Protecting quantum resources via qubit frequency modulation

Protecting quantum resources via qubit frequency modulation

Ali Mortezapour mortezapour@guilan.ac.ir Department of Physics, University of Guilan, P. O. Box 41335–1914, Rasht, Iran    Rosario Lo Franco rosario.lofranco@unipa.it Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy Dipartimento di Fisica e Chimica, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy
July 6, 2019
Abstract

Finding strategies to preserve quantum resources in open systems is nowadays a main requirement for reliable quantum-enhanced technologies. We address this issue by considering structured cavities embedding qubits whose transition frequency is modulated by a driving field. We first focus on a single-qubit system to determine the optimal modulation parameters and the qubit-cavity coupling regime allowing a long-time preservation of both coherence and quantum Fisher information. We relate this behavior to the inhibition of the qubit effective decay rate rather than to stronger memory effects (non-Markovianity) of the system. We then use these findings to efficiently control resources like entanglement, discord and coherence in a system of two separated qubits. We discuss the feasibility of the system and show that individual qubit frequency modulation can produce quantum resource lifetimes orders of magnitude longer than their natural (uncontrolled) decay times. These results provide new useful insights towards efficient experimental strategies against decoherence.

pacs:
03.67.Mn, 03.65.Ud, 03.65.Yz

I Introduction

Quantum coherence, stemming from the superposition principle, is the key concept that distinguishes quantum mechanics from classical mechanics. It is well-known that quantum coherence among different bodies of a multipartite systems is the basic ingredient for the formation of quantum correlations Yao et al. (2015); Streltsov et al. (2015); Hu et al. (2016); Radhakrishnan et al. (2016); Ma et al. (2016); Malvezzi et al. (2016); Li and Lin (2016); Chen et al. (2016); Hu and Fan (2017); Hu et al. (2017). Among these, entanglement represents the part due to non-separability of states, utilized in many paradigmatic quantum information processes such as teleportation Bennett et al. (1993); Pirandola et al. (2015); Lo Franco and Compagno (), quantum error correction Laflamme et al. (1996); Plenio et al. (1997), quantum key distribution Ekert (1991) and quantum dense coding Bennett and Wiesner (1992). Useful quantum correlations beyond entanglement, occurring even for separable states, have been identified by the so-called discord Ollivier and Zurek (2001); Henderson and Vedral (2001), which is in turn employed for specific quantum information tasks Datta et al. (2008); Lanyon et al. (2008); Modi et al. (2012). As a very fundamental trait, quantum coherence itself can be quantified Aberg (2006); Chitambar and Gour (2016); Napoli et al. (2016); Rana et al. (2016); Yu et al. (2016); Streltsov et al. (2017) and exploited as a resource in various quantum protocols Aberg (2006); Baumgratz et al. (2014); Winter and Yang (2016); Chitambar and Hsieh (2016); Streltsov et al. (2017). One of the cutting-edge fields of application of these quantum features is, for instance, quantum metrology, aiming at achieving high precision measurements characterized by the quantum Fisher information Giovannetti et al. (2011); Escher et al. (2011); Joo et al. (2011); Liu et al. (2013); Chaves et al. (2013); Zhang et al. (2013); Demkowicz-Dobrzanski and Maccone (2014); Alipour et al. (2014); Lu et al. (2015); Correa et al. (2015); Baumgratz and Datta (2016); Liu and H.Yuan (2017), with recent developments in open quantum systems Alipour et al. (2014); Tsang (2013); Correa et al. (2015); Li et al. (2015); Nichols et al. (2016); Wang et al. (2017).

