd(ω).Theproper dephasing rate associated with
(133)Φd(t)=Ae-t/td
is
(134)Rd=Atd.
In the presence of a constantV0 [cw V(t)], it is modified into
(135)Rd=AtdV02td2+1.
For a sufficiently strong field, the dephasing rate Rd can be suppressed by the factor 1/(V0td)2≪1. 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 τ-periodic π-pulses [2, 82, 84] is an analog of the above “parity kicks.” Using the analog of  (121), such pulses can be shown to suppress Rd approximately according to (135) with V0=π/τ. This BB method requires pulsed fields with Rabi frequencies ≫1/τ, that is, much stronger fields than the cw field in (135). Using td~10-7 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(136)ϵ(t)=∑k‍ϵke-iνkt,
where νk(k=0,±1,…) are arbitrary discrete frequencies such that
(137)|νk-νk′|≥Ω∀k≠k′,
where Ω is the minimal spectral interval.One can define the long-time limit of the quasi-periodic modulation, when(138)Ωt≫1,t≫tc,
where tc is the bath-memory (correlation) time, defined as the inverse of the largest spectral interval over which G(ω) and G(-ω) change appreciably near the relevant frequencies ωa+νk. In this limit, the average decay rate is given by (Figure 5(a)) as
(139)R↑=2π∑k‍|λk|2G(ωa+νk),(140)λk=ϵkϵc,ϵc=∑k‍|ϵk|2.Spectral representation of the bath coupling,G(ω), and the modulation, Ft(ω). (a) General quasi-periodic modulation, with peaks at ωk. (b) On-off modulation, with 1/τ1 repetition rate for τ1≪τ0. (c) Impulsive phase modulation, (π-pulses), ϕ=π. (d) Monochromatic modulation, or impulsive phase modulation, with small phase shifts, ϕ≪1, and 1/τ repetition rate.
(a)
(b)
(c)
(d)


### 7.4.1. Phase Modulation (PM) of the Coupling

Monochromatic Perturbation.  Let
(141)ϵ(t)=ϵ0e-iΔt.
Then
(142)R↑=2πG(ωa+Δ),
where Δ=const. 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 R relative to the Golden Rule decay rate, that is, the decay rate without any perturbation as
(143)RGR=2πG(ωa).Equation (40) provides the maximal change of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(ω) incurred by the width of Ft(ω): the modified R can even vanish, if the shifted frequency ωa+Δ is beyond the cutoff frequency of the coupling, where G(ω)=0 (Figure 5(d)). This would accomplish the goal of dynamical decoupling [81–87, 118, 162]. Conversely, the increase of R 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 νk 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 ofπ-phases; that is, each pulse has a total area of π, whose effects are similar to time-reversal or the spin-echo technique [94]. Thus, the free evolution propagator after the control π-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 amountϕ at times τ,2τ,…. Such modulation can be achieved by a train of identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed AC Stark shifts of ωa. When ϕ=π, this modulation corresponds to dynamical-decoupling (DD) pulses.For sufficiently long times (see (138)), one can use (139), with
(144)νk=2kπτ-ϕτ,|λk|2=4sin2(ϕ/2)(2kπ-ϕ)2.Forsmall phase shifts, ϕ≪1, the k=0 peak dominates,
(145)|λ0|2≈1-ϕ212,
whereas
(146)|λk|2≈ϕ24π2k2(k≠0).
In this case, one can retain only the k=0 term in (139), unless G(ω) is changing very fast with frequency. Then the modulation acts as a constant shift (Figure 5(d)) as
(147)Δ=-ϕτ.As|ϕ| increases, the difference between the k=0 and k=1 peak heights diminishes, vanishing for ϕ=±π. Then
(148)|λ0|2=|λ1|2=4π2,
that is, Ft(ω) for ϕ=±π contains two identical peaks symmetrically shifted in opposite directions (Figure 5(c)) (the other peaks |λk|2 decrease with k as (2k-1)-2, totaling 0.19).The foregoing features allow one to adjust the modulation parameters for a given scenario to obtain anoptimal decrease or increase of R. Thus, the phase-modulation (PM) scheme with a small ϕ is preferable near a continuum edge (Figure 5(d)), since it yields a spectral shift in the required direction (positive or negative). The adverse effect of k≠0 peaks in Ft(ω) then scales as ϕ2 and hence can be significantly reduced by decreasing |ϕ|. On the other hand, if ωa is near a symmetric peak of G(ω), R is reduced more effectively for ϕ≃π, as in [80, 81], since the main peaks of Ft(ω) at ω0 and ω1 then shift stronger with τ-1 than the peak at ω0=-ϕ/τ for ϕ≪1.


### 7.4.3. Optimal DD

One of the recent extensions of standard DD is the optimal DD, introduced by Uhrig [95]. While the standard DD applies an equidistant sequence of π-pulses, optimal DD finds the temporal spacing of the pulses such that dephasing is minimized. In [95], the optimal temporal spacing was found to satisfy a simple analytic equation and the resulting modulation power spectrum for nπ-pulses during time t can be approximated by
(149)Ft(ω)≈2t(n+1)2Jn+12(ωt/2)(ωt/2)2,
where J is the Bessel function. This optimal DD sequence was derived by making the first n derivatives of Ft(ω) vanish. This requires the modulation spectrum to have higher frequency components, by minimizing the lower frequency peaks observed in the standard DD. However, it is important to note that this optimization is done irrespective of the coupling spectrum G(ω). Hence, this modulation is optimal only if there is weaker coupling to higher frequencies of the bath, which can be easily seen in (121) by noting that the coupling spectrum G(ω) must have lower values at higher frequencies for the optimal DD to be effective. By contrast, we analyze below an optimal modulation based on our universal formula [104] that takes the coupling spectrum into account.Other Extensions. We considered here only two forms of dynamical decoupling that can be easily incorporated into the general framework. However, DD has many extensions that go beyond what has been discussed above. An important extension is the universal DD sequence [119] that combats all forms of coupling to a bath, namely, both amplitude and phase noises. The general formalism presented above has not yet addressed this issue, but attempts are currently being made to generalize the treatment.Another important extension is known as concatenated DD [90] that treats increasingly higher order corrections of the noise, where each concatenation level of the control pulses reduces the previous level’s induced errors. This powerful protocol cannot be easily incorporated into our formalism since it goes beyond the second-order approximation used in our derivation of the universal formula.In general, two major differences make our universal formula approach advantageous compared to the DD approach. The first difference is that we relax the DD assumption that the control fields must be either very short or very strong. In our formalism, the control fields are consideredconcurrently with the coupling to the bath, hence allowing a much wider variety of pulse sequences, ranging from continuous modulation all the way to DD sequences. The second difference relates to the consideration of the bath spectrum. Dynamical decoupling suggests using the same pulse sequence (be it periodic, optimized, or concatenated), no matter what the shape of the bath spectrum. By contrast, our formalism explicitly considers the bath spectrum and allows optimal tailoring of the modulation to a given bath spectrum (see below).


## 7.4.1. Phase Modulation (PM) of the Coupling

Monochromatic Perturbation.  Let
(141)ϵ(t)=ϵ0e-iΔt.
Then
(142)R↑=2πG(ωa+Δ),
where Δ=const. 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 R relative to the Golden Rule decay rate, that is, the decay rate without any perturbation as
(143)RGR=2πG(ωa).Equation (40) provides the maximal change of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(ω) incurred by the width of Ft(ω): the modified R can even vanish, if the shifted frequency ωa+Δ is beyond the cutoff frequency of the coupling, where G(ω)=0 (Figure 5(d)). This would accomplish the goal of dynamical decoupling [81–87, 118, 162]. Conversely, the increase of R 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 νk 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 ofπ-phases; that is, each pulse has a total area of π, whose effects are similar to time-reversal or the spin-echo technique [94]. Thus, the free evolution propagator after the control π-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 amountϕ at times τ,2τ,…. Such modulation can be achieved by a train of identical, equidistant, narrow pulses of nonresonant radiation, which produce pulsed AC Stark shifts of ωa. When ϕ=π, this modulation corresponds to dynamical-decoupling (DD) pulses.For sufficiently long times (see (138)), one can use (139), with
(144)νk=2kπτ-ϕτ,|λk|2=4sin2(ϕ/2)(2kπ-ϕ)2.Forsmall phase shifts, ϕ≪1, the k=0 peak dominates,
(145)|λ0|2≈1-ϕ212,
whereas
(146)|λk|2≈ϕ24π2k2(k≠0).
In this case, one can retain only the k=0 term in (139), unless G(ω) is changing very fast with frequency. Then the modulation acts as a constant shift (Figure 5(d)) as
(147)Δ=-ϕτ.As|ϕ| increases, the difference between the k=0 and k=1 peak heights diminishes, vanishing for ϕ=±π. Then
(148)|λ0|2=|λ1|2=4π2,
that is, Ft(ω) for ϕ=±π contains two identical peaks symmetrically shifted in opposite directions (Figure 5(c)) (the other peaks |λk|2 decrease with k as (2k-1)-2, totaling 0.19).The foregoing features allow one to adjust the modulation parameters for a given scenario to obtain anoptimal decrease or increase of R. Thus, the phase-modulation (PM) scheme with a small ϕ is preferable near a continuum edge (Figure 5(d)), since it yields a spectral shift in the required direction (positive or negative). The adverse effect of k≠0 peaks in Ft(ω) then scales as ϕ2 and hence can be significantly reduced by decreasing |ϕ|. On the other hand, if ωa is near a symmetric peak of G(ω), R is reduced more effectively for ϕ≃π, as in [80, 81], since the main peaks of Ft(ω) at ω0 and ω1 then shift stronger with τ-1 than the peak at ω0=-ϕ/τ for ϕ≪1.


## 7.4.3. Optimal DD

One of the recent extensions of standard DD is the optimal DD, introduced by Uhrig [95]. While the standard DD applies an equidistant sequence of π-pulses, optimal DD finds the temporal spacing of the pulses such that dephasing is minimized. In [95], the optimal temporal spacing was found to satisfy a simple analytic equation and the resulting modulation power spectrum for nπ-pulses during time t can be approximated by
(149)Ft(ω)≈2t(n+1)2Jn+12(ωt/2)(ωt/2)2,
where J is the Bessel function. This optimal DD sequence was derived by making the first n derivatives of Ft(ω) vanish. This requires the modulation spectrum to have higher frequency components, by minimizing the lower frequency peaks observed in the standard DD. However, it is important to note that this optimization is done irrespective of the coupling spectrum G(ω). Hence, this modulation is optimal only if there is weaker coupling to higher frequencies of the bath, which can be easily seen in (121) by noting that the coupling spectrum G(ω) must have lower values at higher frequencies for the optimal DD to be effective. By contrast, we analyze below an optimal modulation based on our universal formula [104] that takes the coupling spectrum into account.Other Extensions. We considered here only two forms of dynamical decoupling that can be easily incorporated into the general framework. However, DD has many extensions that go beyond what has been discussed above. An important extension is the universal DD sequence [119] that combats all forms of coupling to a bath, namely, both amplitude and phase noises. The general formalism presented above has not yet addressed this issue, but attempts are currently being made to generalize the treatment.Another important extension is known as concatenated DD [90] that treats increasingly higher order corrections of the noise, where each concatenation level of the control pulses reduces the previous level’s induced errors. This powerful protocol cannot be easily incorporated into our formalism since it goes beyond the second-order approximation used in our derivation of the universal formula.In general, two major differences make our universal formula approach advantageous compared to the DD approach. The first difference is that we relax the DD assumption that the control fields must be either very short or very strong. In our formalism, the control fields are consideredconcurrently with the coupling to the bath, hence allowing a much wider variety of pulse sequences, ranging from continuous modulation all the way to DD sequences. The second difference relates to the consideration of the bath spectrum. Dynamical decoupling suggests using the same pulse sequence (be it periodic, optimized, or concatenated), no matter what the shape of the bath spectrum. By contrast, our formalism explicitly considers the bath spectrum and allows optimal tailoring of the modulation to a given bath spectrum (see below).


## 7.5. Optimal Decoherence Control of a Qubit

The aforementioned arsenal of the modulation schemes is not general and neither it nor DD [82, 90, 95, 119, 163, 164] have shown optimality with respect to the decoherence suppression, for any given coupling spectrum. Here we apply variational principles to our universal-formula dynamical control of decoherence in order to find the optimal modulation for any given decoherence process. We first derive an equation for the optimal, energy-constrained control by modulation that minimizes decoherence, for any given bath-coupling spectrum and then numerically solve this equation and compare the optimal modulation to energy-constrained dynamical-decoupling pulses.The objective is to minimize the average decoherence rate,R(t), given a bath-coupling spectrum, G(ω), by finding the optimal phase modulation, ϵ(t)=eiϕ(t) under an energy constraint, respectively, given by
(150)R(t)=2π∫-∞∞‍dωG(ω)Ft(ω),G(ω)=(2π)-1Re∫-∞∞‍dtΦ(t)eiωt,Ft(ω)=|ϵt(ω)|2t,ϵt(ω)=12π∫0t‍dt1eiϕ(t1)eiωt1,∫0T‍dt|ϕ˙(t)|2=E.


### 7.5.1. Calculus of Variations

### 7.5.2. Derivation of the Euler-Lagrange Equation

The boundary conditions for the accumulated phase areϕ(0)=ϕ˙(0)=0, which results in a smooth solution and accounts for turning the control field on at t=0. Eliminating λ, we find that the optimal control field shape is the solution to the following equation:
(151)ϕ¨(t)=-EZ[t,ϕ(t)]∫0T‍dt1|∫0t1‍dt2Z[t2,ϕ(t2)]|2,
where λ is the Lagrange multiplier, and
(152)Z[t,ϕ(t)]=1T∫0T‍dt1Φ~(|t-t1|)0000000×sin[ϕ(t)-ϕ(t1)+Δ(t-t1)].We may compare the optimal dephasing rate to the one obtained by the popular periodic DD control (bang bang) procedure. But to make the comparison meaningful, we impose the same energy constraint. Finite-duration periodic DD against pure dephasing is the “bang bang” application ofnπ-pulses and is given in our setting by
(153)Ω(t)={πνjτ≤t<jτ+ν,j=0,…,n-1,0otherwise,
where ν<τ is the width of each pulse and τ is the interval between pulses. The energy constraint E and the total modulation duration T=nτ+ν are related via n=νE/π2. In the frequency domain, the spectral modulation intensity can be described by a series of peaks, where the two main peaks are at ±π/τ. Thus, the peaks are shifted in proportion to the energy invested in the modulation.(a) Single-Peak Resonant Dephasing Spectrum. This simple dephasing spectrum describes a common scenario where Φ(t)=e-t/tcγ/tc, where γ is the long-time dephasing rate [R(t→∞)=2πγ] and tc is the noise correlation time. Figure 6(a) shows R(T), normalized to the bare (unmodulated) dephasing rate, as a function of the energy constraint. As expected, the more energy is available for modulation, the lower is the dephasing rate. For low energies, the optimal modulation significantly outperforms DD, while at higher energies this difference disappears. These results can be understood from Figure 7(a), by noticing that the two central DD peaks have significant overlap with G(ω) at the low energy value shown. As E is increased at fixed T, the DD peaks move farther apart and have less overlap with G(ω), leading to improved performance. Applying the linearized EL equation with DD as initial guess yields only mild improvements (not shown). The explanation for the superior performance of the optimal modulation is also evident from Figure 7(a): since higher frequencies have lower coupling strength in this case, the optimal control “reshapes” so as to maximize its weight in the high-frequency range, to the extent permitted by the energy constraint. The modulation can be well approximated by Ω(t)=a[1+e-t/T(t/T-1)], where a is determined by the energy constraint, which fits the results in Figure 6(a).Average modified final decoherence rateR(T), normalized with respect to the unmodulated rate, as a function of energy constraint. DD—dash, magenta. Optimal modulation—solid, blue. Insets: optimal modulation Ω(t) for different energy constraints. (a) Single-peak resonant dephasing spectrum (inset: E=20). (b) Single-peak off-resonant spectrum (inset: E=50). (c) 1/f spectrum (inset: E=30). (d) Multipeaked spectrum (inset: E=30).
(a)
(b)
(c)
(d)Dephasing spectrumG(ω) (solid, red), optimal (dot-dash, green), and DD (dash, magenta) modulation spectra FT(ω), in arbitrary units (a.u.). Same parameters as in the insets of Figure 6.
(a)
(b)
(c)
(d)(b) Single-Peak Off-Resonant Dephasing Spectrum. This dephasing spectrum describes a variation on the aforementioned scenario, where the spectral peak is shifted (Δ≠0 in Φ(t)=Φ~(t)eiΔt), for example, coupling to a nonresonant bath. With no other constraints, the optimal modulation is trivially similar to the one of the resonant spectrum, with a shifted energy-constraint Enon-res=Eres+Δ. To demonstrate how this formalism can accommodate additional constraints, we impose a positivity constraint, ϕ˙(t)≥0 (positive field amplitude) and obtain nontrivial behavior of R(T) as a function of the energy constraint—see Figure 6(b). Here we used the linearized EL equation with the DD modulation (153) as an initial guess. For both the DD and optimal modulations, we observe an initial increase in the dephasing rate as a function of energy, followed by a decrease. For DD, this can be interpreted as a manifestation of the initial anti-Zeno effect and the subsequent quantum Zeno effect [100, 165]. Because of the positivity constraint, modulations with low energy (both DD and optimal) couple to more resonant modes of the bath (Figure 7(b)) and thus do worse than the unmodulated case. The DD modulation is optimal for small energy constraints, and hence the decoherence rates of DD and our optimal solution coincide. This is because the DD peaks do not overlap with the off-resonant spectral peak. However, as the positive-frequency main DD peak (Figure 7(b)) nears the off-resonant spectral peak, with increased energy, the optimal modulation diverges from the DD modulation and “reshapes” itself so as to couple to higher modes of the bath. In the time domain (Figure 6(b) inset), this is seen as a smoothing of the abrupt DD modulation. At even higher energy constraints, there is once more no improvement by the optimal modulation over DD, yet there is an improvement over the unmodulated case. Over the entire range of E, the optimal modulation results in a much flatter R(T) than DD, which is an indicator of its robustness. While DD is strongly influenced by the off-resonant peak, the optimal modulation exploits the energy available to find the minimal overlap with the dephasing spectrum.(c)
1
/
f
Dephasing Spectrum. The ubiquitous 1/f dephasing spectrum that describes a variety of experiments—for example, charge noise in superconducting qubits [166]—is given in our notation by G(ω)∝1/ω, with cutoffs ωmin and ωmax. Figure 6(c) shows that, as expected, the more the energy available for modulation, the lower the dephasing rate. Notice that due to the cut-off ωmin, a low-energy modulation does worse than the unmodulated case. Furthermore, since, as in case (a), higher frequencies now have lower coupling strength, the optimal control “reshapes” so as to have as high a weight in the high frequency range as the energy constraint allows (Figure 7(c)). This is expressed in the time domain (Figure 6(c) inset) as the initial increase in the modulation strength (t<50). The later decrease in modulation strength can be attributed to the lower cutoff, where the optimal modulation benefits from lower frequencies, that is, lower modulation amplitudes. Upon comparing the 1/f case to the Lorentzian spectrum, Figure 6(a), we observe a similar optimal initial chirped modulation in the time domain.Despite the differences in the long-time behavior (due to the lower cutoff in the1/f case), these two examples allow us to generalize to any dephasing spectrum with a monotonically decreasing system-bath coupling strength as a function of frequency. The optimal modulation for such spectra will be an energy-constrained chirped modulation, with modifications due to other spectral characteristics, for example, cutoffs.(d) Multipeaked Dephasing Spectrum. This describes the most general scenario, where there can be several resonances and noise correlation times. Figure 6(d) shows R(T) as a function of the energy constraint. Once again, because DD does not account for the dephasing spectrum, its performance is much worse than our (universal-formula) optimal modulation, whose “reshaping” results in monotonically improving performance: the peaks of our optimal modulation are predominantly anticorrelated with the peaks of G(ω).


## 7.5.1. Calculus of Variations

## 7.5.2. Derivation of the Euler-Lagrange Equation

The boundary conditions for the accumulated phase areϕ(0)=ϕ˙(0)=0, which results in a smooth solution and accounts for turning the control field on at t=0. Eliminating λ, we find that the optimal control field shape is the solution to the following equation:
(151)ϕ¨(t)=-EZ[t,ϕ(t)]∫0T‍dt1|∫0t1‍dt2Z[t2,ϕ(t2)]|2,
where λ is the Lagrange multiplier, and
(152)Z[t,ϕ(t)]=1T∫0T‍dt1Φ~(|t-t1|)0000000×sin[ϕ(t)-ϕ(t1)+Δ(t-t1)].We may compare the optimal dephasing rate to the one obtained by the popular periodic DD control (bang bang) procedure. But to make the comparison meaningful, we impose the same energy constraint. Finite-duration periodic DD against pure dephasing is the “bang bang” application ofnπ-pulses and is given in our setting by
(153)Ω(t)={πνjτ≤t<jτ+ν,j=0,…,n-1,0otherwise,
where ν<τ is the width of each pulse and τ is the interval between pulses. The energy constraint E and the total modulation duration T=nτ+ν are related via n=νE/π2. In the frequency domain, the spectral modulation intensity can be described by a series of peaks, where the two main peaks are at ±π/τ. Thus, the peaks are shifted in proportion to the energy invested in the modulation.(a) Single-Peak Resonant Dephasing Spectrum. This simple dephasing spectrum describes a common scenario where Φ(t)=e-t/tcγ/tc, where γ is the long-time dephasing rate [R(t→∞)=2πγ] and tc is the noise correlation time. Figure 6(a) shows R(T), normalized to the bare (unmodulated) dephasing rate, as a function of the energy constraint. As expected, the more energy is available for modulation, the lower is the dephasing rate. For low energies, the optimal modulation significantly outperforms DD, while at higher energies this difference disappears. These results can be understood from Figure 7(a), by noticing that the two central DD peaks have significant overlap with G(ω) at the low energy value shown. As E is increased at fixed T, the DD peaks move farther apart and have less overlap with G(ω), leading to improved performance. Applying the linearized EL equation with DD as initial guess yields only mild improvements (not shown). The explanation for the superior performance of the optimal modulation is also evident from Figure 7(a): since higher frequencies have lower coupling strength in this case, the optimal control “reshapes” so as to maximize its weight in the high-frequency range, to the extent permitted by the energy constraint. The modulation can be well approximated by Ω(t)=a[1+e-t/T(t/T-1)], where a is determined by the energy constraint, which fits the results in Figure 6(a).Average modified final decoherence rateR(T), normalized with respect to the unmodulated rate, as a function of energy constraint. DD—dash, magenta. Optimal modulation—solid, blue. Insets: optimal modulation Ω(t) for different energy constraints. (a) Single-peak resonant dephasing spectrum (inset: E=20). (b) Single-peak off-resonant spectrum (inset: E=50). (c) 1/f spectrum (inset: E=30). (d) Multipeaked spectrum (inset: E=30).
(a)
(b)
(c)
(d)Dephasing spectrumG(ω) (solid, red), optimal (dot-dash, green), and DD (dash, magenta) modulation spectra FT(ω), in arbitrary units (a.u.). Same parameters as in the insets of Figure 6.
(a)
(b)
(c)
(d)(b) Single-Peak Off-Resonant Dephasing Spectrum. This dephasing spectrum describes a variation on the aforementioned scenario, where the spectral peak is shifted (Δ≠0 in Φ(t)=Φ~(t)eiΔt), for example, coupling to a nonresonant bath. With no other constraints, the optimal modulation is trivially similar to the one of the resonant spectrum, with a shifted energy-constraint Enon-res=Eres+Δ. To demonstrate how this formalism can accommodate additional constraints, we impose a positivity constraint, ϕ˙(t)≥0 (positive field amplitude) and obtain nontrivial behavior of R(T) as a function of the energy constraint—see Figure 6(b). Here we used the linearized EL equation with the DD modulation (153) as an initial guess. For both the DD and optimal modulations, we observe an initial increase in the dephasing rate as a function of energy, followed by a decrease. For DD, this can be interpreted as a manifestation of the initial anti-Zeno effect and the subsequent quantum Zeno effect [100, 165]. Because of the positivity constraint, modulations with low energy (both DD and optimal) couple to more resonant modes of the bath (Figure 7(b)) and thus do worse than the unmodulated case. The DD modulation is optimal for small energy constraints, and hence the decoherence rates of DD and our optimal solution coincide. This is because the DD peaks do not overlap with the off-resonant spectral peak. However, as the positive-frequency main DD peak (Figure 7(b)) nears the off-resonant spectral peak, with increased energy, the optimal modulation diverges from the DD modulation and “reshapes” itself so as to couple to higher modes of the bath. In the time domain (Figure 6(b) inset), this is seen as a smoothing of the abrupt DD modulation. At even higher energy constraints, there is once more no improvement by the optimal modulation over DD, yet there is an improvement over the unmodulated case. Over the entire range of E, the optimal modulation results in a much flatter R(T) than DD, which is an indicator of its robustness. While DD is strongly influenced by the off-resonant peak, the optimal modulation exploits the energy available to find the minimal overlap with the dephasing spectrum.(c)
1
/
f
Dephasing Spectrum. The ubiquitous 1/f dephasing spectrum that describes a variety of experiments—for example, charge noise in superconducting qubits [166]—is given in our notation by G(ω)∝1/ω, with cutoffs ωmin and ωmax. Figure 6(c) shows that, as expected, the more the energy available for modulation, the lower the dephasing rate. Notice that due to the cut-off ωmin, a low-energy modulation does worse than the unmodulated case. Furthermore, since, as in case (a), higher frequencies now have lower coupling strength, the optimal control “reshapes” so as to have as high a weight in the high frequency range as the energy constraint allows (Figure 7(c)). This is expressed in the time domain (Figure 6(c) inset) as the initial increase in the modulation strength (t<50). The later decrease in modulation strength can be attributed to the lower cutoff, where the optimal modulation benefits from lower frequencies, that is, lower modulation amplitudes. Upon comparing the 1/f case to the Lorentzian spectrum, Figure 6(a), we observe a similar optimal initial chirped modulation in the time domain.Despite the differences in the long-time behavior (due to the lower cutoff in the1/f case), these two examples allow us to generalize to any dephasing spectrum with a monotonically decreasing system-bath coupling strength as a function of frequency. The optimal modulation for such spectra will be an energy-constrained chirped modulation, with modifications due to other spectral characteristics, for example, cutoffs.(d) Multipeaked Dephasing Spectrum. This describes the most general scenario, where there can be several resonances and noise correlation times. Figure 6(d) shows R(T) as a function of the energy constraint. Once again, because DD does not account for the dephasing spectrum, its performance is much worse than our (universal-formula) optimal modulation, whose “reshaping” results in monotonically improving performance: the peaks of our optimal modulation are predominantly anticorrelated with the peaks of G(ω).


