Quantum speed limit for a relativistic electron in a uniform magnetic field
QQuantum speed limit for a relativistic electron in a uniform magnetic field
D. V. Villamizar ∗ and E. I. Duzzioni † Departamento de F´ısica, Universidade Federal de Santa Catarina, Santa Catarina, CEP 88040-900, Brazil (Dated: October 12, 2018)We analyze the influence of relativistic effects on the minimum evolution time between two or-thogonal states of a quantum system. Defining the initial state as an homogeneous superpositionbetween two Hamiltonian eigenstates of an electron in a uniform magnetic field, we obtain a relationbetween the minimum evolution time and the displacement of the mean radial position of the elec-tron wavepacket. The quantum speed limit time is calculated for an electron dynamics describedby Dirac and Schroedinger-Pauli equations considering different parameters, such as the strengthof magnetic field and the linear momentum of the electron in the axial direction. We highlightthat when the electron undergoes a region with extremely strong magnetic field the relativistic andnon-relativistic dynamics differ substantially, so that the description given by Schroedinger-Pauliequation enables the electron traveling faster than c , which is prohibited by Einstein’s theory ofrelativity. This approach allows a connection between the abstract Hilbert space and the space-timecoordinates, besides the identification of the most appropriate quantum dynamics used to describethe electron motion. PACS numbers:I. INTRODUCTION
The question
How fast can quantum information beprocessed? was tackled at first time in 1945 by Man-delstam and Tamm [1]. They developed a criterion tofind the minimum time for a closed quantum system withlimited energy uncertainty ∆ H to change the expectationvalue of a given operator by the standard of this operator.Such result was supported later by [2–4]. On the otherhand, Margolus and Levitin [5] attributed the speed of aquantum evolution between two orthogonal states to themean energy of the system (cid:104) ˆ H (cid:105) . In Ref. [6] it is assumedthat the minimum evolution time has the following ex-pression T min = max { π (cid:126) / H, π (cid:126) / (cid:104) ˆ H (cid:105) − E ) } , where E is the lowest energy of one of the states of the super-position. A unified version of the MT and ML boundswere presented in [7]. Recent developments on this sub-ject extended these ideas to include initial mixed statesand open quantum system dynamics, obtaining realisticbounds for the speed of quantum processes [8–15].The answer to the former question is very importantfor many areas of quantum physics, including quantuminformation and computation [16, 17], quantum metrol-ogy [18], optimal control theory [19], and quantum ther-modynamics [20].Although the achievement of an exact expression forthe quantum speed limit is of fundamental importanceto attain precisely the minimum time of a quantum pro-cess, the correct description of the dynamics of the sys-tem of interest is as important as the former. Regardingthis point we observe that for an accurate description ofa system dynamics it is necessary to take into accountrelativistic effects. In the case of spin 1 / ∗ [email protected] † [email protected] very well quantum mechanics and special relativity. It re-produces accurately the spectrum of the hydrogen atom,provides a natural description of the electron spin, andthe existence of antimatter [21]. The correction to theenergy of atomic levels due to fine structure is a beautyexample of relativistic effects in low energy quantum sys-tems. Such correction is very small, about five orders ofmagnitude smaller than the energy values predicted bythe non-relativistic Schr¨odinger equation, nevertheless,experimentally it is observable [22].The target of this work is to encompass relativistic ef-fects on the quantum speed limit. For this purpose weanalyze the transition between two