exact solution for the dynamical decoupling of a qubit...

Exact solution for the dynamical decoupling of a qubit with telegraph noise

Joakim Bergli1,* and Lara Faoro2,†

1Physics Department, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway2Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey 08854, USA

�Received 4 September 2006; revised manuscript received 30 November 2006; published 21 February 2007�

We study the dissipative dynamics of a qubit that is afflicted by classical random telegraph noise and issubject to dynamical decoupling. We derive exact formulas for the qubit dynamics at arbitrary working pointsin the limit of infinitely strong control pulses �bang-bang control�, and we investigate in great detail theefficiency of the dynamical decoupling techniques for both Gaussian and non-Gaussian �slow� noise at qubitpure dephasing and at optimal point. We demonstrate that control sequences can be successfully implementedas diagnostic tools to infer spectral properties of a few fluctuators interacting with the qubit. The analysis isextended in order to include the effect of noise in the pulses, and we give upper bounds on the noise levels thatcan be tolerated in the pulses while still achieving an efficient dynamical decoupling performance.

DOI: 10.1103/PhysRevB.75.054515 PACS number�s�: 03.65.Yz, 03.67.Pp, 05.40.�a

I. INTRODUCTION

Small superconducting circuits containing Josephsonjunctions have recently received a great deal of attention aspromising candidates for scalable quantum bits �qubits�. Un-fortunately, despite the last few years of remarkable experi-mental and theoretical breakthroughs,1–5 none of these cir-cuits satisfies the requirement that the coherence times mustbe considerably longer than the gate operation times.6 Thus,a major challenge for the coming years is to achieve controlof the decoherence afflicting these devices.

It is generally recognized that in superconducting qubits,the main sources of noise are the external circuitry, the mo-tion of trapped vortices in the superconductors, and the mo-tion of background charges �BCs� in associated dielectricsand oxides. While the noise originating from the externalcircuitry can be reduced and practically eliminated by im-proving the circuit design and while the motion of trappedvortices can be suppressed by making the superconductorfilms sufficiently narrow, it is currently not known how tosuppress and control charge noise. Indeed, moving BCs ofunknown origin are clearly responsible for the relativelyshort coherence time in the smaller Josephson charge qubits.1

In addition, BCs limit the performances of the flux qubits,3

the phase qubits,5 and the hybrid charge-flux qubits2 becausethey induce fluctuations in the critical current of the Joseph-son junctions and lead to qubit dephasing.

It is firmly established now that in all superconductingqubits both the spectrum of charge noise and critical currentfluctuations display a 1/ f behavior at low frequencies. Spe-cifically, charge noise spectra Sq�f�=�2 / f have been directlymeasured up to frequencies f�103 Hz.1,2,5,7–9 The intensityof the noise varies in the small range �= �10−3–10−4�e, andbecause its magnitude is greatly reduced by the echotechniques,10,11 it is commonly believed that a 1 / f noise doesnot extend above fm�1 MHz. Similarly, it has been shownthat critical current fluctuations display a 1/ f behavior at f�10 Hz, although very little is known at higherfrequencies.12–14

Both charge noise and critical current fluctuations can bephenomenologically explained by modeling the environment

as a collection of discrete bistable fluctuators, representingcharged impurities hopping between different locations in thesubstrate or in the tunnel barrier. Because the hopping time isexponential in the height of the barrier separating differentimpurity states, one expects a wide distribution of the relax-ation times for the fluctuators. Such wide distributions natu-rally lead to the low frequency 1/ f dependence.

The discrete nature of the environment has important con-sequences for the qubit dephasing. As was shown in Ref. 15,the effect of a bath of fluctuators on the qubit is very differ-ent from that of a continuum of linearly coupled oscillatormodes, which is the subject of the Caldeira-Leggett dissipa-tion theory.16 The discreteness of the environment becomesespecially important for the slow modes which have switch-ing rates that are longer or comparable to the operationaltime of the qubits. Qubit interaction with slow fluctuatorsmight lead to non-Markovian errors that are difficult tocorrect17 and to a fast initial qubit dephasing.18

It has been recently suggested that dynamical decouplingtechniques19 could be very promising strategies for suppress-ing the noise generated by slow fluctuators in superconduct-ing qubits.20–23 Unfortunately, because the behavior of thecharge noise power spectrum is known only in a very limitedinterval of frequencies, it is still not obvious whether it ispossible to develop efficient dynamical decoupling schemesfor these devices.

Indeed, besides the 1/ f low frequency part of the chargenoise spectrum, so far it is only known that charge noisedisplays an Ohmic behavior at very high frequencies�7 GHz� f�100 GHz�.24 The noise spectrum at intermedi-ate frequencies can only be inferred from its effects on thequbit, and so its detailed behavior is not known. This missingpart is very important for the development of dynamical de-coupling techniques because the presence of noise at the timescales corresponding to the time interval between echopulses limits the error suppression capability of these tech-niques.

Interestingly, it has been argued that dynamical decou-pling techniques might also be used to extract information onthis missing part of the noise spectrum, as a sort of diagnos-tic tools to infer spectral properties of the charge fluctuators

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coupled to the qubit.22,23 Several numerical results pointedout that the interplay between the applied pulses and thequbit dynamics might generate a rich behavior in the dissi-pative dynamics depending on the characteristics of the fluc-tuators and the working point of the qubit. This conjecture isalso supported by a few analytical results. For example, inRef. 22 the authors derived the analytical solution for theideal �i.e., with no errors in the pulses� dynamical decouplingof a qubit coupled to many fluctuators at pure dephasing,while in Ref. 23 the analytical solution for the ideal decou-pling of a qubit coupled to a fluctuator at a generic workingpoint has been sketched. However, we believe that fromthese analytical results, it still remains difficult to estimatethe effective diagnostic capabilities of the dynamical decou-pling techniques. Moreover, in a realistic setup, one shouldexpect that the dynamical decoupling is not ideal, and itshould be desirable to extend this analysis to include errorsin the pulses.

This paper aims �i� to simplify the previous analyticalsolutions of the dynamical decoupling at pure dephasing, �ii�to find the analytical solution of the dissipative qubit dynam-ics at the optimal working point, and �iii� to extend theseanalytical results to the case of nonideal control pulses. Wehope that this approach and our analytical solutions might berelevant in the design and interpretation of future experi-ments.