## 8. Multipartite Decoherence Control and Quantum Computing

Multipartite decoherence control, for many qubits coupled to thermal baths, is a much more challenging task than single-qubit control since (i) entanglement between the qubits is typically more vulnerable and more rapidly destroyed by the environment than single qubit coherence [73, 74, 108–113] (ii) the possibility of cross decoherence, whereby qubits are coupled to each other through the baths, considerably complicates the control. We have analyzed this situation and extended [107, 123] the decoherence control approach to multipartite scenarios, where the qubits are either coupled to zero- or finite-temperature baths.Although our strategy applies in general to any number of excitations, simple closed-form solutions are obtainable for asingle initial excitation of the system. Accordingly, the complete wave function is
(154)|Ψ(t)〉=∑k‍α~0k(t)|k〉⨂j=1N|g〉j+∑j=1N‍∑n=1Mj‍α~j,n(t)|n〉j|vac〉⨂j′≠j|g〉j′.
We will denote the first and second RHS terms in (154) as the bath and system wave function, |ΨB(t)〉 and |ΨS(t)〉, respectively. In order to solve the Schrödinger equation, we shall follow the same procedure used for a single qubit in Section 3 (see (9)–(13)).In order to analyze the time evolution of the wave function, written as a column vectorα(t)={αj,n(t)}, it is expedient to express it in the interaction picture
(155)αj,n(t)=eiωj,nt+i∫0t‍dτδj,n(τ)α~n(t).
The Schrödinger equation for the coupled {αj,n(t)} and {α0k(t)} amplitudes in (154) may be reduced, upon eliminating the {α0k(t)} amplitudes and transforming to the interaction picture, to the following exact integrodifferential equation:
(156)α˙j,n(t)=∫0t‍dt′∑j′,n′‍Φjj′,nn′(t-t′)ϵj,n(t)ϵj′,n′*(t′)×eiωj,nt-iωj′,n′t′αj′,n′(t′).
Here Φ(t) is the reservoir-response matrix, given by
(157)Φjj′,nn′(t)=ħ-2∑k‍μk,j,nμk,j′,n′*e-iωkt,ϵj,n(t)=ϵ~j,n(t)ei∫0t‍dτδj,n(τ)
which accounts for both the modulation and the Stark shift.Assuming that the amplitudes are slowly varying on the time scale of the bath response [99] and using the matrix representation, this equation has the general solution
(158)α(t)=T+e-tR(t)α(0),
where T+ is the time-ordering operator and R(t)={Rjj′,nn′(t)} is the dynamically modified decoherence matrix, determined by the following convolution (see (23)–(27)):
(159)Rjj′,nn′(t)=2π∫-∞∞‍dωGjj′,nn′(ω)Ft,jj′,nn′(ω),(160)Gjj′,nn′(ω)=π-1∫0∞‍dtΦjj′,nn′(t)eiωt,(161)Ft,jj′,n′′(ω)=ϵt,j,n(ω-ωj,n)ϵt,j′,n′*(ω-ωj′n′)t,(162)ϵt,j,n(ω)=∫0t‍dτϵj,n(τ)eiωτ.
Here Gjj′,nn′(ω) is the coupling spectrum matrix given by the environment and Ft,jj′,nn′(ω) is the dynamical modulation matrix, which we design at will to suppress the decoherence. This general solution holds for dynamically modified relaxation of singly excited multipartite entangled states into a zero-temperature bath.It is important to note that both the coupling spectrum matrix,Gjj′,nn′(ω), and the dynamical modulation matrix, Ft,jj′,nn′(ω), are complex and not real, as in the single-qubit scenario. This occurs because, opposed to the single-qubit scenario, the modulation matrix off-diagonal term may be complex due to phase-relations between different modulations of the relevant particles. Hence, one cannot separate the real and imaginary parts of the coupling spectrum matrix and obtain a real decoherence matrix. Nevertheless, the diagonal elements of the decoherence matrix are easily separable to real decay rates, similar to (26), and imaginary Lamb shifts.In the following, we shall use as illustration an impulsive phase modulation, that is, a sequence of pulsed Stark shifts caused by fields whose amplitudes satisfy [99]
(163)ϵj,n(t)=ei[t/τj,n]θj,n,(164)ϵt,j,n(ω)=(eiωτj,n-1)(ei(θj,n+ωτj,n)[t/τj,n]-1)iω(ei(θj,n+ωτj,n)-1).
Here [⋯] denotes the integer part, and τj,n and θj,n are the pulse duration and the phase change for level n of particle j, respectively. In the limit of weak pulses, of area |θj,n|≪π, (164) yields ϵt,j,n(ω)≅ϵt,j,nδ(ω-Δj,n), where Δj,n=θj,n/τj,n is the effective spectral shift caused by the applied pulses.It is expedient to rewrite the fidelity of the evolving system,f(t)=|〈ΨS(0)|ΨS(t)〉|2, as a product of two factors
(165)f(t)=fp(t)fc(t),fp(t)=|A(t)|2=∑j=1N‍∑n=1Mj‍|αj,n(t)|2,fc(t)=|∑j=1N∑n=1Mjαj,n*(0)αj,n(t)|2|A(t)|2,
where fc(t) is the autocorrelation function, |〈ΨS(0)|ΨS(t)〉|2, normalized by the total excitation probability |A(t)|2. Thus 1-fp(t) measures population loss from any |n〉j, whereas 1-fc(t) is a measure of correlation preservation: fc=1 when the initial multipartite correlations are completely preserved. As shown below, population and correlation preservation can be independently controlled in the model of (154).In the absence of dynamical control,fc(t) decays much faster than fp(t) and is much more sensitive to the asymmetry between local particle-bath couplings. Thus, for initial Bell singlet and triplet states, |Ψ(0)〉=1/2(|g〉A|n〉B±|n〉a|g〉B), which do not experience cross decoherence but only different local decoherence rates, αA(B)(t)=1/2e-tRA(B)(t). We find fp(t)=(e-2tRA(t)+e-2tRB(t))/2; fc(t)=(1+C(t))/2=1/2+e-tΔR(t)/(1+e-2tΔR(t)), where ΔR(t)=RA(t)-RB(t) and C(t) is the concurrence [167].We shall first deal withN identical qubits  and thus ignore the n subscript, that is, set ωj≡ω0. We also require that at any chosen time t=T, the AC Stark shifts in (158) satisfy  ∫0T‍dτδj(τ)=2πm, where m=0,±1,…. This requirement ensures that modulations only affect the decoherence matrix (159)  but do not change the relative phases of the entangled qubits when their states are probed or manipulated by logic operations at t=T.Without any modulations, decoherence in this scenario has no inherent symmetry. Our point is that one cansymmetrize the decoherence by appropriate modulations. The key is that different, “local,” phase-locked modulations applied to the individual particles, according to (161), can be chosen to cause controlled interference and/or spectral shifts between the particles’ couplings to the bath. The Ft,jj′(ω) matrices (cf. (161)) can then satisfy 2N requirements at all times and be tailored to impose the advantageous symmetries described below. By contrast, a “global” (identical) modulation, characterized by Ft,jj′(ω)=|ϵt(ω)|2, is not guaranteed to satisfy N≫1 symmetrizing requirements at all times (Figure 9(a)).The most desirable symmetry is that ofidentical coupled particles (ICP), which would emerge if all the modulated particles could acquire the same dynamically modified decoherence and cross decoherence yielding the following N×N fully symmetrized decoherence matrix
(166)Rjj′ICP(t)=R(t)∀j,j′.
ICP would then give rise to a (N-1)-dimensional decoherence-free subspace (DFS), the entire single-excitation sector except the totally symmetric entangled state. An initial state in this DFS [115] would have f(t)=1 for all times, meaning that it would neither lose its population nor its initial correlations (or entanglement).However, it is generally impossible to ensure this symmetry, since it amounts to satisfyingN(N-1)/2 conditions using N modulating fields. Even if we accidently succeed with N particles, the success is not scalable to N+1 or more particles. Moreover, the ability to impose the ICP symmetry by local modulation fails completely if not all particles are coupled to all other particles through the bath, that is, if some Gjj′(ω) elements vanish.A more limited symmetry that we mayensure for N qubits is that of independent identical particles (IIP). This symmetry is formed when spectral shifts and/or interferences imposed by N modulations cause the N different particles to acquire the same single-particle decoherence r(t) and experience no cross decoherence. To this end, we may choose in (163) ϵt,j(ω)≃ϵt,jδ(ω-Δj). The spectral shifts Δj can be different enough to couple each particle to a different spectral range of bath modes so that their cross coupling vanishes as
(167)Rjj′(t)=ϵt,j*ϵt,j′∫dωG(ω)δ(ω-Δj)δ(ω-Δj′)⟶0.
Here, the vanishing of Gjj′(ω) for some j,j′ is not a limitation. The N single-particle decoherence rates can be equated by an appropriate choice of N parameters {Δj} as (Figure 8)
(168)Rjj′IIP(t)=|ϵt,j|2Gjj(Δj)=δjj′R(t),
where δjj′ is Kronecker’s delta (Figure 9(b)). The IIP symmetry results in complete correlation preservation, that is fc(t)=1, but still permits population loss, f(t)=fp(t)=e-2Rer(t). If the single-particle r(t) may be dynamically suppressed, that is, if the spectrally shifted bath response Gjj(ωj+Δj) is small enough, this f(t) will be kept close to 1 (Figure 10).Figure 8
Fidelity of the IIP symmetry for two TLS coupled to zero-temperature baths. The initial state is entangled,|ψ(0)〉=1/2(|g〉A|e〉B+|e〉A|g〉B).Two two-level particles in a cavity, coupled to the cavity modes (thin lines) and subject to local control fields (thick lines). ((a), (b)) Frequency domain overlap of coupling spectrum (dotted) and modulation matrix elements (solid), resulting in modified decoherence matrix elements (shaded), for (a) Global modulation (ICP symmetry) and (b) Cross-decoherence elimination (IIP symmetry). (c) General modulation scheme.
(a)
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(c)Fidelity as a function of time (in units of(10γ)-1): (a) Overall fidelity, (b) Correlation preservation, and (c) Population preservation. ((d)–(f)) Decoherence matrix elements at time t=100. (d) Global π-phase flips impose no symmetry (red) (τj,n=1.1,θj,n/π=1.0). (e) Independent identical particles (IIP) symmetry (blue) (τj,n=(0.75,0.85,0.95,1.05),θj,n/π=(0.834,0.806,0.836,0.82)); (f) Independent identical trapping (IIT) symmetry (green) (τj,n=(0.85,0.85,1.05,1.05),θj,n/π=(0.924,0.9,0.945,0.91)). The parameters are  ω1=0.5,ω2=0.6, γ=0.1, k0rmin=1.0, tj,n=(0.7,1.0,1.06,1.1), rj=(0.0,0.1),  and ηn/π=(0.0,0.1).
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—
(f)
—If IIP symmetry is imposed, the particles become effectively independent in terms of their coupling to the bath. Hence, collective coupling to the bath, which is a prerequisite for a DFS,is not formed in this case. In order to impose decoherence-free conditions, we extend the treatment to multilevel particles. We will show that the use of auxiliary levels and local modulations can result in a decoherence-free singly excitedNqubit system, in a general and realistic coupling scenario.To this end, we invoke three-level particles with excited states|1〉 and |2〉. In a single three-level particle, an external field can impose an intraparticle DFS if |1〉 and |2〉 decay at the same rate to the ground state |0〉; this DFS consists of |0〉 and the “dark” or trapping state [78], the anti-symmetric superposition of the two excited states 1/2(|1〉-|2〉). However, this intraparticle DFS would be destroyed in a system of N three-level particles that are coupled to the bath and/or experience cross decoherence. In order to remedy this problem, let us consider local modulations acting differently on the two excited levels within each particle. Such modulations can be tailored to impose what may be dubbed “independent identical trapping” (IIT) symmetry. This means that all particles acquire identical trapping states and become (effectively) independent, without cross decoherence. As discussed above, N spectral shifts Δj determined by |ϵt,j(ω)|2 suffice to eliminate cross decoherence, (167). Under these conditions, N destructively interfering pairs of local fields will cause trapping in each particle (Figure 9(c)). This would result in an N-dimensional DFS, composed of N particles sharing single excitation with anti symmetrically superposed excited states. Under the IIT symmetry, the decoherence matrix is block diagonal, each block corresponding to particle j with levels n,n′ as
(169)Rjj′,nn′IIT(t)=δjj′Rj(t).
Both disentanglement and population loss are nearly completely eliminated within the multipartite subspace, resulting in F(t) very close to 1 (Figure 10).As an example, the IIT recipe will be analyzed for two three-level particles, where each level of each particle experiences different coupling to the bath, and cross decoherence exists. Let us take the initial state of particlesA,B to be |ΨS(0)〉=(|-〉A|0〉B±|0〉A|-〉B)/2, where (j=A,B)|-〉j=(|1〉j-|2〉j)/2. The goal will be to keep this state intact, |ΨS(t)〉≃|ΨS(0)〉, by preventing cross decoherence and imposing intraparticle destructive interference. The bath response matrix will be taken to be Φjj′,nn′(t)=γdndn′(e-t2/4tj,n2e-t2/4tj′,n′2/k0(rmin+rjj′)) where γ is the single-particle unperturbed decay rate (satisfying Fermi’s Golden Rule [78]), dn=cosηn, with ηn being the angle of transition dipole, tj,n is the correlation time of level n of particle j, k0 is the wavevector of the resonant transition |0〉↔|1(2)〉, and rmin is the distance below which the particles are identically coupled to the bath and rjj′=|rj-rj′|, where rj is the position of particle j. This model may describe the distance-dependent cross-decoherence of either radiatively or vibrationally relaxing atoms or ions at different sites in a trap, lattice or cavity [133, 168]. The parameters of the pulse sequence in (163) for creating the IIP or IIT symmetries were chosen such that the faster decaying particle experienced the stronger and/or faster pulses; that is, θj,n/τj,n were adapted to the bath response parameters Φjj,nn of each particle. The pulse sequences were chosen to be different enough for each particle in order to eliminate the desired cross-decoherence terms, as per (167). Individual pulses were selected to obey the other IIP or IIT requirements described above. The very high fidelity achievable by IIT is seen in Figure 10.


### 8.1. Bipartite Entanglement and Decoherence Control

We shall henceforth analyze the case of bipartite systems,N=2, and quantify entanglement by the often-used measure of concurrence C [167], given by
(170)C(t)=max{0,λ1-λ2-λ3-λ4},
where λi are the positive eigenvalues, in decreasing order, of the matrix ρ(σy⊗σy)ρ*(σy⊗σy) where σy is the Y-Pauli matrix and the conjugation occurs in the computational basis {|gg〉,|ge〉,|eg〉,|ee〉}.In order to explore the effects of dynamical modulations on the concurrence, we first define the following decoherence rate parameters.(i)
Decoherence rate magnitude is defined by νe(g)(t)=∑j=1,2‍|Rjje(g)(t)|, where for the case of νe(g)(t)=0, we have no decoherence, with complete preservation of information and entanglement.(ii)
Decoherence rate asymmetry is defined as ζ(t)=|R11e(t)-R22e(t)|/νe(t), where ζ(t)=0 is the fully symmetric case, and ζ(t)>0 measures the asymmetry. The decoherence rate asymmetry expresses the difference between the two systems with respect to the modulated coupling to the bath. Thus, protecting one system and neglecting the other can increase the asymmetry while preserving the decoherence rate magnitude.(iii)
Cross decoherence is defined as μ(t)=|R12e(t)+R21e(t)|/νe(t), where μ(t)=1 denotes maximal cross-decoherence, and μ(t)=0 denotes completely separable, uncorrelated systems. The cross-decoherence expresses the “cross talk” between the two systems via the bath modes.(iv)
Decoherence rate effective temperature is defined as α(t)=1/ln(νe(t)/νg(t)), where for unmodulated memoryless Markovian identical baths α=kBT/ħωaWhat are the effects of dynamical control on the decoherence rate parameters? The decoherence rate magnitude, being the sum of all individual qubit decoherence rates, is best reduced by control schemes similar to those of single-qubit decoherence control, that is, reducing the overlap of the system-bath coupling and modulation spectra [80, 81, 84, 97]. The decoherence rate asymmetry, on the other hand, is best controlled by differently modulating the two TLS. One can increase the asymmetry by accelerating the decoherence of one qubit and reducing the other’s [85, 97, 99, 121, 153], or decreasing it by equating the individual-qubit spectral overlap of the bath-coupling and modulation spectra.Cross decoherence can be easily eliminated by choosing the appropriate modulations for each qubit. Thus, choosing for each qubit such modulations that they are effectively coupled to different bath modes can completely eliminate cross decoherence [123].The effective temperature cannot be independently controlled, as it is mainly governed by the bath temperature. However, ultrafast modulations can have very high spectral components, such thatωj,k+ωa<0, and can influence the ratio of upward (Rjjg) and downward (Rjje) transitions. Only the absence of the rotating-wave approximation in our analysis has allowed the elucidation of this effect, as “negative” frequencies are made accessible by modulations.


#### 8.1.1. Disentanglement and Its Resuscitation

Next, we analyze the disentanglement of initial fully entangled bipartite states, that is, the Bell-states,|Φ±〉=1/2(|e〉1|e〉2±|g〉1|g〉2), |Ψ±〉=1/2(|e〉1|g〉2±|g〉1|e〉2). The long-time limit concurrences of the Bell states for zero effective temperature are found to be
(171)C(|Φ±〉)μ=0=e-tν(1-(1-e-tν(1-ζ))(1-e-tν(1+ζ))),C(|Ψ±〉)ζ=0=e-tν(1±μ).
This demonstrates that an initial (pure) Bell state’s entanglement decays exponentially for zero-temperature baths  but does not vanish in finite time (the general case for μ,ζ≠0 is analytically solvable, but too cumbersome to be presented here). Conversely, coupling the two qubits to finite-temperature baths does result in entanglement sudden death [109, 112], that is, the concurrence vanishes in finite time, which we shall dub time of death (TOD), tTOD.Figure11 presents the TOD of an initial Bell state as a function of effective temperature and either decoherence asymmetry (Figure 11(a)) or cross decoherence (Figure 11(b)). Equation (171) and Figure 11 imply that the TOD can be postponed by applying modulations that (i) reduce the decoherence magnitude, (ii) increase the asymmetry for |Φ±〉, (iii) reduce (increase) the cross decoherence for |Ψ+〉 (|Ψ-〉), or, (iv) decrease the effective temperature. While the reduction of effective temperature is a formidable challenge, asymmetry and cross decoherence are better controlled via modulations. Thus, for an initial |Φ+〉, increasing the decoherence asymmetry of an otherwise symmetric system guarantees the same prolonged time of death as that of a colder environment (Figure 11(a) inset).Time of death (TOD) as a function of decoherence rate parameters. (a) TOD of an initial|Φ+〉 as a function of effective temperature (α) and decoherence rate asymmetry (ζ), μ=0. Inset: effective temperature resulting in νtTOD=4.6 as a function of asymmetry. (b) TOD of an initial |Ψ-〉 as a function of effective temperature (α) and cross-decoherence (μ), ζ=0.
(a)
(b)This Bell state disentanglement analysis results in recipes for optimal control schemes of quantum information protocols, for example, quantum teleportation. We have shown that the singlet is always optimal, but it is not always experimentally accessible. For such a case, one should use the|Φ±〉 states and dynamically modulate each qubit so as to eliminate the cross decoherence if present [71] and maximize the decoherence asymmetry, thus achieving maximal entanglement preservation.Moving from the long-time limit analysis, the analysis reveals that thetime-dependent decoherence rates may exhibit oscillations that can be controlled via the local modulations applied on the individual qubits. These controlled oscillations and their effects on the partial resuscitation of entanglement are the main results of this section. We show that one can control the frequency as well as the magnitude of oscillations by choosing the appropriate modulations and thus resuscitate entanglement at specific desired times.To demonstrate these effects, we shall investigate two distinct situations: (a) distant systems coupled to separate baths (ΦT,jj′(t)=0,j≠j′), whereby an initially entangled state is locally controlled and does not experience any cross decoherence (Figure 12(a)) and (b) adjacent systems coupled to the same bath, whereby local modulations may control the time dependent qubits’ cross decoherence (Figure 12(c)).(a) Schematic drawing of two locally modulated atoms in separated cavities, coupled to the cavity modes. (b) Concurrence as a function of time. Unmodulated (black) and modulated (blue) scenarios, with no cross decoherence,μ=0. Inset: decoherence rate magnitude ν(t) (solid) and asymmetry ζ(t) (dashed) as a function of time. Parameters: a=0.75, ωa=20/tc, ω0=0.8ωa, β=1/ωa, γunmod=0.032/tc, γmod=2.65γunmod, τ1,2/tc={0.175,0.1}, and ϕ1,2=π/10. (c) Schematic drawing of two locally modulated atoms in the same cavity, coupled to the cavity modes. (d) Concurrence as a function of time. Unmodulated (black) and modulated (blue) scenarios, with cross decoherence. Inset: cross-decoherence μ(t) (dotted) and asymmetry ζ(t) (dashed) as a function of time. Parameters: a=0.75, ωa=20/tc, ω0=ωa, β=10/ωa, γunmod=0.032/tc, γmod=1.1γunmod, τ1,2/tc={0.175,0.1}, ϕ1,2=π/10, and ν(t≫tc)=0.28/tc.
(a)
(b)
(c)
(d)Distant systems were extensively studied under various conditions [74, 108, 109, 112, 169, 170]. Here, we generalize these results under the universal dynamical control of decoherence approach and present new control schemes that facilitate entanglement sudden death and its controlled partial resuscitation (ESD-CPR).Both the decoherence rate magnitude and asymmetry exhibit controlled oscillations on non-Markovian time scales (Figure12(b) inset). This can be best observed by considering the most general quasistationary form of modulation, that is, quasi-periodic modulation, ϵj+(t)=∑k‍ϵj,ke-iωj,kt, where ωj,k are arbitrary discrete frequencies with minimal spectral distance Ω=min(ωj,k-ωj′,k′) for all j≠j′,k≠k′. In this case, the modulation spectrum, Ft,jj(ω), is a sum of terms peaked at ω=ωj,k that oscillate on non-Markovian time-scales [100, 105].Thus, local modulations can control not only the magnitude of disentanglement but also its period of oscillation, which may give rise to entanglement sudden death controlled partial resuscitation (ESD-CPR). To demonstrate this, we use the Werner states as the initial partially entangled states, given byρ(0)=(1/3)(2|Ψ+〉〈Ψ+|+(1-a)|gg〉〈gg|+a|ee〉〈ee|), where 0≤a≤1 [74, 113]. We numerically solve the modified Bloch equations for non-Markovian short time scales, and local (individual-qubit) impulsive-phase modulation, ϵj(t)=eϕj[t/τj], where [⋯] is the integer-part. We consider finite-temperature baths, with a Lorentzian bath-coupling spectra G0,jj(ω)=γj(Γj2/(Γj2+(ω-ω0)2)), where γj is the coupling strength of TLS j to the bath, ω0 is the center of the Lorentzian, Γj=1/tcj, and G0,jj′(ω)=0. Here γj=γ and Γj=Γ are taken to be equal for the two TLS, thus having no asymmetry for the unmodulated scenario (ζ(t)=0).While the modulation can greatly decrease the decoherence rate magnitude, and thus maintain entanglement for longer durations, we have already shown that its effect is similar to that of the single-qubit decoherence case and is of less interest here. Hence, we present renormalized results such that the long-time limit ofν(t) is equal for all scenarios to that of the unmodulated case (Figure 12(b) inset). This is achieved by effectively changing the coupling strength γ for each scenario.Figure12(a) presents the case with no cross decoherence, for example, two atoms in remote cavities. While the concurrence of the unmodulated scenario with zero asymmetry rapidly dies, the non-Markovian oscillation of the asymmetry due to the modulation results in ESD-CPR (Figure 12(b)). The rise of concurrence from the dead is substantial and can occur only at a time comparable to the correlation time of the bath tc. During this time, the entanglement between the two TLS is “transferred” to the bath modes, which still retain some memory. The oscillations of the asymmetry facilitate the return of entanglement back to the two TLS. Notably, ESD-CPR is achieved here by local modulations; that is, the two TLS are not coupled through the control fields [108] nor through the bath modes [73, 110]. This novel result, of controlling entanglement oscillations via local modulations, exemplifies the importance of a unified theory of non-Markovian time-scales multipartite decoherence control.Next, we analyze the case of systems experiencing cross decoherence due to coupling to the same bath (Figure12(c)). While this scenario has been less studied [73, 110], it demonstrates long (Markovian) time-scale controlled oscillations that result in ESD-CPR. To investigate this scenario, we set G0,jj′(ω)=G0,jj(ω)G0,j′j′(ω) and define cross decoherence as μ(t)=|R12e(t)+R21e(t)|/νe(t), where μ(t)=1 denotes maximal cross decoherence, and μ(t)=0 denotes effectively separate systems. The cross decoherence expresses the locally modulated “crosstalk” between the two systems via the bath modes (Figure 12(c)).As was previously discussed, the cross decoherence may vanish due to an imposed coupling of the two TLS to different bath modes. This becomes apparent for quasi-periodic modulation in the long-time limit,Ωt≫1, where the cross-modulation spectrum, Ft,jj′(ω) becomes a series of δ(ωj,k-ωj′,k′). However, this long-time limit for the cross decoherence terms is reached only after the inverse of the minimal spectral distance of the two modulations, that is, t≫1/min(ωj,k-ωj′≠j,k′). This limit can be very long for similar, yet not equal, modulations, thus enabling a long oscillatory behavior of the cross decoherence (Figure 12(d) inset).Figure12(d) presents the effects of cross decoherence on concurrence. While it has been shown that some states (e.g., singlet) may benefit from cross decoherence [71], the Werner state’s concurrence dies faster due to this “cross-talk.” Applying different modulations on each TLS results in the reduction of cross decoherence and the increase of asymmetry, both beneficial for entanglement preservation. Furthermore, oscillations of the cross decoherence are long-lasting as the two modulations’ peaks are close to one another, and can periodically kill and resuscitate the concurrence even in Markovian time scales (t≫tc).This phenomenon can only be investigated by a general model that incorporates both arbitrary time-dependent control and the entire range of cross decoherence, from separate to adjacent systems. The universal dynamical disentanglement approach presented here is, to the best of our knowledge, the first that meets these requirements.To conclude, the ESD-CPR scenario discussed has many aspects: (a) non-Markovian time-scale behavior [112, 169–171], (b) asymmetry between the two decoherence channels [108, 109, 169–172], (c) cross-decoherence of the two-qubits [73, 110, 171, 172], (d) finite temperature effects [109, 112], and (e) time-dependent control [108, 112, 170]. The universal dynamical control of disentanglement presented here generalizes all these aspects, as it describes arbitrary dynamical time-dependent control. Only by combining all these aspects into one formalism, one can fully exploit the control attained by locally modulating entangled systems, namely, oscillations of entanglement with controlled frequencies and amplitudes.While postponing ESD by reducing the decoherence rate requires control schemes similar to those proposed for single-qubits, facilitating ESD-CPR by dynamically controlling the asymmetry and cross decoherence requires more elaborate local modulations. Based on these results, new control strategies for disentanglement reduction of quantum information processing can be suggested. By tailoring optimal control schemes for each scenario, one can locally control the entangled systems and thereby facilitate ESD-CPR at specific times. This may prove beneficial for many quantum computation and communication protocols.


### 8.2. Multipartite Decoherence Control

Here we address the problem of maintaining the fidelity of a multipartite entangled state, after it has been prepared [124, 173, 174]. In the context of one-way quantum computing [175], fidelity protection is more important than preventing disentanglement [126], since maintaining the specific initial state is essential for the success of the computation. Designing high-fidelity gates that are resistant to decoherence was addressed by us elsewhere [125] and its scaling with the number of qubits is beyond the scope of this paper.We pose and answer the basic questions. How does decoherence scale with the number of qubits and is it state dependent? Can dynamical control combat decoherence and its scaling with the number of qubits as this number becomes large? What time scales are required for such control?Fidelity in such systems can be protected by very general, not necessarily pulsed, dynamical control of decoherence [99, 100, 104, 118, 123–126, 174, 176]. In this approach, fields applied to the system can maintain its coherence, as long as their effects are faster than the correlation (memory) time of the environment, that is, they act on non-Markovian time scales. Thus far, single-qubit dynamic decoherence control has received more attention [99, 100, 104] than multipartite control [123, 174].We show that by locally (specifically) modulating each qubit, one can drastically suppress the decoherence and its scaling through modifying the strength and number of effective decoherence channels. Our analysis yields the surprising result that a modulation which actsafter the multipartite coherence time has decayed can still restore coherence to acceptable levels, if it is within the memory time of the bath, irrespective of the number of particles.


#### 8.2.1. Scenarios

Our system is comprised ofN qubits whose ground and excited states are |g〉j and |e〉j, respectively, with j=1,…,N and energy separation ħωa (Figure 1). Each qubit may be differently (weakly) coupled to a bath (henceforth we take ħ=1) as
(172)H(t)=HS+HB+HI+HC(t),HS=ωa∑j=1N‍|e〉jj〈e|,HI=∑j=1N‍S^jB^j,
where HS, HB, and HI are the system, bath, and interaction Hamiltonians, respectively, HC(t) is the time-dependent control Hamiltonian, with B^j being the jth qubit bath operator and S^j the system-bath coupling operator of qubit j. We neglect direct interactions among bath modes and focus on scenarios where two or more qubits collectively couple to common bath modes (see Figure 13).Figure 13
Multipartite systems (black) coupled to their environments (red) (color online). (a) Relaxation of identical qubits via coupling to thesame bath modes causes cross decoherence. (b) Different AC-Stark shifts on individual qubits couple them to different bath modes, eliminating cross decoherence. (c) Different Rabi driving of dephasing qubits is analogous to (b).Two kinds of decoherence can be similarly treated: (i)relaxation [123], caused by a system-bath coupling Hamiltonian that is off diagonal in the energy basis, with S^j=|e〉jj〈g|+|g〉jj〈e| being the X-Pauli matrix of the jth qubit. Relaxation involves temperature-dependent level-population change, as well as erasure (scrambling) of the phase relations. Such a process is controllable if each qubit is subject to a strong but off-resonant driving field, ħηj(t), that causes an AC Stark shift (Figure 1(b)), that is, energy modulation of the system Hamiltonian [174], as
(173)HC(t)=∑j=1N‍ηj(t)|e〉jj〈e|.
(ii) Dephasing [174] is caused by a system-bath coupling Hamiltonian that is diagonal in the energy basis, with S^j=|e〉jj〈e|-|g〉jj〈g| being the Z-Pauli matrix of the jth qubit. Dephasing does not change level populations but only erases (scrambles) the phases. Such a process is controllable by near-resonant Rabi driving fields, Vj(t), that cause transitions between qubit levels,
(174)HC(t)=∑j=1N‍(Vj(t)e-iωat|e〉jj〈g|+h.c.).We first consider the unmodulated scenario, for which the ME becomes(175)ρ^˙=∑j,j′=1N‍[rjj′(t)σ^jxρ^σ^j′x-rjj′(t)σ^j′xσ^jxρ^00000-rjj′*(t)ρ^σ^j′xσ^jx+rjj′*(t)σ^j′xρ^σ^jx],
where rjj′(t) are the instantaneous decoherence rates. One can formally integrate (175) and get
(176)ρ^(t)=ρ^(0)+∑j,j′=1N‍∫0t‍dt′[rjj′(t′)σ^jxρ^(t)σ^j′x000000000000-rjj′(t′)σ^j′xσ^jxρ^(t)000000000000-rjj′*(t′)ρ^(t)σ^j′xσ^jx000000000000+rjj′*(t′)σ^j′xρ^(t)σ^jx].We are interested in the fidelity,f(t)=Tr(ρ(0)1/2ρ(t)ρ(0)1/2), of any initially pure state |ψ〉. To second-order in the system-bath coupling, we obtain from (176) the short-time form of f(t) as
(177)fψ(t)=1-tΓψ(t;N),(178)Γψ(t;N)=∑j,j′=1N‍Rjj′(t)(〈ψ|σ^jxσ^j′x|ψ〉0000000000-〈ψ|σ^jx|ψ〉〈ψ|σ^j′x|ψ〉),(179)Rjj′(t)=2t∫0t‍dt′rjj′(t′),
where Γψ(t;N) is the infidelity rate and Rjj′(t) are elements of the (time-averaged) decoherence-rate matrix discussed below. While (177) and (178) are valid for all initial pure states, we henceforth focus our analysis on specific entangled multipartite states and discuss their decoherence scalability.


### 8.3. Decoherence Scalability

#### 8.3.1. Multipartite Entangled States

We treat decoherence scaling withN for several distinct multipartite entangled states with clear N-dependence.  (i)
GHZ states,|GHZ〉=(1/2)(⨂j|e〉j+⨂j|g〉j).(ii)
W-states,|W〉=(1/N)∑j=1N‍(|e〉j⨂j≠j′|g〉j′).(iii)
Bright states,|B〉=|J=N/2,m=0〉, where |J,m〉 are the Dicke states [177], with J being the total angular momentum and m the excitation number.(iv)
Dark states, given by|D〉=⨂j=2,4,...,2N|Ψ-〉j-1,j for even N, where |Ψ-〉j,j+1=(1/2)(|g〉j|e〉j+1-|e〉j|g〉j+1).Equation (178) shows that the infidelity rate depends on the specific multipartite state and on the decoherence rates Rjj′(t). The time dependence of these rates is revealed on non-Markovian time scales, yet they are constant on long (Markov) time scales. The diagonal elements of the decoherence-rate matrix, Rjj(t), correspond to single-qubit decoherence, determined by the coupling of individual qubits (labeled by j) to their respective environments (baths). On the other hand, the off-diagonal elements, Rjj′(t) with j≠j′, represent cross-decoherence rates, that is, decoherence rates induced on the jth qubit by the j′th qubit. Such cross-decoherence rates are present only for qubits coupled to a common bath or if their dephasing noise is correlated. More explicitly, cross decoherence occurs because two qubits are coupled to the same bath (noise) mode (Figure 1(a)). In the second-order approximation, this means that qubit j interacts with a certain bath (or noise) mode and then the same mode interacts with qubit j′ (cf. (178)). In the dephasing scenario, cross dephasing stems from the cross correlation of the stochastic noise functions of qubits j and j′. Thus, for example, ions subject to the same fluctuating magnetic field will experience cross dephasing.In the absence of cross decoherence, when each qubit is locally coupled to an individual bathRjj′(t)=0, for all j≠j′, the first state-dependent term of the infidelity rate in (178) scales linearly with N, for all states. The second term then appears only for unique states, for example, W-state, and can only lower the N scalability (Table 1), resulting in (at most) linear dependence on N for all states in systems without cross decoherence.Table 1
N-scalability of entangled states decoherence for relaxation and dephasing scenarios, with and without cross decoherence.
Relaxation
Dephasing
No X-deco.
X-deco.
No X-deco.
X-deco.
GHZ
N
N
N
N
2
W-state
N
3
N
-
2
4
(
N
-
1
)
/
N
0
Bright state
N
N
+
N
2
/
2
N
0
Dark state
N
0
N
0By contrast, the infidelity rate is much morestate-dependent in systems with cross decoherence, since these systems have additional decoherence channels that may interfere, either constructively or destructively, depending on the initial entangled state via the phases of 〈ψ|σ^jxσ^j′x|ψ〉. Hence, the N scalability under cross decoherence can either mount to N2 or vanish completely, a situation unattainable in the absence of cross decoherence.More specifically, under global dephasing, the GHZ state becomes a bright state in the|±〉 basis and hence experiences constructive interference of all decoherence channels, so that its dephasing scales quadratically with N. The W-state and bright state, on the other hand, do not experience global dephasing since they are composed of a symmetric superposition of identically excited states and hence the dephasing becomes a global phase factor and does not influence decoherence. Remarkably, the dark state considered here is dark for global relaxation and dephasing, due to the invariance of the |Ψ-〉 state under the basis rotation from |e(g)〉 to |±〉. For higher-order approximation this may not be valid.