orthogonal states ofan electron in a uniform magnetic field according to Diracequation [21] and compare it to the non-relativistic de-scription given by Schroedinger-Pauli equation [23, 24].Defining the electron initial state as an homogeneous su-perposition of two eigenstates of the Hamiltonian, theMadelstam-Tamm and Margolus-Levitin bounds becomeequivalent [25]. Therefore, in some sense, our results areindependent on the expression used to calculate the min-imum transition time. For some states the electron meanradial position is initially different from its final one. Theratio between such average radial displacement and theminimum evolution time furnishes the average speed inwhich the electron travels in space-time in the radial di-rection. Such speed is important for two reasons: i) itenables to find what kind of initial superposition stateprovides the greatest spatial displacement in the shortertime; and ii) for speeds higher than the speed of light invacuum c it works as a criterion to invalidate the equa-tion used to describe the electron dynamics. As expected,the Schroedinger-Pauli equation will be the only one toviolate this criterion.This paper is organized as follows, in section II webreafly describe the relativistic and non-relativistic dy-namics of an electron in a uniform magnetic field by theDirac and Schroedinger-Pauli equations, respectively. In a r X i v : . [ qu a n t - ph ] S e p section III we show an analysis of a particular case ofan initial superposition state which gives us enough in-formation about both quantum mechanical descriptionsand used it in base to realize in section IV a numericalcalculation for looking for fastest superposition states. Insection V follows our conclusion. II. MODEL AND FRAMEWORK
For didactic reasons we briefly review the non-relativistic and relativistic dynamics of an electron in auniform magnetic field, respectively. The Pauli Hamilto-nian is H = 12 m (cid:0) (cid:126)p + e (cid:126)A (cid:1) + em (cid:126)B · (cid:126)S, (1)with (cid:126)p being the linear mechanical momentum, e the ab-solute value of the electron charge, and m the electronrest mass. The magnetic vector potential (cid:126)A is expressedby the symmetric Landau gauge (cid:126)A = ( (cid:126)B × (cid:126)r ) /
2, where (cid:126)B = B ˆ z is the magnetic field oriented in z direction and (cid:126)r is the vector position of the electron. The eigenstatesof the Hamiltonian (1) are [23, 24] ψ n,m l ,m s ,p ( (cid:37), ϕ, z ) = F n,m l ( (cid:37), ϕ ) e ipz/ (cid:126) Γ m s , (2)where the radial wavefunction is F n,m l ( (cid:37), ϕ ) = ( − (cid:16) n −| ml | (cid:17) (cid:18) n − | m l | (cid:19) ! (cid:115) π (cid:18) n + | m l | (cid:19) ! (cid:18) n − | m l | (cid:19) ! × β (cid:0) β(cid:37) (cid:1) | m l | L | m l | (cid:16) n −| ml | (cid:17) ( β (cid:37) ) e − β (cid:37) / e im l ϕ , (3)with L | m l | (cid:16) n −| ml | (cid:17) ( β (cid:37) ) being the generalized Laguerrepolynomials, L | m l | (cid:16) n −| ml | (cid:17) = (cid:16) n −| ml | (cid:17) (cid:88) j =0 ( − j (cid:16) n −| m l | (cid:17) + | m l | (cid:16) n −| m l | (cid:17) − j (cid:0) β(cid:37) (cid:1) j j ! . (4)Here β ≡ (cid:113) eB (cid:126) is the inverse of the characteristic length of the harmonic oscillator, the indexes n = 0 , , , ... and m l = − n, − n + 2 , ..., n − , n refer to the eigenstates F n,m l ( (cid:37), ϕ ) of the 2-dimensional harmonic oscillator inthe plane perpendicular to the orientation of the mag-netic field and also to the coupling between the magneticfield and the orbital angular momentum. Γ m s representsthe eigenstates of the spin opetator S z with eigenvalues (cid:126) m s , so that the index m s = {− / , +1 / } . p is theprojection of the linear momentum in z direction. The corresponding eigenvalues of Hamiltonian (1) are E n,m l ,m s ,p = p m + (cid:126) ω (cid:0) n + m l + 2 m s + 1 (cid:1) , ω ≡ eB m . (5)By its turn, the relativistic dynamics of the electron isgiven by Dirac equation, which one is expressed as i (cid:126) ∂∂t ψ ( (cid:126)r, t ) = (cid:0) c(cid:126)α · (cid:126) Π + βm c (cid:1) ψ ( (cid:126)r, t ) , (6)where (cid:126) Π = (cid:126)p + e (cid:126)A is the linear canonical momentum. Weare using the Bjorken-Drell convention to represent the γ matrices, here denoted by (cid:126)α and β . The Dirac Hamil-tonian eigenstates are spinors with four components, inwhich the two upper components have positive energyand are described by Eq.