A remark is needed at this point. In this paper, we shallstudy only the dissipative dynamics of the qubit coupled to asingle classical fluctuator that is subject to dynamical decou-pling. There are several reasons for this choice. First, thereare interesting experimental situations where there is evi-dence of a single fluctuator dominating the qubitdynamics.10,25,26 In this case, the analytical solutions derivedin the following sections are sufficient to interpret the experi-mental data. Second, we believe that, even from this simplesolvable model, one can obtain some important insights andintuitions on the slow frequency noise and especially on itsnon-Gaussian nature. Third, at pure dephasing, it is trivial toextend our analytical results to many fluctuators interactingwith the qubit because the total response is simply the linearcombination of the contributions from each individualfluctuator.26

On the other hand, the fact that we are considering aclassical fluctuator implies that our discussion will mostlikely be relevant to superconducting qubits made out of aCooper pair box or, more generally, to those superconductingcircuits containing ultrasmall junctions �having an area�0.05 �m2� where the presence of few dominant classicalfluctuators has been clearly observed in critical currentfluctuations.27,28

However, it is relevant to mention here that experimentsalso in phase qubits5,29 and, very recently, in flux qubits30

show the presence of single charged impurities stronglycoupled to the qubit and seriously affecting its dynamics. Itwould be very interesting to see whether dynamical decou-pling sequences could help the characterization of the noiseeven in these cases. However, since it is reasonable to expectthat those charged impurities are quantum two level systemspresent in the insulating barrier of the Josephson junctions,31

the analysis discussed here should be extended in order to

take into account the quantum nature of the fluctuator. Thiswill be the subject of future work.

This paper is organized as follows. In Sec. II, we shallbriefly recall the physics of the Cooper pair box, we shallderive the total Hamiltonian of the qubit interacting with abath of fluctuators and with an applied external control field,and we shall finally explain how sequences of pulses can begenerated in this device. In Sec. III, we shall present analyti-cal results for the dynamics of the qubit subjected to an idealdynamical decoupling �i.e., with no errors in the pulses� atdifferent qubit working points. Specifically, in Sec. III A, weshall consider the qubit at pure dephasing and we shallgreatly simplify the analytical result given in Ref. 22,thereby immediately elucidating the efficacy of the dynami-cal decoupling as a diagnostic tool. In Sec. III B, we shallprovide the analytical solution for the dissipative dynamicsof the qubit subjected to dynamical decoupling at the optimalpoint. This result will allow us to comment on the evidenceof the anti-Zeno effect in this regime22,23 and to explain howthe latter can be conveniently avoided. In Sec. IV, we shalldiscuss dynamical decoupling sequences with imperfectpulses. The consequences of imperfect bang-bang pulseshave been already discussed, for the case of a single fluctua-tor, in Ref. 32. Here, by using our approach, we are able toderive bounds on the pulse noise that can be tolerated by thedevice while still achieving dynamical decoupling. Finally,conclusions are discussed in Sec. V.

II. COOPER PAIR BOX, CHARGE NOISE, ANDCONTROL

A Cooper pair box1,2,33 is a small superconducting islandconnected to a reservoir via two Josephson junctions �seeFig. 1�.

The box is characterized by two energies: theJosephson coupling energy EJ��ext�=�EJ1

2 +EJ22 +2EJ1EJ2 cos��ext /�0� and the charging

FIG. 1. �Color online� Sketch of a Cooper pair box afflicted bycharge noise present in the amorphous materials of the substrate andthe junction barriers.

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energy Ec=e2 /2C. EJi, where i=1,2, are the Josephson en-ergies and �ext and �0 are, respectively, the external mag-netic flux piercing the superconducting loop and the fluxquantum. C is the total capacitance of the box and e denotesthe electron charge.

Under the assumptions that �i� the charging energy Ec ismuch larger than the Josephson coupling EJ, �ii� the super-conducting gap � of the box is larger than its charging en-ergy �to avoid quasiparticle excitations�, and �iii� the tem-perature T is small compared to all these energy scales, it ispossible to describe the dynamics of the Cooper pair box asa two state system, whose Hamiltonian in the charge basis��0�,�1�� reads

HS =Ec

2z +

EJ��ext�2

x, �1�

where x,z are the Pauli matrices. Both the bias Ec=Ec�1−Ng� and the Josephson energy can be easily tuned by vary-ing, respectively, the dimensionless gate charge Ng=CgVg /2e and the magnetic field �ext threading the loopcontaining the island.

The Hamiltonian given in Eq. �1� describes the dynamicsof an ideal Josephson charge qubit. However, it is known thatmany charged impurities are present in the amorphous mate-rials both of the substrate �on which the Cooper pair box isgrown� and of the Josephson barriers. These charges, by in-teracting with the qubit, are responsible for its decoherence.

In order to write a Hamiltonian for a realistic �with chargenoise� Cooper pair box, one usually assumes that, irrespec-tive of the exact microscopic nature, the charge impuritiesare charges that tunnel between two positions. Each charge isdescribed as a quantum two level system �TLS� that is con-trolled by the Hamiltonian,34–36

HTLS = ��z + t�x + Henv. �2�

Here � is the energy difference between the two tunnelingminima, t is the tunneling amplitude, and �z,x are the Paulimatrices acting on the impurity states. Each TLS is subject todissipation due to its own bath with Hamiltonian Henv. In thispicture, the interaction between several charged impuritiesand the charge states of the Cooper pair box can be written as

HSB = i

vi�ziz. �3�

By adding Eqs. �1�–�3�, we finally obtain the Hamiltonian ofa nonideal �with charge noise� Josephson charge qubit.

Fortunately, experiments performed on Cooper pair boxesindicate that this rather complicated model of the chargenoise can be simplified. In fact, it has been observed thatcharge impurities that are responsible for the low frequencynoise behave as essentially classical; i.e., they becomeequivalent to classical fluctuators each characterized by aswitching rate i and the coupling to the qubit vi. In thisapproximation, the interacting Hamiltonian given in Eq. �3�simplifies to the following qubit-fluctuator Hamiltonian

HSBcl =

ivi�i�t�z, �4�

and the quantum TLS dynamics given in Eq. �2� is substi-tuted by the classical noise �i�t�, representing a symmetricrandom telegraph process �RTP� switching between values of±1 with rate i.

37 If we assume a distribution P� ��1/ at � M for the switching rates, we find for the total noise��t�=ivi�i�t� acting on the qubit a 1 / f spectrum S��=A / ��, where �� M, with coefficient A proportional to thenumber of fluctuators per noise decade nd and their typicalcoupling to the qubit v�2.38,39 Note that we assume theswitchings to be symmetric; that is, we assume the switchingrate from the upper to the lower and from the lower to theupper TLS state to be the same. This is valid if the tempera-ture is larger than the TLS level spacing, so that there is nopopulation difference between the two states. This restrictioncan be relaxed,15 but for simplicity we shall work in thesymmetric �high temperature� limit.

Therefore, in this classical picture, the realistic Hamil-tonian of a Cooper pair box afflicted by the 1/ f charge noiseis given by the sum of Eqs. �1� and �4�. In the qubit eigen-states ��0� , �1��, we find that

H =�E

2�z +

��t�2

�cos ��z − sin ��x� , �5�

where �E=�Ec2+EJ

2, �=arctan�EJ /Ec� defines the qubitworking point and �x,z are the Pauli matrices in the rotatedbasis.