#### 8.3.2. Decoherence Scaling and the Zeno Regime

One may view decoherence scalability from a complementary aspect, that of the coherence time of the system. In this section, we consider an unmodulated scenario with identical coupling to all qubits, that is,(180)Γψ(t;N)=cψ(N)R(t),
where cψ(N) is the state-dependent scalability. We define the coherence time as the time at which the fidelity (177), or alternatively the error, e(t)=1-f(t), reaches a certain threshold as
(181)tcohΓψ(tcoh;N)=1-fthreshold≡ethreshold.
This coherence time is an important measure, since its ratio to the time it takes to perform a single quantum operation (e.g., a two-qubit gate) [3] determines how many such operations can be performed successfully, that is, the complexity of the quantum computation.The coherence time is usually identified with the inverse decoherence rate, since in the Markovian time-regime, the decoherence rate is constant, that isR(t)=R, yielding
(182)tcoh(N)|Markov=ethresholdcψ(N)R.
For example, linearly scaling systems cψ(N)=N, such as GHZ relaxation without cross decoherence, have Markovian coherence time that scales inversely with N. However, for large systems, the fidelity threshold (181) is reached according to (177) after a time that may be much shorter than the correlation (memory) time of the bath, tcorr, so that the non-Markov time dependence of R(t) starts playing a role. For still larger N, this threshold will be reached within the extremely short Zeno time-regime, tZeno [86, 97, 99, 102, 178, 179], in which the decoherence rate scales linearly with t, RZeno(t)=tR˙ (see below). Hence, for sufficiently large systems, such that
(183)cψ(N)≥ethresholdtZeno2R˙
the coherence time is always that obtained in the Zeno time-regime as
(184)tcoh(N)|Zeno=ethresholdcψ(N)R˙.
If cψ(N)=N, the Zeno coherence time scales as the inverse square root of N. This paradoxical result, namely, the inevitable improvement of the decoherence scaling law for sufficiently large systems, has been thus far overlooked. This result is a manifestation of the non-Markovian effects in decoherence dynamics.


#### 8.3.3. Dynamically Modified Decoherence Rates

From (178), it is evident that the decoherence rates Rjj′(t) are an important factor in the scaling of decoherence. Their time dependence stems from the non-Markovian master equation (ME) [126]. The same non-Markovian ME governs dynamical decoherence control. The resulting (time-averaged) decoherence rates then assume the form of spectral overlaps of the bath coupling spectrum matrix, Gjj′(ω), and the modulation power-spectrum matrix, Ft,jj′(ω) [174] as
(185)Rjj′(t)=2πRe∫-∞∞‍dωGjj′(ω)Ft,jj′(ω),(186)Ft,jj′(ω)=1tϵt,j(ω)ϵt,j′*(ω),
where Gjj′(ω) is the Fourier transform of the system-bath correlation matrix Φjj′(ω), and Ft,jj′(ω) is the time-dependent normalized modulation spectrum, where ϵt,j(ω)=(1/2π)∫0t‍dt′ϵj+(t′)eiωt′ is the finite-time Fourier transform of the modulation function. In the relaxation scenario, the modulation spectral function represents the effective change in qubit j’s energy splitting or AC Stark shift (Figure 13(b)). In the dephasing scenario, it determines the time-dependent Rabi splitting of the |±〉 states (Figure 13(c)). In both scenarios, the spectral overlap Gjj′(ω)ϵt,j*(ω)ϵt,j′ω) in (185) represents the effective coupling between the systems and the bath, whereby the dynamical control (modulation) determines which spectral bath modes the qubits are coupled to.There is an important distinction between the controllability of single-qubit decoherence rates,Rjj(t), and that of their cross-decoherence counterpart, Rjj′(t). Namely, the single-qubit dynamically modified Rjj(t) can only be reduced by decreasing the overlap between the modulation spectra |ϵt,j(ω)|2 and the bath coupling spectra, Gjj(ω), both positive. No matter how fast the modulation is, |ϵt,j(ω)|2 cannot decrease the single-qubit decoherence rate, if the bath coupling spectrum is spectrally flat (Markovian). By contrast, cross-decoherence rates Rjj′(t) can be completely eliminated by choosing spectrally distinct local modulations whose spectral overlap ϵt,j*(ω)ϵt,j′(ω) vanishes, irrespective of the bath coupling spectrum, Gjj′(ω) even if it is Markovian (spectrally flat) (Figures 2(a) and 2(b)). The qubits effectively couple to orthogonal bath modes. Hence, by modulating each qubit differently, one can effectively decouple the qubits, as if they were interacting with separate baths.


#### 8.3.4. Dynamically Modified Decoherence Scaling

How do these modulations affect decoherence scaling? Let us examine two limits of the general scenario more closely.(A)
Consider first the limit oflocally decohering qubits. Due to the absence of a common bath, modulations cannot induce cross decoherence. Hence, they cannot change the scaling law, which is identical for most multipartite entangled states, namely, Γψ(t;N)∝N (Table 1). Nevertheless, modulations can still improve the infidelity rate, by decreasing the overlap of each qubit’s bath coupling and modulation spectra.(B)
The other limiting case is that of collective decoherence of qubits coupled to a common bath, but locally controlled. By exploiting the spectral distinction ofN different local modulations, one can completely eliminate cross decoherence and effectively render the system similar to the aforementioned locally decohering qubits, in which for most states, decoherence scales linearly with N. Such local modulations may be advantageous for the relaxing bright state, or dephasing GHZ-state, by reducing its infidelity rate scaling by a factor of N.On the other hand, one may enhance cross decoherence by local modulations: if the qubits are coupled to the same bath but with different strengths, then the modulations can be chosen to compensate for their unequal coupling strengths. Such enhanced cross decoherence can drastically improve the coherence time by rendering certain states completely dark: without modulation, partially dark states with unequal coupling have infidelity rate that scales asN, whereas under modulations that impose destructive interference of the decoherence channels, such states can acquire much longer coherence time by becoming fully dark.


#### 8.3.5. Effective Modulation Time Scales

When do such local dynamic modulations become effective? Let us recall the relevant time scales. The longest one is the coherence time of a single-qubit without modulation,tcoh(1)≅1/Γψ. Another is the correlation (memory) time of the bath, tcorr. It is the decay time of the correlation function, which is given by the product of the noise amplitude at time t with the noise amplitude at time t+τ, averaged over realizations. In the frequency domain 1/tcorr is the width of the noise-coupling spectrum and in dephasing noise this is the noise bandwidth.In the single qubit case, the weak-coupling assumption (see below) implies that the correlation time is shorter than the single-qubit coherence time [70] tcorr≪tcoh(1) or, equivalently, noise bandwidth much larger than dephasing rate [174]. In the unmodulated N-qubit scenario, this assumption is given by tcorr≪tcoh(N), whereas in the modulated case, it is the modulated coherence time that sets the limit for the weak coupling regime, tcorr≪tcohmod(N). Hence, even if for a large (unmodulated) system the weak coupling assumption does not hold, modulations that drastically increase the coherence time, for example, by eliminating cross decoherence, can make the system enter the valid regime for which the analysis holds.Only modulations faster than the bath-(noise-) correlation time can affect the single-qubit decoherence ratesRjj(t). Yet a crucial insight transpires: if each qubit decoheres individually (cross decoherence is eliminated), the (unmodulated) N-particle coherence time does not determine whether the control is effective. Neither does the infidelity rate (178). Rather, the modulation period, τ, must be shorter than tcorr, but can be much larger than tcoh(1)/N: it is unaffected by scaling!We can now summarize the conditions for an effective increase of the coherence time of entangled multipartite systems by modulations with a periodτ as
(187)tcoh(N)<︸aτ<︸btcorr<︸ctcoh(1)<︸dtcohmod(N).
Here, (a) lays out the striking result that effective modulation can be slower than the coherence time of the system; (b) is the major requirement for effective modulation; (c) is the single-qubit weak-coupling assumption; (d) shows the main result, that is, the possibility of attaining (via effective modulations) multipartite coherence time tcohmod(N) that is longer than the single-qubit coherence time.


#### 8.3.6. Application to an Ion Trap

Our analysis is applicable to a broad variety of experimental setups. As an example, consider a string of ions in a linear trap [142, 180, 181], in which the qubits are encoded by two internal states of each ion (|g(e)〉j) (Figure 14(c)). Fluctuating magnetic fields give rise to a collective fluctuation of qubit energies, thereby causing cross dephasing. Modulations that locally address each ion (j=1,…,N) may be realized by laser fields whose amplitude varies spatially and differently affects the jth “carrier” frequency ωj(t) of the transition |g〉j↔|e〉j: a gradient of the laser field amplitude along the ion chain would suffice to achieve this.Figure 14
Fidelity as a function of time for entangled states experiencing collective relaxation under different modulations compared to that of a single qubit without modulation (blue, dashed) (color online): bright-state,N=10 without modulation (red, dot-dashed); GHZ with N=10 without modulation (green, dotted); bright-state with N=10 under asymmetric modulation (black, solid). (a) Decoherence rates for N=3 with asymmetric modulation. Single-qubit decoherence rates, Rjj(t) (black, solid), undergo oscillations then attain constant nonzero values at t≫tcorr; cross-decoherence rates Rjj′(t) (blue, dashed) also oscillate but then decay to zero at t≫tcorr. (b) For qubits j and j′, the overlap of the bath coupling spectrum Gjj′(ω) (red area) and the modulation spectra ϵt,j(ω) and ϵt,j′(ω) (green areas) determine the decoherence rate. If the qubits undergo (asymmetric) different modulations ϵj(ω)≠ϵj′(ω), their lack of overlap eliminates cross decoherence. (c) Addressing individual ions in a trap by different control fields with different amplitudes induces asymmetric Rabi flippings and thereby eliminates cross decoherence.Each local modulation may decrease theindividual decoherence rate of each qubit by at least an order of magnitude, depending on the noise spectrum and available modulation frequencies. More importantly, local modulations may eliminate cross dephasing between ions, thereby increasing the coherence time of bright or GHZ states by a factor of N. For an ion trap with tcoh(1)=14.7s, (R≥10-2s-1) [181], local modulations at intervals τ≅1ms can drastically increase the coherence times of GHZ or bright states, for example, tcohmod(10)≅tcoh(1)≅10s (Figure 14).


### 8.4. Validity of the Weak-Coupling Approximation

#### 8.4.1. Thermal Baths

For coupling to a thermal bath, the first-order term is proportional toTr〈HIBρB〉, where HIB is the bath part of the coupling Hamiltonian and ρB is the density matrix of the bath’s thermal state. In the scenario considered here, HIB has only off-diagonal terms in the bath eigenbasis. For example, HIB=∑k‍(λkak+h.c.) for coupling to an oscillator bath, where λk is the coupling coefficient to mode k and ak is the annihilation operator. This form renders the first-order term zero.


#### 8.4.2. Validity of the ME

The master equation, (98), is valid as long as the density matrix does not change much within the correlation (memory) time of the bath, tcorr. In the entangled multipartite scenarios discussed here, this condition becomes tcorr∑j,j′=1N‍Rjj′(tcorr)≪1.


#### 8.4.3. Validity of the Fidelity

The fidelity derived above, (177) requires a stricter condition t∑j,j′=1N‍Rjj′(t)≪1. Several important issues are raised by this approximation. The first is that it is N-dependent, meaning the time in which the approximation holds becomes shorter for larger systems. Furthermore, the N-dependency of the approximation depends on whether there is cross decoherence. With cross decoherence, the time within which the approximation holds depends on N2, whereas without cross decoherence it depends on N.In Figure14, one can see that the single-qubit fidelity is within the validity domain of the approximation, whereas the fidelity of entangled states with N=10 qubits without modulation becomes smaller and exit the validity regime rapidly. However, by applying different modulations on the qubits, one can not only reduce each qubit’s decoherence rate, but also  eliminate the cross decoherence and increase the fidelity by N=10-fold, thus bringing it within the validity region of the weak-coupling approximation.


### 8.5. Fixed-Time Approach

Assume that a quantity of interest (“score”) can be written as a real-valued functionP(t)=P[ϱ^(t)] of the system state ϱ^(t) at a given time t. This might be, for example, a measure of performance of some input-output device that is supposed to operate within a predefined cycle/gate time t. Depending on the physical problem and model chosen, extensions and generalizations are conceivable, such as a comparison of the outcome for different t (on a time scale set by a constraint) [138], a time average P=∫‍dτf(τ)P[ϱ^(τ)] with some probability density f(τ) [182], or a maximum P(t)=maxτ∈[0,t]P[ϱ^(τ)] [183]. Here, we restrict ourselves to the “fixed-time” definition as given above. Our goal is to generate, by means of classical control fields applied to the system, a time dependence of the system Hamiltonian within the interval 0≤τ≤t that adjusts P(t) to a desired value. In particular, this can be an optimum (i.e., maximum or minimum) of the possible values of P. Assume that the initial system state ϱ^(0) is given. The change to the “score” over time t, P=P(t)-P(0), is then given by the first-order Taylor expansion in a chosen basis
(188)P≈∑mn‍∂P∂ϱmnΔϱmn=Tr(P^Δϱ^),
where Δϱ^=ϱ^(t)-ϱ^(0) and the expansion coefficients
(189)(∂P∂ϱmn)t=0≡(P^)nm
are the matrix elements (in the chosen basis) of a Hermitian operator P^, which is the gradient of P[ϱ^(t)] with respect to ϱ^ at t=0; that is, we may formally write P^=(∇ϱ^P)t=0T=(∂P/∂ϱ^)t=0T. In what follows, it is P^ which contains all information on the score variable. Note that the transposition applied in (189) simply allows to express the sum over the Hadamard (i.e., entrywise) matrix product in (188) as a trace of the respective operator product P^Δϱ^.Let us illustrate this in two examples. IfP is the expectation value of an observable (i.e., Hermitian operator) Q^, so that P=Tr(ϱ^Q^), then (189) just reduces to this observable  P^=Q^. If P is the state purity, P=Tr(ϱ^2), then (189) becomes proportional to the state, P^=2ϱ^(0). Note that the score P is supposed to reflect the environment (bath) effects and not the internal system dynamics.Equation (188) implies that Δϱ^ and with it P are small. Hence Δϱ^ must refer to the interaction picture and a weak interaction, while P[ϱ^(t)] should not be affected by the internal dynamics (so that no separate time dependence emerges in (188), which is not included in the chain-rule derivative). In the examples above, this is obvious for state purity, whereas an observable Q^ might be thought of as coevolving with the internal dynamics.Our starting point for what follows is simply the relationP=Tr(P^Δϱ^) with some Hermitian P^, whose origin is not relevant.


### 8.6. Role of Averaged Interaction Energy

Equation (188) expresses the score P as an overlap between the gradient P^ and the change of system state Δϱ^. In order to find expressions for P in terms of physically insightful quantities, we decompose the total Hamiltonian into system, bath, and interaction parts,
(190)H^(t)=H^S(t)+H^B+H^I,
and consider the von Neumann equation of the total (system and environment) state in the interaction picture,
(191)∂∂tϱ^tot(t)=-i[H^I(t),ϱ^tot(t)],
(H^I(t)=U^F†(t)H^I(S)(t)U^F(t), (S) denoting the Schrödinger picture and U^F(t)=T+e-i∫0t‍dτ[H^S(τ)+H^B]). Its solution can be written as Dyson (state) expansion
(192)ϱ^tot(t)=ϱ^tot(0)+(-i)∫0t‍dt1[H^I(t1),ϱ^tot(0)]+(-i)2∫0t‍dt1∫0t1‍dt2[H^I(t1),[H^I(t2),ϱ^tot(0)]]+⋯,
which can be obtained either by an iterated integration of (191) or from its formal solution
(193)ϱ^tot(t)=U^I(t)ϱ^tot(0)U^I†(t),U^I(t)=T+e-i∫0t‍dt′H^I(t′),
by applying the Magnus (operator) expansion
(194)U^I(t)=e-itH^eff(t),(195)H^eff(t)=1t∫0t‍dt1H^I(t1)-i2t∫0t‍dt1∫0t1‍dt2[H^I(t1),H^I(t2)]+⋯,
expanding in (193) the exponential U^I=U^I(H^eff) and sorting the terms according to their order in H^I.We assume that, initially, the system is brought in contact with its environment (rather than being in equilibrium with it), which corresponds to factorizing initial conditionsϱ^tot(0)=ϱ^(0)⊗ϱ^B. The environment is in a steady state ϱ^B, [ϱ^B,H^B]=0, so it is more adequate to speak of a “bath.” Tracing over the bath in (192) then gives the change of system state Δϱ^=ϱ^(t)-ϱ^(0) over time t, which we must insert into (188). We further assume a vanishing bath expectation value of the interaction Hamiltonian,
(196)〈H^I〉B≡TrB(ϱ^BH^I)=0^.
As a consequence, the “drift” term corresponding to the first order in (192) vanishes, and we only consider the second order term as the lowest nonvanishing order approximation. Finally, we assume that the initial system state commutes with P^,
(197)[ϱ^(0),P^]=0.
In the language of control theory, Tr[ϱ^(0)P^] is a “kinematic critical point” [184] if (197) holds, since Tr[eiH^ϱ^(0)e-iH^P^]=Tr[ϱ^(0)P^]+iTr(H^[ϱ^(0),P^])+𝒪(H^2) for a small arbitrary system Hamiltonian H^. Since we consider ϱ^ in the interaction picture, (197) means that the score is insensitive (in first order) to a bath-induced unitary evolution (i.e., a generalized Lamb shift) [140]. The purpose of this assumption is only to simplify the expressions, but it is not essential. Physically, one may think of a fast auxiliary unitary transformation that is applied initially in order to diagonalize the initial state in the eigenbasis of P^.To the lowest (i.e., second) order, we then evaluate (188) for the score change as
(198)P=t2〈[H^,P^]H^〉,H^=1t∫0t‍dτH^I(τ),
where 〈·〉=Tr[ϱ^tot(0)(·)]. This expresses the change of score in terms of the interaction Hamiltonian, averaged in the interaction picture over the time interval of interest. Our scheme is summarized in Figure 15.Figure 15
Our control scheme (color online): a system is brought in contact with a bath over a fixed time interval0≤τ≤t during which the time dependence of the system Hamiltonian H^S(τ) is chosen such that a given system variable P is adjusted to a desired value at the final time t.


### 8.7. Spectral Overlap

Alternatively, (198) can be written as an overlap of system and bath matrices, which allows a more direct physical interpretation. To do so, we assume d dimensional Hilbert space  and expand the interaction Hamiltonian as a sum of products of system and bath operators,
(199)H^I=∑j=1d2-1‍S^j⊗B^j,
in such a way that 〈B^j〉=0, which ensures that (196) is satisfied (otherwise we may shift B^j′=B^j-〈B^j〉I^, H^S′=H^S+∑j‍〈B^j〉S^j). Considering (199) in the interaction picture and expanding S^j(t)=∑k‍ϵjk(t)S^k in terms of (Hermitian, traceless, orthonormalized to Tr(S^jS^k)=dδjk) basis operators, S^j defines a (real orthogonal) rotation matrix ϵ(t) in the system’s Hilbert space, with elements
(200)ϵjk(t)=〈S^j(t)S^k〉id,
where 〈·〉id=Tr[d-1I^(·)]. These elements of the matrix ϵ(t) may be regarded as the dynamical correlation functions of the basis operators. Analogously, we define a bath correlation matrix Φ(t) with elements
(201)Φjk(t)=〈B^j(t)B^k〉B.
It contains the entire description of the bath behavior in our approximation. Finally, we define a Hermitian matrix Γ with elements
(202)Γkj=〈[S^j,P^]S^k〉,
where 〈·〉=Tr[ϱ^(0)(·)]. The matrix Γ may be understood as a representation of the gradient P^ with respect to the chosen basis operators S^j. Finally, we define the bath and (finite-time) system spectra according to
(203)G(ω)=∫-∞∞‍dteiωtΦ(t),ϵt(ω)=12π∫0t‍dτeiωτϵ(τ).
This allows to express the score, (198), as the matrix overlap
(204)P=∬0tdt1dt2Tr[ϵT(t1)Φ(t1-t2)ϵ(t2)Γ](205)=∫-∞∞‍dωTr[ϵt†(ω)G(ω)ϵt(ω)Γ](206)=t∫-∞∞‍dωTr[Ft(ω)G(ω)].
In (206), we have used the cyclic property of the trace to write the spectral overlap in a more compact form by combining the rotation matrix spectra ϵt(ω) and the gradient representation Γ to a system spectral matrix
(207)Ft(ω)=1tϵt(ω)Γϵt†(ω).
Analogously, (204) can be written in a more compact form by introducing the matrix
(208)R(t1,t2)=ϵT(t1)Φ(t1-t2)ϵ(t2).Equation (206) is a generalization of [105] and demonstrates that the change P over a given time t is determined by the spectral overlap between system and bath dynamics, analogously to DCM [185], or the measurement induced quantum Zeno and anti-Zeno control of open systems [97, 102]. The bath-spectral matrix G(ω) must be positive semidefinite for all ω. If the same holds for the matrix Ft(ω), then P is always positive. Below we will consider such a case where P reflects a gate error and the goal is then to minimize this error. The spectral overlap (206) can be made as small as desired by a rapid modulation of the system, such that the entire weight of the system spectrum is shifted beyond that of the bath, which is assumed to vanish for sufficiently high frequencies. Since this fast modulation may cause unbounded growth of the system energy, a meaningful posing of the problem requires a constraint.In general,Ft(ω) is Hermitian but need not necessarily be positive semi-definite, depending on the choice of score as encoded in Γ. This reflects the fact that P can increase or decrease over t. Depending on the application, our goal can therefore also be to maximize P with positive and negative sign. In what follows, we will consider the question how to find a system dynamics that optimizes the score.


### 8.8. Euler-Lagrange Optimization

#### 8.8.1. Role of Control, Score, and Constraint

Our considerations in the previous section suggest to define our control problem in terms of a triple(f,P,E) consisting of a control f, a score P, and a constraint E.The control is a set of real parametersfl, which have been combined to a vector f. These can either be timings, amplitudes, and/or phases of a given number of discrete pulses or can describe a time-continuous modulation of the system. Here, we focus on time-dependent control, where the fl(τ) parametrize the system Hamiltonian as H^S=H^S[f(τ)], or the unitary evolution operator U^(τ)=T+e-i∫0τ‍dτ′H^S(τ′)≡U^[f(τ)]. A direct parametrization of U^ avoids the need of time-ordered integration of its exponent. The U^(τ) thus obtained [176] can be then used to calculate the system Hamiltonian H^S(τ)=i[(∂/∂t)U^(τ)]U^†(τ).Two explicit examples of the scoreP pertain to the fidelity of P^ with a given pure state FΨ=〈Ψ|ϱ^|Ψ〉 (for which P^=|Ψ〉〈Ψ|), or to the von Neumann entropy which we can approximate (for nearly pure states) by the linear entropy, S=-kTr(ϱ^lnϱ^)≈SL=k[1-Tr(ϱ^2)] (for which P^=-2kϱ^(0)). The latter score can be used to maximize the fidelity with the maximally mixed state ϱ^~I^ (for which SL becomes maximum), or to maximize the concurrence C|ΨAB〉=2(1-Trϱ^A2), ϱ^A=TrB|ΨAB〉〈ΨAB|, as a measure of entanglement of a pure state |ΨAB〉 of a bipartite system.If aconstraint is required to ensure the existence of a finite (physical) solution, its choice should depend on the most critical source of error. An example is the average speed with which the controls change, E=∫0t‍dτf˙2(τ), which depend on the control bandwidth in the spectral domain. A parametrization-independent alternative is the mean square of the modulation energy, E=∫0t‍dτ〈(ΔH^)2(τ)〉id, where 〈·〉id refers to a maximally mixed state and hence to a state-independent norm, and ΔH^ is the difference between the modulated and unmodulated (natural) system Hamiltonians.


#### 8.8.2. A Projected Gradient Search

We want to find controlsf that optimize a score P(f) subject to a constraint E(f). A numerical local optimization can be visualized in parameter space as shown in Figure 16.(a) Optimization of the scoreP(f) subject to a constraint E(f) in control space {f} by walking along the component δP⊥ of the gradient δP orthogonal to δE. (b) Resulting iteration consisting in steps determined by a small parameter ϵ which yields a local solution depending on the starting point f0 (color online).
(a)
(b)We start at some initial pointf0 for which E(f0) is the desired value of the constraint. Simply following the gradient δP would maximize or minimize P but would also change E. To optimize P while keeping E constant, we therefore move along the projection of δP orthogonal to δE, that is, along δP⊥=δP-((δP·δE)/(δE)2)δE. Since the gradients depend on f, the iteration consists of small steps fn+1=fn±ϵδP⊥(fn), ϵ≪1. Assuming that neither δP nor δE vanishes, the iteration will come to a halt where δP⊥ vanishes, because the gradients are parallel,
(209)δP=λδE.
This condition constitutes the Euler-Lagrange (EL) equation of the extremal problem, with the proportionality constant λ being the Lagrange multiplier. Its concrete form depends on the choice of P and E. Since the solutions of the EL optimization represent local optima of the constrained P, we may repeat the search with randomly chosen f0 a number of times and select the best solution. The gradients at each point fn may be obtained either from a calculation based on prior knowledge of the bath or experimentally from data measured in real time. A discretization of the time interval 0≤τ≤t then reduces the variational δ to a finite-dimensional vector gradient ∇.


### 8.9. Optimal Gate Protection

#### 8.9.1. Gate Error as Average Fidelity Decline

A particular application of our formalism is decoherence protection of a given quantum operation by bath-optimal minimal-energy control (BOMEC) [176, 185]. Consider the implementation of a predetermined quantum gate, that is, unitary operation within a given “gate time” t. It is sufficient to consider a pure input state |Ψ〉. In the interaction picture with respect to the desired gate operation and in the absence of bath effects, we should therefore observe at time t the initial state |Ψ〉. The quantity of interest is here the fidelity 〈Ψ|ϱ^(t)|Ψ〉, and we use the projector P^=ϱ^(0)=|Ψ〉〈Ψ| as the gradient operator, so that (197) is satisfied and (198) gives the fidelity change as the score
(210)P=〈Ψ|Δϱ^|Ψ〉=-t2〈〈Ψ|H^2|Ψ〉-〈Ψ|H^|Ψ〉2〉B,
which is given by H^ defined in (198).Since a quantum gate is supposed to act on an unknown input state, we need to get rid of the dependence on|Ψ〉. One possibility is to perform a uniform average over all |Ψ〉. We may apply, (see [186, 187])
(211)〈Ψ|A^|Ψ〉〈Ψ|B^|Ψ〉¯=TrA^B^+TrA^TrB^d(d+1)
which gives the average
(212)P¯=-t2dd+1〈H^2〉id,
where 〈·〉id=Tr[d-1I^⊗ϱ^B(·)]. In (212), we have used TrSH^=0^, which corresponds to TrS^j=0 in Section 8.7. Because of this and because of (196), 〈H^〉B=〈H^I〉B=0^, we have 〈H^〉id=0, and (212) also describes the variance Var(H^)=〈H^2〉id-〈H^〉id2. On the other hand, P^=-2kϱ^(0) that gives the change ΔS of entropy S=-kTr(ϱ^lnϱ^) is (up to a proportionality factor of -2k) the same as the P^ used here to give the change of fidelity, and we have ΔS=-2kP. If we define a gate errorℰ as the average fidelity decline, ℰ=-P¯, with P¯ given in (212), we can summarize the following proportionalities: gate error ≡ average fidelity decline ~ average entropy increase (purity decline) ΔS¯~ square (variance) of the average interaction energy H^ as
(213)ℰ≡-P¯=ΔS¯2k=t2dd+1〈H^2〉id=t2dd+1Var(H^).
In the matrix representation of Section 8.7, the average over the initial states in the matrix Γ defined in (202) gives Γ¯=-(d/(d+1))I (using Tr(S^jS^k)=dδjk and TrS^j=0), so that
(214)ℰ=dd+1∫-∞∞‍dωTr[ϵt(ω)ϵt†(ω)G(ω)]
in agreement with [176] (except a different normalization Tr(S^jS^k)=2δjk leading there to a prefactor (2/(d+1))). From the requirement that ℰ≥0 must hold for any positive semi-definite matrix ϵt(ω)ϵt†(ω), we conclude that G(ω) must be a positive semi-definite matrix for any ω. The task of BOMEC is then to find a system evolution U^(τ) (cf. the control examples in the previous section) that minimizes ℰ, subject to the boundary condition that the final U^(t) is the desired gate.


### 8.10. Comparison of BOMEC with DD

It is appropriate to compare the effect of dynamical decoupling (DD) [82, 188] with that of BOMEC. DD does not change with the bath spectrum G(ω). With an increasing number of pulses, DD shifts the weight of the system spectrum F(ω) towards higher frequencies, until the overlap equation (206) has become sufficiently small. This is illustrated for two different numbers of pulses of periodic DD (PDD, pulses periodic in time), in the upper row of Figure 17 in the case of a 1D single qubit modulation (i.e., all pulses are given by an arbitrary but fixed Pauli matrix).System modulation spectraF(ω) (scaled to 1 with respect to their maximum value, scaling of ω as in Figure 18) generated by two methods of DD. Upper row: periodic dynamical decoupling with nπ-pulses (PDDn), lower row: Uhrig dynamical decoupling with the same number of π-pulses (UDDn), compared for n=11 (left column) and n=19 (right column) pulses (color online).
(a)
(b)Aperiodic DD such as UDD [188] suppresses low-frequency components (to the left of the main peak) in the system spectrum, which retain the system-bath coupling even if the main peak of the system spectrum has been shifted beyond the bath cut-off frequency (Figure 17). The plots indicate that this suppression of low frequency components is achieved at the price of a smaller shift of the main peak; that is, shifting the main peak beyond a given cutoff requires more pulses in UDD than in PDD. Note that optimized DD sequences with improved asymptotics exist [189], which we will not consider here.System modulation spectra obtained with BOMEC are shown in Figure18.Figure 18
BOMEC minimization of the gate error for a single qubitπ-gate caused by pure dephasing with a given bath spectrum (G(ω), bold red line, scaled to 0.61 with respect to its maximum value, scaling of ω with respect to the chosen bath resonance, i.e., in units of the inverse bath correlation time). The Z-components of the obtained system modulation spectra Fi(ω) are shown for energy constraints Ei=0.1+4(i-1), i=1,2,…,101, (thin lines, blue to green) individually scaled to 1 (color online).The plot refers to a qubit subject to pure dephasing (i.e.,Z-coupling) by a bath whose spectrum G(ω) has a Lorentzian peak and low-frequency tail. The BOMEC optimizes U^(τ) simultaneously for 3D Pauli matrix couplings to the bath (Z, Y, and X). The resulting system spectrum F(ω) is shown for different energy constraints E which are increased in small and equal steps. For low E, F(ω) has a single peak on the left of the bath peak. Increasing E causes a second peak of F(ω) to emerge on the right of the bath peak, which continues to grow, while the peak on the left diminishes, until for high E, only the right peak remains. Figure 18 hence demonstrates that the spectrum F(ω) generated by BOMEC changes continuously as E increases but avoids overlap with the maxima of G(ω) irrespective of E. BOMEC can therefore be superior to all forms of DD including UDD, especially if the bath has high cutoff but bandgaps at low frequencies.