Open quantum systems undergo decoherence due to the interaction with the surrounding environment, which usually destroys coherence and correlations thus limiting their practical use in quantum information Aolita et al. (2015) and metrology Li et al. (2015); Chin et al. (2012); Chaves et al. (2013); Macieszczak (2015); Smirne et al. (2016). The dynamics of open quantum systems can be classified in Markovian (memoryless) and non-Markovian (memory-keeping) regimes Breuer and Petruccione (2002); de Vega and Alonso (2017). Markovian regime is typically associated to the lack of memory effects, a situation where the information irreversibly flows from the system to the environment. Differently, non-Markovian regime implies that the past history of the system affects the present one: such a memory effect results in an information feedback from the environment to the system Breuer and Petruccione (2002); de Vega and Alonso (2017); López et al. (2010); Chiuri et al. (2012); Caruso et al. (2014); Reina et al. (2014); Haseli et al. (2014); Lo Franco and Compagno (2017); Aolita et al. (2015); Leggio et al. (2015); Rivas et al. (2014); Costa-Filho et al. (2017). In this sense, non-Markovianity has been quantified by a variety of measures Rivas et al. (2014); Wolf et al. (2008); Breuer et al. (2009); Rivas et al. (2010); Lu et al. (2010); Luo et al. (2012); Lorenzo et al. (2013); Chrúscínski and Maniscalco (2014); Bylicka et al. (2014); Addis et al. (2014a); Breuer et al. (2016) and regarded as a precious resource for certain applications Rivas et al. (2014); Vasile et al. (2011); Bylicka et al. (2014). Although memory effects permit coherence, entanglement and discord of noninteracting subsystems to partially revive after disappearing during the time evolution de Vega and Alonso (2017); Aolita et al. (2015); Rivas et al. (2010); Bellomo et al. (2007, 2008); Xiao et al. (2010); Man et al. (2015a), these revivals eventually decay. Decoherence remains one of the main drawbacks to overcome towards the implementation of quantum-enhanced technologies, which require qubits with long-lived quantum features. To this aim, many strategies have been proposed in order to harnessing and protecting quantum resources in systems of qubits under different environmental conditions Lo Franco et al. (2013a); Scala et al. (2008); Duan and Guo (1997); Viola and Lloyd (1998); Protopopescu et al. (2003); Facchi et al. (2005); Hartmann et al. (2007); Branderhorst et al. (2008); Lo Franco et al. (2013b); Maniscalco et al. (2008); Tan et al. (2010); Tong et al. (2010); Scala et al. (2011); Xue et al. (2012); D’Arrigo et al. (2014a); Orieux et al. (2015); Man et al. (2012); Bellomo and Antezza (2013a, b); Xu et al. (2013); Aaronson et al. (2013a, b); Jing and Wu (2013); Addis et al. (2014b); Lo Franco et al. (2014); Cianciaruso et al. (2015); Lo Franco et al. (2012); Bromley et al. (2015); Man et al. (2014, 2015b, 2015c); D’Arrigo et al. (2013); D’Arrigo et al. (2014b); Yan et al. (2015); Yang et al. (2016); Lo Franco (2016); Silva et al. (2016); Wang and Fang (2016); Mortezapour et al. (2017); Campos Venuti et al. (2017); Çakmak et al. (2017); Mortezapour et al. ().

In this paper, we address this issue by investigating the influence of frequency modulation in controlling the dynamics of quantum resources. Generally, a quantum system is frequency modulated when its energy levels are shifted by an external driving. Frequency modulation in an atomic qubit can be performed by applying an external off-resonant field Noel et al. (1998); Zhang et al. (2003); Silveri et al. (2017). Moreover, the most recent experimental progress in fabrication and control of quantum circuit devices enables frequency modulation in superconducting Josephson qubits (artificial atoms) Silveri et al. (2017); Nakamura et al. (2001); Oliver et al. (2005); Tuorila et al. (2010); Li et al. (2013), which are the preferred building blocks in current quantum computer prototypes Trabesinger (2017). It has been reported that external control of the qubit frequency can induce sidebands transitions Beaudoin et al. (2012); Strand et al. (2013), modify its fluorescence spectrum Ficek et al. (2001) and population dynamics Janowicz et al. (2008); Janowicz (2000); Zhou et al. (2009); Deng et al. (2015), as well as amplify non-Markovianity Poggi et al. (2017). Motivated by these considerations, we first study a frequency modulated qubit inside a structured cavity with Lorentzian spectral density to determine the values of parameters of the external field and of the qubit-cavity coupling which efficiently protect the initial coherence. We also analyze how the choice of these parameters plays a role in preserving quantum Fisher information so to enhance the precision of phase estimation in the system during the evolution. We then extend the analysis to a system of two separated noninteracting qubits for establishing at which extent entanglement, discord and coherence can be shielded from decay thanks to individual control of the qubit frequencies. The investigation of such simple systems constitutes the essential step for acquiring useful insights which can then be straightforwardly generalized to a system of many separated qubits for scalability.