The state of the qubit given in Eq. �5� can be interpreted

as the state of a pseudospin in a static magnetic field B�

= �0,0 ,�E� along z with noise both in the x and z directions.Manipulations of the Cooper pair box state can be conve-niently achieved by tuning the device at the optimal point,i.e., �=� /2, and by applying microwave pulses Ng�t�=Ng cos��rft+�� on the gate at frequency �rf, at or near thepseudospin precession frequency �E. Here � is the phase ofthe microwave with respect to a reference carrier.

In a frame rotating at �rf, the dynamics of the qubit andthe control field are described by the following effectivemagnetic field

B� �t� = �Ez +��t��x cos � + y sin �� , �6�

where ��t�=2EcNg�1N �0�� is the amplitude and N denotesthe number of excess Cooper pairs on the island.40

Different pulse sequences can then be generated by modu-lating the amplitude ��t� in Eq. �6�. Specifically, usual dy-namical decoupling sequences19 are realized first by setting

��t� = k=1

N

�rf��t − k�� − ��t − k� − �P�� ,

where N is the number of pulses, �P is the length of eachpulse, ��x� denotes the Heaviside step function, and � is thetime interval between each pulse, and then by choosing dif-ferent values for the constant A=2�rf�P and the phase �.For example, rotations by � around the x and y axes, �x and�y pulses, are achieved by fixing A=� and by setting, re-

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spectively, �=0 and �=� /2. Notice that the so-called bang-bang control, characterized by pulses that are infinitely shortand strong so that there is no qubit dynamics during the timeof the pulse, is achieved in the limit �P→0 for a fixed valueof the constant A. In this paper, we always consider only thislimiting case. In superconducting qubits, typical values forthe energy splitting are �E 5–10 GHz, while the ampli-tude of the applied microwave field �rf might extend up to250 MHz.

III. IDEAL DYNAMICAL DECOUPLING: EXACTSOLUTIONS

In this section, we analytically solve the dynamics of aCooper pair box coupled to a single fluctuator and subject toan ideal �i.e., without errors in the pulses� dynamical decou-pling both at pure dephasing and at the optimal workingpoint for the qubit.

The analytical solutions are derived by following the rea-soning given in Ref. 41. The starting point is the Hamiltoniangiven in Eq. �5� that describes a qubit interacting with asingle fluctuator. The charge noise is then given by ��t�=v��t�, where v is the coupling strength of the fluctuator and��t� denotes a symmetric RTP characterized by the switchingrate .

It may be useful to recall here that, in general, dependingon the ratio between v and , we can distinguish betweentwo different types of fluctuators: those characterized by v� �named weakly coupled or Gaussian fluctuators� andthose with v� �named strongly coupled fluctuators� thatare responsible for the non-Gaussian features of the noise.42

The effect of weak and strong fluctuators on the qubit dy-namics is dramatically different. In fact, while the weak fluc-tuators are fast and produce essentially the same effects onthe qubit as a bath of harmonic oscillators, thereby leading tothe exponential decay of the coherent qubit oscillations atlong times, the strongly coupled fluctuators, being slow, areresponsible for memory effects in the qubit dynamics, for afast initial decoherence of the qubit signal, and for the inho-mogeneous broadening of the qubit spectrum.

Due to the time dependence of the noise generated by thefluctuator, the Cooper pair box behaves as a pseudospin in avariable magnetic field. Therefore, solving its dissipative dy-namics is a nontrivial task. Fortunately, because of thebistable nature of the fluctuator, the problem can be simpli-fied. In fact, at different times, depending on the state of thefluctuator, the qubit vector state precesses on the Blochsphere around one of the two effective static magnetic fields:

B� ±= �±v sin � ,0 ,�E±v cos ��. As a result, a convenientstrategy to study the decoherence of the qubit is to follow theevolution of the state vector on the Bloch sphere. In thispicture, solving the dissipative dynamics of the qubit reducesto calculating the probability p�x , t� that at time t the statevector reaches a point x= �x ,y ,z� on the sphere.

A. Dynamical decoupling at pure dephasing

At pure dephasing, �=0, the analytical solutions for thedephasing rate �2 of a qubit interacting with a single fluc-

tuator are well known both in the case of free-inductiondecay26,41,42 and when sequences of � pulses are applied tothe qubit.22,23 Here we derive these results by using a differ-ent approach that allows us �i� to greatly simplify the previ-ous analytical solutions, �ii� to better elucidate the diagnosticcapability of the dynamical decoupling, and, finally, �iii� tosuggest an easy experiment that might be performed in orderto extract information on the microscopic nature of the fluc-tuator.

At pure dephasing, the qubit acquires an extra randomphase shift ��t�=v�0

t dt��t�� due to the interaction with thefluctuator. The dephasing rate �2 is then given by the decayrate of the average value:

ei�� =� d�p��,t�ei�. �7�

In order to evaluate it, we need to calculate the probabilitydistribution p�� , t�.

Let us first introduce the small time steps tS� t /M anddiscretize the time integral given in Eq. �7�. The randomphase shift can be written as ��t�=vtSn=1

M �n, where �n

��ntS� and the integration over time can be thought of as arandom walk process, where at each time step the randomwalker moves a step xS=vtS in a direction depending on thecurrent position of the fluctuator.

To first order in tS, the probability that the fluctuator doesnot switch, i.e., that the next step is made in the same direc-tion as the previous one, is given by �= P0�tS�=1− tS. Simi-larly, �= P1�tS�= tS is the probability that one switch occurs,i.e., that the step is made in the opposite direction of theprevious one. Higher numbers of switches are negligible tothe first order.

Let us denote by m the number of steps from the origin�i.e., the position of the random walker is x=mxS� and by nthe number of temporal steps �i.e., the dimensional time istn=ntS�. Let us calculate the discretized probability Pn�m�that the random walker reaches the position m at time step n.It is useful to introduce the probabilities Pn

+�m� and Pn−�m�

for the random walker to reach, respectively, the position mcoming from the left and that from the right. These probabili-ties satisfy the following master equations:

Pn+1+ �m� = �Pn

+�m − 1� + �Pn−�m − 1� ,

Pn+1− �m� = �Pn

+�m + 1� + �Pn−�m + 1� , �8�

and are such that Pn�m�= Pn+�m�+ Pn

−�m�.In the limit of infinitesimal steps tS, these probabilities

reduce to

Pn+�m� = p�mxS,ntS� = p+��,t� ,

Pn−�m� = p�mxS,ntS� = p−��,t� .

Accordingly, the master equations given in Eqs. �8� can bewritten as

�tp��,t� = Mdephp��,t� , �9�

where we have introduced the matrix


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Mdeph = �− − v��

− + v�� 10�

and we have defined

p��,t� = � p−��,t�p+��,t�� .