### 8.11. Purity Control of a Qubit

To give an example of the opposite case, where the goal is to maximize the system-bath coupling, we apply our approach of constrained optimization to the linear entropySL=2[1-Tr(ϱ^2)] of a qubit (note that here SL has been normalized to 1 by setting the coefficient k=d/(d-1)=2, cf. Section 8.8). We assume an initial mixture
(215)ϱ^(0)=p|1〉〈1|+(1-p)|0〉〈0|
of a ground (excited) state |0〉 (|1〉), where 0≤p≤0.5 is related to SL by p=(1-1-SL)/2. With S^j=σ^j denoting for d=2 the Pauli matrices, (215) can be written in terms of H^0=(ω0/2)σ^3 as ϱ^(0)=(e-βH^0/Tr(e-βH^0))=(|1〉〈1|/(1+eβω0))+(|0〉〈0|/(1+e-βω0)), where β=ln(p-1-1)/ω0 is the inverse temperature. Purity and temperature are hence related via the energy scale ω0. Our goal is a constrained optimization of ΔSL, that is, P^=-4ϱ^(0) in (189). Unlike the gate error equation (213), ΔSL can be negative or positive, which can be understood as cooling or heating, respectively.The time evolutions resulting from a minimization ofΔSL for the initial state equation  (215) are illustrated in Figure 19.  The optimization refers to a bath spectrum as shown in Figure 20. The fj shown in Figure 19(a) are defined by U^(τ)=e-(i/2)f3(τ)σ^3e-(i/2)f2(τ)σ^2e-(i/2)f1(τ)σ^3, whereas the ωj shown in Figure 19(b) are given by H^S(τ)=∑j‍ωj(τ)σ^j. The chosen constraint E=(1/2)∫0t‍dτTr(HS-H^0)2(τ) can be written in terms of the fj as E=(1/4)∫0t‍dt1[f˙12+f˙22+(f˙3-ω0)2+2f˙1(f˙3-ω0)cosf2]. (Note that we have set ħ=1, so that the dimension of energy becomes inverse time and hence corresponds to that of angular frequency. By measuring time in units of some characteristic correlation time of the bath chosen in the plots, time, energy, and angular frequency have been made dimensionless.) The overlap between the evolving system state ϱ^(τ) (in the Schrödinger picture) and the ground state |0〉 shown in Figure 19(c) indicates the fast unitary system modulation through short time population inversions without significantly altering the state purity as verified in Figure 19(d). This can be visualized as fast π-rotations of the state inside the Bloch sphere, which, together with smaller rotations, here result in the final reduction of SL(t) seen in Figure 19(d). Figure 19(d) also confirms that for the chosen time and coupling strength, differences between various methods of approximation are small.Evolution within0≤τ≤t for an optimized cooling with initial p=0.25 of (a) effective and (b) instantaneous controls (1 red, 2 green, and 3 blue graphs show x, y, and z component), (c) ground state overlap of the system state, and (d) linear entropy (upper red, middle green, and lower blue graphs show numerical integration of the time-convolutionless equation [190], of the Nakajima-Zwanzig equation, and second-order approximation of the solution as given in (192), respectively, which are all indistinguishable in (c)) (system-bath coupling strength κ=10-2, ω0=2π/t, t=10 in units of effective bath correlation time, E=100) (color online).
(a)
(b)
(c)
(d)Figure 20
Cooling and heating of a TLS by minimization (left: (a), (b), (c)) and maximization (right: (g), (h), (i)) of the change of linear entropy for a given bath spectrum (Gj, red) with j=1,2,3 denoting x, y, and z components. The optimized system spectra (Fj, green) are shown for an energy constraint E=102 (left) and E=103 (right) as contrasted to the unmodulated Hamiltonian H^0 (i.e., for E=0.001, 0.05, 0.1, …, 0.45, 0.499). All graphs are individually scaled to 1 with respect to their maximum value, and scaling of ω is the same as in Figure 18 (color online).In contrast to gate protection, no initial-state averaging is performed here, that is, (215) is known. Consequently, as Figure 20 shows,  the relevant components Fj≡(Ft)jj(ω) of the system modulation spectrum contributing to the spectral overlap Equation (206) depend on the initial state ϱ^(0) via the matrix Γ Equation (202). (We assume an uncorrelated bath, that is, Gjk(ω)=0 for j≠k and Gj≡Gjj(ω).) This influence is clearly visible in case of a constant (unmodulated, i.e., free) Hamiltonian (middle column), for which we set ω0=2π/t with the final t being in the order of the bath correlation time. Cooling (heating) is achieved by realizing negative (positive) ΔSL via maximum negative (positive) spectral overlap, as shown in the left (right) column of Figure 20. This is the opposite to system-bath decoupling, where the goal is to minimize the overlap.The plots illustrate the role of the energy constraintE: increasing E allows to establish overlap with higher frequency components of the bath spectrum. This also suggests that for a bath spectrum with a finite frequency cutoff, increasing E beyond a certain saturation value will not lead to further improvement of the optimization. In the time domain, increasing E leads to more rapid changes in the physical Hamiltonian, however, requiring higher resolution of the numerical treatment. On the contrary, for attempted cooling (heating) by minimization (maximization) of ΔSL, a given E may be too small to lead to negative (positive) ΔSL. The obtained ΔSL may then be understood as “reduced heating” (reduced cooling) as compared to a ΔSL obtained with an unmodulated H^0. This is shown in Figure 21.   The figure also illustrates once more that the ϱ^(0) dependence of the spectra shown in Figure 20 is accompanied by a ϱ^(0) dependency of the achievable change ΔSL for a given bath. For a maximally mixed state in particular (p=0.5), the matrix Γ equation (202) vanishes and with it ΔSL.Figure 21
Change of linear entropy in units of the system-bath coupling strength obtained by minimization (attempted cooling) and maximization (attempted heating) ofΔSL under different constraints E=0,1,100 as a function of the initial p for a bath as shown in Figure 20 (color online).A possibility to achieve negative (positive)ΔSL by its minimization (maximization) even for weak modulation (i.e., for small E) is to adapt the temperature of the bath such that for an undriven system Hamiltonian H^0, no change is observed, ΔSL=0, which is a necessary condition for a system-bath equilibrium. This would require nonunitary system modulation, for example, the effect of repeated measurements [191–193].


### 8.12. Bound Estimation

Our goal is to give constraint-independent upper and lower bounds for the maximum changeP=Tr(P^Δϱ^) that can be achieved with a given bath and P^ under the condition (197). We assume that ϵ, Γ, and G(ω) are quadratic (d2-1)-dimensional matrices. If max[TrG(ω)]<∞, we can estimate P by using that Tr(AB)≤Tr(A)Tr(B) for positive semi-definite matrices A, B and applying Hölder’s inequality in the form of ∫-∞∞‍dω|f(ω)g(ω)|≤sup{|g(ω)|}∫-∞∞‍dω|f(ω)|. Decomposing Γ=Γ1−Γ2 into positive semi-definite matrices Γi and making use of (1/t)∫-∞∞‍dωϵt†(ω)ϵt(ω)=I^, we thus get
(216)-P2≤P≤P1,Pi=tsup[TrG(ω)]TrΓi.
This reveals that for given t and G(ω), the bounds Pi depend on ϱ^(0) and P^ via Γ.


## 8.1. Bipartite Entanglement and Decoherence Control

We shall henceforth analyze the case of bipartite systems,N=2, and quantify entanglement by the often-used measure of concurrence C [167], given by
(170)C(t)=max{0,λ1-λ2-λ3-λ4},
where λi are the positive eigenvalues, in decreasing order, of the matrix ρ(σy⊗σy)ρ*(σy⊗σy) where σy is the Y-Pauli matrix and the conjugation occurs in the computational basis {|gg〉,|ge〉,|eg〉,|ee〉}.In order to explore the effects of dynamical modulations on the concurrence, we first define the following decoherence rate parameters.(i)
Decoherence rate magnitude is defined by νe(g)(t)=∑j=1,2‍|Rjje(g)(t)|, where for the case of νe(g)(t)=0, we have no decoherence, with complete preservation of information and entanglement.(ii)
Decoherence rate asymmetry is defined as ζ(t)=|R11e(t)-R22e(t)|/νe(t), where ζ(t)=0 is the fully symmetric case, and ζ(t)>0 measures the asymmetry. The decoherence rate asymmetry expresses the difference between the two systems with respect to the modulated coupling to the bath. Thus, protecting one system and neglecting the other can increase the asymmetry while preserving the decoherence rate magnitude.(iii)
Cross decoherence is defined as μ(t)=|R12e(t)+R21e(t)|/νe(t), where μ(t)=1 denotes maximal cross-decoherence, and μ(t)=0 denotes completely separable, uncorrelated systems. The cross-decoherence expresses the “cross talk” between the two systems via the bath modes.(iv)
Decoherence rate effective temperature is defined as α(t)=1/ln(νe(t)/νg(t)), where for unmodulated memoryless Markovian identical baths α=kBT/ħωaWhat are the effects of dynamical control on the decoherence rate parameters? The decoherence rate magnitude, being the sum of all individual qubit decoherence rates, is best reduced by control schemes similar to those of single-qubit decoherence control, that is, reducing the overlap of the system-bath coupling and modulation spectra [80, 81, 84, 97]. The decoherence rate asymmetry, on the other hand, is best controlled by differently modulating the two TLS. One can increase the asymmetry by accelerating the decoherence of one qubit and reducing the other’s [85, 97, 99, 121, 153], or decreasing it by equating the individual-qubit spectral overlap of the bath-coupling and modulation spectra.Cross decoherence can be easily eliminated by choosing the appropriate modulations for each qubit. Thus, choosing for each qubit such modulations that they are effectively coupled to different bath modes can completely eliminate cross decoherence [123].The effective temperature cannot be independently controlled, as it is mainly governed by the bath temperature. However, ultrafast modulations can have very high spectral components, such thatωj,k+ωa<0, and can influence the ratio of upward (Rjjg) and downward (Rjje) transitions. Only the absence of the rotating-wave approximation in our analysis has allowed the elucidation of this effect, as “negative” frequencies are made accessible by modulations.


### 8.1.1. Disentanglement and Its Resuscitation

Next, we analyze the disentanglement of initial fully entangled bipartite states, that is, the Bell-states,|Φ±〉=1/2(|e〉1|e〉2±|g〉1|g〉2), |Ψ±〉=1/2(|e〉1|g〉2±|g〉1|e〉2). The long-time limit concurrences of the Bell states for zero effective temperature are found to be
(171)C(|Φ±〉)μ=0=e-tν(1-(1-e-tν(1-ζ))(1-e-tν(1+ζ))),C(|Ψ±〉)ζ=0=e-tν(1±μ).
This demonstrates that an initial (pure) Bell state’s entanglement decays exponentially for zero-temperature baths  but does not vanish in finite time (the general case for μ,ζ≠0 is analytically solvable, but too cumbersome to be presented here). Conversely, coupling the two qubits to finite-temperature baths does result in entanglement sudden death [109, 112], that is, the concurrence vanishes in finite time, which we shall dub time of death (TOD), tTOD.Figure11 presents the TOD of an initial Bell state as a function of effective temperature and either decoherence asymmetry (Figure 11(a)) or cross decoherence (Figure 11(b)). Equation (171) and Figure 11 imply that the TOD can be postponed by applying modulations that (i) reduce the decoherence magnitude, (ii) increase the asymmetry for |Φ±〉, (iii) reduce (increase) the cross decoherence for |Ψ+〉 (|Ψ-〉), or, (iv) decrease the effective temperature. While the reduction of effective temperature is a formidable challenge, asymmetry and cross decoherence are better controlled via modulations. Thus, for an initial |Φ+〉, increasing the decoherence asymmetry of an otherwise symmetric system guarantees the same prolonged time of death as that of a colder environment (Figure 11(a) inset).Time of death (TOD) as a function of decoherence rate parameters. (a) TOD of an initial|Φ+〉 as a function of effective temperature (α) and decoherence rate asymmetry (ζ), μ=0. Inset: effective temperature resulting in νtTOD=4.6 as a function of asymmetry. (b) TOD of an initial |Ψ-〉 as a function of effective temperature (α) and cross-decoherence (μ), ζ=0.
(a)
(b)This Bell state disentanglement analysis results in recipes for optimal control schemes of quantum information protocols, for example, quantum teleportation. We have shown that the singlet is always optimal, but it is not always experimentally accessible. For such a case, one should use the|Φ±〉 states and dynamically modulate each qubit so as to eliminate the cross decoherence if present [71] and maximize the decoherence asymmetry, thus achieving maximal entanglement preservation.Moving from the long-time limit analysis, the analysis reveals that thetime-dependent decoherence rates may exhibit oscillations that can be controlled via the local modulations applied on the individual qubits. These controlled oscillations and their effects on the partial resuscitation of entanglement are the main results of this section. We show that one can control the frequency as well as the magnitude of oscillations by choosing the appropriate modulations and thus resuscitate entanglement at specific desired times.To demonstrate these effects, we shall investigate two distinct situations: (a) distant systems coupled to separate baths (ΦT,jj′(t)=0,j≠j′), whereby an initially entangled state is locally controlled and does not experience any cross decoherence (Figure 12(a)) and (b) adjacent systems coupled to the same bath, whereby local modulations may control the time dependent qubits’ cross decoherence (Figure 12(c)).(a) Schematic drawing of two locally modulated atoms in separated cavities, coupled to the cavity modes. (b) Concurrence as a function of time. Unmodulated (black) and modulated (blue) scenarios, with no cross decoherence,μ=0. Inset: decoherence rate magnitude ν(t) (solid) and asymmetry ζ(t) (dashed) as a function of time. Parameters: a=0.75, ωa=20/tc, ω0=0.8ωa, β=1/ωa, γunmod=0.032/tc, γmod=2.65γunmod, τ1,2/tc={0.175,0.1}, and ϕ1,2=π/10. (c) Schematic drawing of two locally modulated atoms in the same cavity, coupled to the cavity modes. (d) Concurrence as a function of time. Unmodulated (black) and modulated (blue) scenarios, with cross decoherence. Inset: cross-decoherence μ(t) (dotted) and asymmetry ζ(t) (dashed) as a function of time. Parameters: a=0.75, ωa=20/tc, ω0=ωa, β=10/ωa, γunmod=0.032/tc, γmod=1.1γunmod, τ1,2/tc={0.175,0.1}, ϕ1,2=π/10, and ν(t≫tc)=0.28/tc.
(a)
(b)
(c)
(d)Distant systems were extensively studied under various conditions [74, 108, 109, 112, 169, 170]. Here, we generalize these results under the universal dynamical control of decoherence approach and present new control schemes that facilitate entanglement sudden death and its controlled partial resuscitation (ESD-CPR).Both the decoherence rate magnitude and asymmetry exhibit controlled oscillations on non-Markovian time scales (Figure12(b) inset). This can be best observed by considering the most general quasistationary form of modulation, that is, quasi-periodic modulation, ϵj+(t)=∑k‍ϵj,ke-iωj,kt, where ωj,k are arbitrary discrete frequencies with minimal spectral distance Ω=min(ωj,k-ωj′,k′) for all j≠j′,k≠k′. In this case, the modulation spectrum, Ft,jj(ω), is a sum of terms peaked at ω=ωj,k that oscillate on non-Markovian time-scales [100, 105].Thus, local modulations can control not only the magnitude of disentanglement but also its period of oscillation, which may give rise to entanglement sudden death controlled partial resuscitation (ESD-CPR). To demonstrate this, we use the Werner states as the initial partially entangled states, given byρ(0)=(1/3)(2|Ψ+〉〈Ψ+|+(1-a)|gg〉〈gg|+a|ee〉〈ee|), where 0≤a≤1 [74, 113]. We numerically solve the modified Bloch equations for non-Markovian short time scales, and local (individual-qubit) impulsive-phase modulation, ϵj(t)=eϕj[t/τj], where [⋯] is the integer-part. We consider finite-temperature baths, with a Lorentzian bath-coupling spectra G0,jj(ω)=γj(Γj2/(Γj2+(ω-ω0)2)), where γj is the coupling strength of TLS j to the bath, ω0 is the center of the Lorentzian, Γj=1/tcj, and G0,jj′(ω)=0. Here γj=γ and Γj=Γ are taken to be equal for the two TLS, thus having no asymmetry for the unmodulated scenario (ζ(t)=0).While the modulation can greatly decrease the decoherence rate magnitude, and thus maintain entanglement for longer durations, we have already shown that its effect is similar to that of the single-qubit decoherence case and is of less interest here. Hence, we present renormalized results such that the long-time limit ofν(t) is equal for all scenarios to that of the unmodulated case (Figure 12(b) inset). This is achieved by effectively changing the coupling strength γ for each scenario.Figure12(a) presents the case with no cross decoherence, for example, two atoms in remote cavities. While the concurrence of the unmodulated scenario with zero asymmetry rapidly dies, the non-Markovian oscillation of the asymmetry due to the modulation results in ESD-CPR (Figure 12(b)). The rise of concurrence from the dead is substantial and can occur only at a time comparable to the correlation time of the bath tc. During this time, the entanglement between the two TLS is “transferred” to the bath modes, which still retain some memory. The oscillations of the asymmetry facilitate the return of entanglement back to the two TLS. Notably, ESD-CPR is achieved here by local modulations; that is, the two TLS are not coupled through the control fields [108] nor through the bath modes [73, 110]. This novel result, of controlling entanglement oscillations via local modulations, exemplifies the importance of a unified theory of non-Markovian time-scales multipartite decoherence control.Next, we analyze the case of systems experiencing cross decoherence due to coupling to the same bath (Figure12(c)). While this scenario has been less studied [73, 110], it demonstrates long (Markovian) time-scale controlled oscillations that result in ESD-CPR. To investigate this scenario, we set G0,jj′(ω)=G0,jj(ω)G0,j′j′(ω) and define cross decoherence as μ(t)=|R12e(t)+R21e(t)|/νe(t), where μ(t)=1 denotes maximal cross decoherence, and μ(t)=0 denotes effectively separate systems. The cross decoherence expresses the locally modulated “crosstalk” between the two systems via the bath modes (Figure 12(c)).As was previously discussed, the cross decoherence may vanish due to an imposed coupling of the two TLS to different bath modes. This becomes apparent for quasi-periodic modulation in the long-time limit,Ωt≫1, where the cross-modulation spectrum, Ft,jj′(ω) becomes a series of δ(ωj,k-ωj′,k′). However, this long-time limit for the cross decoherence terms is reached only after the inverse of the minimal spectral distance of the two modulations, that is, t≫1/min(ωj,k-ωj′≠j,k′). This limit can be very long for similar, yet not equal, modulations, thus enabling a long oscillatory behavior of the cross decoherence (Figure 12(d) inset).Figure12(d) presents the effects of cross decoherence on concurrence. While it has been shown that some states (e.g., singlet) may benefit from cross decoherence [71], the Werner state’s concurrence dies faster due to this “cross-talk.” Applying different modulations on each TLS results in the reduction of cross decoherence and the increase of asymmetry, both beneficial for entanglement preservation. Furthermore, oscillations of the cross decoherence are long-lasting as the two modulations’ peaks are close to one another, and can periodically kill and resuscitate the concurrence even in Markovian time scales (t≫tc).This phenomenon can only be investigated by a general model that incorporates both arbitrary time-dependent control and the entire range of cross decoherence, from separate to adjacent systems. The universal dynamical disentanglement approach presented here is, to the best of our knowledge, the first that meets these requirements.To conclude, the ESD-CPR scenario discussed has many aspects: (a) non-Markovian time-scale behavior [112, 169–171], (b) asymmetry between the two decoherence channels [108, 109, 169–172], (c) cross-decoherence of the two-qubits [73, 110, 171, 172], (d) finite temperature effects [109, 112], and (e) time-dependent control [108, 112, 170]. The universal dynamical control of disentanglement presented here generalizes all these aspects, as it describes arbitrary dynamical time-dependent control. Only by combining all these aspects into one formalism, one can fully exploit the control attained by locally modulating entangled systems, namely, oscillations of entanglement with controlled frequencies and amplitudes.While postponing ESD by reducing the decoherence rate requires control schemes similar to those proposed for single-qubits, facilitating ESD-CPR by dynamically controlling the asymmetry and cross decoherence requires more elaborate local modulations. Based on these results, new control strategies for disentanglement reduction of quantum information processing can be suggested. By tailoring optimal control schemes for each scenario, one can locally control the entangled systems and thereby facilitate ESD-CPR at specific times. This may prove beneficial for many quantum computation and communication protocols.


## 8.1.1. Disentanglement and Its Resuscitation

Next, we analyze the disentanglement of initial fully entangled bipartite states, that is, the Bell-states,|Φ±〉=1/2(|e〉1|e〉2±|g〉1|g〉2), |Ψ±〉=1/2(|e〉1|g〉2±|g〉1|e〉2). The long-time limit concurrences of the Bell states for zero effective temperature are found to be
(171)C(|Φ±〉)μ=0=e-tν(1-(1-e-tν(1-ζ))(1-e-tν(1+ζ))),C(|Ψ±〉)ζ=0=e-tν(1±μ).
This demonstrates that an initial (pure) Bell state’s entanglement decays exponentially for zero-temperature baths  but does not vanish in finite time (the general case for μ,ζ≠0 is analytically solvable, but too cumbersome to be presented here). Conversely, coupling the two qubits to finite-temperature baths does result in entanglement sudden death [109, 112], that is, the concurrence vanishes in finite time, which we shall dub time of death (TOD), tTOD.Figure11 presents the TOD of an initial Bell state as a function of effective temperature and either decoherence asymmetry (Figure 11(a)) or cross decoherence (Figure 11(b)). Equation (171) and Figure 11 imply that the TOD can be postponed by applying modulations that (i) reduce the decoherence magnitude, (ii) increase the asymmetry for |Φ±〉, (iii) reduce (increase) the cross decoherence for |Ψ+〉 (|Ψ-〉), or, (iv) decrease the effective temperature. While the reduction of effective temperature is a formidable challenge, asymmetry and cross decoherence are better controlled via modulations. Thus, for an initial |Φ+〉, increasing the decoherence asymmetry of an otherwise symmetric system guarantees the same prolonged time of death as that of a colder environment (Figure 11(a) inset).Time of death (TOD) as a function of decoherence rate parameters. (a) TOD of an initial|Φ+〉 as a function of effective temperature (α) and decoherence rate asymmetry (ζ), μ=0. Inset: effective temperature resulting in νtTOD=4.6 as a function of asymmetry. (b) TOD of an initial |Ψ-〉 as a function of effective temperature (α) and cross-decoherence (μ), ζ=0.
(a)
(b)This Bell state disentanglement analysis results in recipes for optimal control schemes of quantum information protocols, for example, quantum teleportation. We have shown that the singlet is always optimal, but it is not always experimentally accessible. For such a case, one should use the|Φ±〉 states and dynamically modulate each qubit so as to eliminate the cross decoherence if present [71] and maximize the decoherence asymmetry, thus achieving maximal entanglement preservation.Moving from the long-time limit analysis, the analysis reveals that thetime-dependent decoherence rates may exhibit oscillations that can be controlled via the local modulations applied on the individual qubits. These controlled oscillations and their effects on the partial resuscitation of entanglement are the main results of this section. We show that one can control the frequency as well as the magnitude of oscillations by choosing the appropriate modulations and thus resuscitate entanglement at specific desired times.To demonstrate these effects, we shall investigate two distinct situations: (a) distant systems coupled to separate baths (ΦT,jj′(t)=0,j≠j′), whereby an initially entangled state is locally controlled and does not experience any cross decoherence (Figure 12(a)) and (b) adjacent systems coupled to the same bath, whereby local modulations may control the time dependent qubits’ cross decoherence (Figure 12(c)).(a) Schematic drawing of two locally modulated atoms in separated cavities, coupled to the cavity modes. (b) Concurrence as a function of time. Unmodulated (black) and modulated (blue) scenarios, with no cross decoherence,μ=0. Inset: decoherence rate magnitude ν(t) (solid) and asymmetry ζ(t) (dashed) as a function of time. Parameters: a=0.75, ωa=20/tc, ω0=0.8ωa, β=1/ωa, γunmod=0.032/tc, γmod=2.65γunmod, τ1,2/tc={0.175,0.1}, and ϕ1,2=π/10. (c) Schematic drawing of two locally modulated atoms in the same cavity, coupled to the cavity modes. (d) Concurrence as a function of time. Unmodulated (black) and modulated (blue) scenarios, with cross decoherence. Inset: cross-decoherence μ(t) (dotted) and asymmetry ζ(t) (dashed) as a function of time. Parameters: a=0.75, ωa=20/tc, ω0=ωa, β=10/ωa, γunmod=0.032/tc, γmod=1.1γunmod, τ1,2/tc={0.175,0.1}, ϕ1,2=π/10, and ν(t≫tc)=0.28/tc.
(a)
(b)
(c)
(d)Distant systems were extensively studied under various conditions [74, 108, 109, 112, 169, 170]. Here, we generalize these results under the universal dynamical control of decoherence approach and present new control schemes that facilitate entanglement sudden death and its controlled partial resuscitation (ESD-CPR).Both the decoherence rate magnitude and asymmetry exhibit controlled oscillations on non-Markovian time scales (Figure12(b) inset). This can be best observed by considering the most general quasistationary form of modulation, that is, quasi-periodic modulation, ϵj+(t)=∑k‍ϵj,ke-iωj,kt, where ωj,k are arbitrary discrete frequencies with minimal spectral distance Ω=min(ωj,k-ωj′,k′) for all j≠j′,k≠k′. In this case, the modulation spectrum, Ft,jj(ω), is a sum of terms peaked at ω=ωj,k that oscillate on non-Markovian time-scales [100, 105].Thus, local modulations can control not only the magnitude of disentanglement but also its period of oscillation, which may give rise to entanglement sudden death controlled partial resuscitation (ESD-CPR). To demonstrate this, we use the Werner states as the initial partially entangled states, given byρ(0)=(1/3)(2|Ψ+〉〈Ψ+|+(1-a)|gg〉〈gg|+a|ee〉〈ee|), where 0≤a≤1 [74, 113]. We numerically solve the modified Bloch equations for non-Markovian short time scales, and local (individual-qubit) impulsive-phase modulation, ϵj(t)=eϕj[t/τj], where [⋯] is the integer-part. We consider finite-temperature baths, with a Lorentzian bath-coupling spectra G0,jj(ω)=γj(Γj2/(Γj2+(ω-ω0)2)), where γj is the coupling strength of TLS j to the bath, ω0 is the center of the Lorentzian, Γj=1/tcj, and G0,jj′(ω)=0. Here γj=γ and Γj=Γ are taken to be equal for the two TLS, thus having no asymmetry for the unmodulated scenario (ζ(t)=0).While the modulation can greatly decrease the decoherence rate magnitude, and thus maintain entanglement for longer durations, we have already shown that its effect is similar to that of the single-qubit decoherence case and is of less interest here. Hence, we present renormalized results such that the long-time limit ofν(t) is equal for all scenarios to that of the unmodulated case (Figure 12(b) inset). This is achieved by effectively changing the coupling strength γ for each scenario.Figure12(a) presents the case with no cross decoherence, for example, two atoms in remote cavities. While the concurrence of the unmodulated scenario with zero asymmetry rapidly dies, the non-Markovian oscillation of the asymmetry due to the modulation results in ESD-CPR (Figure 12(b)). The rise of concurrence from the dead is substantial and can occur only at a time comparable to the correlation time of the bath tc. During this time, the entanglement between the two TLS is “transferred” to the bath modes, which still retain some memory. The oscillations of the asymmetry facilitate the return of entanglement back to the two TLS. Notably, ESD-CPR is achieved here by local modulations; that is, the two TLS are not coupled through the control fields [108] nor through the bath modes [73, 110]. This novel result, of controlling entanglement oscillations via local modulations, exemplifies the importance of a unified theory of non-Markovian time-scales multipartite decoherence control.Next, we analyze the case of systems experiencing cross decoherence due to coupling to the same bath (Figure12(c)). While this scenario has been less studied [73, 110], it demonstrates long (Markovian) time-scale controlled oscillations that result in ESD-CPR. To investigate this scenario, we set G0,jj′(ω)=G0,jj(ω)G0,j′j′(ω) and define cross decoherence as μ(t)=|R12e(t)+R21e(t)|/νe(t), where μ(t)=1 denotes maximal cross decoherence, and μ(t)=0 denotes effectively separate systems. The cross decoherence expresses the locally modulated “crosstalk” between the two systems via the bath modes (Figure 12(c)).As was previously discussed, the cross decoherence may vanish due to an imposed coupling of the two TLS to different bath modes. This becomes apparent for quasi-periodic modulation in the long-time limit,Ωt≫1, where the cross-modulation spectrum, Ft,jj′(ω) becomes a series of δ(ωj,k-ωj′,k′). However, this long-time limit for the cross decoherence terms is reached only after the inverse of the minimal spectral distance of the two modulations, that is, t≫1/min(ωj,k-ωj′≠j,k′). This limit can be very long for similar, yet not equal, modulations, thus enabling a long oscillatory behavior of the cross decoherence (Figure 12(d) inset).Figure12(d) presents the effects of cross decoherence on concurrence. While it has been shown that some states (e.g., singlet) may benefit from cross decoherence [71], the Werner state’s concurrence dies faster due to this “cross-talk.” Applying different modulations on each TLS results in the reduction of cross decoherence and the increase of asymmetry, both beneficial for entanglement preservation. Furthermore, oscillations of the cross decoherence are long-lasting as the two modulations’ peaks are close to one another, and can periodically kill and resuscitate the concurrence even in Markovian time scales (t≫tc).This phenomenon can only be investigated by a general model that incorporates both arbitrary time-dependent control and the entire range of cross decoherence, from separate to adjacent systems. The universal dynamical disentanglement approach presented here is, to the best of our knowledge, the first that meets these requirements.To conclude, the ESD-CPR scenario discussed has many aspects: (a) non-Markovian time-scale behavior [112, 169–171], (b) asymmetry between the two decoherence channels [108, 109, 169–172], (c) cross-decoherence of the two-qubits [73, 110, 171, 172], (d) finite temperature effects [109, 112], and (e) time-dependent control [108, 112, 170]. The universal dynamical control of disentanglement presented here generalizes all these aspects, as it describes arbitrary dynamical time-dependent control. Only by combining all these aspects into one formalism, one can fully exploit the control attained by locally modulating entangled systems, namely, oscillations of entanglement with controlled frequencies and amplitudes.While postponing ESD by reducing the decoherence rate requires control schemes similar to those proposed for single-qubits, facilitating ESD-CPR by dynamically controlling the asymmetry and cross decoherence requires more elaborate local modulations. Based on these results, new control strategies for disentanglement reduction of quantum information processing can be suggested. By tailoring optimal control schemes for each scenario, one can locally control the entangled systems and thereby facilitate ESD-CPR at specific times. This may prove beneficial for many quantum computation and communication protocols.