(2), while the two lower compo-nents with negative energy are given by c(cid:126)σ · (cid:126) Π E + m c ψ n,m l ,m s ,p ( (cid:37), ϕ, z ) . (7)The quantity E represents the eigenenergies E n,m l ,m s ,j,p = j (cid:113) m c + p c + eB (cid:126) c (cid:0) n + m l +2 m s +1 (cid:1) , (8)with j = { + , −} indicating the sign of the energy. Formore details about this solution see Ref. [21].The electron initial state is assumed to have +1 / z direction and a gaussian wavepacket in the same spatial direction with standard devia-tion d and expectation value p for the linear momentumoperator ˆ p z , ψ z ( z ) = 1 (cid:0) πd (cid:1) / e − z / d e ip z/ (cid:126) . (9)Our idea is to establish a connection between the quan-tum speed limit and the speed in which the electron wavepacket moves through the space-time. For this purposewe consider the initial state of the system in x − y planeas a homogeneous superposition of two radial eigenstates F n,m l ( (cid:37), ϕ ) in different Landau energy levels. After thetime of evolution T min the state of the system is orthog-onal to the initial one, so that the mean radial positionof the electron wave packet experiences a displacement.In the next sections we analyze the relativistic effectson T min , besides the dependence of the electron’s dis-placement on the initial superposition state and on therelativistic and non-relativistic descriptions of quantummechanics. III. QUANTUM SPEED LIMIT FOR ANELECTRON UNDER RELATIVISTIC ANDNON-RELATIVISTIC QUANTUM DYNAMICS
We start analyzing the non-relativistic case of super-position between the radial eigenstates F , ( (cid:37), ϕ ) and F , ( (cid:37), ϕ ), ψ ( (cid:126)r,
0) = 1 √ F , ( (cid:37), ϕ ) + F , ( (cid:37), ϕ )] ψ z ( z )Γ +1 / . (10)After the evolution from ψ ( (cid:126)r,
0) to ψ ( (cid:126)r, T min ), we obtainthe quantities required to evaluate the quantum speedlimit criteria, ∆ H = (cid:104) H (cid:105)− E = eB (cid:126) m . (11)Thus the minimum evolution time is, T min = πm eB . (12)As we are considering basically the change in the radialpart of the system state, we will analyze the radial dis-placement of the electron, which enable us to set theexpectation value of the linear momentum in the axialdirection as p = 0. Thus, the mean radial position atany time is given by the expression, (cid:104) (cid:37) (cid:105) t = 12 (cid:104) (cid:104) , | (cid:37) | , (cid:105) + (cid:104) , | (cid:37) | , (cid:105) +2 (cid:104) , | (cid:37) | , (cid:105) cos (cid:0) E t (cid:1)(cid:105) . (13)By using the Dirac notation, we nominated each eigen-state by its quantum numbers n and m l , and E = eB/m is a constant with dimension of frequency. Therefore,the maximum radial displacement of the electron’s meanposition is, (cid:12)(cid:12) (cid:104) (cid:37) (cid:105) T min −(cid:104) (cid:37) (cid:105) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:104) , | (cid:37) | , (cid:105) (cid:2) cos( E T min ) − (cid:3)(cid:12)(cid:12)(cid:12) . = (cid:114) π (cid:126) eB . (14)The expression above shows us the relevancyof the crossed term D S ( (cid:37) ) = (cid:104) , | (cid:37) | , (cid:105) =2 π (cid:82) ∞ F † , ( (cid:37), ϕ ) (cid:37)F , ( (cid:37), ϕ ) (cid:37)d(cid:37) for the electron’s dis-placement. From Eqs. (2), (3), and (14) we observe thatfor a non null radial displacement of the electron, theinitial superposition state must be built by eigenstateswith the same quantum numbers of spin m s and orbitalangular