The solutions of Eq. �9� can be easily guessed asp=Cei��−�t, and we get the following dispersion equation:

− i�C = MdephC , �11�

where the matrix

Mdeph = �− − iv�

− + iv�� 12�

has the following eigenvalues,

�± = − i ± i �1 −v2

2�2 = − i ± i ��, �13�

and corresponding normalized eigenvectors,

v± =1

�1 + �− iv � ± ��2�− i

v � ± ��

1� . �14�

It follows that the general solution to Eq. �9� reads

p��,t� = �−�

� d�

2��A+v+e−i�+t + A−v−e−i�−t�ei��

= �−�

� d�

2�p�t�ei��, �15�

where the constants A± are determined by the initial condi-tions

p�0� = A+v+ + A−v−, �16�

or, by using a more compact notation, A±= p�0� ·v±.By introducing the matrices

U = �e−i�+t 0

0 e−i�−t � and V = �v+�v−� , �17�

we can conveniently write Eq. �15� as follows:

p��,t� = �−�

� d�

2��t�p�0��ei��, �18�

where ��t�=VUVT. Then, by substituting Eq. �18� into Eq.�7�, we finally find that the average phase shift reads

ei�� = �−�

�

d�� d�

2��t�p�0��ei��+1��

= �−�

�

d��t�p�0�� + 1� = ��t�p�0��=−1.

�19�

In order to evaluate the dephasing rate �2, it is convenient to

expand the vector p�0� in terms of the eigenvectors of ��t�.It is then a matter of simple algebra to find that the resultingdecay of the average phase shift given in Eq. �19� is indeedgiven by the sum of two exponentials with coefficients �2

±

=−ln��±� / t, where

�± = e− t± �t �20�

are the eigenvalues of the matrix ��t� and we define ��−1=�1−v2 / 2.

The long time decay is given by the component that de-cays slower, i.e., the one that corresponds to the larger eigen-value,

�2 = − ln�max��+,�−��/t = �1 − R�� . �21�

This result agrees with the one given in Refs. 26 and 41–43.A similar approach can be used in order to evaluate the

dephasing rate �2dd of the qubit that is subject to dynamical

decoupling. In particular, let us assume that we apply to thequbit a sequence of �x pulses. It is easy to realize that theeffect of each �x pulse is simply to interchange the constantsA+ and A−. As a result, when a sequence of N �x pulses isapplied, the probability given in Eq. �18� reduces to

p��,t� = �−�

� d�

2��dd�t�p�0��ei��, �22�

where

�dd�t� = x�� ¯ x�� 23�

and t=N� is the total time.Once this probability is known, it is straightforward to

evaluate first the average phase shift acquired by the qubit,

ei�� = �−�

�

d�� d�

2��dd�t�p�0��ei��+1��

= �−�

�

d��dd�t�p�0�� + 1� = ��dd�t�p�0��=−1,

�24�

and then the corresponding dephasing rate,

�2dd = − ln�max��+

N,�−N��/t . �25�

Here �± are the eigenvalues of x��. Depending on thenature of the fluctuator �i.e., weak or strong�, the eigenvalues

�± can be written as a function of both the fluctuator and thecontrol parameters.

Let us first consider that the fluctuator coupled to the qu-bit is weak, i.e., v� . We find that

�± =e− �

��sinh �� ±�cosh2 �� −

v2

2� . �26�

Since the eigenvalue �+ is the maximum one for each choiceof the parameters, we find that the dephasing rate is given by

�2dd = − ln��+

N�/t = − ln��+�/� . �27�

In particular, to the lowest order in v / , we find that


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�2dd =

v2

2 �1 −

1

�tanh �� . �28�

From this simple analytical result, it is evident that in orderto achieve an efficient suppression of the noise generated bya weak fluctuator, we need to choose the time interval be-tween two pulses ��1/ . Namely, the efficiency of the dy-namical decoupling is related to the switching rate of theweak fluctuator.

In Fig. 2, we plot �2dd�� given in Eq. �28�, and we com-

pare our analytical solution with the result of a numericalsimulation where the time evolution of the qubit was fol-lowed for a given realization of the fluctuator RTP and aver-ages taken over many such realizations. Numerical and ana-lytical results are in good agreement. Moreover, as oneshould expect, by reducing the time interval between thepulses, the dephasing rate �2

dd vanishes.Let us now consider the case of a strongly coupled fluc-

tuator. For v� , the constant � becomes complex ��= i��and we find that the eigenvalues now read

�± =e− �

��sin �� ±�v2

2 − cos2 �� . �29�

Different from the expressions given in Eq. �26�, here, thetwo eigenvalues �± are oscillating in the time interval � be-tween the pulses. It follows that there are now two relevantdephasing times:

�2dd± = − ln��±

N�/t . �30�

To the lowest order in /v, we find that

�2dd± = − ln��±�/� = �1 ±

1

v�sin v�� . �31�

Notice that, by looking at Eq. �31�, it is quite straightforwardto realize that for the case of a strong fluctuator, an efficientsuppression of the noise is achieved by setting ��1/v.Therefore, the efficiency of the dynamical decoupling is re-lated to the coupling strength v of the strong fluctuator.

In Fig. 3, we plot the dephasing rates �2dd±�� given in Eq.

�30� and we compare them with numerics.As evidenced by Fig. 3, the results of the numerical simu-

lations clearly show that, depending on the time interval �between the pulses, the dephasing rate �2

dd of the qubit isdominated differently by the two dephasing rates �2

dd±.In order to understand this behavior, it is useful to write

the average phase shift given in Eq. �24� as follows:

ei�� = w+�+N + w−�−

N, �32�

where we have introduced the weights w±. These can becalculated as follows. First, we write the initial state p�0�,given in Eq. �16�, as a linear superposition of the eigenvec-tors u± of the matrix x��:

p�0� = i=±

Biui.

Then, we solve the following linear system of equations inorder to find the coefficients Bi:

p�0� · u j = i

Biui · u j . �33�

For example, in the numerical simulations shown in Fig. 3,the fluctuator was initially assumed at equilibrium, i.e.,p�0�= �1,1� /2. By following the above procedure, we findthat the weights read

w± =1

2±

� cosh ��

2�cosh2 �� −v2

2

. �34�

In particular, to the lowest order in /v, we obtain thatw±��= 1

2 �1±cos v��, and we can immediately explain the os-cillating behavior displayed in Fig. 3.

An important consideration is relevant at this point. Bycomparing Figs. 2 and 3, it is immediate to realize that thetwo plots are qualitatively different. One can then exploit thisdifference in order to distinguish between a weak and astrong fluctuator coupled to the qubit. For example, in anexperiment, once a single fluctuator has been singled out in adevice, one could measure systematically the dephasing time

FIG. 2. Dephasing rate of a qubit interacting with a weak fluc-tuator as a function of the time interval � of the bang-bang pulses.The analytical solution given in Eq. �27� is represented by the solidline, while the black dots indicate the numerical results. The valuesfor the fluctuator parameters are v=0.1 and =1 �in units of �E�.The dephasing rate �2=v2 /2 is given in Eq. �28�.