The paper is organized as follows. In Sec. II we describe the evolution of the frequency-modulated open qubit, giving the dynamics of coherence and quantum Fisher information (QFI). In Sec. III we discuss the dynamics of quantum resources in the two-qubit system for some classes of initially correlated states. Finally, in Sec. IV we summarize the main conclusions.

Ii Single-qubit system

Figure 1: A qubit, whose transition frequency is controlled by an external drive, is embedded inside a high- leaky cavity.

The system consists of a qubit (two-level system) coupled to a zero-temperature reservoir formed by the quantized modes of a high- cavity. The excited and ground states of the qubit are labeled by and , respectively. It is assumed that the transition frequency of the qubit is modulated sinusoidally by an external driving field, as depicted in Fig. 1. The Hamiltonian of the system is

(1)

where is the qubit Hamiltonian ()

(2)

with and denoting the modulation amplitude and frequency, respectively, and the Pauli operator along the z-direction. is the reservoir Hamiltonian with () being the creation (annihilation) operator of the cavity field mode . describes the interaction between the qubit and the cavity modes which, in the dipole and rotating-wave approximation, can be written as

(3)

where is the coupling constant between the qubit and mode while () represents the raising (lowering) operator for the qubit. Moving to a non-uniformly rotating frame (interaction picture) by the unitary transformation

(4)

the effective Hamiltonian is

(5)

We notice that shows how the system behaves as if the same frequency modulation was applied to each qubit-cavity coupling constant . This is relevant to our purpose since, under this condition, it is known that decoherence can be softened Agarwal (1999). Making use of Jacobi-Anger expansion, the exponential factors in Eq. (5) can be written as

(6)

where is the -th Bessel function of the first kind.

We assume the qubit is initially in a coherent superposition and the reservoir modes in the vacuum state , so that the overall initial state is

(7)

Hence, at any later time the quantum state of the whole system can be written as

(8)

where is the cavity state with a single photon in mode and is its probability amplitude. Using the time-dependent Schröِdinger equation, the differential equations for the probability amplitudes and are, respectively,

(9)

and

(10)

Solving Eq. (10) formally and substituting the solution into Eq. (9), one obtains

(11)

where the kernel , that is the correlation function including the memory effects, is

(12)

In the continuous limit, the kernel above becomes

(13)

where is the spectral density of reservoir modes. We choose a Lorentzian spectral density, which is typical of a structured cavity Breuer and Petruccione (2002), whose form is

(14)

where indicates the spectral width of the coupling and is related to the reservoir correlation time via . On the other hand, represents the decay rate of the excited state of the qubit in the Markovian limit of flat spectrum (i.e., the spontaneous emission decay rate) and it is linked to the qubit relaxation time by Breuer and Petruccione (2002). Qubit-cavity weak coupling occurs for (); the opposite condition () thus identifies strong coupling. The larger the cavity quality factor, the smaller the spectral width .

With such a spectral density, the kernel of Eq. (13) becomes

(15)

Substituting it into Eq. (11), one gets

(16)

Calculating from this equation, the reduced density matrix of the qubit in the basis is given by

(17)

Taking the derivative of Eq. (16) with respect to time, we have

(18)

where plays the role of a time-dependent Lamb shift and can be interpreted as a time-dependent decay rate Breuer and Petruccione (2002).