## 8.2. Multipartite Decoherence Control

Here we address the problem of maintaining the fidelity of a multipartite entangled state, after it has been prepared [124, 173, 174]. In the context of one-way quantum computing [175], fidelity protection is more important than preventing disentanglement [126], since maintaining the specific initial state is essential for the success of the computation. Designing high-fidelity gates that are resistant to decoherence was addressed by us elsewhere [125] and its scaling with the number of qubits is beyond the scope of this paper.We pose and answer the basic questions. How does decoherence scale with the number of qubits and is it state dependent? Can dynamical control combat decoherence and its scaling with the number of qubits as this number becomes large? What time scales are required for such control?Fidelity in such systems can be protected by very general, not necessarily pulsed, dynamical control of decoherence [99, 100, 104, 118, 123–126, 174, 176]. In this approach, fields applied to the system can maintain its coherence, as long as their effects are faster than the correlation (memory) time of the environment, that is, they act on non-Markovian time scales. Thus far, single-qubit dynamic decoherence control has received more attention [99, 100, 104] than multipartite control [123, 174].We show that by locally (specifically) modulating each qubit, one can drastically suppress the decoherence and its scaling through modifying the strength and number of effective decoherence channels. Our analysis yields the surprising result that a modulation which actsafter the multipartite coherence time has decayed can still restore coherence to acceptable levels, if it is within the memory time of the bath, irrespective of the number of particles.


### 8.2.1. Scenarios

Our system is comprised ofN qubits whose ground and excited states are |g〉j and |e〉j, respectively, with j=1,…,N and energy separation ħωa (Figure 1). Each qubit may be differently (weakly) coupled to a bath (henceforth we take ħ=1) as
(172)H(t)=HS+HB+HI+HC(t),HS=ωa∑j=1N‍|e〉jj〈e|,HI=∑j=1N‍S^jB^j,
where HS, HB, and HI are the system, bath, and interaction Hamiltonians, respectively, HC(t) is the time-dependent control Hamiltonian, with B^j being the jth qubit bath operator and S^j the system-bath coupling operator of qubit j. We neglect direct interactions among bath modes and focus on scenarios where two or more qubits collectively couple to common bath modes (see Figure 13).Figure 13
Multipartite systems (black) coupled to their environments (red) (color online). (a) Relaxation of identical qubits via coupling to thesame bath modes causes cross decoherence. (b) Different AC-Stark shifts on individual qubits couple them to different bath modes, eliminating cross decoherence. (c) Different Rabi driving of dephasing qubits is analogous to (b).Two kinds of decoherence can be similarly treated: (i)relaxation [123], caused by a system-bath coupling Hamiltonian that is off diagonal in the energy basis, with S^j=|e〉jj〈g|+|g〉jj〈e| being the X-Pauli matrix of the jth qubit. Relaxation involves temperature-dependent level-population change, as well as erasure (scrambling) of the phase relations. Such a process is controllable if each qubit is subject to a strong but off-resonant driving field, ħηj(t), that causes an AC Stark shift (Figure 1(b)), that is, energy modulation of the system Hamiltonian [174], as
(173)HC(t)=∑j=1N‍ηj(t)|e〉jj〈e|.
(ii) Dephasing [174] is caused by a system-bath coupling Hamiltonian that is diagonal in the energy basis, with S^j=|e〉jj〈e|-|g〉jj〈g| being the Z-Pauli matrix of the jth qubit. Dephasing does not change level populations but only erases (scrambles) the phases. Such a process is controllable by near-resonant Rabi driving fields, Vj(t), that cause transitions between qubit levels,
(174)HC(t)=∑j=1N‍(Vj(t)e-iωat|e〉jj〈g|+h.c.).We first consider the unmodulated scenario, for which the ME becomes(175)ρ^˙=∑j,j′=1N‍[rjj′(t)σ^jxρ^σ^j′x-rjj′(t)σ^j′xσ^jxρ^00000-rjj′*(t)ρ^σ^j′xσ^jx+rjj′*(t)σ^j′xρ^σ^jx],
where rjj′(t) are the instantaneous decoherence rates. One can formally integrate (175) and get
(176)ρ^(t)=ρ^(0)+∑j,j′=1N‍∫0t‍dt′[rjj′(t′)σ^jxρ^(t)σ^j′x000000000000-rjj′(t′)σ^j′xσ^jxρ^(t)000000000000-rjj′*(t′)ρ^(t)σ^j′xσ^jx000000000000+rjj′*(t′)σ^j′xρ^(t)σ^jx].We are interested in the fidelity,f(t)=Tr(ρ(0)1/2ρ(t)ρ(0)1/2), of any initially pure state |ψ〉. To second-order in the system-bath coupling, we obtain from (176) the short-time form of f(t) as
(177)fψ(t)=1-tΓψ(t;N),(178)Γψ(t;N)=∑j,j′=1N‍Rjj′(t)(〈ψ|σ^jxσ^j′x|ψ〉0000000000-〈ψ|σ^jx|ψ〉〈ψ|σ^j′x|ψ〉),(179)Rjj′(t)=2t∫0t‍dt′rjj′(t′),
where Γψ(t;N) is the infidelity rate and Rjj′(t) are elements of the (time-averaged) decoherence-rate matrix discussed below. While (177) and (178) are valid for all initial pure states, we henceforth focus our analysis on specific entangled multipartite states and discuss their decoherence scalability.


## 8.2.1. Scenarios

Our system is comprised ofN qubits whose ground and excited states are |g〉j and |e〉j, respectively, with j=1,…,N and energy separation ħωa (Figure 1). Each qubit may be differently (weakly) coupled to a bath (henceforth we take ħ=1) as
(172)H(t)=HS+HB+HI+HC(t),HS=ωa∑j=1N‍|e〉jj〈e|,HI=∑j=1N‍S^jB^j,
where HS, HB, and HI are the system, bath, and interaction Hamiltonians, respectively, HC(t) is the time-dependent control Hamiltonian, with B^j being the jth qubit bath operator and S^j the system-bath coupling operator of qubit j. We neglect direct interactions among bath modes and focus on scenarios where two or more qubits collectively couple to common bath modes (see Figure 13).Figure 13
Multipartite systems (black) coupled to their environments (red) (color online). (a) Relaxation of identical qubits via coupling to thesame bath modes causes cross decoherence. (b) Different AC-Stark shifts on individual qubits couple them to different bath modes, eliminating cross decoherence. (c) Different Rabi driving of dephasing qubits is analogous to (b).Two kinds of decoherence can be similarly treated: (i)relaxation [123], caused by a system-bath coupling Hamiltonian that is off diagonal in the energy basis, with S^j=|e〉jj〈g|+|g〉jj〈e| being the X-Pauli matrix of the jth qubit. Relaxation involves temperature-dependent level-population change, as well as erasure (scrambling) of the phase relations. Such a process is controllable if each qubit is subject to a strong but off-resonant driving field, ħηj(t), that causes an AC Stark shift (Figure 1(b)), that is, energy modulation of the system Hamiltonian [174], as
(173)HC(t)=∑j=1N‍ηj(t)|e〉jj〈e|.
(ii) Dephasing [174] is caused by a system-bath coupling Hamiltonian that is diagonal in the energy basis, with S^j=|e〉jj〈e|-|g〉jj〈g| being the Z-Pauli matrix of the jth qubit. Dephasing does not change level populations but only erases (scrambles) the phases. Such a process is controllable by near-resonant Rabi driving fields, Vj(t), that cause transitions between qubit levels,
(174)HC(t)=∑j=1N‍(Vj(t)e-iωat|e〉jj〈g|+h.c.).We first consider the unmodulated scenario, for which the ME becomes(175)ρ^˙=∑j,j′=1N‍[rjj′(t)σ^jxρ^σ^j′x-rjj′(t)σ^j′xσ^jxρ^00000-rjj′*(t)ρ^σ^j′xσ^jx+rjj′*(t)σ^j′xρ^σ^jx],
where rjj′(t) are the instantaneous decoherence rates. One can formally integrate (175) and get
(176)ρ^(t)=ρ^(0)+∑j,j′=1N‍∫0t‍dt′[rjj′(t′)σ^jxρ^(t)σ^j′x000000000000-rjj′(t′)σ^j′xσ^jxρ^(t)000000000000-rjj′*(t′)ρ^(t)σ^j′xσ^jx000000000000+rjj′*(t′)σ^j′xρ^(t)σ^jx].We are interested in the fidelity,f(t)=Tr(ρ(0)1/2ρ(t)ρ(0)1/2), of any initially pure state |ψ〉. To second-order in the system-bath coupling, we obtain from (176) the short-time form of f(t) as
(177)fψ(t)=1-tΓψ(t;N),(178)Γψ(t;N)=∑j,j′=1N‍Rjj′(t)(〈ψ|σ^jxσ^j′x|ψ〉0000000000-〈ψ|σ^jx|ψ〉〈ψ|σ^j′x|ψ〉),(179)Rjj′(t)=2t∫0t‍dt′rjj′(t′),
where Γψ(t;N) is the infidelity rate and Rjj′(t) are elements of the (time-averaged) decoherence-rate matrix discussed below. While (177) and (178) are valid for all initial pure states, we henceforth focus our analysis on specific entangled multipartite states and discuss their decoherence scalability.


## 8.3. Decoherence Scalability

### 8.3.1. Multipartite Entangled States

We treat decoherence scaling withN for several distinct multipartite entangled states with clear N-dependence.  (i)
GHZ states,|GHZ〉=(1/2)(⨂j|e〉j+⨂j|g〉j).(ii)
W-states,|W〉=(1/N)∑j=1N‍(|e〉j⨂j≠j′|g〉j′).(iii)
Bright states,|B〉=|J=N/2,m=0〉, where |J,m〉 are the Dicke states [177], with J being the total angular momentum and m the excitation number.(iv)
Dark states, given by|D〉=⨂j=2,4,...,2N|Ψ-〉j-1,j for even N, where |Ψ-〉j,j+1=(1/2)(|g〉j|e〉j+1-|e〉j|g〉j+1).Equation (178) shows that the infidelity rate depends on the specific multipartite state and on the decoherence rates Rjj′(t). The time dependence of these rates is revealed on non-Markovian time scales, yet they are constant on long (Markov) time scales. The diagonal elements of the decoherence-rate matrix, Rjj(t), correspond to single-qubit decoherence, determined by the coupling of individual qubits (labeled by j) to their respective environments (baths). On the other hand, the off-diagonal elements, Rjj′(t) with j≠j′, represent cross-decoherence rates, that is, decoherence rates induced on the jth qubit by the j′th qubit. Such cross-decoherence rates are present only for qubits coupled to a common bath or if their dephasing noise is correlated. More explicitly, cross decoherence occurs because two qubits are coupled to the same bath (noise) mode (Figure 1(a)). In the second-order approximation, this means that qubit j interacts with a certain bath (or noise) mode and then the same mode interacts with qubit j′ (cf. (178)). In the dephasing scenario, cross dephasing stems from the cross correlation of the stochastic noise functions of qubits j and j′. Thus, for example, ions subject to the same fluctuating magnetic field will experience cross dephasing.In the absence of cross decoherence, when each qubit is locally coupled to an individual bathRjj′(t)=0, for all j≠j′, the first state-dependent term of the infidelity rate in (178) scales linearly with N, for all states. The second term then appears only for unique states, for example, W-state, and can only lower the N scalability (Table 1), resulting in (at most) linear dependence on N for all states in systems without cross decoherence.Table 1
N-scalability of entangled states decoherence for relaxation and dephasing scenarios, with and without cross decoherence.
Relaxation
Dephasing
No X-deco.
X-deco.
No X-deco.
X-deco.
GHZ
N
N
N
N
2
W-state
N
3
N
-
2
4
(
N
-
1
)
/
N
0
Bright state
N
N
+
N
2
/
2
N
0
Dark state
N
0
N
0By contrast, the infidelity rate is much morestate-dependent in systems with cross decoherence, since these systems have additional decoherence channels that may interfere, either constructively or destructively, depending on the initial entangled state via the phases of 〈ψ|σ^jxσ^j′x|ψ〉. Hence, the N scalability under cross decoherence can either mount to N2 or vanish completely, a situation unattainable in the absence of cross decoherence.More specifically, under global dephasing, the GHZ state becomes a bright state in the|±〉 basis and hence experiences constructive interference of all decoherence channels, so that its dephasing scales quadratically with N. The W-state and bright state, on the other hand, do not experience global dephasing since they are composed of a symmetric superposition of identically excited states and hence the dephasing becomes a global phase factor and does not influence decoherence. Remarkably, the dark state considered here is dark for global relaxation and dephasing, due to the invariance of the |Ψ-〉 state under the basis rotation from |e(g)〉 to |±〉. For higher-order approximation this may not be valid.


### 8.3.2. Decoherence Scaling and the Zeno Regime

One may view decoherence scalability from a complementary aspect, that of the coherence time of the system. In this section, we consider an unmodulated scenario with identical coupling to all qubits, that is,(180)Γψ(t;N)=cψ(N)R(t),
where cψ(N) is the state-dependent scalability. We define the coherence time as the time at which the fidelity (177), or alternatively the error, e(t)=1-f(t), reaches a certain threshold as
(181)tcohΓψ(tcoh;N)=1-fthreshold≡ethreshold.
This coherence time is an important measure, since its ratio to the time it takes to perform a single quantum operation (e.g., a two-qubit gate) [3] determines how many such operations can be performed successfully, that is, the complexity of the quantum computation.The coherence time is usually identified with the inverse decoherence rate, since in the Markovian time-regime, the decoherence rate is constant, that isR(t)=R, yielding
(182)tcoh(N)|Markov=ethresholdcψ(N)R.
For example, linearly scaling systems cψ(N)=N, such as GHZ relaxation without cross decoherence, have Markovian coherence time that scales inversely with N. However, for large systems, the fidelity threshold (181) is reached according to (177) after a time that may be much shorter than the correlation (memory) time of the bath, tcorr, so that the non-Markov time dependence of R(t) starts playing a role. For still larger N, this threshold will be reached within the extremely short Zeno time-regime, tZeno [86, 97, 99, 102, 178, 179], in which the decoherence rate scales linearly with t, RZeno(t)=tR˙ (see below). Hence, for sufficiently large systems, such that
(183)cψ(N)≥ethresholdtZeno2R˙
the coherence time is always that obtained in the Zeno time-regime as
(184)tcoh(N)|Zeno=ethresholdcψ(N)R˙.
If cψ(N)=N, the Zeno coherence time scales as the inverse square root of N. This paradoxical result, namely, the inevitable improvement of the decoherence scaling law for sufficiently large systems, has been thus far overlooked. This result is a manifestation of the non-Markovian effects in decoherence dynamics.


### 8.3.3. Dynamically Modified Decoherence Rates

From (178), it is evident that the decoherence rates Rjj′(t) are an important factor in the scaling of decoherence. Their time dependence stems from the non-Markovian master equation (ME) [126]. The same non-Markovian ME governs dynamical decoherence control. The resulting (time-averaged) decoherence rates then assume the form of spectral overlaps of the bath coupling spectrum matrix, Gjj′(ω), and the modulation power-spectrum matrix, Ft,jj′(ω) [174] as
(185)Rjj′(t)=2πRe∫-∞∞‍dωGjj′(ω)Ft,jj′(ω),(186)Ft,jj′(ω)=1tϵt,j(ω)ϵt,j′*(ω),
where Gjj′(ω) is the Fourier transform of the system-bath correlation matrix Φjj′(ω), and Ft,jj′(ω) is the time-dependent normalized modulation spectrum, where ϵt,j(ω)=(1/2π)∫0t‍dt′ϵj+(t′)eiωt′ is the finite-time Fourier transform of the modulation function. In the relaxation scenario, the modulation spectral function represents the effective change in qubit j’s energy splitting or AC Stark shift (Figure 13(b)). In the dephasing scenario, it determines the time-dependent Rabi splitting of the |±〉 states (Figure 13(c)). In both scenarios, the spectral overlap Gjj′(ω)ϵt,j*(ω)ϵt,j′ω) in (185) represents the effective coupling between the systems and the bath, whereby the dynamical control (modulation) determines which spectral bath modes the qubits are coupled to.There is an important distinction between the controllability of single-qubit decoherence rates,Rjj(t), and that of their cross-decoherence counterpart, Rjj′(t). Namely, the single-qubit dynamically modified Rjj(t) can only be reduced by decreasing the overlap between the modulation spectra |ϵt,j(ω)|2 and the bath coupling spectra, Gjj(ω), both positive. No matter how fast the modulation is, |ϵt,j(ω)|2 cannot decrease the single-qubit decoherence rate, if the bath coupling spectrum is spectrally flat (Markovian). By contrast, cross-decoherence rates Rjj′(t) can be completely eliminated by choosing spectrally distinct local modulations whose spectral overlap ϵt,j*(ω)ϵt,j′(ω) vanishes, irrespective of the bath coupling spectrum, Gjj′(ω) even if it is Markovian (spectrally flat) (Figures 2(a) and 2(b)). The qubits effectively couple to orthogonal bath modes. Hence, by modulating each qubit differently, one can effectively decouple the qubits, as if they were interacting with separate baths.


### 8.3.4. Dynamically Modified Decoherence Scaling

How do these modulations affect decoherence scaling? Let us examine two limits of the general scenario more closely.(A)
Consider first the limit oflocally decohering qubits. Due to the absence of a common bath, modulations cannot induce cross decoherence. Hence, they cannot change the scaling law, which is identical for most multipartite entangled states, namely, Γψ(t;N)∝N (Table 1). Nevertheless, modulations can still improve the infidelity rate, by decreasing the overlap of each qubit’s bath coupling and modulation spectra.(B)
The other limiting case is that of collective decoherence of qubits coupled to a common bath, but locally controlled. By exploiting the spectral distinction ofN different local modulations, one can completely eliminate cross decoherence and effectively render the system similar to the aforementioned locally decohering qubits, in which for most states, decoherence scales linearly with N. Such local modulations may be advantageous for the relaxing bright state, or dephasing GHZ-state, by reducing its infidelity rate scaling by a factor of N.On the other hand, one may enhance cross decoherence by local modulations: if the qubits are coupled to the same bath but with different strengths, then the modulations can be chosen to compensate for their unequal coupling strengths. Such enhanced cross decoherence can drastically improve the coherence time by rendering certain states completely dark: without modulation, partially dark states with unequal coupling have infidelity rate that scales asN, whereas under modulations that impose destructive interference of the decoherence channels, such states can acquire much longer coherence time by becoming fully dark.


### 8.3.5. Effective Modulation Time Scales

When do such local dynamic modulations become effective? Let us recall the relevant time scales. The longest one is the coherence time of a single-qubit without modulation,tcoh(1)≅1/Γψ. Another is the correlation (memory) time of the bath, tcorr. It is the decay time of the correlation function, which is given by the product of the noise amplitude at time t with the noise amplitude at time t+τ, averaged over realizations. In the frequency domain 1/tcorr is the width of the noise-coupling spectrum and in dephasing noise this is the noise bandwidth.In the single qubit case, the weak-coupling assumption (see below) implies that the correlation time is shorter than the single-qubit coherence time [70] tcorr≪tcoh(1) or, equivalently, noise bandwidth much larger than dephasing rate [174]. In the unmodulated N-qubit scenario, this assumption is given by tcorr≪tcoh(N), whereas in the modulated case, it is the modulated coherence time that sets the limit for the weak coupling regime, tcorr≪tcohmod(N). Hence, even if for a large (unmodulated) system the weak coupling assumption does not hold, modulations that drastically increase the coherence time, for example, by eliminating cross decoherence, can make the system enter the valid regime for which the analysis holds.Only modulations faster than the bath-(noise-) correlation time can affect the single-qubit decoherence ratesRjj(t). Yet a crucial insight transpires: if each qubit decoheres individually (cross decoherence is eliminated), the (unmodulated) N-particle coherence time does not determine whether the control is effective. Neither does the infidelity rate (178). Rather, the modulation period, τ, must be shorter than tcorr, but can be much larger than tcoh(1)/N: it is unaffected by scaling!We can now summarize the conditions for an effective increase of the coherence time of entangled multipartite systems by modulations with a periodτ as
(187)tcoh(N)<︸aτ<︸btcorr<︸ctcoh(1)<︸dtcohmod(N).
Here, (a) lays out the striking result that effective modulation can be slower than the coherence time of the system; (b) is the major requirement for effective modulation; (c) is the single-qubit weak-coupling assumption; (d) shows the main result, that is, the possibility of attaining (via effective modulations) multipartite coherence time tcohmod(N) that is longer than the single-qubit coherence time.


### 8.3.6. Application to an Ion Trap

Our analysis is applicable to a broad variety of experimental setups. As an example, consider a string of ions in a linear trap [142, 180, 181], in which the qubits are encoded by two internal states of each ion (|g(e)〉j) (Figure 14(c)). Fluctuating magnetic fields give rise to a collective fluctuation of qubit energies, thereby causing cross dephasing. Modulations that locally address each ion (j=1,…,N) may be realized by laser fields whose amplitude varies spatially and differently affects the jth “carrier” frequency ωj(t) of the transition |g〉j↔|e〉j: a gradient of the laser field amplitude along the ion chain would suffice to achieve this.Figure 14
Fidelity as a function of time for entangled states experiencing collective relaxation under different modulations compared to that of a single qubit without modulation (blue, dashed) (color online): bright-state,N=10 without modulation (red, dot-dashed); GHZ with N=10 without modulation (green, dotted); bright-state with N=10 under asymmetric modulation (black, solid). (a) Decoherence rates for N=3 with asymmetric modulation. Single-qubit decoherence rates, Rjj(t) (black, solid), undergo oscillations then attain constant nonzero values at t≫tcorr; cross-decoherence rates Rjj′(t) (blue, dashed) also oscillate but then decay to zero at t≫tcorr. (b) For qubits j and j′, the overlap of the bath coupling spectrum Gjj′(ω) (red area) and the modulation spectra ϵt,j(ω) and ϵt,j′(ω) (green areas) determine the decoherence rate. If the qubits undergo (asymmetric) different modulations ϵj(ω)≠ϵj′(ω), their lack of overlap eliminates cross decoherence. (c) Addressing individual ions in a trap by different control fields with different amplitudes induces asymmetric Rabi flippings and thereby eliminates cross decoherence.Each local modulation may decrease theindividual decoherence rate of each qubit by at least an order of magnitude, depending on the noise spectrum and available modulation frequencies. More importantly, local modulations may eliminate cross dephasing between ions, thereby increasing the coherence time of bright or GHZ states by a factor of N. For an ion trap with tcoh(1)=14.7s, (R≥10-2s-1) [181], local modulations at intervals τ≅1ms can drastically increase the coherence times of GHZ or bright states, for example, tcohmod(10)≅tcoh(1)≅10s (Figure 14).


## 8.3.1. Multipartite Entangled States

We treat decoherence scaling withN for several distinct multipartite entangled states with clear N-dependence.  (i)
GHZ states,|GHZ〉=(1/2)(⨂j|e〉j+⨂j|g〉j).(ii)
W-states,|W〉=(1/N)∑j=1N‍(|e〉j⨂j≠j′|g〉j′).(iii)
Bright states,|B〉=|J=N/2,m=0〉, where |J,m〉 are the Dicke states [177], with J being the total angular momentum and m the excitation number.(iv)
Dark states, given by|D〉=⨂j=2,4,...,2N|Ψ-〉j-1,j for even N, where |Ψ-〉j,j+1=(1/2)(|g〉j|e〉j+1-|e〉j|g〉j+1).Equation (178) shows that the infidelity rate depends on the specific multipartite state and on the decoherence rates Rjj′(t). The time dependence of these rates is revealed on non-Markovian time scales, yet they are constant on long (Markov) time scales. The diagonal elements of the decoherence-rate matrix, Rjj(t), correspond to single-qubit decoherence, determined by the coupling of individual qubits (labeled by j) to their respective environments (baths). On the other hand, the off-diagonal elements, Rjj′(t) with j≠j′, represent cross-decoherence rates, that is, decoherence rates induced on the jth qubit by the j′th qubit. Such cross-decoherence rates are present only for qubits coupled to a common bath or if their dephasing noise is correlated. More explicitly, cross decoherence occurs because two qubits are coupled to the same bath (noise) mode (Figure 1(a)). In the second-order approximation, this means that qubit j interacts with a certain bath (or noise) mode and then the same mode interacts with qubit j′ (cf. (178)). In the dephasing scenario, cross dephasing stems from the cross correlation of the stochastic noise functions of qubits j and j′. Thus, for example, ions subject to the same fluctuating magnetic field will experience cross dephasing.In the absence of cross decoherence, when each qubit is locally coupled to an individual bathRjj′(t)=0, for all j≠j′, the first state-dependent term of the infidelity rate in (178) scales linearly with N, for all states. The second term then appears only for unique states, for example, W-state, and can only lower the N scalability (Table 1), resulting in (at most) linear dependence on N for all states in systems without cross decoherence.Table 1
N-scalability of entangled states decoherence for relaxation and dephasing scenarios, with and without cross decoherence.
Relaxation
Dephasing
No X-deco.
X-deco.
No X-deco.
X-deco.
GHZ
N
N
N
N
2
W-state
N
3
N
-
2
4
(
N
-
1
)
/
N
0
Bright state
N
N
+
N
2
/
2
N
0
Dark state
N
0
N
0By contrast, the infidelity rate is much morestate-dependent in systems with cross decoherence, since these systems have additional decoherence channels that may interfere, either constructively or destructively, depending on the initial entangled state via the phases of 〈ψ|σ^jxσ^j′x|ψ〉. Hence, the N scalability under cross decoherence can either mount to N2 or vanish completely, a situation unattainable in the absence of cross decoherence.More specifically, under global dephasing, the GHZ state becomes a bright state in the|±〉 basis and hence experiences constructive interference of all decoherence channels, so that its dephasing scales quadratically with N. The W-state and bright state, on the other hand, do not experience global dephasing since they are composed of a symmetric superposition of identically excited states and hence the dephasing becomes a global phase factor and does not influence decoherence. Remarkably, the dark state considered here is dark for global relaxation and dephasing, due to the invariance of the |Ψ-〉 state under the basis rotation from |e(g)〉 to |±〉. For higher-order approximation this may not be valid.


## 8.3.2. Decoherence Scaling and the Zeno Regime

One may view decoherence scalability from a complementary aspect, that of the coherence time of the system. In this section, we consider an unmodulated scenario with identical coupling to all qubits, that is,(180)Γψ(t;N)=cψ(N)R(t),
where cψ(N) is the state-dependent scalability. We define the coherence time as the time at which the fidelity (177), or alternatively the error, e(t)=1-f(t), reaches a certain threshold as
(181)tcohΓψ(tcoh;N)=1-fthreshold≡ethreshold.
This coherence time is an important measure, since its ratio to the time it takes to perform a single quantum operation (e.g., a two-qubit gate) [3] determines how many such operations can be performed successfully, that is, the complexity of the quantum computation.The coherence time is usually identified with the inverse decoherence rate, since in the Markovian time-regime, the decoherence rate is constant, that isR(t)=R, yielding
(182)tcoh(N)|Markov=ethresholdcψ(N)R.
For example, linearly scaling systems cψ(N)=N, such as GHZ relaxation without cross decoherence, have Markovian coherence time that scales inversely with N. However, for large systems, the fidelity threshold (181) is reached according to (177) after a time that may be much shorter than the correlation (memory) time of the bath, tcorr, so that the non-Markov time dependence of R(t) starts playing a role. For still larger N, this threshold will be reached within the extremely short Zeno time-regime, tZeno [86, 97, 99, 102, 178, 179], in which the decoherence rate scales linearly with t, RZeno(t)=tR˙ (see below). Hence, for sufficiently large systems, such that
(183)cψ(N)≥ethresholdtZeno2R˙
the coherence time is always that obtained in the Zeno time-regime as
(184)tcoh(N)|Zeno=ethresholdcψ(N)R˙.
If cψ(N)=N, the Zeno coherence time scales as the inverse square root of N. This paradoxical result, namely, the inevitable improvement of the decoherence scaling law for sufficiently large systems, has been thus far overlooked. This result is a manifestation of the non-Markovian effects in decoherence dynamics.


## 8.3.3. Dynamically Modified Decoherence Rates

From (178), it is evident that the decoherence rates Rjj′(t) are an important factor in the scaling of decoherence. Their time dependence stems from the non-Markovian master equation (ME) [126]. The same non-Markovian ME governs dynamical decoherence control. The resulting (time-averaged) decoherence rates then assume the form of spectral overlaps of the bath coupling spectrum matrix, Gjj′(ω), and the modulation power-spectrum matrix, Ft,jj′(ω) [174] as
(185)Rjj′(t)=2πRe∫-∞∞‍dωGjj′(ω)Ft,jj′(ω),(186)Ft,jj′(ω)=1tϵt,j(ω)ϵt,j′*(ω),
where Gjj′(ω) is the Fourier transform of the system-bath correlation matrix Φjj′(ω), and Ft,jj′(ω) is the time-dependent normalized modulation spectrum, where ϵt,j(ω)=(1/2π)∫0t‍dt′ϵj+(t′)eiωt′ is the finite-time Fourier transform of the modulation function. In the relaxation scenario, the modulation spectral function represents the effective change in qubit j’s energy splitting or AC Stark shift (Figure 13(b)). In the dephasing scenario, it determines the time-dependent Rabi splitting of the |±〉 states (Figure 13(c)). In both scenarios, the spectral overlap Gjj′(ω)ϵt,j*(ω)ϵt,j′ω) in (185) represents the effective coupling between the systems and the bath, whereby the dynamical control (modulation) determines which spectral bath modes the qubits are coupled to.There is an important distinction between the controllability of single-qubit decoherence rates,Rjj(t), and that of their cross-decoherence counterpart, Rjj′(t). Namely, the single-qubit dynamically modified Rjj(t) can only be reduced by decreasing the overlap between the modulation spectra |ϵt,j(ω)|2 and the bath coupling spectra, Gjj(ω), both positive. No matter how fast the modulation is, |ϵt,j(ω)|2 cannot decrease the single-qubit decoherence rate, if the bath coupling spectrum is spectrally flat (Markovian). By contrast, cross-decoherence rates Rjj′(t) can be completely eliminated by choosing spectrally distinct local modulations whose spectral overlap ϵt,j*(ω)ϵt,j′(ω) vanishes, irrespective of the bath coupling spectrum, Gjj′(ω) even if it is Markovian (spectrally flat) (Figures 2(a) and 2(b)). The qubits effectively couple to orthogonal bath modes. Hence, by modulating each qubit differently, one can effectively decouple the qubits, as if they were interacting with separate baths.


