Revista Mexicana de Física Sociedad Mexicana de Físicarmf@smf2.fciencias.unam.mx ISSN (Versión impresa): 0035-001XMÉXICO

2007 S. Cruz y Cruz / B. Mielnik

QUANTUM CONTROL WITH PERIODIC SEQUENCES OF NON RESONANT PULSES

Revista Mexicana de Física, agosto, año/vol. 53, número especial 4 Sociedad Mexicana de Física

Distrito Federal, México pp. 37-41

Red de Revistas Científicas de América Latina y el Caribe, España y Portugal

Universidad Autónoma del Estado de México

http://redalyc.uaemex.mx

REVISTA MEXICANA DE FISICA S53 (4) 37–41 AGOSTO 2007

Quantum control with periodic sequences of non resonant pulses

S. Cruz y Cruza,b and B. MielnikaaDepartamento de Fısica, Cinvestav,

Apartado Postal 14-740 07000 Mexico D.F., Mexico,bCiencias Basicas, UPIITA-IPN,

Av. IPN 2580, 07340 Mexico D.F., Mexico,e-mail: sara@fis.cinvestav.mx, bogdan@fis.cinvestav.mx

Recibido el 1 de mayo de 2006; aceptado el 1 de noviembre de 2006

The resonant quantum control techniques are vulnerable to cumulative errors specially if the manipulation operations involve many individualsteps. It is shown that non trivial operations can be induced by non resonant fields challenging too simplified control models.

Keywords:Quantum control; time-dependent Hamiltonians; periodical systems.

El control cuantico resonante es sensible a la acumulacion de errores, especialmente si el proceso involucra muchas operaciones individuales.Se muestra que, a traves de campos externos no resonantes, se pueden inducir mecanismos no triviales de control que se contraponen a losmodelos simplificados convencionales.

Descriptores:Control cuantico; Hamiltonianos dependientes del tiempo; sistemas periodicos.

PACS: 03.65.Ta; 32.80.Qk

1. Introduction

The dynamics of quantum systems, represented by time de-pendent HamiltoniansH(t), is encoded in the family of uni-tary operatorsU(t, t0) describing the evolution of the systemin the time interval[t0, t] (~ = 1):

d

dtU(t, t0) = −iH(t)U(t, t0), U(t0, t0) = 1. (1)

The essential problem in the control theory is the gener-ation of arbitrary unitary operations causing a desired evolu-tion of the quantum system. The main mathematical dilemmais to determine the proper HamiltonianH(t) which generatesa givenU(t, t0) according to (1).

These evolution operators must have exponential repre-sentations in terms of Hermitian operatorsHeff(t, t0), calledtheeffective Hamiltoniansof the system in[t0, t], defined bythe Baker-Campbell-Hausdorff (BCH) formulae [1–3]:

U(t, t0) = e−i(t−t0)Heff (t,t0). (2)

The construction of the effective Hamiltonians (2) for a givenquantum system is, in general, a quite complicated task,even in the case of finite dimensional systems. The theo-retical group techniques may, in some cases, provide the de-siredU(t, t0) as a product of a long sequence of elementarysteps [4–6]; however, such formal solutions are not alwaysthe most convenient from the practical point of view.

One of the typical tools of quantum control is to inducethe dynamical operations by means of soft, patiently repeatedpulses. Suppose,e.g., an initially stationary system (repre-sented byH0) is being manipulated by an external field. Ifthe strengthε of this field is small enough, and if the proba-bility of causing radiative transitions is negligible, the properdescription is then semiclassical,i.e., the stationary system

H0 has quantum mechanical properties while the manipula-tion partV (t) typically depends on certain classical parame-ters or fields:

H(t) = H0 + εV (t). (3)

One of the most interesting exact solutions of (1-3) general-izing therotating wave approximationarises if

V (t) = e−iΩtV eiΩt, (4)

with V a time independent operator and[Ω,H0] = 0. Thecorresponding evolutionU(t) = U(t, 0) becomes :

U(t) = e−iΩte−i[H0−Ω+εV ]t (5)

(we put t0 = 0), where the first factor is the unitary trans-formation to the “rotating frame” and the second one can beinterpreted as the evolution operator in theΩ-frame. By tak-ing Ω = H0 one obtains the simplest exact description of theresonant process (the generalized Rabi rotations).