The evolved density matrix of the qubit shall be utilized for obtaining the dynamics of the quantum properties of interest, such as coherence, quantum Fisher information and non-Markovianity. Before displaying the dynamics of these quantities, we recall their definitions and give their time-dependent expressions in the following.

ii.1 Coherence

Coherence of a quantum state characterizes the property of superposition among the basis states of the system and can be regarded as a basis-dependent resource by itself Streltsov et al. (2017). Among the various bona-fide measures of coherence, we choose here the so-called -norm defined as Baumgratz et al. (2014)

(19)

where is the density matrix of an arbitrary quantum state and the minimum is taken over the set of incoherent states . Such a measure, depending only on the off-diagonal elements () of the quantum state, is clearly related to the fundamental property of quantum interference.

For of Eq. (18) and assuming in the initial state of Eq. (7), we have (maximum initial coherence) and a time-dependent qubit coherence .

ii.2 Quantum Fisher information

Quantum metrology exploits quantum-mechanical effects to reach high precision measurements. In a typical metrological procedure, one first encodes the parameter of interest on a probe state by means of a unitary process . Thus, the output state is . The output state is then measured by a set of positive operator valued measurements and the value of finally estimated from the outcomes. It is known that the precision in estimating is limited by the quantum Cramer-Rao bound inequality Helstrom (1976); Holevo (1982)

(20)

where is the standard deviation associated to the variable and is the quantum Fisher information (QFI) defined as , with being the so-called symmetric logarithmic derivative determined by with Helstrom (1976); Holevo (1982). From the spectral decomposition of the -dependent density matrix , where and are, respectively, eigenvalues and eigenstates, the QFI is known to have the analytical expression Zhang et al. (2013)

(21)

For our dynamical system, we consider a phase-estimation problem where acts on the initial maximally coherent state of the qubit Giovannetti et al. (2011) and successively let the system evolve under the dissipative noise and frequency modulation. The initial overall qubit-cavity state is therefore and the evolved reduced density of the qubit has the form of Eq. (17) with a -dependence in the off-diagonal elements. Using Eq. (21) we get , that is the coherence squared. The associated minimum estimation error is therefore .

ii.3 Non-Markovianity

In order to discuss the non-Markovian character of our system we employ, among the various quantifiers, the well-known measure based on the dynamics of the trace distance between two initially different states and of the qubit, identifying information backflows. It is defined as Chrúscínski and Maniscalco (2014)

(22)

where , with being the trace distance () Yao et al. (2015). In Eq. (22) the integration is taken over time intervals when and the maximization is made over all the possible pairs of initial states and . It is noteworthy that the trace distance is related to the distinguishability between quantum states, whereas its time derivative () means a flow of information between the system and its environment. While Markovian processes satisfy for all pairs of initial states at any time , non-Markovian ones admit at least a pair of initial states such that for some time intervals, so that the information flows from the environment back to the system Chrúscínski and Maniscalco (2014).

For our system, which experiences a dissipative dynamics describable as an amplitude damping channel, the evolved qubit density matrix has the form of Eq. (17). Under this condition, it is known that the non-Markovianity measure is maximized for the choice of the initial orthogonal states of the qubit Addis et al. (2014a) and assumes the expression

(23)

where is the effective time-dependent decay rate defined after Eq. (18).

ii.4 Time evolution of single-qubit resources

Figure 2: Qubit coherence as a function of scaled time for different values of the modulation frequency . The values of other parameters are: , (weak coupling), . Solid-blue line in panel (a) corresponds to the situation in which frequency modulation is off.

We start the quantitative analysis by studying the time evolution of coherence under a weak coupling regime (), plotted in Fig. 2 for different values of the modulation frequency and fixed modulation amplitude . This evolution is compared to the case when the external driving is off (, ), for which coherence disappears at time . As shown, under the given condition for , coherence survives for times longer than when is smaller. Presence of oscillations in the dynamics (see Fig. 2(b)) indicates the manifestation of non-Markovian effects under weak coupling regime Poggi et al. (2017). However, frequency modulation can even produce negative effects for large values of the modulation frequency, , which are such that the coherence disappears at times .