FIG. 3. Dephasing rates of a qubit interacting with a strongfluctuator as a function of the time interval � of the bang-bangpulses. Solid lines represent the analytical solutions given in Eqs.�30�, while black dots are the numerical results. The values for thefluctuator parameters are v=0.1 and =0.01 �in units of �E�. FromEq. �31�, we have �2= .


054515-6

of the qubit after cycles of pulses characterized by differenttimes � and one could observe if there is an oscillatory be-havior or not in the data. The efficiency of the dynamicaldecoupling will then provide further information on the cou-pling strength of the fluctuator v or on its switching rate .

However, in order to completely characterize the fluctua-tor, both parameters v and must be known.

B. Dynamical decoupling at the optimal point

In this section we study the dissipative dynamics of aCooper pair box tuned at the optimal point, i.e., �=� /2, thatis subject to control sequences of � pulses. By using a simi-lar approach to the one discussed in the previous section, wefind the analytical solution for the qubit decoherence rate.

Contrary to the case of pure dephasing, at the optimalpoint, the performance of the dynamical decoupling dependsboth on the characteristics of the charged fluctuators coupledto the qubit and on the qubit energy level splitting. In par-ticular, quite worryingly, it has been noticed that the qubitdecoherence can be even accelerated if the cycle time of thecontrol is long with respect to the period of the free evolutionof the qubit.22,23 This acceleration is sometimes referred to inthe literature as the anti-Zeno effect.44

In the following, we shall indeed see that this accelerationappears only when sequences of �x pulses are used to flip thequbit state and can be conveniently avoided by flipping thestate with �y pulses.

To this aim, let us introduce the time evolution operator

U�t�=Te−i�0t H�t��dt�, where H�t� is the time dependent Hamil-

tonian given in Eq. �5�, and let us study the time evolution ofthe qubit when a cycle of �x and �y pulses is applied to it.Let � be the time interval between the pulses and let usconsider first the case of pure dephasing.

Since the time evolution operator U�t� depends only onthe qubit operator z, at pure dephasing, we see that thenoise coming from fluctuators that are static during the timeinterval � can be completely eliminated by applying either acycle of �x or �y pulses. In fact, mathematically, we obtainthat

xU��xU�� = yU��yU�� = 1 , �35�

where 1 is the identity.On the contrary, at the optimal point, the evolution opera-

tor U�t� depends on both the qubit operators z �because ofthe interaction with the fluctuators� and x �because of theJosephson coupling energy EJ�. Mathematically, we find that

xU��xU�� 1 and yU��yU�� = 1 . �36�

Namely, only by flipping the qubit state around the y axis canwe completely suppress the noise due to fluctuators that arestatic in the time interval �. Therefore, at the optimal point,the performance of the dynamical decoupling changes re-markably depending on whether sequences of �x or �ypulses are used to decouple the qubit.

These considerations are clearly illustrated in Figs. 4 and5. In Fig. 4, we numerically study the dissipative dynamicsof a Cooper pair box that is coupled to a weak fluctuator. Thequbit dephasing rate �2

dd is displayed as a function of the

time interval � between cycles of �y pulses �upper panel� and�x pulses �lower panel�.

Notice that only when �y pulses are applied to the qubit isthe dephasing rate �2

dd monotonously reduced as the flippingrate is increased. On the contrary, for sequences of �x pulses,there are values of � where the dephasing rate �2

dd increasesremarkably as compared to the rate when there are no controlflips �i.e., anti-Zeno effect22,23�.

A similar analysis is performed in Fig. 5 where the dissi-pative dynamics of the qubit coupled to a strong fluctuator isillustrated. Again, we find that the dephasing rate �2

dd dis-plays both the anti-Zeno effect and the rapid oscillationswhen sequences of �x pulses are applied to the qubit �lowerpanel�. However, both the acceleration and the oscillationsdisappear when control sequences of �y pulses are insteadused �upper panel�.

Note that when control sequences of �x pulses are used inboth Figs. 4 and 5, the efficiency of the dynamical decou-pling is related to the qubit energy scale EJ. In this case, infact, we need to choose time interval ��1/EJ in order toobtain an efficient suppression of the noise due to the fluc-tuator. For sequences of �y pulses, the efficiency of the dy-namical decoupling is determined either by the switchingrate ��1/ �if �EJ� or by energy scale ��1/EJ �if �EJ� for the weak fluctuator and by ��1/ for the strongfluctuator. This will be discussed after we have derived theanalytical expressions below.

FIG. 4. Dissipative dynamics of the dynamical decoupling ap-plied to a Cooper pair box coupled to a weak fluctuator at theoptimal point �EJ=10, v=0.1, =1�. From Eq. �55�, we have �2

= v2 /EJ2. The lines are added to guide the eye and do not represent

any analytical result.


054515-7

As a final remark, let us notice that the different perfor-mances of sequences of �x and �y pulses can be easily ex-plained in the context of the effective Hamiltonian theory45

by stating that while �x pulses provide decoupling only tothe first order in the Magnus expansion of the dephasing rate,�y pulses give exact decoupling to all orders.

Having clarified this point, let us now proceed to derivethe analytical solution for the dissipative dynamics of thequbit at the optimal point. In view of the above consider-ations, in the following, we shall only consider control se-quences of �y pulses.

The analytical solution is derived by following the samereasoning discussed in the previous section. Here, the maindifference is that the two effective static magnetic fieldsaround which the state of the Cooper pair box precesses nowread B±= �±v ,0 ,EJ�. As a consequence, at the optimal point,the qubit does not only acquire a simple random phase due tothe interaction with the charged fluctuator, but its state willalso spread over the entire Bloch sphere.

In order to calculate the probability p�x , t� that the qubitvector state reaches the point x= �x ,y ,z� on the Bloch sphereat a certain time t, first, we split the probability p�x , t�= p+�x , t�+ p−�x , t�, and then we derive the master equationsfor p+�x , t� and p−�x , t�:

p+�x,t + tS� = �p+�U+−1x,t� + �p−�U+

−1x,t� ,

p−�x,t + tS� = �p+�U−−1x,t� + �p−�U−

−1x,t� , �37�

where �=1− tS is the probability for the fluctuator to re-main in the same state at the time tS, while �= tS denotesthe probability that it switches. Notice that, in Eqs. �37�, wehave introduced the operator U±=etSB±·R, where R= �Rx ,Ry ,Rz� and the rotation matrices are defined as follows:

Rx = �0 0 0

0 0 − 1

0 1 0�, Ry = � 0 0 1

0 0 0

− 1 0 0� ,

Rz = �0 − 1 0

1 0 0

0 0 0� .