## 8.3.4. Dynamically Modified Decoherence Scaling

How do these modulations affect decoherence scaling? Let us examine two limits of the general scenario more closely.(A)
Consider first the limit oflocally decohering qubits. Due to the absence of a common bath, modulations cannot induce cross decoherence. Hence, they cannot change the scaling law, which is identical for most multipartite entangled states, namely, Γψ(t;N)∝N (Table 1). Nevertheless, modulations can still improve the infidelity rate, by decreasing the overlap of each qubit’s bath coupling and modulation spectra.(B)
The other limiting case is that of collective decoherence of qubits coupled to a common bath, but locally controlled. By exploiting the spectral distinction ofN different local modulations, one can completely eliminate cross decoherence and effectively render the system similar to the aforementioned locally decohering qubits, in which for most states, decoherence scales linearly with N. Such local modulations may be advantageous for the relaxing bright state, or dephasing GHZ-state, by reducing its infidelity rate scaling by a factor of N.On the other hand, one may enhance cross decoherence by local modulations: if the qubits are coupled to the same bath but with different strengths, then the modulations can be chosen to compensate for their unequal coupling strengths. Such enhanced cross decoherence can drastically improve the coherence time by rendering certain states completely dark: without modulation, partially dark states with unequal coupling have infidelity rate that scales asN, whereas under modulations that impose destructive interference of the decoherence channels, such states can acquire much longer coherence time by becoming fully dark.


## 8.3.5. Effective Modulation Time Scales

When do such local dynamic modulations become effective? Let us recall the relevant time scales. The longest one is the coherence time of a single-qubit without modulation,tcoh(1)≅1/Γψ. Another is the correlation (memory) time of the bath, tcorr. It is the decay time of the correlation function, which is given by the product of the noise amplitude at time t with the noise amplitude at time t+τ, averaged over realizations. In the frequency domain 1/tcorr is the width of the noise-coupling spectrum and in dephasing noise this is the noise bandwidth.In the single qubit case, the weak-coupling assumption (see below) implies that the correlation time is shorter than the single-qubit coherence time [70] tcorr≪tcoh(1) or, equivalently, noise bandwidth much larger than dephasing rate [174]. In the unmodulated N-qubit scenario, this assumption is given by tcorr≪tcoh(N), whereas in the modulated case, it is the modulated coherence time that sets the limit for the weak coupling regime, tcorr≪tcohmod(N). Hence, even if for a large (unmodulated) system the weak coupling assumption does not hold, modulations that drastically increase the coherence time, for example, by eliminating cross decoherence, can make the system enter the valid regime for which the analysis holds.Only modulations faster than the bath-(noise-) correlation time can affect the single-qubit decoherence ratesRjj(t). Yet a crucial insight transpires: if each qubit decoheres individually (cross decoherence is eliminated), the (unmodulated) N-particle coherence time does not determine whether the control is effective. Neither does the infidelity rate (178). Rather, the modulation period, τ, must be shorter than tcorr, but can be much larger than tcoh(1)/N: it is unaffected by scaling!We can now summarize the conditions for an effective increase of the coherence time of entangled multipartite systems by modulations with a periodτ as
(187)tcoh(N)<︸aτ<︸btcorr<︸ctcoh(1)<︸dtcohmod(N).
Here, (a) lays out the striking result that effective modulation can be slower than the coherence time of the system; (b) is the major requirement for effective modulation; (c) is the single-qubit weak-coupling assumption; (d) shows the main result, that is, the possibility of attaining (via effective modulations) multipartite coherence time tcohmod(N) that is longer than the single-qubit coherence time.


## 8.3.6. Application to an Ion Trap

Our analysis is applicable to a broad variety of experimental setups. As an example, consider a string of ions in a linear trap [142, 180, 181], in which the qubits are encoded by two internal states of each ion (|g(e)〉j) (Figure 14(c)). Fluctuating magnetic fields give rise to a collective fluctuation of qubit energies, thereby causing cross dephasing. Modulations that locally address each ion (j=1,…,N) may be realized by laser fields whose amplitude varies spatially and differently affects the jth “carrier” frequency ωj(t) of the transition |g〉j↔|e〉j: a gradient of the laser field amplitude along the ion chain would suffice to achieve this.Figure 14
Fidelity as a function of time for entangled states experiencing collective relaxation under different modulations compared to that of a single qubit without modulation (blue, dashed) (color online): bright-state,N=10 without modulation (red, dot-dashed); GHZ with N=10 without modulation (green, dotted); bright-state with N=10 under asymmetric modulation (black, solid). (a) Decoherence rates for N=3 with asymmetric modulation. Single-qubit decoherence rates, Rjj(t) (black, solid), undergo oscillations then attain constant nonzero values at t≫tcorr; cross-decoherence rates Rjj′(t) (blue, dashed) also oscillate but then decay to zero at t≫tcorr. (b) For qubits j and j′, the overlap of the bath coupling spectrum Gjj′(ω) (red area) and the modulation spectra ϵt,j(ω) and ϵt,j′(ω) (green areas) determine the decoherence rate. If the qubits undergo (asymmetric) different modulations ϵj(ω)≠ϵj′(ω), their lack of overlap eliminates cross decoherence. (c) Addressing individual ions in a trap by different control fields with different amplitudes induces asymmetric Rabi flippings and thereby eliminates cross decoherence.Each local modulation may decrease theindividual decoherence rate of each qubit by at least an order of magnitude, depending on the noise spectrum and available modulation frequencies. More importantly, local modulations may eliminate cross dephasing between ions, thereby increasing the coherence time of bright or GHZ states by a factor of N. For an ion trap with tcoh(1)=14.7s, (R≥10-2s-1) [181], local modulations at intervals τ≅1ms can drastically increase the coherence times of GHZ or bright states, for example, tcohmod(10)≅tcoh(1)≅10s (Figure 14).


## 8.4. Validity of the Weak-Coupling Approximation

### 8.4.1. Thermal Baths

For coupling to a thermal bath, the first-order term is proportional toTr〈HIBρB〉, where HIB is the bath part of the coupling Hamiltonian and ρB is the density matrix of the bath’s thermal state. In the scenario considered here, HIB has only off-diagonal terms in the bath eigenbasis. For example, HIB=∑k‍(λkak+h.c.) for coupling to an oscillator bath, where λk is the coupling coefficient to mode k and ak is the annihilation operator. This form renders the first-order term zero.


### 8.4.2. Validity of the ME

The master equation, (98), is valid as long as the density matrix does not change much within the correlation (memory) time of the bath, tcorr. In the entangled multipartite scenarios discussed here, this condition becomes tcorr∑j,j′=1N‍Rjj′(tcorr)≪1.


### 8.4.3. Validity of the Fidelity

The fidelity derived above, (177) requires a stricter condition t∑j,j′=1N‍Rjj′(t)≪1. Several important issues are raised by this approximation. The first is that it is N-dependent, meaning the time in which the approximation holds becomes shorter for larger systems. Furthermore, the N-dependency of the approximation depends on whether there is cross decoherence. With cross decoherence, the time within which the approximation holds depends on N2, whereas without cross decoherence it depends on N.In Figure14, one can see that the single-qubit fidelity is within the validity domain of the approximation, whereas the fidelity of entangled states with N=10 qubits without modulation becomes smaller and exit the validity regime rapidly. However, by applying different modulations on the qubits, one can not only reduce each qubit’s decoherence rate, but also  eliminate the cross decoherence and increase the fidelity by N=10-fold, thus bringing it within the validity region of the weak-coupling approximation.


## 8.4.1. Thermal Baths

For coupling to a thermal bath, the first-order term is proportional toTr〈HIBρB〉, where HIB is the bath part of the coupling Hamiltonian and ρB is the density matrix of the bath’s thermal state. In the scenario considered here, HIB has only off-diagonal terms in the bath eigenbasis. For example, HIB=∑k‍(λkak+h.c.) for coupling to an oscillator bath, where λk is the coupling coefficient to mode k and ak is the annihilation operator. This form renders the first-order term zero.


## 8.4.2. Validity of the ME

The master equation, (98), is valid as long as the density matrix does not change much within the correlation (memory) time of the bath, tcorr. In the entangled multipartite scenarios discussed here, this condition becomes tcorr∑j,j′=1N‍Rjj′(tcorr)≪1.


## 8.4.3. Validity of the Fidelity

The fidelity derived above, (177) requires a stricter condition t∑j,j′=1N‍Rjj′(t)≪1. Several important issues are raised by this approximation. The first is that it is N-dependent, meaning the time in which the approximation holds becomes shorter for larger systems. Furthermore, the N-dependency of the approximation depends on whether there is cross decoherence. With cross decoherence, the time within which the approximation holds depends on N2, whereas without cross decoherence it depends on N.In Figure14, one can see that the single-qubit fidelity is within the validity domain of the approximation, whereas the fidelity of entangled states with N=10 qubits without modulation becomes smaller and exit the validity regime rapidly. However, by applying different modulations on the qubits, one can not only reduce each qubit’s decoherence rate, but also  eliminate the cross decoherence and increase the fidelity by N=10-fold, thus bringing it within the validity region of the weak-coupling approximation.


## 8.5. Fixed-Time Approach

Assume that a quantity of interest (“score”) can be written as a real-valued functionP(t)=P[ϱ^(t)] of the system state ϱ^(t) at a given time t. This might be, for example, a measure of performance of some input-output device that is supposed to operate within a predefined cycle/gate time t. Depending on the physical problem and model chosen, extensions and generalizations are conceivable, such as a comparison of the outcome for different t (on a time scale set by a constraint) [138], a time average P=∫‍dτf(τ)P[ϱ^(τ)] with some probability density f(τ) [182], or a maximum P(t)=maxτ∈[0,t]P[ϱ^(τ)] [183]. Here, we restrict ourselves to the “fixed-time” definition as given above. Our goal is to generate, by means of classical control fields applied to the system, a time dependence of the system Hamiltonian within the interval 0≤τ≤t that adjusts P(t) to a desired value. In particular, this can be an optimum (i.e., maximum or minimum) of the possible values of P. Assume that the initial system state ϱ^(0) is given. The change to the “score” over time t, P=P(t)-P(0), is then given by the first-order Taylor expansion in a chosen basis
(188)P≈∑mn‍∂P∂ϱmnΔϱmn=Tr(P^Δϱ^),
where Δϱ^=ϱ^(t)-ϱ^(0) and the expansion coefficients
(189)(∂P∂ϱmn)t=0≡(P^)nm
are the matrix elements (in the chosen basis) of a Hermitian operator P^, which is the gradient of P[ϱ^(t)] with respect to ϱ^ at t=0; that is, we may formally write P^=(∇ϱ^P)t=0T=(∂P/∂ϱ^)t=0T. In what follows, it is P^ which contains all information on the score variable. Note that the transposition applied in (189) simply allows to express the sum over the Hadamard (i.e., entrywise) matrix product in (188) as a trace of the respective operator product P^Δϱ^.Let us illustrate this in two examples. IfP is the expectation value of an observable (i.e., Hermitian operator) Q^, so that P=Tr(ϱ^Q^), then (189) just reduces to this observable  P^=Q^. If P is the state purity, P=Tr(ϱ^2), then (189) becomes proportional to the state, P^=2ϱ^(0). Note that the score P is supposed to reflect the environment (bath) effects and not the internal system dynamics.Equation (188) implies that Δϱ^ and with it P are small. Hence Δϱ^ must refer to the interaction picture and a weak interaction, while P[ϱ^(t)] should not be affected by the internal dynamics (so that no separate time dependence emerges in (188), which is not included in the chain-rule derivative). In the examples above, this is obvious for state purity, whereas an observable Q^ might be thought of as coevolving with the internal dynamics.Our starting point for what follows is simply the relationP=Tr(P^Δϱ^) with some Hermitian P^, whose origin is not relevant.


## 8.6. Role of Averaged Interaction Energy

Equation (188) expresses the score P as an overlap between the gradient P^ and the change of system state Δϱ^. In order to find expressions for P in terms of physically insightful quantities, we decompose the total Hamiltonian into system, bath, and interaction parts,
(190)H^(t)=H^S(t)+H^B+H^I,
and consider the von Neumann equation of the total (system and environment) state in the interaction picture,
(191)∂∂tϱ^tot(t)=-i[H^I(t),ϱ^tot(t)],
(H^I(t)=U^F†(t)H^I(S)(t)U^F(t), (S) denoting the Schrödinger picture and U^F(t)=T+e-i∫0t‍dτ[H^S(τ)+H^B]). Its solution can be written as Dyson (state) expansion
(192)ϱ^tot(t)=ϱ^tot(0)+(-i)∫0t‍dt1[H^I(t1),ϱ^tot(0)]+(-i)2∫0t‍dt1∫0t1‍dt2[H^I(t1),[H^I(t2),ϱ^tot(0)]]+⋯,
which can be obtained either by an iterated integration of (191) or from its formal solution
(193)ϱ^tot(t)=U^I(t)ϱ^tot(0)U^I†(t),U^I(t)=T+e-i∫0t‍dt′H^I(t′),
by applying the Magnus (operator) expansion
(194)U^I(t)=e-itH^eff(t),(195)H^eff(t)=1t∫0t‍dt1H^I(t1)-i2t∫0t‍dt1∫0t1‍dt2[H^I(t1),H^I(t2)]+⋯,
expanding in (193) the exponential U^I=U^I(H^eff) and sorting the terms according to their order in H^I.We assume that, initially, the system is brought in contact with its environment (rather than being in equilibrium with it), which corresponds to factorizing initial conditionsϱ^tot(0)=ϱ^(0)⊗ϱ^B. The environment is in a steady state ϱ^B, [ϱ^B,H^B]=0, so it is more adequate to speak of a “bath.” Tracing over the bath in (192) then gives the change of system state Δϱ^=ϱ^(t)-ϱ^(0) over time t, which we must insert into (188). We further assume a vanishing bath expectation value of the interaction Hamiltonian,
(196)〈H^I〉B≡TrB(ϱ^BH^I)=0^.
As a consequence, the “drift” term corresponding to the first order in (192) vanishes, and we only consider the second order term as the lowest nonvanishing order approximation. Finally, we assume that the initial system state commutes with P^,
(197)[ϱ^(0),P^]=0.
In the language of control theory, Tr[ϱ^(0)P^] is a “kinematic critical point” [184] if (197) holds, since Tr[eiH^ϱ^(0)e-iH^P^]=Tr[ϱ^(0)P^]+iTr(H^[ϱ^(0),P^])+𝒪(H^2) for a small arbitrary system Hamiltonian H^. Since we consider ϱ^ in the interaction picture, (197) means that the score is insensitive (in first order) to a bath-induced unitary evolution (i.e., a generalized Lamb shift) [140]. The purpose of this assumption is only to simplify the expressions, but it is not essential. Physically, one may think of a fast auxiliary unitary transformation that is applied initially in order to diagonalize the initial state in the eigenbasis of P^.To the lowest (i.e., second) order, we then evaluate (188) for the score change as
(198)P=t2〈[H^,P^]H^〉,H^=1t∫0t‍dτH^I(τ),
where 〈·〉=Tr[ϱ^tot(0)(·)]. This expresses the change of score in terms of the interaction Hamiltonian, averaged in the interaction picture over the time interval of interest. Our scheme is summarized in Figure 15.Figure 15
Our control scheme (color online): a system is brought in contact with a bath over a fixed time interval0≤τ≤t during which the time dependence of the system Hamiltonian H^S(τ) is chosen such that a given system variable P is adjusted to a desired value at the final time t.


## 8.7. Spectral Overlap

Alternatively, (198) can be written as an overlap of system and bath matrices, which allows a more direct physical interpretation. To do so, we assume d dimensional Hilbert space  and expand the interaction Hamiltonian as a sum of products of system and bath operators,
(199)H^I=∑j=1d2-1‍S^j⊗B^j,
in such a way that 〈B^j〉=0, which ensures that (196) is satisfied (otherwise we may shift B^j′=B^j-〈B^j〉I^, H^S′=H^S+∑j‍〈B^j〉S^j). Considering (199) in the interaction picture and expanding S^j(t)=∑k‍ϵjk(t)S^k in terms of (Hermitian, traceless, orthonormalized to Tr(S^jS^k)=dδjk) basis operators, S^j defines a (real orthogonal) rotation matrix ϵ(t) in the system’s Hilbert space, with elements
(200)ϵjk(t)=〈S^j(t)S^k〉id,
where 〈·〉id=Tr[d-1I^(·)]. These elements of the matrix ϵ(t) may be regarded as the dynamical correlation functions of the basis operators. Analogously, we define a bath correlation matrix Φ(t) with elements
(201)Φjk(t)=〈B^j(t)B^k〉B.
It contains the entire description of the bath behavior in our approximation. Finally, we define a Hermitian matrix Γ with elements
(202)Γkj=〈[S^j,P^]S^k〉,
where 〈·〉=Tr[ϱ^(0)(·)]. The matrix Γ may be understood as a representation of the gradient P^ with respect to the chosen basis operators S^j. Finally, we define the bath and (finite-time) system spectra according to
(203)G(ω)=∫-∞∞‍dteiωtΦ(t),ϵt(ω)=12π∫0t‍dτeiωτϵ(τ).
This allows to express the score, (198), as the matrix overlap
(204)P=∬0tdt1dt2Tr[ϵT(t1)Φ(t1-t2)ϵ(t2)Γ](205)=∫-∞∞‍dωTr[ϵt†(ω)G(ω)ϵt(ω)Γ](206)=t∫-∞∞‍dωTr[Ft(ω)G(ω)].
In (206), we have used the cyclic property of the trace to write the spectral overlap in a more compact form by combining the rotation matrix spectra ϵt(ω) and the gradient representation Γ to a system spectral matrix
(207)Ft(ω)=1tϵt(ω)Γϵt†(ω).
Analogously, (204) can be written in a more compact form by introducing the matrix
(208)R(t1,t2)=ϵT(t1)Φ(t1-t2)ϵ(t2).Equation (206) is a generalization of [105] and demonstrates that the change P over a given time t is determined by the spectral overlap between system and bath dynamics, analogously to DCM [185], or the measurement induced quantum Zeno and anti-Zeno control of open systems [97, 102]. The bath-spectral matrix G(ω) must be positive semidefinite for all ω. If the same holds for the matrix Ft(ω), then P is always positive. Below we will consider such a case where P reflects a gate error and the goal is then to minimize this error. The spectral overlap (206) can be made as small as desired by a rapid modulation of the system, such that the entire weight of the system spectrum is shifted beyond that of the bath, which is assumed to vanish for sufficiently high frequencies. Since this fast modulation may cause unbounded growth of the system energy, a meaningful posing of the problem requires a constraint.In general,Ft(ω) is Hermitian but need not necessarily be positive semi-definite, depending on the choice of score as encoded in Γ. This reflects the fact that P can increase or decrease over t. Depending on the application, our goal can therefore also be to maximize P with positive and negative sign. In what follows, we will consider the question how to find a system dynamics that optimizes the score.


## 8.8. Euler-Lagrange Optimization

### 8.8.1. Role of Control, Score, and Constraint

Our considerations in the previous section suggest to define our control problem in terms of a triple(f,P,E) consisting of a control f, a score P, and a constraint E.The control is a set of real parametersfl, which have been combined to a vector f. These can either be timings, amplitudes, and/or phases of a given number of discrete pulses or can describe a time-continuous modulation of the system. Here, we focus on time-dependent control, where the fl(τ) parametrize the system Hamiltonian as H^S=H^S[f(τ)], or the unitary evolution operator U^(τ)=T+e-i∫0τ‍dτ′H^S(τ′)≡U^[f(τ)]. A direct parametrization of U^ avoids the need of time-ordered integration of its exponent. The U^(τ) thus obtained [176] can be then used to calculate the system Hamiltonian H^S(τ)=i[(∂/∂t)U^(τ)]U^†(τ).Two explicit examples of the scoreP pertain to the fidelity of P^ with a given pure state FΨ=〈Ψ|ϱ^|Ψ〉 (for which P^=|Ψ〉〈Ψ|), or to the von Neumann entropy which we can approximate (for nearly pure states) by the linear entropy, S=-kTr(ϱ^lnϱ^)≈SL=k[1-Tr(ϱ^2)] (for which P^=-2kϱ^(0)). The latter score can be used to maximize the fidelity with the maximally mixed state ϱ^~I^ (for which SL becomes maximum), or to maximize the concurrence C|ΨAB〉=2(1-Trϱ^A2), ϱ^A=TrB|ΨAB〉〈ΨAB|, as a measure of entanglement of a pure state |ΨAB〉 of a bipartite system.If aconstraint is required to ensure the existence of a finite (physical) solution, its choice should depend on the most critical source of error. An example is the average speed with which the controls change, E=∫0t‍dτf˙2(τ), which depend on the control bandwidth in the spectral domain. A parametrization-independent alternative is the mean square of the modulation energy, E=∫0t‍dτ〈(ΔH^)2(τ)〉id, where 〈·〉id refers to a maximally mixed state and hence to a state-independent norm, and ΔH^ is the difference between the modulated and unmodulated (natural) system Hamiltonians.


### 8.8.2. A Projected Gradient Search

We want to find controlsf that optimize a score P(f) subject to a constraint E(f). A numerical local optimization can be visualized in parameter space as shown in Figure 16.(a) Optimization of the scoreP(f) subject to a constraint E(f) in control space {f} by walking along the component δP⊥ of the gradient δP orthogonal to δE. (b) Resulting iteration consisting in steps determined by a small parameter ϵ which yields a local solution depending on the starting point f0 (color online).
(a)
(b)We start at some initial pointf0 for which E(f0) is the desired value of the constraint. Simply following the gradient δP would maximize or minimize P but would also change E. To optimize P while keeping E constant, we therefore move along the projection of δP orthogonal to δE, that is, along δP⊥=δP-((δP·δE)/(δE)2)δE. Since the gradients depend on f, the iteration consists of small steps fn+1=fn±ϵδP⊥(fn), ϵ≪1. Assuming that neither δP nor δE vanishes, the iteration will come to a halt where δP⊥ vanishes, because the gradients are parallel,
(209)δP=λδE.
This condition constitutes the Euler-Lagrange (EL) equation of the extremal problem, with the proportionality constant λ being the Lagrange multiplier. Its concrete form depends on the choice of P and E. Since the solutions of the EL optimization represent local optima of the constrained P, we may repeat the search with randomly chosen f0 a number of times and select the best solution. The gradients at each point fn may be obtained either from a calculation based on prior knowledge of the bath or experimentally from data measured in real time. A discretization of the time interval 0≤τ≤t then reduces the variational δ to a finite-dimensional vector gradient ∇.


## 8.8.1. Role of Control, Score, and Constraint

Our considerations in the previous section suggest to define our control problem in terms of a triple(f,P,E) consisting of a control f, a score P, and a constraint E.The control is a set of real parametersfl, which have been combined to a vector f. These can either be timings, amplitudes, and/or phases of a given number of discrete pulses or can describe a time-continuous modulation of the system. Here, we focus on time-dependent control, where the fl(τ) parametrize the system Hamiltonian as H^S=H^S[f(τ)], or the unitary evolution operator U^(τ)=T+e-i∫0τ‍dτ′H^S(τ′)≡U^[f(τ)]. A direct parametrization of U^ avoids the need of time-ordered integration of its exponent. The U^(τ) thus obtained [176] can be then used to calculate the system Hamiltonian H^S(τ)=i[(∂/∂t)U^(τ)]U^†(τ).Two explicit examples of the scoreP pertain to the fidelity of P^ with a given pure state FΨ=〈Ψ|ϱ^|Ψ〉 (for which P^=|Ψ〉〈Ψ|), or to the von Neumann entropy which we can approximate (for nearly pure states) by the linear entropy, S=-kTr(ϱ^lnϱ^)≈SL=k[1-Tr(ϱ^2)] (for which P^=-2kϱ^(0)). The latter score can be used to maximize the fidelity with the maximally mixed state ϱ^~I^ (for which SL becomes maximum), or to maximize the concurrence C|ΨAB〉=2(1-Trϱ^A2), ϱ^A=TrB|ΨAB〉〈ΨAB|, as a measure of entanglement of a pure state |ΨAB〉 of a bipartite system.If aconstraint is required to ensure the existence of a finite (physical) solution, its choice should depend on the most critical source of error. An example is the average speed with which the controls change, E=∫0t‍dτf˙2(τ), which depend on the control bandwidth in the spectral domain. A parametrization-independent alternative is the mean square of the modulation energy, E=∫0t‍dτ〈(ΔH^)2(τ)〉id, where 〈·〉id refers to a maximally mixed state and hence to a state-independent norm, and ΔH^ is the difference between the modulated and unmodulated (natural) system Hamiltonians.


## 8.8.2. A Projected Gradient Search

We want to find controlsf that optimize a score P(f) subject to a constraint E(f). A numerical local optimization can be visualized in parameter space as shown in Figure 16.(a) Optimization of the scoreP(f) subject to a constraint E(f) in control space {f} by walking along the component δP⊥ of the gradient δP orthogonal to δE. (b) Resulting iteration consisting in steps determined by a small parameter ϵ which yields a local solution depending on the starting point f0 (color online).
(a)
(b)We start at some initial pointf0 for which E(f0) is the desired value of the constraint. Simply following the gradient δP would maximize or minimize P but would also change E. To optimize P while keeping E constant, we therefore move along the projection of δP orthogonal to δE, that is, along δP⊥=δP-((δP·δE)/(δE)2)δE. Since the gradients depend on f, the iteration consists of small steps fn+1=fn±ϵδP⊥(fn), ϵ≪1. Assuming that neither δP nor δE vanishes, the iteration will come to a halt where δP⊥ vanishes, because the gradients are parallel,
(209)δP=λδE.
This condition constitutes the Euler-Lagrange (EL) equation of the extremal problem, with the proportionality constant λ being the Lagrange multiplier. Its concrete form depends on the choice of P and E. Since the solutions of the EL optimization represent local optima of the constrained P, we may repeat the search with randomly chosen f0 a number of times and select the best solution. The gradients at each point fn may be obtained either from a calculation based on prior knowledge of the bath or experimentally from data measured in real time. A discretization of the time interval 0≤τ≤t then reduces the variational δ to a finite-dimensional vector gradient ∇.


## 8.9. Optimal Gate Protection

### 8.9.1. Gate Error as Average Fidelity Decline

A particular application of our formalism is decoherence protection of a given quantum operation by bath-optimal minimal-energy control (BOMEC) [176, 185]. Consider the implementation of a predetermined quantum gate, that is, unitary operation within a given “gate time” t. It is sufficient to consider a pure input state |Ψ〉. In the interaction picture with respect to the desired gate operation and in the absence of bath effects, we should therefore observe at time t the initial state |Ψ〉. The quantity of interest is here the fidelity 〈Ψ|ϱ^(t)|Ψ〉, and we use the projector P^=ϱ^(0)=|Ψ〉〈Ψ| as the gradient operator, so that (197) is satisfied and (198) gives the fidelity change as the score
(210)P=〈Ψ|Δϱ^|Ψ〉=-t2〈〈Ψ|H^2|Ψ〉-〈Ψ|H^|Ψ〉2〉B,
which is given by H^ defined in (198).Since a quantum gate is supposed to act on an unknown input state, we need to get rid of the dependence on|Ψ〉. One possibility is to perform a uniform average over all |Ψ〉. We may apply, (see [186, 187])
(211)〈Ψ|A^|Ψ〉〈Ψ|B^|Ψ〉¯=TrA^B^+TrA^TrB^d(d+1)
which gives the average
(212)P¯=-t2dd+1〈H^2〉id,
where 〈·〉id=Tr[d-1I^⊗ϱ^B(·)]. In (212), we have used TrSH^=0^, which corresponds to TrS^j=0 in Section 8.7. Because of this and because of (196), 〈H^〉B=〈H^I〉B=0^, we have 〈H^〉id=0, and (212) also describes the variance Var(H^)=〈H^2〉id-〈H^〉id2. On the other hand, P^=-2kϱ^(0) that gives the change ΔS of entropy S=-kTr(ϱ^lnϱ^) is (up to a proportionality factor of -2k) the same as the P^ used here to give the change of fidelity, and we have ΔS=-2kP. If we define a gate errorℰ as the average fidelity decline, ℰ=-P¯, with P¯ given in (212), we can summarize the following proportionalities: gate error ≡ average fidelity decline ~ average entropy increase (purity decline) ΔS¯~ square (variance) of the average interaction energy H^ as
(213)ℰ≡-P¯=ΔS¯2k=t2dd+1〈H^2〉id=t2dd+1Var(H^).
In the matrix representation of Section 8.7, the average over the initial states in the matrix Γ defined in (202) gives Γ¯=-(d/(d+1))I (using Tr(S^jS^k)=dδjk and TrS^j=0), so that
(214)ℰ=dd+1∫-∞∞‍dωTr[ϵt(ω)ϵt†(ω)G(ω)]
in agreement with [176] (except a different normalization Tr(S^jS^k)=2δjk leading there to a prefactor (2/(d+1))). From the requirement that ℰ≥0 must hold for any positive semi-definite matrix ϵt(ω)ϵt†(ω), we conclude that G(ω) must be a positive semi-definite matrix for any ω. The task of BOMEC is then to find a system evolution U^(τ) (cf. the control examples in the previous section) that minimizes ℰ, subject to the boundary condition that the final U^(t) is the desired gate.


## 8.9.1. Gate Error as Average Fidelity Decline

A particular application of our formalism is decoherence protection of a given quantum operation by bath-optimal minimal-energy control (BOMEC) [176, 185]. Consider the implementation of a predetermined quantum gate, that is, unitary operation within a given “gate time” t. It is sufficient to consider a pure input state |Ψ〉. In the interaction picture with respect to the desired gate operation and in the absence of bath effects, we should therefore observe at time t the initial state |Ψ〉. The quantity of interest is here the fidelity 〈Ψ|ϱ^(t)|Ψ〉, and we use the projector P^=ϱ^(0)=|Ψ〉〈Ψ| as the gradient operator, so that (197) is satisfied and (198) gives the fidelity change as the score
(210)P=〈Ψ|Δϱ^|Ψ〉=-t2〈〈Ψ|H^2|Ψ〉-〈Ψ|H^|Ψ〉2〉B,
which is given by H^ defined in (198).Since a quantum gate is supposed to act on an unknown input state, we need to get rid of the dependence on|Ψ〉. One possibility is to perform a uniform average over all |Ψ〉. We may apply, (see [186, 187])
(211)〈Ψ|A^|Ψ〉〈Ψ|B^|Ψ〉¯=TrA^B^+TrA^TrB^d(d+1)
which gives the average
(212)P¯=-t2dd+1〈H^2〉id,
where 〈·〉id=Tr[d-1I^⊗ϱ^B(·)]. In (212), we have used TrSH^=0^, which corresponds to TrS^j=0 in Section 8.7. Because of this and because of (196), 〈H^〉B=〈H^I〉B=0^, we have 〈H^〉id=0, and (212) also describes the variance Var(H^)=〈H^2〉id-〈H^〉id2. On the other hand, P^=-2kϱ^(0) that gives the change ΔS of entropy S=-kTr(ϱ^lnϱ^) is (up to a proportionality factor of -2k) the same as the P^ used here to give the change of fidelity, and we have ΔS=-2kP. If we define a gate errorℰ as the average fidelity decline, ℰ=-P¯, with P¯ given in (212), we can summarize the following proportionalities: gate error ≡ average fidelity decline ~ average entropy increase (purity decline) ΔS¯~ square (variance) of the average interaction energy H^ as
(213)ℰ≡-P¯=ΔS¯2k=t2dd+1〈H^2〉id=t2dd+1Var(H^).
In the matrix representation of Section 8.7, the average over the initial states in the matrix Γ defined in (202) gives Γ¯=-(d/(d+1))I (using Tr(S^jS^k)=dδjk and TrS^j=0), so that
(214)ℰ=dd+1∫-∞∞‍dωTr[ϵt(ω)ϵt†(ω)G(ω)]
in agreement with [176] (except a different normalization Tr(S^jS^k)=2δjk leading there to a prefactor (2/(d+1))). From the requirement that ℰ≥0 must hold for any positive semi-definite matrix ϵt(ω)ϵt†(ω), we conclude that G(ω) must be a positive semi-definite matrix for any ω. The task of BOMEC is then to find a system evolution U^(τ) (cf. the control examples in the previous section) that minimizes ℰ, subject to the boundary condition that the final U^(t) is the desired gate.