2. The Resonant control

A frequently used mechanism in control theory is to generatethe evolution operations by taking full advantage of the sen-sitivity of the system to pulses of some particular frequencies(the resonant control technique). In this case, the externalfield is generally of the form:

V (t) =12

[eiωtV + e−iωtV †] . (6)

Such harmonic, monochromatic field isresonantto H0, inthe conventional sense, ifω = ω0, with ω0 = El − Em

the difference of any pair ofH0 eigenvalues. Notice that

38 S. CRUZ Y CRUZ AND B. MIELNIK

U0(T0) = e−iH0T0 , T0 = 2π/ω0, will be, in this case, pro-portional to1 in the spectral subspacesHl, Hm correspond-ing to this pair of energy levels. In turn, the evolution opera-torU(T0) will then be proportional toe−iεV T0 : a small “rota-tion” in Hl⊕Hm which becomes cumulative if the operationis applied over many oscillation periods of the external field.A perturbative description is therefore not proper in this case,even ifε is small. To the contrary, this cumulative effect donot occur in the non resonant caseω 6= ω0: the external fieldwill only produce a slightprecessionof the system around itsbasic evolutionU0(t) = e−iH0t [7,8].

2.1. The Rabi rotations

In order to provide a clear picture of this situation, considerthe simplest case of a two-level system (aqubit) with corre-sponding states and energies|1〉, |2〉 andE1, E2 respectively,perturbed by the monochromatic, harmonic field (6) in thedipole approximation:

H(t) = E1|1〉〈1|+ E2|2〉〈2|

+ε

2[eiωt|1〉〈2|+ e−iωt|2〉〈1|] . (7)

The state of the system, at an arbitrary timet, is

|φ(t)〉 = a1(t)|1〉+ a2(t)|2〉, (8)

wherea1(t), a2(t) are the respective probability amplitudesof finding the system in the states|1〉, |2〉 (under the initialconditiona1(0) = 1, a2(0) = 0) at that moment:

|a2(t)|2 =ε2

ω21

sin2 ω1

2t, |a1(t)|2 = 1− |a2(t)|2, (9)

with ω1 =√

ε2 + (ω0 − ω)2 [9, 10]. In the resonant case(ω = ω0), (8) reduces to

|φ(t)〉 = cosε

2t|1〉+ sin

ε

2t|2〉, (10)

which represents a rotation of the state of the system between|1〉, |2〉, with an angular frequencyε/2. If the field is weakenough, the process is slow and free of radiative jumps. Thisphenomenon is known as “Rabi rotation” [9]. When the pro-cess is continued for a complete period of time, the evolu-tion operatorU(T0) coincides withe−iεV T0 , indicating thatthe rotations can be accumulated by applying the operation anumber of times. The system can then be softly manipulated:once in its ground state|1〉, it slowly collects informationfrom the exterior and makes a transition to the excited state|2〉; next, it returns to|1〉 and so on.

To the contrary, if the frequency of the driving field isnon resonant, the basic evolution operatorU0(t) produces aviolent draft of the system along its natural evolution orbitand the precession effects are hardly visible. Even so, thetransitions|1〉 ↔ |2〉 are not completely suppressed in thisfrequency regime. In fact, the maximum probability (which

is reached within a timet = π/ω1) of such a transition is,in this case,ε2/ω2

1 and it is non zero forω 6= ω0. Figure 1shows this maximum probability as a function ofω for dif-ferent values ofε. There is a marked peak in the resonancewhich is sharper as the strength of the field tends to zero.The mean width of the curve is2ε, telling the transitions oc-cur just in the resonance regime if the intensity of the field issmall enough.

