Circumventing Detector Backaction on a Quantum Cyclotron
EEvading Detector Backaction on a Quantum Cyclotron
X. Fan
1, 2, ∗ and G. Gabrielse † Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Center for Fundamental Physics, Northwestern University, Evanston, Illinois 60208, USA (Dated: August 6, 2020)The state of a one-particle quantum cyclotron can be detected by coupling it to a simple har-monic oscillator. The resulting quantum nondemolition (QND) detection comes at the price of adetector backaction that broadens the resonance lineshapes. The first quantum calculation of thecoupled system shows that detection backaction can be evaded by resolving the quantum state ofthe detection oscillator.
The electron magnetic moment in Bohr magnetons, de-termined to 3 parts in 10 , is the most precisely deter-mined property of an elementary particle [1, 2]. A bettermeasurement is currently of great interest because of anintriguing, 2.4 standard deviation discrepancy [3] withthe most precise prediction [4] of the Standard Model ofparticle physics (SM). Tests of the prediction are criticalbecause these would check important elements of the SM– Dirac theory [5], quantum electrodynamics through thetenth order [4, 6, 7], hadronic contributions [8–10] andpossible weak interaction contributions [10–14]. The SMprediction requires a measured fine structure constant, α , as an input. This constant is determined from themeasured Rydberg constant [15–17], atomic mass ratios[18–20], and atom recoil masses [21, 22]. The intriguingdiscrepancy has stimulated new theoretical investigationsinto possible physics beyond the SM [23–27].A quantum cyclotron [28] is a single trapped electronthat occupies only the ground and first excited states ofits cyclotron motion. Quantum non-demolition (QND)detection of the state of this quantum cyclotron madepast measurements possible. However, a detector back-action produced a broad cyclotron lineshape that lim-ited the accuracy with which the cyclotron oscillationfrequency could be determined. In this Letter, we presentthe possibility of evading the detector backaction to pro-duce extremely narrow lineshapes for better measure-ments. A steady-state solution to a master equation pro-vides the first quantum treatment of a quantum cyclotroncoupled to a detection oscillator with a QND coupling.For promising quantum measurement conditions becom-ing available, the predicted lineshape differs dramaticallyfrom a previous prediction that assumed a classical detec-tion oscillator [29, 30]. Calculation details, comparisonsto Brownian motion lineshapes, and applications to otherelements of electron and positron magnetic moment mea-surements are in a longer report [31].The Hamiltonian for the quantum cyclotron [32, 33]with angular cyclotron frequency, ω c , H c = (cid:126) ω c (cid:0) a † c a c + (cid:1) , (1) ∗ [email protected] † [email protected] has the form of a simple harmonic oscillator. The energyeigenvalues are a ladder of equally spaced Landau levels[32], (cid:126) ω c ( n c + 1 /
2) with n c = 0 , , , · · · . The raising andlowering operators, a † c and a c , in terms of position andmomentum operators differs from that for a simple har-monic oscillator, of course, because circular rather thanlinear motion is described. For the same reason, the po-sition representation of energy eigenstates | n c (cid:105) are as-sociated Laguerre polynomials rather than the Hermitepolynomials for a simple harmonic oscillator.QND detection of the quantum cyclotron state is possi-ble if it is appropriately coupled to a harmonic oscillatorthat has a Hamiltonian, H z = (cid:126) ω z (cid:0) a † z a z + (cid:1) , (2)energy eigenstates | n z (cid:105) , and eigenvalues (cid:126) ω z ( n z + 1 / n z = 0 , , · · · . For an electron in the electrostaticquadrupole potential of a Penning trap, this detectionmotion is the axial oscillation of the electron along themagnetic field direction. The raising and lowering op-erators, a † z and a z , in terms of position and momentumoperators are in every quantum mechanics textbook, asare the energy eigenstates in the position representation.The uncoupled cyclotron-plus-axial Hamiltonian H = H c + H z , (3)has | n c , n z (cid:105) = | n c (cid:105) | n z (cid:105) as energy eigenstates, and E ( n c , n z ) = (cid:126) ω c ( n c + ) + (cid:126) ω z ( n z + ) (4)as energy eigenvalues. The representation in Fig. 1 is notto scale since ω c is typically 1000 times larger than ω z .A QND coupling [34–36] of the two oscillators, V = (cid:126) δ c (cid:0) a † c a c + (cid:1) ( a † z a z + ) , . (5)commutes with H and does not change the system state.In a Penning trap with uniform field B ˆ z , the shift δ c = (cid:126) eB / ( m ω z ) , (6)comes from adding a magnetic bottle gradient [37]∆ B = B (cid:0) z − ( x + y ) (cid:1) . (7)The Eq. (5) coupling pertains when two rapidly oscillat-ing terms average to zero. a r X i v : . [ phy s i c s . a t o m - ph ] A ug ・・・・・・
10 001221 n c n z ω c ω z ω z ω z ω z FIG. 1. Lowest energy levels for the combined quantum cy-clotron and axial detection oscillator (not to scale).