## 8.10. Comparison of BOMEC with DD

It is appropriate to compare the effect of dynamical decoupling (DD) [82, 188] with that of BOMEC. DD does not change with the bath spectrum G(ω). With an increasing number of pulses, DD shifts the weight of the system spectrum F(ω) towards higher frequencies, until the overlap equation (206) has become sufficiently small. This is illustrated for two different numbers of pulses of periodic DD (PDD, pulses periodic in time), in the upper row of Figure 17 in the case of a 1D single qubit modulation (i.e., all pulses are given by an arbitrary but fixed Pauli matrix).System modulation spectraF(ω) (scaled to 1 with respect to their maximum value, scaling of ω as in Figure 18) generated by two methods of DD. Upper row: periodic dynamical decoupling with nπ-pulses (PDDn), lower row: Uhrig dynamical decoupling with the same number of π-pulses (UDDn), compared for n=11 (left column) and n=19 (right column) pulses (color online).
(a)
(b)Aperiodic DD such as UDD [188] suppresses low-frequency components (to the left of the main peak) in the system spectrum, which retain the system-bath coupling even if the main peak of the system spectrum has been shifted beyond the bath cut-off frequency (Figure 17). The plots indicate that this suppression of low frequency components is achieved at the price of a smaller shift of the main peak; that is, shifting the main peak beyond a given cutoff requires more pulses in UDD than in PDD. Note that optimized DD sequences with improved asymptotics exist [189], which we will not consider here.System modulation spectra obtained with BOMEC are shown in Figure18.Figure 18
BOMEC minimization of the gate error for a single qubitπ-gate caused by pure dephasing with a given bath spectrum (G(ω), bold red line, scaled to 0.61 with respect to its maximum value, scaling of ω with respect to the chosen bath resonance, i.e., in units of the inverse bath correlation time). The Z-components of the obtained system modulation spectra Fi(ω) are shown for energy constraints Ei=0.1+4(i-1), i=1,2,…,101, (thin lines, blue to green) individually scaled to 1 (color online).The plot refers to a qubit subject to pure dephasing (i.e.,Z-coupling) by a bath whose spectrum G(ω) has a Lorentzian peak and low-frequency tail. The BOMEC optimizes U^(τ) simultaneously for 3D Pauli matrix couplings to the bath (Z, Y, and X). The resulting system spectrum F(ω) is shown for different energy constraints E which are increased in small and equal steps. For low E, F(ω) has a single peak on the left of the bath peak. Increasing E causes a second peak of F(ω) to emerge on the right of the bath peak, which continues to grow, while the peak on the left diminishes, until for high E, only the right peak remains. Figure 18 hence demonstrates that the spectrum F(ω) generated by BOMEC changes continuously as E increases but avoids overlap with the maxima of G(ω) irrespective of E. BOMEC can therefore be superior to all forms of DD including UDD, especially if the bath has high cutoff but bandgaps at low frequencies.


## 8.11. Purity Control of a Qubit

To give an example of the opposite case, where the goal is to maximize the system-bath coupling, we apply our approach of constrained optimization to the linear entropySL=2[1-Tr(ϱ^2)] of a qubit (note that here SL has been normalized to 1 by setting the coefficient k=d/(d-1)=2, cf. Section 8.8). We assume an initial mixture
(215)ϱ^(0)=p|1〉〈1|+(1-p)|0〉〈0|
of a ground (excited) state |0〉 (|1〉), where 0≤p≤0.5 is related to SL by p=(1-1-SL)/2. With S^j=σ^j denoting for d=2 the Pauli matrices, (215) can be written in terms of H^0=(ω0/2)σ^3 as ϱ^(0)=(e-βH^0/Tr(e-βH^0))=(|1〉〈1|/(1+eβω0))+(|0〉〈0|/(1+e-βω0)), where β=ln(p-1-1)/ω0 is the inverse temperature. Purity and temperature are hence related via the energy scale ω0. Our goal is a constrained optimization of ΔSL, that is, P^=-4ϱ^(0) in (189). Unlike the gate error equation (213), ΔSL can be negative or positive, which can be understood as cooling or heating, respectively.The time evolutions resulting from a minimization ofΔSL for the initial state equation  (215) are illustrated in Figure 19.  The optimization refers to a bath spectrum as shown in Figure 20. The fj shown in Figure 19(a) are defined by U^(τ)=e-(i/2)f3(τ)σ^3e-(i/2)f2(τ)σ^2e-(i/2)f1(τ)σ^3, whereas the ωj shown in Figure 19(b) are given by H^S(τ)=∑j‍ωj(τ)σ^j. The chosen constraint E=(1/2)∫0t‍dτTr(HS-H^0)2(τ) can be written in terms of the fj as E=(1/4)∫0t‍dt1[f˙12+f˙22+(f˙3-ω0)2+2f˙1(f˙3-ω0)cosf2]. (Note that we have set ħ=1, so that the dimension of energy becomes inverse time and hence corresponds to that of angular frequency. By measuring time in units of some characteristic correlation time of the bath chosen in the plots, time, energy, and angular frequency have been made dimensionless.) The overlap between the evolving system state ϱ^(τ) (in the Schrödinger picture) and the ground state |0〉 shown in Figure 19(c) indicates the fast unitary system modulation through short time population inversions without significantly altering the state purity as verified in Figure 19(d). This can be visualized as fast π-rotations of the state inside the Bloch sphere, which, together with smaller rotations, here result in the final reduction of SL(t) seen in Figure 19(d). Figure 19(d) also confirms that for the chosen time and coupling strength, differences between various methods of approximation are small.Evolution within0≤τ≤t for an optimized cooling with initial p=0.25 of (a) effective and (b) instantaneous controls (1 red, 2 green, and 3 blue graphs show x, y, and z component), (c) ground state overlap of the system state, and (d) linear entropy (upper red, middle green, and lower blue graphs show numerical integration of the time-convolutionless equation [190], of the Nakajima-Zwanzig equation, and second-order approximation of the solution as given in (192), respectively, which are all indistinguishable in (c)) (system-bath coupling strength κ=10-2, ω0=2π/t, t=10 in units of effective bath correlation time, E=100) (color online).
(a)
(b)
(c)
(d)Figure 20
Cooling and heating of a TLS by minimization (left: (a), (b), (c)) and maximization (right: (g), (h), (i)) of the change of linear entropy for a given bath spectrum (Gj, red) with j=1,2,3 denoting x, y, and z components. The optimized system spectra (Fj, green) are shown for an energy constraint E=102 (left) and E=103 (right) as contrasted to the unmodulated Hamiltonian H^0 (i.e., for E=0.001, 0.05, 0.1, …, 0.45, 0.499). All graphs are individually scaled to 1 with respect to their maximum value, and scaling of ω is the same as in Figure 18 (color online).In contrast to gate protection, no initial-state averaging is performed here, that is, (215) is known. Consequently, as Figure 20 shows,  the relevant components Fj≡(Ft)jj(ω) of the system modulation spectrum contributing to the spectral overlap Equation (206) depend on the initial state ϱ^(0) via the matrix Γ Equation (202). (We assume an uncorrelated bath, that is, Gjk(ω)=0 for j≠k and Gj≡Gjj(ω).) This influence is clearly visible in case of a constant (unmodulated, i.e., free) Hamiltonian (middle column), for which we set ω0=2π/t with the final t being in the order of the bath correlation time. Cooling (heating) is achieved by realizing negative (positive) ΔSL via maximum negative (positive) spectral overlap, as shown in the left (right) column of Figure 20. This is the opposite to system-bath decoupling, where the goal is to minimize the overlap.The plots illustrate the role of the energy constraintE: increasing E allows to establish overlap with higher frequency components of the bath spectrum. This also suggests that for a bath spectrum with a finite frequency cutoff, increasing E beyond a certain saturation value will not lead to further improvement of the optimization. In the time domain, increasing E leads to more rapid changes in the physical Hamiltonian, however, requiring higher resolution of the numerical treatment. On the contrary, for attempted cooling (heating) by minimization (maximization) of ΔSL, a given E may be too small to lead to negative (positive) ΔSL. The obtained ΔSL may then be understood as “reduced heating” (reduced cooling) as compared to a ΔSL obtained with an unmodulated H^0. This is shown in Figure 21.   The figure also illustrates once more that the ϱ^(0) dependence of the spectra shown in Figure 20 is accompanied by a ϱ^(0) dependency of the achievable change ΔSL for a given bath. For a maximally mixed state in particular (p=0.5), the matrix Γ equation (202) vanishes and with it ΔSL.Figure 21
Change of linear entropy in units of the system-bath coupling strength obtained by minimization (attempted cooling) and maximization (attempted heating) ofΔSL under different constraints E=0,1,100 as a function of the initial p for a bath as shown in Figure 20 (color online).A possibility to achieve negative (positive)ΔSL by its minimization (maximization) even for weak modulation (i.e., for small E) is to adapt the temperature of the bath such that for an undriven system Hamiltonian H^0, no change is observed, ΔSL=0, which is a necessary condition for a system-bath equilibrium. This would require nonunitary system modulation, for example, the effect of repeated measurements [191–193].


## 8.12. Bound Estimation

Our goal is to give constraint-independent upper and lower bounds for the maximum changeP=Tr(P^Δϱ^) that can be achieved with a given bath and P^ under the condition (197). We assume that ϵ, Γ, and G(ω) are quadratic (d2-1)-dimensional matrices. If max[TrG(ω)]<∞, we can estimate P by using that Tr(AB)≤Tr(A)Tr(B) for positive semi-definite matrices A, B and applying Hölder’s inequality in the form of ∫-∞∞‍dω|f(ω)g(ω)|≤sup{|g(ω)|}∫-∞∞‍dω|f(ω)|. Decomposing Γ=Γ1−Γ2 into positive semi-definite matrices Γi and making use of (1/t)∫-∞∞‍dωϵt†(ω)ϵt(ω)=I^, we thus get
(216)-P2≤P≤P1,Pi=tsup[TrG(ω)]TrΓi.
This reveals that for given t and G(ω), the bounds Pi depend on ϱ^(0) and P^ via Γ.


## 9. Discussion

A comprehensive approach for dynamical protection of quantum information processing (QIP) systems from the deleterious effects of decoherence was presented. Below, we discuss several theoretical and practical implications of this general approach. We first examine the temporal and spectral domains in order to increase our understanding of the underlying physical mechanisms of dynamical decoherence control. A practical approach, whereby one uses the universal decoherence control formalism to first reconstruct the noise spectra and then optimize the modulation to combat decoherence in experimental scenarios is then discussed. We next consider a theoretical approach, whereby one gains insight into the mechanism and dynamics of entanglement, which is both an essential resource for QIP and the main reason for decoherence.


### 9.1. Temporal and Spectral Aspects

The universal dynamical decoherence control formula was first developed for single qubit decay due to interaction with a zero-temperature bath in Section3 [97, 102] and was later extended to decay and dephasing due to interaction with finite-temperature baths in Section 7 [100, 124] and finally to multipartite systems in Section 8. One can examine the universal dynamical decoherence and disentanglement formalism presented above from two complementary perspectives, namely, the time and spectral domains.


#### 9.1.1. Temporal Domain

In the temporal domain, one can express the gist of the formalism by the following equation [104, 107]:
(217)Rij(t)=∫0t‍dt′Φij(t-t′)ϵi(t)ϵj*(t′),
where Rii(t) is the decoherence rate (either due to decay or dephasing) of the ith qubit, Rij(t) is the average cross-decoherence of qubits i and j, Φii(t) is the bath response (or memory) function due to coupling of qubit i, Φij(t) is the cross-response function due to coupling of qubits i and j, and ϵi(t) is the temporal modulation function of qubit i.The bath response function is usually associated with a characteristic correlation or memory time,tc, which separates the non-Markovian (t≪tc) and the Markovian (t≫tc) temporal regimes, for example, Φ(t)∝e-t/tc. Within the bath memory time, the bath modes oscillate coherently and in unison and maintain “memory” of their interaction with the system, whereas after the correlation time has passed, the modes lose their coherent oscillations and “forget” their prior interactions [178].The temporal modulation function usually has two factors, namely, a system factor of the formeiωat, where ħωa is the energy separation of the two-level system, and a modulation or control factor that can represent either amplitude (ϵ~(t)) or phase (ϕ(t)) modulation, ϵ(t)=ϵ~(t)eiωat+iϕ(t).In order to affect the system-bath coupling and control, or modulate the decoherence due to this coupling, one must dynamically modulate the system faster than the correlation time. Slower modulation will have no effect on the loss of coherence and will thus not be able to control it. Modulating the system faster than the correlation time can effectively “reset” the clock. Applying a modulation sequence repeatedly can thus drastically change the decoherence and impose a continued coherent evolution of the system-bath coupling [86, 178].Moreover, during the short non-Markovian time regime, the decoherence rates are time dependent and may even acquire a negative sign [178]. This can result in novel behavior such as rapid cooling of a system coupled to a finite temperature bath, below the bath temperature. By repeatedly disturbing the free evolution by frequent measurements or phase modulations, one can maintain this behavior of the time-dependent decoherence rate over long times and thus accumulate the short-time cooling.Another novel feature that is apparent from (217) is the effect of local modulations on cross decoherence, where local modulations refer to different modulation of each qubit, ϵi(t)≠ϵj(t). Due to the phase difference of the local modulations, one can exploit interference effects between two decoherence channels. As shown in Section 8 [123], by applying different modulation to each qubit, faster than the cross correlation time, one can create destructive interference and thus eliminate the cross decoherence. On the other hand, by applying tailored modulations to different qubits, one can create constructive interference that can generate a decoherence-free subspace (DFS).


#### 9.1.2. Spectral Domain

In the spectral domain, one can express the same decoherence rate as in (217) by the following equation:
(218)Rij(t)=2π∫-∞∞‍dωGij(ω)Ft,ij(ω),(219)Ft,ij(ω)=1tϵt,i(ω)ϵt,j*(ω),
where Gii(ω) is the coupling spectrum of the ith qubit to the decohering bath (either a decay-inducing bath, a dephasing-inducing bath, or the spectrum of the classical proper-dephasing), Gij(ω) is the cross-coupling spectrum of qubits i and j, and ϵt,i(ω) is the ith qubit time-dependent modulation spectrum.The temperature-dependent coupling spectrum is the Fourier transform of the bath response function in (217), and it usually has a certain width proportional to the inverse of the correlation time. The time-dependent modulation spectrum is the finite-time Fourier transform of the modulation function, ϵi(t).Equation (218) identifies the source of decoherence as the coupling of the time-dependent modulated system to the bath modes. Thus, for an unmodulated system in the long-time regime, the single-qubit modulation power spectrum given by Ft,ii(ω)=|ϵt,i(ω)|2 becomes a δ-function centered at the system energy-separation, ωa. This Golden Rule result can be viewed as the coupling between the system and the corresponding mode of the bath, with coupling strength G(ωa).However, one can analyze more complex scenarios using (218). The modulation spectrum is a time-dependent function and for an unmodulated system it becomes a sinc function with a time-dependent width. For short times, the modulation spectrum is wide, expressing the time-energy uncertainty, whereby the system energy separation is not well defined and the system is effectively coupled to many bath modes. Depending on the coupling spectrum, G(ω), the decoherence rate oscillates and can acquire negative values, until, after the correlation time, it settles to the Golden Rule value, expressing the well-defined energy separation.Another important insight gained from the frequency-domain analysis is the optimal shape of the modulation that reduces the decoherence rate. Given a specific coupling spectrum, one should choose a modulation so as to decrease the overlap between the modulation and coupling spectra [104]. This optimal modulation means that the system is effectively coupled to bath modes with lower coupling strength. Furthermore, since the coupling spectrum has a width proportional to the inverse correlation time, in order to reduce decoherence one should drive the system with a higher frequency than this width, that is, faster than the correlation time. One can also choose to increase the decoherence (i.e., impose the anti-Zeno effect) by applying a modulation that would couple the system to more strongly-coupled bath modes.Finally, cross decoherence can be understood from the spectral-domain analysis as the coupling of two systems via common bath modes; that is, (218) is the overlap of three functions, namely, cross coupling spectrum and the individual modulation spectra of the two systems. For example, if two systems couple to different modes, then in the absence of modulations, they will not experience any cross decoherence. Hence, in order to impose cross decoherence, one should modulate the systems in such a way that they effectively couple to the same modes with the same strength. On the other hand, if one wishes to eliminate cross decoherence, one should apply local modulations, such that the modulation spectra have different peaks, which would result in the two systems coupling to different modes and thus experiencing no cross decoherence.


### 9.2. A User Manual for Decoherence Control

The goal of any practical modulation scheme is to reduce, or, if possible, eliminate, all the elements of the decoherence matrix in (218) (see [123] for an alternative solution). However, in order to obtain the optimal modulation [104], one must first know the system-bath coupling spectra of the qubits in question. This information is usually not available a priori and thus most experimentalists have resorted to the suboptimal dynamical-decoupling (or bang-bang) modulation, which does not require this knowledge.Instead, we suggest a practical and general two-step approach to optimally combat decoherence.(1)
Use known, prescribed modulations to reveal the unknown decoherence parameters, that is, the system-bath coupling spectra.(2)
Tailor optimal modulations for the obtained parameters.


#### 9.2.1. Use of Modulations to Reconstruct Coupling Spectra

Equation (218) can be expressed via three matrices: (i) The coupling spectra matrix, Gij(ω), is usually unknown and is the objective of this analysis. (ii) The modulation power spectra matrix, Ft,ij(ω)=(1/t)ϵt,i(ω)ϵt,j*(ω), can be calculated for each specific known modulation. (iii) The average decoherence matrix, Rij(t), can be experimentally obtained via measurements of the decohering system.Thus, the protocol outlined here is to use a parameterized modulation arsenal (see [124] for a comprehensive analysis of modulation pulses)  and measure the decoherence rate for each such modulation. This results in a vector of decoherence rates that depends on the modulation parameters. By using simple numerical methods, one can then find the coupling spectrum that best fits the calculated (Ft,ij(ω)) and the measured (Rij(t)) quantities.For example, one can apply to a single qubit an impulsive phase modulation,ϵ(t)=ei[t/τ]ϕ, with varying phase ϕ and pulse intervals τ. This can be applied as a time-dependent AC-Stark shift to combat decay or a time-dependent resonant field to combat dephasing [100, 107]. This modulation scheme has the following power spectrum:
(220)Fnτ(ω)=2sin2(ωτ/2)sin2[n(ϕ+ωτ)/2]πnτω2sin2[(ϕ+ωτ)/2],
where n=[t/τ].For each modulation parameters pair,{ϕ,τ}, one experimentally measure the average decoherence rate of the qubit, R(t) (either decay of an initially excited state or dephasing of an initial superposition of ground and excited states). Then one runs a simple numerical algorithm that finds the coupling spectrum, G(ω), that best fits the observed results.An alternative approach to obtain the coupling spectrum is to use the fact that within the bath response (memory) time, the decoherence rate changes with time, even if no modulation is applied. Thus, for non-Markovian time scales and no modulation, the “modulation” power spectrum is given by a sinc function. Measuring the excited-state population (for decay control) or coherence (for dephasing control) for short times compared to the bath response time should reveal a nonexponential, nonmonotonic decoherence and thus divulge the underlying coupling spectrum.Furthermore, if one has a multipartite system, and thus a decoherence matrix, one must perform multiple modulation schemes that address the different qubits, on top of performing the aforementioned single-qubit scheme for all the qubits. This is essential to ascertain the cross-coupling spectra of all the possible qubit pairs. As discussed in [107, 123, 126], these cross-coupling spectra are extremely important in reducing disentanglement and allow, in certain circumstances, a complete elimination of decoherence.To summarize, the methods of applying different modulations or measuring decoherence at different times effectively sample the frequency space,ω, and thus address different bath modes and allow measuring their coupling strength to the system. Thereafter, one can numerically obtain the coupling spectra that best fits the measured data.


#### 9.2.2. Optimal Modulation for Decoherence Control

After one obtains the system-bath coupling spectra by applying specific, parameterized modulation schemes, one can finally tailor the specific modulation that would optimally reduce or eliminate decoherence. However, two aspects should be considered carefully, namely, the specific QIP application used and the specific modulation constraints.As shown in [107, 125, 126, 183], each QIP application has its own figure-of-merit that assesses the effectiveness of the dynamical decoherence control scheme. In general, we have shown that this effectiveness is related to the decoherence matrix of the form given by (218). Yet, if some of its elements cannot be drastically reduced, one can still have a dramatic effect and increase the appropriate figure of merit.We have shown in Section8 [107] that if the systems in question have three levels, one can completely eliminate decoherence and disentanglement by imposing a special symmetry using the appropriate modulation. Thus, even if drastic reduction of all the decoherence matrix elements is not possible, then by using local modulations, one may equate the intraparticle elements, eliminate the interparticle elements, and code the quantum information in the ground and antisymmetric “dark” state of the two excited levels and consequently completely preserve coherence and entanglement.Furthermore, we have shown [125, 137] that, in multiqubit quantum computation, it is vital to consider all possible modulations and fields, and not only those that directly perform a quantum logic gate. Thus, while usually one only requires the optimization of the fields applied to perform the gate, we have stressed that the gate fidelity can be greatly increased if one also applies and optimizes all other possible fields.However, these schemes can only be applied after obtaining an approximate form of the system-bath coupling spectra, without which one cannot tailor the local modulations that equate or eliminate specific elements of the decoherence matrix.In any optimization problem, one must consider the constraints of the modulations. In [104], we have considered an energy constraint of the resonant field applied in order to optimally reduce dephasing and have derived the appropriate Euler-Lagrange equation. However, an experimental system may have additional constraints, such as the dynamical range of the modulator, other resonances within the system, heating, and so forth. Each such scenario must be taken into account when considering the available repertoire of modulation parameters. Usually, one will have to resort to numerical optimization given the system-bath coupling spectra matrix and the possible modulations.To summarize, after obtaining the general shape of the system-bath coupling spectra matrix of the multipartite system by applying parameterized, known modulations, one can optimize the appropriate QIP-dependent figure of merit while obeying the constraints of the available modulations.


### 9.3. Entanglement: Decoherence Interplay

Entanglement is the main resource of quantum information processing, without which quantum computation will not be faster than its classical counterpart [3] and quantum communication protocols will not work [194–196]. Moreover, as shown in [194–196], the impact of nonmaximally entangled quantum channels is identical to a nonmaximally entangled Bell measurement. Thus, entanglement is crucial for all nonlocal stages of quantum protocols, namely, the quantum channel preparation as well as bipartite Bell measurement.However, entanglement is also the main source of decoherence, when the decohering agent is a quantum bath (as opposed to classical proper dephasing [100, 107, 125]). We have shown here that a similar formula for dynamical decoherence control, (218) is obtained for decay and dephasing due to coupling to a bath of harmonic oscillators. In these scenarios, as well as the scenario detailed in [178], the source of decoherence is the entanglement of the system to the bath’s degrees of freedom. Because the latter are inaccessible to both modulation and probing, they are traced out and this results in loss of coherence of the system.Nevertheless, not all is lost when interacting with a bath. During short times, much shorter than the bath memory times, the system and bath still interact coherently and the effects of the bath can be remedied by applying the appropriate modulations. This can understood from (218) by noting that the width of G(ω) is proportional to the inverse of the memory time. Thus, one can apply rapidly repeating modulations, corresponding to a multipeaked modulation power spectrum, where the peaks are positioned at the high frequency range. The spectral overlap between the system-bath coupling spectrum and the modulation power spectrum can be greatly reduced if the modulations are temporally separated by a shorter interval than the bath correlation time.These frequent modulations can be viewed as driving the system to higher frequencies in which the bath modes are dramatically more weakly coupled than those modes coupled without modulation. This reduced coupling results in less entanglement with the bath modes and thus less decoherence.A novel interplay between entanglement as a QIP resource and entanglement as the source of decoherence was detailed in Section8 [126]. Two entangled qubits were analyzed, each coupled to a bath via common modes. The non-Markovian time scale was considered, as well as dynamical modulations. It was shown how the entanglement of the qubits could vanish after a finite time (entanglement sudden death—ESD), but later restored by non-Markovian modulation-induced oscillations of the system-bath coherence.Thus, an exchange between two-qubit entanglement and system-bath entanglement may take place and be dynamically controlled by modulations. If the systems couple to two baths that have common modes, one can observe the transfer of coherence and buildup of entanglement between the two systems via these modes. If, on the other hand, the baths were completely separate, the coherence transfer between each system and its bath can modulate the amount of system-system entanglement, first lost and then regained.Another striking example of entanglement-mediated dynamics was presented in [178]. There we considered a two-level system that first reaches the state of equilibrium with a bath of harmonic oscillators, a state in which large system-bath entanglement is present, but no further dynamics. Then, a quantum probe is coupled to the system and performs a quantum nondemolition (QND) measurement of the system’s excited-level population. This QND measurement results in system-probe entanglement and the consequent loss of system-bath coherence, or vanishing of the off-diagonal elements of the system-bath density matrix. After this impulsive, nonselective, QND measurement, the state is no longer in equilibrium and begins to evolve.We have shown that immediately after the measurement the system and bath always “heat up”, that is, get excited. Remarkably, for certain system-bath coupling spectra, one can also observe a system that has lower excited-state population than the equilibrium state, that is, a purer system. This occurs despite the fact that the system has effectively recoupled with the bath and has become entangled with it.This interplay between system-bath entanglement, and system-probe entanglement and its effect on system cooling is still under active investigation by our group.Entanglement in open systems is a fascinating concept that is not yet fully understood. It is essential for QIP, yet it is also the origin of decoherence. The interplay between system-system entanglement, system-probe entanglement and system-bath entanglement on non-Markovian time-scales is an intriguing field that deserves more attention.


## 9.1. Temporal and Spectral Aspects

The universal dynamical decoherence control formula was first developed for single qubit decay due to interaction with a zero-temperature bath in Section3 [97, 102] and was later extended to decay and dephasing due to interaction with finite-temperature baths in Section 7 [100, 124] and finally to multipartite systems in Section 8. One can examine the universal dynamical decoherence and disentanglement formalism presented above from two complementary perspectives, namely, the time and spectral domains.


### 9.1.1. Temporal Domain

In the temporal domain, one can express the gist of the formalism by the following equation [104, 107]:
(217)Rij(t)=∫0t‍dt′Φij(t-t′)ϵi(t)ϵj*(t′),
where Rii(t) is the decoherence rate (either due to decay or dephasing) of the ith qubit, Rij(t) is the average cross-decoherence of qubits i and j, Φii(t) is the bath response (or memory) function due to coupling of qubit i, Φij(t) is the cross-response function due to coupling of qubits i and j, and ϵi(t) is the temporal modulation function of qubit i.The bath response function is usually associated with a characteristic correlation or memory time,tc, which separates the non-Markovian (t≪tc) and the Markovian (t≫tc) temporal regimes, for example, Φ(t)∝e-t/tc. Within the bath memory time, the bath modes oscillate coherently and in unison and maintain “memory” of their interaction with the system, whereas after the correlation time has passed, the modes lose their coherent oscillations and “forget” their prior interactions [178].The temporal modulation function usually has two factors, namely, a system factor of the formeiωat, where ħωa is the energy separation of the two-level system, and a modulation or control factor that can represent either amplitude (ϵ~(t)) or phase (ϕ(t)) modulation, ϵ(t)=ϵ~(t)eiωat+iϕ(t).In order to affect the system-bath coupling and control, or modulate the decoherence due to this coupling, one must dynamically modulate the system faster than the correlation time. Slower modulation will have no effect on the loss of coherence and will thus not be able to control it. Modulating the system faster than the correlation time can effectively “reset” the clock. Applying a modulation sequence repeatedly can thus drastically change the decoherence and impose a continued coherent evolution of the system-bath coupling [86, 178].Moreover, during the short non-Markovian time regime, the decoherence rates are time dependent and may even acquire a negative sign [178]. This can result in novel behavior such as rapid cooling of a system coupled to a finite temperature bath, below the bath temperature. By repeatedly disturbing the free evolution by frequent measurements or phase modulations, one can maintain this behavior of the time-dependent decoherence rate over long times and thus accumulate the short-time cooling.Another novel feature that is apparent from (217) is the effect of local modulations on cross decoherence, where local modulations refer to different modulation of each qubit, ϵi(t)≠ϵj(t). Due to the phase difference of the local modulations, one can exploit interference effects between two decoherence channels. As shown in Section 8 [123], by applying different modulation to each qubit, faster than the cross correlation time, one can create destructive interference and thus eliminate the cross decoherence. On the other hand, by applying tailored modulations to different qubits, one can create constructive interference that can generate a decoherence-free subspace (DFS).