momentum m l , besides E T min (cid:54) = sπ , with s even.Then the average speed of the mean radial position ofthe electron from its initial state to the orthogonal oneis, ¯ v = 1 m (cid:114) eB (cid:126) π . (15) On the other hand, in the relativistic description with p = 0, the minimum evolution time is T min = π (cid:126) (cid:112) m c +4 eB (cid:126) c − (cid:112) m c +2 eB (cid:126) c . (16)At follows we write the two spinors that compose theevolved state of the system U , = N , F , ( (cid:37), ϕ )0 cpF , ( (cid:37), ϕ )( E , + m c )2 i (cid:126) cβF , ( (cid:37), ϕ )( E , + m c ) e ipz/ (cid:126) , (17)and U , = N , F , ( (cid:37), ϕ )0 cpF , ( (cid:37), ϕ )( E , + m c )2 i (cid:126) cβ √ F , ( (cid:37), ϕ )( E , + m c ) e ipz/ (cid:126) , (18)where N , and N , are normalization constants, and E , e E , are positive eigenvalues given by Eq. (8).Therefore, the superposition state evolves in time as, ψ ( (cid:126)r, t )= 1 √ (cid:90) ∞−∞ α ( p ) (cid:104) U , e − iE , t/ (cid:126) + U , e − iE , t/ (cid:126) (cid:105) dp, (19)with α ( p ) being the coefficient of expansion of the Gaus-sian wave packet defined in Eq.(9). Now we are able tocalculate the radial displacement of the electron’s meanposition in the relativistic case, which one is made nu-merically [26]. In FIG. (1) we plot the average speedof the electron’s wave packet when moving from the ini-tial to final state during the time interval T min underboth relativistic and non-relativistic quantum dynamics.We noticed in the non-relativistic case that there is amagnetic field strong enough to yield ¯ v ≥ c given by B ≥ . × T, which contradicts the Einstein’s the-ory of relativity. Naturally, it is impossible to achieve thisintensity of magnetic field in a laboratory on the Earth,but not in special neutron stars, called magnetar [27].Conversely, the Dirac’s theory for the electron predictsthe asymptotic value of ¯ v (cid:119) . c . To attain this valuewe first need to evaluate the radial displacement of theelectron mean position, which one depends on the crossedterm D S ( (cid:37) ) = 2 πU † , (cid:37)U , and on the minimum evolutiontime T min as in Eq. (16) . In the limit case B → ∞ theexpressions for the eigenenergies can be approximated by E , ≈ c (cid:126) β, E , ≈ √ c (cid:126) β. (20) FIG. 1. Average radial speed as function of the external mag-netic field according to both quantum dynamics. which renders, T min ≈ π cβ ( √ − . (21)Inside this approximation, the spinor normalization con-stants become N , = N , = 1 / √ (cid:12)(cid:12) (cid:104) (cid:37) (cid:105) T min −(cid:104) (cid:37) (cid:105) (cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:90) (cid:37)D S ( (cid:37) ) d(cid:37) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , = √ π β (cid:18)
1+ 32 √ (cid:19) . (22)Differently from the non-relativistic case, now the dis-placement in time and space have the same dependenceon the magnetic field, as shown in Eqs. (21) and (22),respectively. Consequently, the maximum value of theaverage speed of the electron in the radial direction is¯ v = c √ π (cid:0) √ (cid:1) ≈ . c. (23)Hence, as expected, the relativistic quantum dynamics isthe most appropriate to describe the electron dynamics inthe presence of high intensity magnetic fields. Through-out the present development we observed that the rel-ativistic theory of quantum mechanics funded by Dirac does not restrict the time interval of a quantum processof being arbitrarily small, as shown in Eq. (16). Makinga comparison between Eqs. (12) and (16) we verify thatthere is a quadratic dependence on the magnetic fieldin the non-relativistic case in relation to the relativis-tic one. Such difference can turn out to be importantfor B ∼ m c /e (cid:126) ∼ IV. THE FASTEST SUPERPOSITIONS