The uncoupled states | n c , n z (cid:105) are also the eigenstatesof the coupled Hamiltonian, H = H c + H z + V. (8)Because of the coupling, the energy eigenvalues E ( n c , n z ) = (cid:126) ω c ( n c + ) + (cid:126) ω z ( n z + )+ (cid:126) δ c ( n c + )( n z + ) , (9)acquire a small term that depends upon both n c and n z .That this coupling provides the desired QND detectioncan be seen by writing the energy eigenvalues as E ( n c , n z ) = (cid:126) ( n c + ) + (cid:126) (cid:101) ω z ( n z + ) . (10)Measuring the effective axial frequency (cid:101) ω z = ω z + δ c ( n c + ) , (11)determines n c without changing the cyclotron state.Quantum jump spectroscopy [28] can then determine thecyclotron frequency that is needed (along with anotherfrequency) to determine the magnetic moment.The undesirable and unavoidable detector backactioncan be seen by writing the energy eigenvalues as E ( n c , n z ) = (cid:126) (cid:101) ω c ( n c + ) + (cid:126) ω z ( n z + ) . (12)This formulation emphasizes that the cyclotron fre-quency shifts from ω c to (cid:101) ω c = ω c + δ c ( n z + ) . (13)The backaction shifts the cyclotron frequency in propor-tional to the axial energy and quantum number. Anydistribution of axial states is thereby turned into an un-desirable distribution of effective cyclotron frequencies.A distribution of axial states arises because the ax-ial oscillator is weakly coupled to its environment, witha coupling constant, γ z . For times larger than 1 /γ z ,this leads to a thermal Boltzmann distribution of axialstates. For T = 0 . ω z / (2 π ) = 200 MHz, the low-est ambient temperature and typical frequency used for measurements[1, 2], the average axial quantum numberis ¯ n z = (cid:20) exp (cid:18) (cid:126) ω z k B T (cid:19) − (cid:21) − ≈ k B T (cid:126) ω z ≈ . (14)For past measurements, the effective axial temperaturewas actually at least 3 to 5 times higher due to theelevated temperature of the electronics used to detectthe axial oscillation and its frequency[2]. The broad cy-clotron linewidth that resulted because of the detectionbackaction limited the accuracy of the measurements.The cyclotron motion also weakly couples to the ther-mal reservoir, with a coupling γ c . A state | n c (cid:105) radiatessynchrotron radiation at a rate n c γ c . In principle, cy-clotron states can also absorb blackbody radiation, but at0.1 K and ω c / (2 π ) = 150 GHz[1], the number of availableblackbody photons is negligible. The average quantumnumber for a Boltzmann distribution of states is¯ n c = (cid:20) exp (cid:18) (cid:126) ω c k B T (cid:19) − (cid:21) − = 1 . × − ≈ . (15)The cyclotron motion thus remains in its n c = 0 groundstate [28] unless a cyclotron driving force is applied.A cyclotron drive adds the Hamiltonian term V c ( t ) = (cid:126) Ω c (cid:104) a † c e − i ( ω c + (cid:15) c ) t + a c e i ( ω c + (cid:15) c ) t (cid:105) . (16)The drive strength is given by the angular Rabi fre-quency, Ω c , and the drive is detuned from resonance at ω c by a detuning (cid:15) c . For measurements, the driving forceprovided by 150 GHz microwaves injected into a trap cav-ity excites the | , n z (cid:105) states to | , n z (cid:105) . Higher cyclotronstates can be neglected because it is less probable to ex-cite from small population in an excited state, but alsobecause a relativistic shift keeps the cyclotron transitionsbetween excited states off resonance from the drive [38].A density operator is required for a system that decaysand is coupled to a thermal bath. The initial state attime t = 0 is the cyclotron ground state and a thermalsuperposition of axial states, ρ (0) = ∞ (cid:88) n z =0 p n z ( T ) | , n z (cid:105)(cid:104) , n z | . (17)The Boltzmann weighting factors are p n ( T ) = (cid:20) − exp (cid:18) − (cid:126) ω z