### 9.1.2. Spectral Domain

In the spectral domain, one can express the same decoherence rate as in (217) by the following equation:
(218)Rij(t)=2π∫-∞∞‍dωGij(ω)Ft,ij(ω),(219)Ft,ij(ω)=1tϵt,i(ω)ϵt,j*(ω),
where Gii(ω) is the coupling spectrum of the ith qubit to the decohering bath (either a decay-inducing bath, a dephasing-inducing bath, or the spectrum of the classical proper-dephasing), Gij(ω) is the cross-coupling spectrum of qubits i and j, and ϵt,i(ω) is the ith qubit time-dependent modulation spectrum.The temperature-dependent coupling spectrum is the Fourier transform of the bath response function in (217), and it usually has a certain width proportional to the inverse of the correlation time. The time-dependent modulation spectrum is the finite-time Fourier transform of the modulation function, ϵi(t).Equation (218) identifies the source of decoherence as the coupling of the time-dependent modulated system to the bath modes. Thus, for an unmodulated system in the long-time regime, the single-qubit modulation power spectrum given by Ft,ii(ω)=|ϵt,i(ω)|2 becomes a δ-function centered at the system energy-separation, ωa. This Golden Rule result can be viewed as the coupling between the system and the corresponding mode of the bath, with coupling strength G(ωa).However, one can analyze more complex scenarios using (218). The modulation spectrum is a time-dependent function and for an unmodulated system it becomes a sinc function with a time-dependent width. For short times, the modulation spectrum is wide, expressing the time-energy uncertainty, whereby the system energy separation is not well defined and the system is effectively coupled to many bath modes. Depending on the coupling spectrum, G(ω), the decoherence rate oscillates and can acquire negative values, until, after the correlation time, it settles to the Golden Rule value, expressing the well-defined energy separation.Another important insight gained from the frequency-domain analysis is the optimal shape of the modulation that reduces the decoherence rate. Given a specific coupling spectrum, one should choose a modulation so as to decrease the overlap between the modulation and coupling spectra [104]. This optimal modulation means that the system is effectively coupled to bath modes with lower coupling strength. Furthermore, since the coupling spectrum has a width proportional to the inverse correlation time, in order to reduce decoherence one should drive the system with a higher frequency than this width, that is, faster than the correlation time. One can also choose to increase the decoherence (i.e., impose the anti-Zeno effect) by applying a modulation that would couple the system to more strongly-coupled bath modes.Finally, cross decoherence can be understood from the spectral-domain analysis as the coupling of two systems via common bath modes; that is, (218) is the overlap of three functions, namely, cross coupling spectrum and the individual modulation spectra of the two systems. For example, if two systems couple to different modes, then in the absence of modulations, they will not experience any cross decoherence. Hence, in order to impose cross decoherence, one should modulate the systems in such a way that they effectively couple to the same modes with the same strength. On the other hand, if one wishes to eliminate cross decoherence, one should apply local modulations, such that the modulation spectra have different peaks, which would result in the two systems coupling to different modes and thus experiencing no cross decoherence.


## 9.1.1. Temporal Domain

In the temporal domain, one can express the gist of the formalism by the following equation [104, 107]:
(217)Rij(t)=∫0t‍dt′Φij(t-t′)ϵi(t)ϵj*(t′),
where Rii(t) is the decoherence rate (either due to decay or dephasing) of the ith qubit, Rij(t) is the average cross-decoherence of qubits i and j, Φii(t) is the bath response (or memory) function due to coupling of qubit i, Φij(t) is the cross-response function due to coupling of qubits i and j, and ϵi(t) is the temporal modulation function of qubit i.The bath response function is usually associated with a characteristic correlation or memory time,tc, which separates the non-Markovian (t≪tc) and the Markovian (t≫tc) temporal regimes, for example, Φ(t)∝e-t/tc. Within the bath memory time, the bath modes oscillate coherently and in unison and maintain “memory” of their interaction with the system, whereas after the correlation time has passed, the modes lose their coherent oscillations and “forget” their prior interactions [178].The temporal modulation function usually has two factors, namely, a system factor of the formeiωat, where ħωa is the energy separation of the two-level system, and a modulation or control factor that can represent either amplitude (ϵ~(t)) or phase (ϕ(t)) modulation, ϵ(t)=ϵ~(t)eiωat+iϕ(t).In order to affect the system-bath coupling and control, or modulate the decoherence due to this coupling, one must dynamically modulate the system faster than the correlation time. Slower modulation will have no effect on the loss of coherence and will thus not be able to control it. Modulating the system faster than the correlation time can effectively “reset” the clock. Applying a modulation sequence repeatedly can thus drastically change the decoherence and impose a continued coherent evolution of the system-bath coupling [86, 178].Moreover, during the short non-Markovian time regime, the decoherence rates are time dependent and may even acquire a negative sign [178]. This can result in novel behavior such as rapid cooling of a system coupled to a finite temperature bath, below the bath temperature. By repeatedly disturbing the free evolution by frequent measurements or phase modulations, one can maintain this behavior of the time-dependent decoherence rate over long times and thus accumulate the short-time cooling.Another novel feature that is apparent from (217) is the effect of local modulations on cross decoherence, where local modulations refer to different modulation of each qubit, ϵi(t)≠ϵj(t). Due to the phase difference of the local modulations, one can exploit interference effects between two decoherence channels. As shown in Section 8 [123], by applying different modulation to each qubit, faster than the cross correlation time, one can create destructive interference and thus eliminate the cross decoherence. On the other hand, by applying tailored modulations to different qubits, one can create constructive interference that can generate a decoherence-free subspace (DFS).


## 9.1.2. Spectral Domain

In the spectral domain, one can express the same decoherence rate as in (217) by the following equation:
(218)Rij(t)=2π∫-∞∞‍dωGij(ω)Ft,ij(ω),(219)Ft,ij(ω)=1tϵt,i(ω)ϵt,j*(ω),
where Gii(ω) is the coupling spectrum of the ith qubit to the decohering bath (either a decay-inducing bath, a dephasing-inducing bath, or the spectrum of the classical proper-dephasing), Gij(ω) is the cross-coupling spectrum of qubits i and j, and ϵt,i(ω) is the ith qubit time-dependent modulation spectrum.The temperature-dependent coupling spectrum is the Fourier transform of the bath response function in (217), and it usually has a certain width proportional to the inverse of the correlation time. The time-dependent modulation spectrum is the finite-time Fourier transform of the modulation function, ϵi(t).Equation (218) identifies the source of decoherence as the coupling of the time-dependent modulated system to the bath modes. Thus, for an unmodulated system in the long-time regime, the single-qubit modulation power spectrum given by Ft,ii(ω)=|ϵt,i(ω)|2 becomes a δ-function centered at the system energy-separation, ωa. This Golden Rule result can be viewed as the coupling between the system and the corresponding mode of the bath, with coupling strength G(ωa).However, one can analyze more complex scenarios using (218). The modulation spectrum is a time-dependent function and for an unmodulated system it becomes a sinc function with a time-dependent width. For short times, the modulation spectrum is wide, expressing the time-energy uncertainty, whereby the system energy separation is not well defined and the system is effectively coupled to many bath modes. Depending on the coupling spectrum, G(ω), the decoherence rate oscillates and can acquire negative values, until, after the correlation time, it settles to the Golden Rule value, expressing the well-defined energy separation.Another important insight gained from the frequency-domain analysis is the optimal shape of the modulation that reduces the decoherence rate. Given a specific coupling spectrum, one should choose a modulation so as to decrease the overlap between the modulation and coupling spectra [104]. This optimal modulation means that the system is effectively coupled to bath modes with lower coupling strength. Furthermore, since the coupling spectrum has a width proportional to the inverse correlation time, in order to reduce decoherence one should drive the system with a higher frequency than this width, that is, faster than the correlation time. One can also choose to increase the decoherence (i.e., impose the anti-Zeno effect) by applying a modulation that would couple the system to more strongly-coupled bath modes.Finally, cross decoherence can be understood from the spectral-domain analysis as the coupling of two systems via common bath modes; that is, (218) is the overlap of three functions, namely, cross coupling spectrum and the individual modulation spectra of the two systems. For example, if two systems couple to different modes, then in the absence of modulations, they will not experience any cross decoherence. Hence, in order to impose cross decoherence, one should modulate the systems in such a way that they effectively couple to the same modes with the same strength. On the other hand, if one wishes to eliminate cross decoherence, one should apply local modulations, such that the modulation spectra have different peaks, which would result in the two systems coupling to different modes and thus experiencing no cross decoherence.


## 9.2. A User Manual for Decoherence Control

The goal of any practical modulation scheme is to reduce, or, if possible, eliminate, all the elements of the decoherence matrix in (218) (see [123] for an alternative solution). However, in order to obtain the optimal modulation [104], one must first know the system-bath coupling spectra of the qubits in question. This information is usually not available a priori and thus most experimentalists have resorted to the suboptimal dynamical-decoupling (or bang-bang) modulation, which does not require this knowledge.Instead, we suggest a practical and general two-step approach to optimally combat decoherence.(1)
Use known, prescribed modulations to reveal the unknown decoherence parameters, that is, the system-bath coupling spectra.(2)
Tailor optimal modulations for the obtained parameters.


### 9.2.1. Use of Modulations to Reconstruct Coupling Spectra

Equation (218) can be expressed via three matrices: (i) The coupling spectra matrix, Gij(ω), is usually unknown and is the objective of this analysis. (ii) The modulation power spectra matrix, Ft,ij(ω)=(1/t)ϵt,i(ω)ϵt,j*(ω), can be calculated for each specific known modulation. (iii) The average decoherence matrix, Rij(t), can be experimentally obtained via measurements of the decohering system.Thus, the protocol outlined here is to use a parameterized modulation arsenal (see [124] for a comprehensive analysis of modulation pulses)  and measure the decoherence rate for each such modulation. This results in a vector of decoherence rates that depends on the modulation parameters. By using simple numerical methods, one can then find the coupling spectrum that best fits the calculated (Ft,ij(ω)) and the measured (Rij(t)) quantities.For example, one can apply to a single qubit an impulsive phase modulation,ϵ(t)=ei[t/τ]ϕ, with varying phase ϕ and pulse intervals τ. This can be applied as a time-dependent AC-Stark shift to combat decay or a time-dependent resonant field to combat dephasing [100, 107]. This modulation scheme has the following power spectrum:
(220)Fnτ(ω)=2sin2(ωτ/2)sin2[n(ϕ+ωτ)/2]πnτω2sin2[(ϕ+ωτ)/2],
where n=[t/τ].For each modulation parameters pair,{ϕ,τ}, one experimentally measure the average decoherence rate of the qubit, R(t) (either decay of an initially excited state or dephasing of an initial superposition of ground and excited states). Then one runs a simple numerical algorithm that finds the coupling spectrum, G(ω), that best fits the observed results.An alternative approach to obtain the coupling spectrum is to use the fact that within the bath response (memory) time, the decoherence rate changes with time, even if no modulation is applied. Thus, for non-Markovian time scales and no modulation, the “modulation” power spectrum is given by a sinc function. Measuring the excited-state population (for decay control) or coherence (for dephasing control) for short times compared to the bath response time should reveal a nonexponential, nonmonotonic decoherence and thus divulge the underlying coupling spectrum.Furthermore, if one has a multipartite system, and thus a decoherence matrix, one must perform multiple modulation schemes that address the different qubits, on top of performing the aforementioned single-qubit scheme for all the qubits. This is essential to ascertain the cross-coupling spectra of all the possible qubit pairs. As discussed in [107, 123, 126], these cross-coupling spectra are extremely important in reducing disentanglement and allow, in certain circumstances, a complete elimination of decoherence.To summarize, the methods of applying different modulations or measuring decoherence at different times effectively sample the frequency space,ω, and thus address different bath modes and allow measuring their coupling strength to the system. Thereafter, one can numerically obtain the coupling spectra that best fits the measured data.


### 9.2.2. Optimal Modulation for Decoherence Control

After one obtains the system-bath coupling spectra by applying specific, parameterized modulation schemes, one can finally tailor the specific modulation that would optimally reduce or eliminate decoherence. However, two aspects should be considered carefully, namely, the specific QIP application used and the specific modulation constraints.As shown in [107, 125, 126, 183], each QIP application has its own figure-of-merit that assesses the effectiveness of the dynamical decoherence control scheme. In general, we have shown that this effectiveness is related to the decoherence matrix of the form given by (218). Yet, if some of its elements cannot be drastically reduced, one can still have a dramatic effect and increase the appropriate figure of merit.We have shown in Section8 [107] that if the systems in question have three levels, one can completely eliminate decoherence and disentanglement by imposing a special symmetry using the appropriate modulation. Thus, even if drastic reduction of all the decoherence matrix elements is not possible, then by using local modulations, one may equate the intraparticle elements, eliminate the interparticle elements, and code the quantum information in the ground and antisymmetric “dark” state of the two excited levels and consequently completely preserve coherence and entanglement.Furthermore, we have shown [125, 137] that, in multiqubit quantum computation, it is vital to consider all possible modulations and fields, and not only those that directly perform a quantum logic gate. Thus, while usually one only requires the optimization of the fields applied to perform the gate, we have stressed that the gate fidelity can be greatly increased if one also applies and optimizes all other possible fields.However, these schemes can only be applied after obtaining an approximate form of the system-bath coupling spectra, without which one cannot tailor the local modulations that equate or eliminate specific elements of the decoherence matrix.In any optimization problem, one must consider the constraints of the modulations. In [104], we have considered an energy constraint of the resonant field applied in order to optimally reduce dephasing and have derived the appropriate Euler-Lagrange equation. However, an experimental system may have additional constraints, such as the dynamical range of the modulator, other resonances within the system, heating, and so forth. Each such scenario must be taken into account when considering the available repertoire of modulation parameters. Usually, one will have to resort to numerical optimization given the system-bath coupling spectra matrix and the possible modulations.To summarize, after obtaining the general shape of the system-bath coupling spectra matrix of the multipartite system by applying parameterized, known modulations, one can optimize the appropriate QIP-dependent figure of merit while obeying the constraints of the available modulations.


## 9.2.1. Use of Modulations to Reconstruct Coupling Spectra

Equation (218) can be expressed via three matrices: (i) The coupling spectra matrix, Gij(ω), is usually unknown and is the objective of this analysis. (ii) The modulation power spectra matrix, Ft,ij(ω)=(1/t)ϵt,i(ω)ϵt,j*(ω), can be calculated for each specific known modulation. (iii) The average decoherence matrix, Rij(t), can be experimentally obtained via measurements of the decohering system.Thus, the protocol outlined here is to use a parameterized modulation arsenal (see [124] for a comprehensive analysis of modulation pulses)  and measure the decoherence rate for each such modulation. This results in a vector of decoherence rates that depends on the modulation parameters. By using simple numerical methods, one can then find the coupling spectrum that best fits the calculated (Ft,ij(ω)) and the measured (Rij(t)) quantities.For example, one can apply to a single qubit an impulsive phase modulation,ϵ(t)=ei[t/τ]ϕ, with varying phase ϕ and pulse intervals τ. This can be applied as a time-dependent AC-Stark shift to combat decay or a time-dependent resonant field to combat dephasing [100, 107]. This modulation scheme has the following power spectrum:
(220)Fnτ(ω)=2sin2(ωτ/2)sin2[n(ϕ+ωτ)/2]πnτω2sin2[(ϕ+ωτ)/2],
where n=[t/τ].For each modulation parameters pair,{ϕ,τ}, one experimentally measure the average decoherence rate of the qubit, R(t) (either decay of an initially excited state or dephasing of an initial superposition of ground and excited states). Then one runs a simple numerical algorithm that finds the coupling spectrum, G(ω), that best fits the observed results.An alternative approach to obtain the coupling spectrum is to use the fact that within the bath response (memory) time, the decoherence rate changes with time, even if no modulation is applied. Thus, for non-Markovian time scales and no modulation, the “modulation” power spectrum is given by a sinc function. Measuring the excited-state population (for decay control) or coherence (for dephasing control) for short times compared to the bath response time should reveal a nonexponential, nonmonotonic decoherence and thus divulge the underlying coupling spectrum.Furthermore, if one has a multipartite system, and thus a decoherence matrix, one must perform multiple modulation schemes that address the different qubits, on top of performing the aforementioned single-qubit scheme for all the qubits. This is essential to ascertain the cross-coupling spectra of all the possible qubit pairs. As discussed in [107, 123, 126], these cross-coupling spectra are extremely important in reducing disentanglement and allow, in certain circumstances, a complete elimination of decoherence.To summarize, the methods of applying different modulations or measuring decoherence at different times effectively sample the frequency space,ω, and thus address different bath modes and allow measuring their coupling strength to the system. Thereafter, one can numerically obtain the coupling spectra that best fits the measured data.


## 9.2.2. Optimal Modulation for Decoherence Control

After one obtains the system-bath coupling spectra by applying specific, parameterized modulation schemes, one can finally tailor the specific modulation that would optimally reduce or eliminate decoherence. However, two aspects should be considered carefully, namely, the specific QIP application used and the specific modulation constraints.As shown in [107, 125, 126, 183], each QIP application has its own figure-of-merit that assesses the effectiveness of the dynamical decoherence control scheme. In general, we have shown that this effectiveness is related to the decoherence matrix of the form given by (218). Yet, if some of its elements cannot be drastically reduced, one can still have a dramatic effect and increase the appropriate figure of merit.We have shown in Section8 [107] that if the systems in question have three levels, one can completely eliminate decoherence and disentanglement by imposing a special symmetry using the appropriate modulation. Thus, even if drastic reduction of all the decoherence matrix elements is not possible, then by using local modulations, one may equate the intraparticle elements, eliminate the interparticle elements, and code the quantum information in the ground and antisymmetric “dark” state of the two excited levels and consequently completely preserve coherence and entanglement.Furthermore, we have shown [125, 137] that, in multiqubit quantum computation, it is vital to consider all possible modulations and fields, and not only those that directly perform a quantum logic gate. Thus, while usually one only requires the optimization of the fields applied to perform the gate, we have stressed that the gate fidelity can be greatly increased if one also applies and optimizes all other possible fields.However, these schemes can only be applied after obtaining an approximate form of the system-bath coupling spectra, without which one cannot tailor the local modulations that equate or eliminate specific elements of the decoherence matrix.In any optimization problem, one must consider the constraints of the modulations. In [104], we have considered an energy constraint of the resonant field applied in order to optimally reduce dephasing and have derived the appropriate Euler-Lagrange equation. However, an experimental system may have additional constraints, such as the dynamical range of the modulator, other resonances within the system, heating, and so forth. Each such scenario must be taken into account when considering the available repertoire of modulation parameters. Usually, one will have to resort to numerical optimization given the system-bath coupling spectra matrix and the possible modulations.To summarize, after obtaining the general shape of the system-bath coupling spectra matrix of the multipartite system by applying parameterized, known modulations, one can optimize the appropriate QIP-dependent figure of merit while obeying the constraints of the available modulations.


## 9.3. Entanglement: Decoherence Interplay

Entanglement is the main resource of quantum information processing, without which quantum computation will not be faster than its classical counterpart [3] and quantum communication protocols will not work [194–196]. Moreover, as shown in [194–196], the impact of nonmaximally entangled quantum channels is identical to a nonmaximally entangled Bell measurement. Thus, entanglement is crucial for all nonlocal stages of quantum protocols, namely, the quantum channel preparation as well as bipartite Bell measurement.However, entanglement is also the main source of decoherence, when the decohering agent is a quantum bath (as opposed to classical proper dephasing [100, 107, 125]). We have shown here that a similar formula for dynamical decoherence control, (218) is obtained for decay and dephasing due to coupling to a bath of harmonic oscillators. In these scenarios, as well as the scenario detailed in [178], the source of decoherence is the entanglement of the system to the bath’s degrees of freedom. Because the latter are inaccessible to both modulation and probing, they are traced out and this results in loss of coherence of the system.Nevertheless, not all is lost when interacting with a bath. During short times, much shorter than the bath memory times, the system and bath still interact coherently and the effects of the bath can be remedied by applying the appropriate modulations. This can understood from (218) by noting that the width of G(ω) is proportional to the inverse of the memory time. Thus, one can apply rapidly repeating modulations, corresponding to a multipeaked modulation power spectrum, where the peaks are positioned at the high frequency range. The spectral overlap between the system-bath coupling spectrum and the modulation power spectrum can be greatly reduced if the modulations are temporally separated by a shorter interval than the bath correlation time.These frequent modulations can be viewed as driving the system to higher frequencies in which the bath modes are dramatically more weakly coupled than those modes coupled without modulation. This reduced coupling results in less entanglement with the bath modes and thus less decoherence.A novel interplay between entanglement as a QIP resource and entanglement as the source of decoherence was detailed in Section8 [126]. Two entangled qubits were analyzed, each coupled to a bath via common modes. The non-Markovian time scale was considered, as well as dynamical modulations. It was shown how the entanglement of the qubits could vanish after a finite time (entanglement sudden death—ESD), but later restored by non-Markovian modulation-induced oscillations of the system-bath coherence.Thus, an exchange between two-qubit entanglement and system-bath entanglement may take place and be dynamically controlled by modulations. If the systems couple to two baths that have common modes, one can observe the transfer of coherence and buildup of entanglement between the two systems via these modes. If, on the other hand, the baths were completely separate, the coherence transfer between each system and its bath can modulate the amount of system-system entanglement, first lost and then regained.Another striking example of entanglement-mediated dynamics was presented in [178]. There we considered a two-level system that first reaches the state of equilibrium with a bath of harmonic oscillators, a state in which large system-bath entanglement is present, but no further dynamics. Then, a quantum probe is coupled to the system and performs a quantum nondemolition (QND) measurement of the system’s excited-level population. This QND measurement results in system-probe entanglement and the consequent loss of system-bath coherence, or vanishing of the off-diagonal elements of the system-bath density matrix. After this impulsive, nonselective, QND measurement, the state is no longer in equilibrium and begins to evolve.We have shown that immediately after the measurement the system and bath always “heat up”, that is, get excited. Remarkably, for certain system-bath coupling spectra, one can also observe a system that has lower excited-state population than the equilibrium state, that is, a purer system. This occurs despite the fact that the system has effectively recoupled with the bath and has become entangled with it.This interplay between system-bath entanglement, and system-probe entanglement and its effect on system cooling is still under active investigation by our group.Entanglement in open systems is a fascinating concept that is not yet fully understood. It is essential for QIP, yet it is also the origin of decoherence. The interplay between system-system entanglement, system-probe entanglement and system-bath entanglement on non-Markovian time-scales is an intriguing field that deserves more attention.


## 10. Conclusions

In this paper, we have expounded our universal approach to the dynamical control of qubits subject to AN and PN, by either off- or on-resonant modulating fields, respectively. It is based on a general non-Markovian master equation valid for weak system-bath coupling and arbitrary modulations, since it does not invoke the rotating wave approximation. The resulting universal convolution formulae provide intuitive clues as to the optimal tailoring of modulation and noise spectra. Our analysis of multiple, field-driven, qubits which are coupled to partly correlated or independent baths or undergo locally varying random dephasing has resulted in the universal formula (129) for coupling to zero-temperature bath, and (61) for Bell-state preservation under local proper dephasing.Our general analysis allows one to come up with an optimal choice between global and local control, based on the observation that the maximal suppression of decoherence is not necessarily the best one. Instead, we demand an optimalphase-relation between different, but synchronous, local modulations of each particle. The merits of local versus global modulations have been shown to be essentially twofold: (i)
Local modulation can effectively  decorrelate the different proper dephasings of the multiple TLS, resulting in equal dephasing rates for all states. For two TLS, we have shown that the singlet and triplet Bell states acquire the same dynamically modified dephasing rate. This should be beneficial compared to the standard global “Bang-Bang” (π-phase flips) if both states are used (intermittently) for information transmission or storage.(ii)
For different couplings to a zero-temperature bath, one can better preserve any initial state by using local modulation which can reduce the decay as well as the mixing with other states, than by using global modulation. It was shown that local modulation which eliminates the cross decoherence terms increases the fidelity more than the global modulation alternative. For two TLS, it was shown that local modulation better preserves an initial Bell state, whether a singlet or a triplet, compared to globalπ-phase “parity kicks.”In this paper, we have expounded our comprehensive approach to the dynamical control of decay and decoherence. Our analysis of dynamically modified coupling between a qubit and a bath has resulted in the universal formula (25) for the dynamically modified decay rate into a zero-temperature bath, as well as its counterparts (121) for excited- and ground-state dynamical decay into finite-temperature baths. This ground-state dynamically induced decay results from RWA violation by ultrafast modulation. For multilevel systems with partly interfering decaying levels (Section 5), the merits of global (common to all levels) versus local (specific to each level) modulations have been compared.The previous ground-breaking works on dynamical decoupling [80–87, 118] are more restrictive than the universal approach detailed in Sections 2–5 of this paper.(i)
In most works [82–87], the coupling between the system and the bath had only proper dephasing elements, that is, HI=ħσz∑λ‍(κλaλ+κλ*aλ†). This coupling means that there is no population decay to the bath and the decoherence is only via the phase. In other pioneering works [80, 81, 118], off-diagonal coupling to a Lorentzian bath was considered within the rotating-wave approximation (RWA). The Hamiltonians in all of these works should be contrasted with the more general Hamiltonians, 1, which account for both dephasing and decay into arbitrary baths, without the RWA.(ii)
The “parity kicks” [80, 118] or “bang-bang” (BB) pulses have usually been considered [81–87] to be so strong and fast that during the pulses there is no coupling to the bath, while between the pulses there is natural evolution of the system coupled to the bath. By contrast, the approach outlined in the present paper allows for modulations that are effected during the coupling to the bath.(iii)
The BB pulses [82] that have been previously proposed flip the phase of the system state by π, so that to achieve suppression of the decoherence. In particular, in [86] it was shown that the use of BB pulses results in suppression of the low-frequency modes of the spectral density, thus effectively suppressing the decoherence due to 1/f-type spectral densities. The BB pulses formalism then yields for Rd(t2N)t2N the same expression (see [83], Equation (4)) as our universal approach, if one chooses in our (129) the specific form [83] Gd(ω)=coth(βω/2)G(ω) and adopts
(221)Ft2N(ω)=2tan2(ωτ2)1-cosωt2Nπω2t2N,which coincides with our result in (52) upon settingϕ=π and n=2N, t2N=2Nτ.(iv)
As shown in Section7.3.2, a new field may be more effective (much less energy consuming) in suppressing proper dephasing as BB pulses.(v)
The pioneering results [80, 108, 118] on dynamical suppression of decay into a Lorentzian or Gaussian bath coincide with ours, advocating ϕ=π phase flips as the most appropriate (48). However, our more general results  and (123) identify small periodic phase shifts ϕ≪1 as being far more suitable near the Debye cut-off frequency of a phonon bath.(vi)
Our innovative treatment ofmultilevel decay via coupling to partly correlated baths in Section 5 gives a flavor for the pros and cons of decay suppression via local versus global phase modulations.(vii)
The ground-breaking theory and experiment [106] concerning decay to the continuum in a periodically tilted washboard potential coincide with our results on amplitude modulation (Section 3.6), but our treatment also covers other parameter ranges, suitable for modulated Josephson junctions [100].In Section6, we have outlined a method complementary to the one in Sections 2–5. We have demonstrated that a “counterintuitively” ordered sequence of abrupt changes of the detuning between an atomic transition and a continuum edge (the photonic band cutoff) is able to protect the atomic state from spontaneous emission more effectively than the intuitively obvious alternative, which is to fix the resonance frequency well within the forbidden bandgap, as far as possible from the continuum edge. This method is effective even under conditions of strong coupling to the continuum, since it is based on phase-dependent changes of the atomic state that is “dressed” by the continuum, rather than on modulation of the “bare” state that is weakly perturbed by the bath [81, 99, 118], as opposed to all previously proposed dynamical modulation strategies [81, 97, 99, 118], including that of Sections 2–5. The ability to maintain high fidelity of quantum states and quantum-logic operations in the presence of decoherence by nonadiabatic interference, contrary to the prevailing adiabatic approach to quantum-state control [1, 11, 12, 129, 142], may pave the way to new methods of controlling decay and decoherence in spectrally structured continua [2, 85, 87, 119].Peculiarity of the Approach. In summary, we have considered a way of finding a time dependence of the system Hamiltonian over a fixed time interval such that a given system observable attains a desired value at the end of this interval. The peculiarity of our approach is that it relies on knowledge of the bath coupling spectrum and adapts the spectrum of the system modulation to it. This allows to adjust the modulation to bandgaps or peaks in the bath coupling spectrum. In contrast to dynamic decoupling of system and bath, which can be achieved by shifting the entire system-modulation spectrum beyond some assumed bath cutoff frequency, an enhancement of the coupling requires more detailed knowledge on the peak positions of the bath spectrum. In this way, our approach may comprise suppression and enhancement of the system-bath coupling in a unified way for executing more general tasks than decoherence suppression. The same approach can also be applied to map out the bath spectrum by measuring the coherence decay rate for a narrow-band modulation centered at different frequencies [197].As far as the controls are concerned, we here consider time-continuous modulation of the system Hamiltonian, which allows for vastly more freedom compared to control that is restricted to stroboscopic pulses as in DD [82, 188, 189]. We do not rely on rapidly changing control fields that are required to approximate stroboscopic π-pulses. These features allow efficient optimization under energy constraint. On the other hand, the generation of a sequence of well-defined pulses may be preferable experimentally. We may choose the pulse timings and/or areas as continuous control-parameters and optimize them with respect to a given bath spectrum. Hence, our approach encompasses both pulsed and continuous modulation as special cases.Open Issues. An open issue of the approach is the inclusion of higher orders in the system-bath coupling, which becomes important for strong or resonant system-bath coupling, so that a perturbative expansion cannot be applied. This may be the case especially when this coupling is to be enhanced in order to achieve a nonunitary operation (e.g., cooling), since in this case an optimization of the coupling may take us out of the domain of validity of the entire approach.Another concern regards the initial conditions. Here we have assumed a factorized initial state of the system and bath. This prevents us from taking into account system-bath interactions that may have occurred prior to that time. In particular, if the system is in equilibrium with the bath, their states are entangled or correlated [191, 198].An immediate problem of both higher order coupling and system-bath correlations is that their consideration requires knowledge of the corresponding parameters. It may be difficult to obtain such data with sufficient experimental precision. Moreover, its consideration renders the theory cumbersome and the intuition gained from the spectral overlap approach presented here is lost. A way out is offered by replacing the “open” iteration loop in Figure16 with a “closed” loop [199], where the calculation of the score, constraint, and their gradients are based on actual measurements performed on the controlled system in real time rather than on prior model assumptions, that is, knowledge of bath properties. Such closed loop control would allow efficient optimization, but at the cost of losing any insight into the physical mechanisms behind the result obtained.From a fundamental point of view, it is interesting to derive analytic bounds of a desired score (under a chosen constraint) and see if this bound can be achieved by means of some (global) optimization, that is, if the bound is tight. The need for a constraint in such optimization is not obvious if the task requires coupling enhancement, especially when the bath spectrum has a single maximum.Our investigation of the scalability of decoherence control has revealed several general, basic insights. The first is that for small systems the coherence time is inversely proportional to the state-dependent scalability,cψ(N), while for sufficiently large systems, this scaling is always inversely proportional to the square root of cψ(N), due to the transition to a Zeno-like behavior of the decoherence rates.We have then shown that whenever qubitscollectively couple to common bath modes (e.g., in ion traps [180, 181] or cavities [200–202]) local modulations of individual qubits can eliminate cross decoherence, thus changing the scalability in a scenario-dependent manner (Table 1). (i) Local distinct modulations can eliminate cross decoherence, causing decoherence scaling to be linear in N for most entangled states. (ii) Imposing cross decoherence equality by local modulations can render states truly dark, regardless of N. (iii) Very large systems may revert from their unmodulated Zeno-like behavior to their Markov-like behavior, by either strong enough or scenario-tailored local modulations. Our major result is that the modulation must be faster than the correlation time of the bath, but can be much slower than the (unmodulated) multipartite coherence time, and still be effective.These results show that dynamical control of decoherence is a promising avenue for combatting multipartite decoherence, since their rate is not required to become faster as the system grows. Coherence-time scalability withN is particularly crucial for one-way quantum computing [175], since the initial state is highly entangled and the computation, performed by measurements, is done in a serial manner. On the fundamental side, these results may pave the way to understanding and controlling “quantumness” in large multipartite systems. These findings identify local dynamic control as a promising path to the protection of quantum information from the environment in large multipartite systems.



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*Source: 783865-2013-09-19.xml*