2.2. A geometric picture

An elementary illustration of the phenomenon previ-ously described [11–13] corresponds to the Hopfπ-mapS3 → S2 [14–16]. The states|φ(t)〉 of the system can beassociated with the points of a unitary two-sphereS2 (thePoincare sphere) with its center at the origin of a frame de-fined by the vectorse1, e2, e3, provided that orthogonalstates correspond to antipodal points [11]. The Hamilto-nian (7), expressed in terms of spin Pauli matrices (take forsimplicity E1 = −ω0/2, E2 = ω0/2), has the form:

H(t) =ω0

2σ3 +

ε

2e−i ω

2 tσ3σ1ei ω2 tσ3 . (11)

(compare with (4)). Ifε = 0, the evolution operator

U(t) = e−iω02 tσ3

causes rotations of a state|φ(t)〉 upon the sphere surface withan angular frequencyω0/2 arounde3 (the natural evolutiontrajectory, see Fig. 2). When the external field is applied,Eq. (5) reads

U(t) = e−i ω2 tσ3e−i 1

2 t[(ω0−ω)σ3+εσ1]. (12)

In the rotating frame,|φ(t)〉 describes a circular trajectoryarounde′3 = (ε/|ω0 − ω|)e1 + e3 with an angular frequencyω1/2 (see Fig. 2). Ifε ¿ |ω0 − ω|, this trajectory practi-cally coincides with the basic one (the angle betweene3 ande′3 is α = arctan ε/|ω0 − ω|), meaning that the whole evo-lution process is confined in a narrow belt of widthδθ = 2αaround the orbit defined byH0. In the inertial framee′3 ro-tates arounde3 with an angular frequencyω and|φ(t)〉 pre-cesses around the natural evolution orbit keeping its positionon the sphere confined again to the same belt (see Fig. 2).

FIGURE 1. The maximum probability of finding the system in theexcited state has a peak which is sharper asε → 0.

Rev. Mex. Fıs. S53 (4) (2007) 37–41

QUANTUM CONTROL WITH PERIODIC SEQUENCES OF NON RESONANT PULSES 39

FIGURE 2. If ε = 0, |φ(t)〉 describes a circular trajectory orthogo-nal toe3 (solid curve). Whenε 6= 0, in the rotating frame,|φ(t)〉circulates arounde′3 (dotted curve). This vector rotates arounde3

from the point of view of the inertial frame, meaning the evolutionis precessing around the unperturbedH0-orbit. In both cases, thetrajectory is confined to a sharp belt of width2α.

FIGURE 3. When the external field is resonant, the system per-forms Rabi rotations.

However, in the exactly resonant caseΩ = H0,

U(T0) = e−i ε2 σ1T0

and |φ(t)〉 rotates arounde1 with an angular frequencyε/2. After one periodT0 the polar positionθ changes byδθ = πε/ω0, and even whenε is small, the process can besuperposed by a number of times untilδθ reaches an arbi-trary value (the Rabi rotations between|1〉, |2〉, see Fig. 3).

The difference between the evolution of Fig. 2 and that ofFig. 3 is usually taken as an argument that ifε/|ω0−ω| ¿ 1,then the resonant control operations upon one qubit (Fig. 3)can be selectively performed with negligible non resonantconsequences for the other ones (Fig. 2). However, the con-trol problem is far from being completely solved.

3. The selectivity breaking in the step by stepoperations

A very important amendment appears for systems composedby a great number of qubits. In this case, the most effi-cient control techniques involves operations developed stepby step, always with a monochromatic field of frequencyproper to manipulate just one part of the system (a qubit or apair of levels) [4–6,17]. Consistently with the mechanism ofFig. 2, it is assumed that the non resonant qubits will ignorethis field. This is indeed true for the proper parametersω, ε,but always under the condition that the external field is ap-plied in a continuous way. It turns out however that, for theinterrupted sequences of pulses, the final response may be, inspite of all, unexpected [18].

To illustrate this phenomenon suppose one has a pair ofnoninteracting qubitsQω, Qω0 with characteristic frequen-ciesω, ω0 respectively. Suppose also that a unitary operationis carried out onQω by means of a harmonic field of strengthε and frequencyω acting during a timeT1. Then, this field isturned off (presumably while other fields are being applied onQω0) for a timeτ1 before it is turned on again, etc. The evo-lution operators in a complete period of time (T = T1 + τ1),are

Uω0(T ) = e−iω02 τ1σ3e−i ω

2 T1σ3e−i ε2 T1[(ω0−ω)σ3+εσ1]

= e−i 12 [ω0τ1+ωT1]σ3e−iT1G, (13)