Figure 3: Qubit coherence as a function of scaled time for different values of the modulation frequency, for a fixed modulation amplitude . Other parameters are: (strong coupling), . Solid-blue line in panel (a) corresponds to the situation in which frequency modulation is off.

This fact suggests to see what happens under a strong coupling regime. Fig. 3 shows the time behavior of coherence under such a condition () for different values of modulation frequency at a fixed modulation amplitude . Interesting results are obtained for , and . The coherence dynamic for and is accompanied with rapid and small oscillations, whereas the latter are observed only in the range for . This implies that non-Markovianity, associated to backflows of information, should be greater for the first two values of . In general, a non-monotonic behavior of non-Markovianity is expected as a function of , since the intensity of oscillations appears to be very sensitive to different values of the modulation frequency (this aspect shall be treated below). We then point out that, although the plot for seems to perform better than those for and within the time range shown in the figures, our calculations instead show that the coherence lasts much longer for these latter values of the modulation frequency. In general, however, reducing to values smaller and smaller, with a fixed nonzero , helps to prolonging the time when coherence vanishes. This happens because the final effect is to simply detune the central frequency of the high- cavity Lorentzian spectral density from the qubit transition frequency Bellomo et al. (2010), as can be deduced from of Eq. (5) for . Nevertheless, this extreme condition is not relevant to our purpose, since we are interested in the modulation of the qubit frequency during the evolution Silveri et al. (2017) and not to a detuning effect.

Figure 4: Qubit coherence as a function of scaled time for different values of the modulation frequency: (dashed green line), (dotted red line), (solid blue line). The panels correspond to various values of the modulation amplitudes: (a) , (b) , (c) , (d) . Other parameters are: (strong coupling), .

Therefore, to gain a more comprehensive understanding of the system evolution, we now study the interplay of the two driving parameters and , which is expected to play a significant role in affecting the decoherence process Agarwal (1999). Fig. 4 displays the dynamics of coherence, under strong coupling (), when the ratio is tuned such as to assume values which make the -th Bessel function of Eq. (6) vanish. As seen in Fig. 4(a), the initial coherence can be strongly protected against the noise by increasing the modulation frequency when is such that vanishes. However, this behavior is not general for larger values of the ratio , as shown in Fig. 4(b)-(d). The interplay between and is thus non-trivial, since it is not sufficient to fix as a zero of the Bessel functions to have long-lasting coherence. In particular, we find that no other settings of and supply a result superseding that occurring for and (solid blue line of Fig. 4(a)).

Figure 5: (a) Quantum Fisher information and (b) optimal phase estimation as a function of scaled time for various values of modulation frequency: (dashed light green line), (dash-dotted dark green line), (dotted red line), (solid blue line). Modulation amplitude is fixed at . Other parameters are: (strong coupling), .

We then examine the effect of frequency modulation on the dynamics of both QFI and optimal phase estimation for the ratio under strong coupling (). From Fig. 5 it is immediately seen, as expected, that maintenance of QFI and a significant improvement in phase estimation can be reached during the evolution by increasing up to the value .

Figure 6: Non-Markovianity as a function of . Values of other parameters are: , (weak coupling), .