By performing the continuous limit tS→0, from Eqs. �37�,we directly obtain the equations governing the time evolu-tion of the Cooper pair box before the first �y pulse is ap-plied:

�tp+ = p− − �B+ · Rx� · �xp+ − p+,

�tp− = p+ − �B− · Rx� · �xp− − p−. �38�

In this picture, to apply a �y pulse to the qubit simply meansto reverse the effective magnetic field. As a result, given Eqs.�38�, it is immediate to write the equations describing thetime evolution of the state of the Cooper pair box after theapplication of the first �y pulse:

�tp+ = p− + �B+ · Rx� · �xp+ − p+,

�tp− = p+ + �B− · Rx� · �xp− − p−. �39�

Clearly, once the second �y pulse is applied, the dissipativedynamics of the box is again described by the set of equa-tions given in Eqs. �38� and so on for all the control cycles.Still, we need to understand how to match the solutions justbefore and after the application of each �y pulse.

Let p±�x ,�→0−� be the solutions of Eqs. �38� just beforethe first pulse, while p±�x ,�→0+� are the solutions just afterthe pulse. Notice that the latter are indeed initial conditionsfor Eqs. �39�. It is then easy to realize that a � rotationaround the y axis leads to the following matching conditions:

p+�x,�→ 0+� = p+�x,�→ 0−� ,

p−�x,�→ 0+� = p−�x,�→ 0−� . �40�

Therefore, we can explicitly write Eqs. �39� as follows:

�tp+ = �EJ�y�x − x�y� + v�z�y − y�z��p+ − p ,

�tp− = �EJ�y�x − x�y� − v�z�y − y�z��p− + p , �41�

where p= p+− p−.

FIG. 5. Dissipative dynamics of the dynamical decoupling ap-plied to a Cooper pair box coupled to a strong fluctuator at theoptimal point �EJ=1, v=0.1, =0.01�. From Eq. �57�, we have �2

= v2 /EJ2.


054515-8

In order to solve them, it is useful to define the followingexpectation values:

k± =� d3xp±�x,t�k, k = x,y,z . �42�

We can then write Eqs. �41� in terms of Eq. �42�, and weobtain the following set of equations:

�tx+ = − �x+ − x−� − EJy+,

�ty+ = − �y+ − y−� + EJx+ − vz+,

�tz+ = − �z+ − z−� + vy+,

�tx− = �x+ − x−� − EJy−,

�ty− = �y+ − y−� + EJx− + vz−,

�tz− = �z+ − z−� − vy−, �43�

having constant coefficients. Moreover, by introducing thevariables X±�x+±x−, Y±�y+±y−, and Z±�z+±z−, Eqs. �43�can be further reduced into the following two independentsets of equations:

�tX+ = − EJY+,

�tY+ = EJX+ − vZ−,

�tZ− = − 2 Z− + vY+ �44�

and

�tX− = − 2 X− − EJY−,

�tY− = − 2 Y− + EJX− − vZ+,

�tZ+ = vY−. �45�

Finally, by solving the system of Eqs. �44�, we can estimatethe dephasing rate �2. To this aim, it is sufficient to preparethe system in an equal superposition of eigenstates, i.e., withZ=0, and then to study the decay of the x and y componentsof the Bloch vector. Namely, we need to derive the decayingbehavior of the variables X+ and Y+.

It is convenient to write the system of Eqs. �44� in matrixform:

�t�X+

Y+

Z−� = M1�X+

Y+

Z−� , �46�

where the matrix M1 reads

M1 = � 0 − EJ 0

EJ 0 − v

0 v − 2 � . �47�

After a �y pulse is applied to the qubit, the matrix M1 isreplaced by the following matrix:

M2 = � 0 EJ 0

− EJ 0 v

0 − v − 2 � . �48�

By resorting to Eqs. �40�, it is straightforward to see that theevolution of the qubit state after one full control cycle can bedescribed by the operator

T = yeM2� ye

M1�, �49�

where we introduced the matrix y =diag�1,1 ,1�.Equation �49� can be further simplified to

T = � yLeM1��2, �50�

where we defined the matrix L=diag�1,−1,1�. Finally, wefind that after an echo cycle,

X+ = i=1

3

wie−�2

dd�i�t, i = 1,2,3, �51�

where

�2dd�i� = − ln��i�/� , �52�

with �i as the eigenvalues of the matrix T. It follows that thedissipative dynamics of the qubit tuned at the optimal pointis controlled by three different dephasing rates.

Let us now discuss the decay rates in the limiting cases ofweak �v! � and strong �v" � fluctuators. In both cases, itcan be proven that, in Eq. �51�, the weight w3=0. As a result,only the two dephasing rates �2

dd�1� and �2dd�2� are relevant for

the qubit dissipative dynamics.Let us first evaluate them for the case of a weak fluctuator.

After some calculations, we find that

�2dd�1� =

v2

EJ2 + 4 2�1 −

EJ2 − 4 2

EJ2 + 4 2

sin�EJ��EJ�

−8 2

EJ2 + 4 2 cos2�EJ�

2� tanh� ��

�� , �53�

�2dd�2� =

v2

EJ2 + 4 2�1 +

EJ2 − 4 2

EJ2 + 4 2

sin�EJ��EJ�

−8 2

EJ2 + 4 2 sin2�EJ�

2� coth� ��

�� . �54�

From both Eqs. �53� and �54�, it is evident that, contrary tothe case of pure dephasing, the energy level splitting of thequbit EJ now plays a major role in the dissipative dynamicsof the qubit.

In order to understand the efficiency of the dynamicaldecoupling, it is useful to discuss the dephasing rates in twolimiting cases.


054515-9

(i) Weak fluctuator with �EJ. In this case, Eqs. �53� and�54� reduce to

�2dd�1� �

v2

EJ2 �1 −

sin�EJ��EJ�

� ,

�2dd�2� �

v2

EJ2 �1 +

sin�EJ��EJ�

−8 2

EJ2 sin2�EJ�

2� coth� ��

�� .

�55�

From Eq. �55�, it is evident that in order to achieve an effi-cient suppression of the noise induced by the fluctuator, weneed to apply sequences of �y pulses with a time intervalbetween the pulses � such that �� /EJ. Once this constraintis satisfied, we find that �2

dd�2��−� v2 /EJ2��2/3�� 2. Since

this represents a negligible term, the decay of the Cooperpair box is effectively dominated by the decay rate �2

dd�1�.As an illustrative example, in Fig. 6, we consider the dis-

sipative dynamics of a qubit that is coupled to a fluctuator inthis regime, and we compare the exact solutions given inEqs. �53� and �54� to the numerics. Notice that for a smalltime interval �, the dissipative dynamics is governed by thelarger decay rate, presumably because a larger weight isgiven to this solution.

(ii) Weak fluctuator with "EJ. In this limit, Eqs. �53�and �54� become

�2dd�1� �

v2

4 �1 +

sin�EJ��EJ�

− 2 cos2�EJ�

2� tanh� ��

�� ,

�2dd�2� �

v2

4 �1 −

sin�EJ��EJ�

� . �56�

The efficiency of the dynamical decoupling can be easilyinferred from the decaying rate �2

dd�2� given in Eq. �56�, and,as in the previous case, the condition to satisfy is �� /EJ.By expanding in the limit EJ�!1, we obtain �2

dd�1�

��v2 /4 ��2− �EJ��2 /6��tanh� �� / �−1�. It follows that inorder to keep the latter negligible, one also has to satisfy thecondition �!1/ . As a result, in this limit, one shouldchoose as condition for the efficient decoupling the strongestone, i.e., �!1/ . This behavior is illustrated in Fig 7.