In the preceding section we verified that the dynam-ics of an electron described by Schroedinger-Pauli equa-tion violates a basic principle of Einstein’s theory of rel-ativity, which states that any object with non-null restmass cannot travel faster than c . For that reason, westudy the dependence of the quantum speed limit for anelectron evolving according to the Dirac theory as func-tion of the initial superposition state. Our main pur-pose here is looking for the maximum radial displace-ment in the shortest time interval. Since the electron’sradial displacement depends strongly on the crossed term D S ( (cid:37) ), its maximum absolute value is attained when theinitial and final states have the same spin orientation( m s = 1 / m l = 0,and the initial superposition state is made of two near-est neighbors eigenstates, i.e., with quantum numbers n and n + 2. In FIG. 1 we observe that ¯ v increases as theintensity of the magnetic field is strengthened. There-fore, in the regime β → ∞ the minimum evolution timebetween two orthogonal states, where the initial superpo-sition state is composed by two eigenstates with positiveenergy (called particle-particle states), is T min ≈ π (cid:2) √ n +4 −√ n + 2 (cid:3) √ cβ , (24)and the crossed term turns out to be, D S ( (cid:37) ) ≈ π(cid:37) (cid:104) F † n, F n +2 , + F † n +1 , F n +3 , (cid:105) . (25)After some steps we get an analytic expression for themaximum radial displacement as function of n (cid:12)(cid:12) (cid:104) (cid:37) (cid:105) T min −(cid:104) (cid:37) (cid:105) (cid:12)(cid:12) = 1 β ( n ) (cid:88) i =0 ( n +22 ) (cid:88) j =0 ( − i + j i ! j ! (cid:18) n n − i (cid:19)(cid:18) n +22 n +22 − j (cid:19) Γ (cid:18) i + j +1+ 12 (cid:19) (cid:113)(cid:0) n +2 (cid:1) (cid:0) n +1 (cid:1) ( i +1)( j +1) (cid:18) i + j +1+ 12 (cid:19) . (26)In FIG.2 we plot the average radial speed of the elec-tron for different even values of n ranging in the interval [0 , v for higher values of n is veryhard, provided that Eq. (26) has many factorials. Theinset of such figure shows the convergence of ¯ v/c to theasymptotic value 0 . ≤ n ≤
132 the value of ¯ v changes inthe fourth decimal place only, which shows that the av-erage radial speed is reaching a constant value less than c for n → ∞ . FIG. 2. Average radial speed of an electron for differ-ent initial superpositions of two positive energy eigenstates( U n, + U n +2 , ) / √ Instead of considering only initial particle-particlestates, we will take into account superpositions ofeigenstates with negative and positive energies (called antiparticle-particle states). The reason we are tacklingthis subject only now is that it is not clear if it is fair tocompare the non-relativistic dynamics, which describesonly particle states, with the relativistic antiparticle dy-namics. Despite that, antiparticle-particle dynamics re-veals the role played by the electron rest mass in theenergy spectrum and thus imposes physical limits on thequantum speed limit [30]. Repeating the same procedureabove to obtain the maximum displacement of the meanradial position of the electron, we find that the two statesof the superposition must have the same spin orienta-tion (spin up), null angular momentum projection alongthe z direction, and must be made of nearest neighborseigenstates with even quantum numbers n . Assumingthe negative energy eigenvalue as the lowest one in mod-ule, according to Eq. (8) we attribute to it the quantumnumber n , while for the positive energy eigenvalue thequantum number n + 2. Thus, the minimum evolutiontime for the particular case n = 0 and p = 0 is given by T min = π (cid:126) (cid:112) m c +4 eB (cid:126) c + (cid:112) m c +2 eB (cid:126) c . (27)This time is shorter than in the particle-particle case (seeEq. (16)) because the energy gap is bigger by a quantitythat is at least the energy of the electron rest mass. Toevaluate the mean radial displacement of the electron, weneed the expression of the negative energy spinor for a general quantum number n and null angular momentum, U − n, ( (cid:126)r ) = N − n, cp ( E n − m c ) F n, √ ic (cid:126) β √ n + 2( E n − m c ) F n +1 , F n, e ipz/ (cid:126) , (28)where N − n, is the normalization constant. In addition,the radial displacement of the electron is proportional tothe absolute value of the crossed term D S = 2 π(cid:37)N − n, N n +2 , cp × (cid:20) E n − m c + 1 E n +2 + m c (cid:21) F † n, F n +2 , , (29)which one is maximized for p ≈ β (cid:126) (cid:29) m c . In FIG. 3 weplot the average radial speed of the electron to changefrom a n dependent initial negative-positive state to afinal one orthogonal to the former in the minimum timeinterval T min . The asymptotic value of ¯ v is 0 . c and lower than the speed in the positive-positive case(see FIG. 2). FIG. 3. Average radial speed of an electron for different initialsuperposition states composed by a positive and a negativeenergy eigenstate (cid:0) U − n, + U n +2 , (cid:1) / √ Making a comparison between FIGs. 2 and 3 we observethat ¯ v for negative-positive states is always less than ¯ v forpositive-positive states. This behavior is clarified in FIG.4, where ¯ v/c is plotted for both cases of initial superpo-sition states as function of p for three different values ofthe magnetic field. If the initial state is negative-positive,then, according to Eq. (29), the displacement of the ra-dial mean position of the electron depends linearly on p ,which justify the null value of ¯ v/c at the origin of FIG.4. For intermediate values of p , we observe that ¯ v/c attains a maximum value for p ≈ β (cid:126) (cid:29) m c , while forgreat values of p (cid:29) β (cid:126) , m c whatever the initial super-position the average radial speed becomes smaller. In thelater case both positive and negative energy eigenstateshave the same expression, and therefore the same radialdisplacement of the electron and T min .In the context of Dirac’s theory, this can be explainedby the fact that each spinor does not describe its ownparticle only, but also its antiparticle by the two termsin the bottom position of the spinor. One of these an-tiparticle terms is relevant to the whole description whenthe electron presents a very high linear momentum orwhen the particle is strongly confined in a region lessthan or equal to its Compton wavelength. In these caseswe could say that the spinor by itself describes a superpo-sition between its particle and its antiparticle [21, 31, 32]. FIG. 4. Average radial speed of an electron for particle-particle (solid lines) and antiparticle-particle (dashed lines)states as function of the expectation value of the linear mo-mentum along the z direction, p . V. CONCLUSIONS
We analyzed the role played by relativistic effects onthe quantum speed limit of a system composed by anelectron in a uniform magnetic field. The relativisticdynamics by itself does not restrict the minimum timeof evolution of being arbitrary small, but imposes con-straints on the average speed at which the electron trav-els along the space-time. As expected, we observed thatthe quantum dynamics described by Schroedinger-Pauliequation enables the electron wave packet traveling fasterthan c , in contradiction to Einstein’s theory of relativ-ity. Such problem is circumvented by the use of Dirac’sequation. The minimum evolution time between two or-thogonal states in the relativistic formulation can be sig-nificantly different from the non-relativistic case. If theinitial state is a homogenous superposition of two Hamil-tonian eigenstates with positive energies, then, the min-imum evolution time is dilated in the laboratory frame.On the other hand, if the Hamiltonian eigenstates havenegative and positive energies, then, the minimum evo-lution time is contracted in the laboratory frame. Thislast result can be useful for quantum computing, since itcan speed up quantum gates, although a precise controlover the creation of particle-antiparticle states is neces-sary [33]. VI. ACKNOWLEDGEMENTS
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