At this stage, one may ask whether the high coherence protection found above is ultimately linked to strong memory effects. The answer requires the knowledge of the degree of non-Markovianity as a function of under different parameter conditions. This is first done under the weak coupling regime () for some modulation amplitudes in Fig. 6. These plots disclose that the frequency modulation process can induce non-Markovian features even in the weak-coupling regime Poggi et al. (2017), which justifies the existence of oscillations in the time evolution of coherence for and (see Fig. 2(b)). Moreover, this induced non-Markovianity can be reinforced by increasing the modulation amplitude, while no memory effect is observed for small values of tending to turn off the frequency modulation. In Fig. 7 we then display the behavior of non-Markovianity as a function of under the strong coupling regime (). Panels (a) and (b) of this figure consider the cases when the ratio corresponds to a zero of the Bessel functions, as in Fig. 4: one sees that non-Markovian characteristics can be amplified only for , with maximum amplification occurring when is the zero of the Bessel function . Panels (c) and (d) of Fig. 7 instead consider fixed modulation amplitudes as changes: it is evident that an increase of enlarges the range of where non-Markovianity can be enriched, albeit this is achieved only when . As a general behavior, all plots of Fig. 7 show that tends to reach approximately the same value as increases, independently of the values of the modulation amplitude . Thus, on the basis of our above analysis of qubit dynamics, a more efficient preservation of coherence cannot be related to a higher degree of non-Markovianity (stronger memory effects).

Figure 7: Non-Markovianity as a function of for: (a) (solid-blue line), (dotted-red line); (b) (solid-blue line); (c) (solid-blue line), (dotted-red line); (d) (solid-blue line), (dotted-red line). The values of other parameters are: (strong coupling), .

To have a deeper physical interpretation behind the effective coherence protection, especially exhibited in Fig. 4(a), we plot in Fig. 8 the time-dependent decay rate (in units of ), appearing in the qubit master equation of Eq. (18), for some relevant values of . From these plots (see, in particular, panel (d)), it is clear that the main cause of long-lasting coherence is the inhibition of the effective decay rate of the qubit.

Summarizing, according to our analysis, the best performance of coherence preservation controlled by qubit frequency modulation occurs under strong coupling and for values of modulation amplitude and modulation frequency given, respectively, by

(24)

which are expressed in units of the spontaneous decay rate of the qubit .

Figure 8: Decay rate as a function of scaled time for different values of the modulation frequency: (a) , (b) , (c) , and (d) . Other parameters are those of Fig. 3(a), that is: , (strong coupling), .

ii.5 Experimental context

We discuss here how the previous results may impact in experimental contexts where quantum computing platforms allow for accurate preparation of initial states and control of the qubit. This is particularly the case of superconducting qubits, which have been imposing as the building blocks of quantum computer prototypes based on architectures of circuit quantum electrodynamics Trabesinger (2017); Castelvecchi (2017).

We know from our analysis above that the coherence of a qubit in a high- cavity naturally (i.e., without frequency modulation) would completely disappear at a time under weak coupling (see Fig. 2(a)) and at (after oscillations) under strong coupling (see Fig. 3(a)). These times are meant as lifetimes of coherent superpositions. For superconducting transmon qubits, which typically have (free space) relaxation times ns Paik et al. (2011), one would obtain s, which is comparable to the current performance achieved in the IBM-Q quantum processor, where a single qubit has average coherence lifetimes of 90 s IBM ().

Let us now see to which extent our results with frequency-modulated (FM) qubit are able to prolong coherence lifetimes. Looking at the time behavior obtained under strong coupling with the optimal driving parameters of Eq. (24), as displayed by the solid (top) curve of Fig. 4(a), our calculations estimate the dimensionless time when coherence vanishes at . This implies a coherence lifetime and thus s, producing an extension of four orders of magnitude compared to the case of uncontrolled qubit.

This achievement seems feasible with the current technology. In fact, quality factors of cavities in circuit quantum electrodynamics can be adjusted to high values such that the cavity spectral bandwidth (or photon decay rate) is smaller than the spontaneous emission decay rate , thus entering the strong coupling regime Paik et al. (2011). Moreover, frequency modulation of individual superconducting qubits is already realized Silveri et al. (2017); Li et al. (2013), with the possibility to suitably fix modulation amplitude and frequency to the desired optimal values.