Let us now consider the case of a strong fluctuatorcoupled to the qubit. The dephasing rates now read

�2dd�1� =

v2

�2 �1 −sin��

� ,

�2dd�2� =

EJ2

�2 � 2�2 + v2�2 cos�� + 1 − 3sin��

��2 + EJ

2 + v2 cos��−�4v2�cot� ��

2� −

2

��2

EJ2 + �2 cot2� ��

2� +

2EJ2 − v2�1 − 3

sin��

��2 + EJ

2 + v2 cos�� , �57�

where �=�EJ2+v2. Since we are only considering noise that

is weak compared to the level splitting of the qubit, v!EJ,and since the fluctuator is strong, i.e., �v, only the case

!EJ is relevant. In order to have an efficient dynamicaldecoupling of �2

dd�1�, it is immediate to realize that we needto set �� /EJ. Once this constraint is satisfied, we find that

FIG. 6. Weak fluctuator �EJ=1, v=0.1, =1�. The lines showEqs. �53� and �54�; points are numerical simulations. From Eq. �53�we have �2= v2 / �EJ

2+4 2�.FIG. 7. Weak fluctuator �EJ=1, v=0.1, =100�. The lines illus-

trate Eqs. �53� and �54� and the black points represent the numericalresults. The inset shows the analytical and numerical solutions forsmall values of �. From Eq. �56�, we have �2=v2 /4 .


054515-10

�2dd�2��0. Therefore, the dissipative dynamics is finally

dominated by �2dd�1�. An example is shown in Fig. 8.

IV. IMPERFECT PULSES

So far, we have assumed that the control pulses are ideal.Namely, they are zero-width � pulses applied exactly alongthe x or y axis of the Bloch sphere. However, in real appli-cations, it is plausible that the duration of the pulses exhibitsrandom fluctuations, leading to error in the rotation angle,and that the effective rotation axis might deviate from the xor y axis.

Clearly, if the pulses are imperfect and the repetition ratebetween the control cycles is high, there is a possibility thatthe dynamical decoupling introduces more dephasing thanwhat it is effectively capable to remove. As a result, giventhe noise characteristics of both the environment and thecontrol pulses, there must exist an optimal pulse rate that setsthe best decoupling performance. This section aims to deter-mine this optimal pulse rate.

Control sequences with imperfect pulses have alreadybeen considered in two recent works.32,46 Our analysis differsfrom the previous studies because �i� it takes into account thequbit dynamics between the control pulses and �ii� by usingthe analytical solutions derived in the previous sections, weare able to provide bounds on the amount of noise that canbe tolerated in the pulses while still achieving efficient de-coupling.

In order to calculate the optimal pulse rate, it is conve-nient to proceed by steps. First, we study the dissipativedynamics for an ideal qubit �without charge noise� that isdynamically decoupled with control sequences of noisypulses, and then we extend this analysis to a qubit afflictedby a charge fluctuator.

A. Dephasing due only to errors in the pulses

In this section, we study the dissipative dynamics of anideal qubit that is subject to dynamical decoupling with noisein the � pulses. Specifically, we assume that the effectiverotation realized by each pulse fluctuates around the ex-

pected values, i.e., �=�±, where is a random variable.In the absence of a fluctuator, the time evolution of the

vector state on the Bloch sphere from the end of one pulse tothe end of the next pulse is then given by

Ui = e��+i�RxeEc�Rz at pure dephasing,

Ui = e��+i�RyeEJ�Rx at optimal point.

After N pulses, we find that the total time evolution readsU=�i=1

N Ui. Let us now assume that the fluctuations i �i=1, . . . ,N� are uncorrelated and that they are Gaussian dis-tributed. The average time evolution can then be easily cal-culated as U�=�iUi�= Ui�N, and the decay rate of the qubitcan be inferred by evaluating the eigenvalues of Ui�. Bothfor the case of pure dephasing and at the optimal point, wefind that, in the limit 2�!1, the eigenvalues are

�0 = − pc,

�± =1

2cos��1 − pc� ±�1

4cos2��1 − pc�2 + pc,

where pc= cos � and �=Ec at pure dephasing and �=EJ atthe qubit optimal point.

It follows that, to the lowest order in 2�, the dissipativedynamics of the qubit is controlled by three possible decay-ing rates �2

dd�i�=−ln��i� /� for i=0,±. Precisely we find that

�2dd�0� =

2�2�

, �58�

�2dd�±� =

2�4�

�1 ± cos �� . �59�

At long times, we expect that the qubit decay is controlled bythe largest dephasing rate given in Eq. �58�.

At small times, the controlled dissipative dynamics ismore complicated. In order to understand this, we considerthe case of a qubit that is tuned at pure dephasing, �E=EJ=1, and is subject to sequences of �x pulses with fluctuatingnoise, −0.12��0.12, uniformly distributed. We solve nu-merically the dissipative dynamics for two different initialpreparations of the qubit: the qubit initial state is preparedalong either the x axis or the y axis of the Bloch sphere. Theresults of our simulations are illustrated in Fig. 9.

Note that, depending on the initial conditions, the dissipa-tive dynamics is dominated by different dephasing rates. Fora qubit initially prepared along the y axis �x axis� of theBloch sphere, the dynamics is dominated by the dephasingrate �2

dd+ ��2dd−�. In order to understand this, we need to

evaluate the weights of the different components.Let us indicate with s0 the initial qubit vector state. It is

convenient to write it as a linear combination of the eigen-vectors vi��vi

0 ,vi+ ,vi

−� of the average operator Ui� describ-ing the evolution of the qubit vector state on the Blochsphere during a control cycle, i.e., s0=�A�vi

�, with �=0,+ ,−. After N control cycles, the qubit vector state reads s= Ui�Ns0=�A��

Nvi� and the coefficient Ai can be easily

found.

FIG. 8. Strong fluctuator �EJ=1, v=0.1, =0.01�. The solidlines show Eqs. �53� and �54� and the black points illustrate numer-ics. From Eq. �57�, we have �2= v2 /EJ

2.


054515-11

In the considered example, when the qubit is initially pre-pared along the y axis of the Bloch sphere, i.e., s0=ey, wefind that the coefficients read A0=0 and A±= �1±cos �� /2.Therefore, we immediately understand why, in Fig. 9, thedephasing rate �2

dd�0� never dominates the dissipative dynam-ics. Moreover, it is clear that, according to the oscillatorybehavior of the coefficient A± displayed in the inset, the dis-sipative qubit dynamics at different time intervals � betweenthe control pulses is alternatively dominated by the twodephasing rates �2

dd±.Note that, in a realistic situation where dynamical decou-

pling sequences are introduced in the protocol during thecomputation, we might not know exactly which is the effec-tive state of the qubit. As a result, from the point of view ofthe efficiency of the control when fluctuating noise in thepulses is considered, it is reasonable to assume that the dis-sipation is always given by the largest decay rate �2

dd�0� at allinterval times � between the pulses.