Uω(T ) = e−i ω2 (T1+τ1)σ3e−i ε

2 T1σ1 . (14)

Let now T1 and τ1 be such that e−iT1G ande−i(1/2)[ω0τ1+ωT1]σ3 generate rotations ofπ-rad on Qω0

states arounde′3 ande3 respectively, whilee−i(ω/2)(T1+τ1)σ3

generates also rotations ofπ-rad onQω states arounde3,then

ω1T1 = (2n + 1)π, (15)

ω0τ1 + ωT1 = (2m + 1)π, (16)

ω (T1 + τ1) = (2l + 1)π, (17)

wheren,m, l ∈ Z+. If the sequence is applied many times,something extraordinary happens:Qω0 (the non resonantqubit!) experiences an effect similar to the Rabi rotations(see Fig. 4).

Rev. Mex. Fıs. S53 (4) (2007) 37–41

40 S. CRUZ Y CRUZ AND B. MIELNIK

FIGURE 4. A sequence of interrupted pulses applied in the propertime intervals, may cause the escape of the system out of the con-finement zone. In the most dramatic case, the system may experi-ment an effect similar to the Rabi rotations (see [18]).

FIGURE 5. If a sequence of resonant pulses are applied in an inter-rupted way in the proper time intervals, the Rabi rotations may becompletely annihilated (following [19]).

In fact, the effective rotation of the state ofQω0 after atimeT is

δθ = 2 arctanε

|ω0 − ω|

in the plane generated bye3 ande′3. After applying severaltimes this “on” and “off” sequence,Qω0 will have escapedfrom the confinement zone. The evolution of the system willbe comparable to the resonant “stationary” one, changing thepolar positionθ of its state at will. What is happening mean-while to Qω? It turns out that the alphabet of the “on-off”pulses is, in the same way, determinant. If the sequence isapplied an even number of times, the Rabi rotations will becompletely annihilated (see Fig. 5).

Now a question arises: is it possible that these phenom-ena, namely, therevival of the non resonant effects and thetotal annihilation of the Rabi rotations, simultaneously hap-pen? The answer isYES! . From (15-17) it follows that

2n + 1ω1

+2(m− l)ω0 − ω

=2l + 1

ω. (18)

The question is now: for which values ofn, l, m this expres-sion is held for all values ofε? There are different families ofcases [19]:

(i) For 0 < ω0 < ω, (18) impliesn < m < l and(2l + 1)/(2m + 1)ω0 < ω < (2l + 1)/(2(m− n))ω0.The right hand end of this interval corresponds toε → 0 while the left hand one to the caseε →∞.

(ii) For0 < ω < ω0, (18) states thatl < m, l−m < n and(2l+1)/(2(n+m+1))ω0 < ω < (2l+1/2m+1)ω0,where now the left hand end corresponds to the caseε → 0 and the right hand one toε →∞.

Time intervalsT1 and τ1 can then be evaluated from theseranges of frequencies, for a chosen triplet(n, l,m). It isworthwhile to mention that, in both cases, the non reso-nant cumulative rotations onQω0 can reach an angular speedω′ ∼ ε/π, which is comparable to that in the Rabi rotations(namelyε/2). It can be shown that this effects could have aresonant reinterpretation whenε → 0; in any other case theyhave a pure non resonant nature [18].

It seems to be that this phenomena become specially sig-nificant only for an unlikely coincidence in the parameters inthese periodical on-off sequences, however, there are someevidences that when applying arbitrary sequences of inter-rupted pulses, the state of the system will perform an erraticmovement indicating that it is out of control [18]. This is avery important challenge,e.g., in designing a quantum com-puter, since after some number of operations, the system willnot be able to store information. Yet, not all the news are bad,since there is an evidence that the off resonant effects not al-ways destabilize the system: if orderly applied, they mightbe also used to maintain the stability [18], permitting then tomanipulate the system in a more precise way.

Acknowledgments

The financial support of CONACyT projects 50766 and49253-F is acknowledged.

Rev. Mex. Fıs. S53 (4) (2007) 37–41

QUANTUM CONTROL WITH PERIODIC SEQUENCES OF NON RESONANT PULSES 41

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