Iii Two-qubit system

After individuating the optimal parameters of the driving external field such as to efficiently shield single-qubit coherence from the detrimental effects of the environment, we now extend our study to a composite system of two separated noninteracting qubit-cavity subsystems, namely A and B. Each qubit-cavity subsystem is structured like that considered in the above Sec. II, with the transition frequency of each qubit individually modulated.

iii.1 Initial states and evolved density matrix

The qubits are initially prepared in the extended Werner-like (EWL) states Bellomo et al. (2008)

(25)

where is a measure of the state purity , being for both states , is the identity matrix and

(26)

represent the pure part of the state in the form of Bell-like states with . This class of states is quite general, including both Bell states and Werner states.

The two subsystems, being separated, evolve independently and are individually governed by the Hamiltonain of Eq. (1), so that the total Hamiltonian is simply . For the EWL initial states above, the density matrix of the two qubits at time , in the standard (tensor product) computational basis , takes the X-type structure Man et al. (2015c)

(27)

where . The elements of this evolved density matrix can be straightforwardly calculated by the well-known method introduced in Ref. Yan et al. (2015) for separated qubits, based on the knowledge of the single-qubit density matrix. We are interested in the dynamics of quantum resources such as entanglement, discord and coherence associated to the evolved two-qubit state . In the following we recall the definitions of their quantifiers.

iii.2 Quantification of quantum resources

Entanglement between subsystems of any bipartite quantum system can be measured by concurrence Yang et al. (2016). For the density matrix given by Eq.( 27), the concurrence is Lo Franco et al. (2013a)

(28)

where and . The two initial EWL states of Eq. (25) have the same concurrence and are thus entangled for .

Discord captures all nonclassical correlations between two qubits beyond entanglement Chen et al. (2016). For a X-structure density matrix, like that of Eq. (27), the analytic expression of discord is Silva et al. (2016)

(29)

where (), with being the eigenvalues of the density matrix , , with , and (the explicit time-dependence of the density matrix elements has been omitted for simplicity). The two initial EWL states have the same discord as a function of and whose explicit expression, obtainable by the previous formulas, is not reported here.

Concerning coherence, the definition given in Eq. (19) is immediately applicable to the two-qubit evolved density matrix of Eq. (27), so to give the two-qubit coherence with respect to the computational basis

(30)

Both the initial EWL states of Eq. (25) have the same coherence and are thus coherent for any provided that . Notice that for , that is the pure part of the EWL states is a (maximally entangled) Bell state, one has .

We are now ready to display the dynamics of the relevant quantities defined above.

iii.3 Time evolution of two-qubit resources

Figure 9: Time evolution of two-qubit coherence (solid blue line), discord (dotted red line) and concurrence (dashed green line) for the initial states (column I) and (column II), with and (Bell states), under (strong coupling). Panels (a), (b): uncontrolled dynamics (no frequency modulation). Panels (c), (d): controlled dynamics (individual frequency modulation) with optimal parameters , .

We assume that the subsystems are identical, that is characterized by the same values of qubit-environment parameters, with the optimal frequency modulation parameters fixed as in Eq. (24), under the strong coupling regime with . Such a choice has the advantage to make us directly focus on the best quantitative protection of two-qubit resources attainable by individual qubit frequency modulation under a given strong coupling condition. To acquire information about the protection efficiency of individual frequency modulation, we report the dynamics of , and starting from the initial EWL states of Eq. (25) with and , which reduces them to the Bell states, respectively, and . Fig. 9 immediately shows that individual qubit control by frequency modulation enables lifetimes of quantum resources (that is, the time when they completely disappear) which are three orders of magnitude longer than the lifetimes when modulation is off. In particular, we find against without frequency modulation. Concerning dynamical details, one sees that while the evolution of coherence is the same for both initial states, discord and entanglement vanish earlier for the initial state compared to . Another difference in the time behavior of quantum resources for the two initial states is that entanglement and coherence coincide at any time for (panels (b) and (d) of Fig. 9), whereas this is not the case for .