B. Dephasing due to error in the pulses and fluctuator noise

Let us now consider that during the dynamical decou-pling, there are both noise coming from the pulses and acharge fluctuator that is coupled to the Cooper pair box.Since the two sources of noise are independent, we expectthat the relaxation rates will just add. We could not prove thisconjecture analytically, but results of numerical simulationsclearly demonstrate that this is always the case.

As an example, in Fig. 10, we illustrate the dissipativedynamics of a qubit that is coupled to a weak fluctuator atpure dephasing. In the figure, we plot the analytical solutionfor the dissipative dynamics when there is noise only in thepulses given in Eq. �58� �green�, the analytical solution forthe dephasing rate due to the interaction of a weak fluctuatorderived in Eq. �28� �blue�, and the total dephasing rate re-sulting from the sum of the two �black�. Notice that there isa good agreement between numerical and analytical results.

By looking at Fig. 10, we see that by decreasing the timeinterval � between the pulses, the dephasing rate initiallydecreases. Then at a particular value of �, it reaches a pro-nounced minimum where the best dynamical decoupling per-formance is achieved. Finally, by further decreasing �, theperformance of the decoupling deteriorates. Since the effi-ciency of the decoupling �i.e., the value of the minimum� isrelated to the level of noise in the pulses, it might be inter-esting to estimate how much noise in the pulses can be tol-erated in order to have an efficient suppression of the noisedue to the fluctuator.

To this aim, one has to find the time interval � betweenthe pulses when dynamical decoupling starts to reduce thedecoherence rate. If the decoherence rate because of thenoise on the flips ��2� /�� at this pulse rate is small com-pared to the one from fluctuators �which will be of the orderof the rate without dynamical decoupling�, there will be arange of � below this point where dynamical decoupling isworking, and it will not be destroyed by the pulse noise.

In Sec. III, we have demonstrated that, depending on thenature of the fluctuator, an efficient dynamical decoupling isachieved for ��1/ if the fluctuator is weakly coupled tothe qubit and for ��1/v if the fluctuator is non-Gaussian. Asa consequence, depending on the type of fluctuator interact-ing with the qubit, we find that the control sequences areefficient when the noise in the pulses satisfies the followingconditions:

2�� v �2

for weak fluctuator,

2��

vfor strong fluctuator.

Notice that the noise tolerated in the pulses is bounded by thesecond power of the ratio v / �1 for a weak fluctuator andonly by the first power of /v�1 for a non-Gaussian fluc-tuator. Therefore, in a qubit afflicted by a strong fluctuator,

FIG. 9. �Color online� The three decay rates �2dd�0� �black, mo-

notonously decreasing�, �2dd�−� �blue, oscillating, starting at 0�, and

�2dd�+� �green, oscillating, diverging at 0� � uniformly distributed

between −0.12 and 0.12 and �=1�. The black points �upper set� andblue points �lower set� are the result of numerical simulation, withthe initial state on the y axis �x axis� of the Bloch sphere and pulsesaround the x axis. The inset shows the weights A+ �black, starting at1� and A− �blue, starting at 0� for the initial state on the y axis. If theinitial state is on the x axis, the weights interchange.

FIG. 10. �Color online� �2 with uniformly distributed between−0.04 and 0.04 and �=1, EJ=0. One weak fluctuator, v=0.1, and =1. Increasing curve �green� — noise only in the pulses; decreas-ing curve �blue� — noise only from the fluctuator; nonmonotonouscurve �black� — the sum of the two noise contributions. Points arenumerical simulation. From Eq. �28�, we have �2=v2 /2 .


054515-12

an efficient dynamical decoupling can be achieved withhigher noise in the pulses.

A similar analysis can be done for a qubit that is tuned atthe optimal point. Using the analytical solutions given inSec. III B, we find that an efficient dynamical decoupling canbe achieved for a weak fluctuator, i.e., v� , when

2�� v2

EJ3 if ! EJ,

2��v2

2 if " EJ.

and for a strong fluctuator, i.e., v� , when

2�� v2

EJ3 .

V. CONCLUSIONS

In this paper, we studied the dissipative dynamics of aCooper pair box �qubit� afflicted by charge noise that is dy-namically decoupled. By focusing on the case where a singlefluctuator is coupled to the qubit, we investigated the effi-ciency of ideal �without errors in the pulses� decoupling se-quences. At qubit pure dephasing, irrespective of whetherone applies sequences of �x or �y pulses, we found that theefficiency of the decoupling is strongly dependent on thecharacteristics of the fluctuator. For the case of a weak�Gaussian� fluctuator, an efficient suppression of the noise isachieved by choosing the time interval between the pulsesless than the switching rate of the fluctuator, while for astrong �non-Gaussian� fluctuator the relevant decoupling rateis determined by the coupling strength of the fluctuator to thequbit. As a result, in this limit, dynamical decoupling tech-niques can be successfully implemented as a diagnostic tool

to infer spectral information on the nature of the fluctuatorcoupled to the qubit.

A different situation is encountered at the qubit optimalpoint, where we showed that the decoupling performancevaries considerably depending on whether control sequencesof �x or �y pulses are used to decouple the qubit. For bothcases of weak and strong fluctuators, we found that the dis-sipative dynamics is dominated by the qubit energy scale.For the special case of weak fluctuator characterized by aswitching rate greater than the qubit energy level splitting,we observed that the efficiency of the dynamical decouplingis achieved by choosing the time interval between the pulsesless than the switching rate of the fluctuator.

We have extended this analysis in order to also includeerrors in the pulses. We showed that, depending on the char-acteristic of the fluctuator and the error in the pulses, theremust exist an optimal pulse rate that sets the best decouplingperformance. Using the analytical solutions for the dissipa-tive qubit dynamics derived both at pure dephasing and atthe optimal point, we found upper bounds on the level ofnoise present in the pulses that can be tolerated while stillachieving an efficient dynamical decoupling.

We believe that an interesting future research direction isto extend this analysis in order to take into account the quan-tum nature of the fluctuator. It would be extremely interest-ing to see whether dynamical decoupling techniques couldbe useful in order to understand the microscopic mechanismsof relaxations at low temperature of the quantum two levelsystems present in the amorphous materials of the barrier andthe substrate.

ACKNOWLEDGMENTS

This work was supported by the National Security Agency�NSA� under the Army Research Office �ARO� �ContractNo. W911NF-06-1-0208� and the Norwegian ResearchCouncil via a StorForsk program.

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exact solution for the dynamical decoupling of a qubit...

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