Finally, we analyze an attenuated strong coupling condition for decreasing values of the purity parameter in the EWL states of Eq. (25), in order to take into account both lower quality factors of cavities and possible imperfections in the initial state preparation. From Figs. 10 and 11, as expected, one sees that the amount of all the quantum resources diminishes for smaller and tends to zero faster. For entanglement is always zero, since . We also observe that two-qubit coherence is the more robust resource of the three ones, entanglement being instead the more fragile. Once again, the general emerging result is that all the quantum resources are efficiently shielded from the noise by qubit frequency modulation, their lifetime being extended of about three orders of magnitude with respect to the case without modulation. This is seen, for instance, in the case of Figs. 10(a) and 11(a) where , against found in absence of frequency modulation.

Figure 10: Two-qubit coherence (solid blue line), discord (dotted red line) and concurrence (dashed green line) for the two-qubit initial state as functions of scaled time for different degrees of purity: (a) , (b) , (c) , and (d) . Other parameters are: , , and (strong coupling).
Figure 11: Two-qubit coherence (solid blue line), discord (dotted red line) and concurrence (dashed green line) for the two-qubit initial state as functions of scaled time for different degrees of purity: (a) , (b) , (c) , and (d) . Other parameters are taken as in Fig. 5.

These results assume experimental significance in the context of circuit quantum electrodynamics, where the required individual control of the subsystems has been already discussed above (Sec. II.5) and a reliable preparation of entangled states of superconducting qubits has been implemented DiCarlo et al. (2010); Rigetti et al. (2012). For example, entangled states with purity and fidelity to ideal Bell states have been generated in the laboratory by using a two-qubit interaction mediated by a cavity bus in a circuit quantum electrodynamics architecture DiCarlo et al. (2010). These states may be approximately described as the EWL states of Eq. (25) with and Lo Franco et al. (2014), which is just the configuration of the initial states we have assumed in our study.

Iv Conclusion

In this work we have investigated in detail the quantitative effect of individual modulation of qubit transition frequency in protecting quantum resources from the detrimental effect of the environment, constituted by leaky high- cavities.

We have first put our attention on a single frequency-modulated qubit embedded in a cavity, determining the optimal modulation parameters (modulation amplitude) and (modulation frequency) and the qubit-cavity coupling regime allowing a long-time preservation of coherence. We have found that, under a strong coupling regime, qubit coherence lifetimes can be extended of orders of magnitude with respect to the case when modulation is off. In particular, when the ratio between the cavity spectral bandwidth and the spontaneous emission rate of the qubit is we have shown that this coherence lifetime extension can be of four orders of magnitude. We have also seen that the same conditions guarantee a maintenance of quantum Fisher information with a consequent relevant advantage in phase estimation processes of the single-qubit state during the evolution. We have individuated the inhibition of the effective decay rate of the qubit as the mechanism underlying the efficient dynamical preservation of quantum coherence, which is therefore not due to stronger non-Markovianity (memory effects) of the system.

We have then exploited the findings for the single-qubit case to efficiently control resources like entanglement, discord and coherence in a composite system of two separated qubits. Such a step has revealed very useful to understand the scalability of the control procedure to individually addressable subsystems. We have discovered that individual qubit frequency modulation is still capable to increase the lifetimes of the desired quantum resources of orders of magnitude compared to their natural (uncontrolled) disappearance times. In fact, albeit the gain in the two-qubit system is weaker than in the single-qubit case, with the optimal parameters one may reach lifetime prolongations of three orders of magnitude.

We have discussed the experimental feasibility of the systems, particularly considering the state-of-art achievements in the context of circuit quantum electrodynamics. Setups of superconducting qubits can indeed modulate the qubit transition frequency by external control, reach strong qubit-cavity coupling conditions and permit high-fidelity initial state preparation. The preservation procedure proposed here therefore appears to be implementable. Moreover, it is straightforwardly applicable to an array of many separated qubit-cavity subsystems. Our results provide novel useful insights for efficient experimental strategies against decoherence towards reliable quantum-enhanced technologies.

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