k B T (cid:19)(cid:21) exp (cid:18) − n (cid:126) ω z k B T (cid:19) . (18)Explicit calculations show that 150 axial states suffice foraxial states in thermal equilibrium at 0.1 K.The time evolution of the density operator is describedby a Lindblad equation [39, 40], dρdt = − i (cid:126) [ H + V + V c , ρ ] − γ c (cid:0) a † c a c ρ − a c ρa † c + ρa † c a c (cid:1) − γ z n z (cid:0) a z a † z ρ − a † z ρa z + ρa z a † z (cid:1) − γ z n z + 1) (cid:0) a † z a z ρ − a z ρa † z + ρa † z a z (cid:1) . (19)The first line describes the driven motion. The seconddescribes the incoherent cyclotron decay. The third andfourth lines describe the incoherent deexcitation and ex-citation of the axial motion by the thermal bath.To efficiently solve the master equation, several trans-formations are made. All terms in Eq. (19) are trans-formed to an interaction picture. For example, (cid:101) ρ = e iH t/ (cid:126) ρe − iH t/ (cid:126) . (20)Since the coupled system starts and remains axially di-agonal, only the probabilities (cid:101) ρ jk ; n z = (cid:104) j, n z | (cid:101) ρ | k, n z (cid:105) areneeded. The indices j and k are 0 or 1, and n z takespositive values as large as needed to describe the thermaldistribution – up to about 150 for ¯ n z = 10, as mentioned.A second transformation, p jk ; n z ≡ ρ jk ; n z e i ( j − k ) (cid:15) c t (21)produces a time-independent equation for the p jk ; n z . Thetime-dependent probabilities we seek to calculate, p jj ; n z = (cid:101) ρ jj ; n z = (cid:104) j, n z | ρ | j, n z (cid:105) (22)are invariant under these transformations.The master equation in terms of vectors (cid:126)p jk , with com-ponents p jk ; n z , is ddt (cid:126)p ( t ) = R (0 , , (cid:126)p ( t ) − Ω c Im [ (cid:126)p ( t )]+ γ c (cid:126)p ( t ) (23a) ddt (cid:126)p ( t ) = R ( (cid:15) c , δ c , γ c ) (cid:126)p ( t ) − i Ω c (cid:126)p ( t ) − (cid:126)p ( t ))(23b) ddt (cid:126)p ( t ) = R (0 , , γ c ) (cid:126)p ( t ) + Ω c Im [ (cid:126)p ( t )] . (23c)The nonzero components of the matrices are R ( (cid:15), δ, γ c ) n,n − = γ z ¯ n z n (24a) R ( (cid:15), δ, γ c ) n,n = i (cid:2) − (cid:15) + ( n + ) δ (cid:3) − γ c − γ z (2¯ n z + 1) n − γ z ¯ n z (24b) R ( (cid:15), δ, γ c ) n,n +1 = γ z (¯ n z + 1)( n + 1) . (24c)Beside the specified arguments and indices, these equa-tions and matrices depend upon the bath temperaturevia ¯ n z , and the axial damping rate, γ z .This vector master equation must be solved for initialconditions (at t = 0) that (cid:126)p has components p n z ( T )(from Eq. (17)) and (cid:126)p = (cid:126)p = 0. The desired resonancelineshape is the probability of a cyclotron excitation, P = ∞ (cid:88) n z =0 p n z ( t d ) , (25)as a function of the drive detuning, (cid:15) c . This lineshapedepends upon the drive strength, Ω c and the time thatthe drive is applied, t d . In general, the vector master equation must be inte-grated numerically from t = 0 to t = t d to determine thelineshape. However, for a weak drive with Ω c (cid:28) γ c (toavoid power broadening) and t d (cid:29) /γ c (to let transientsdamp out), there is a steady state for which the drivencyclotron excitation balances the emission of synchrotronradiation. This steady-state solution suffices to demon-strate the possibility of detector backaction evasion.To obtain the steady-state solution, the derivatives inEq. (23) are set to zero. The three equations are summedover all axial states and simplified using ∞ (cid:88) n z =0 p n z ( t ) ≈ ∞ (cid:88) n z =0 p n z ( T ) = 1 (26) ∞ (cid:88) n z =0 p n z ( t ) (cid:28) ∞ (cid:88) n z =0 ( R (0 , , γ c ) (cid:126)p ) n z = − γ c ∞ (cid:88) n z =0 p n z , (28)The first two simplifications pertain for a weak drive andmake terms involving (cid:126)p negligible compared to those in-volving (cid:126)p . The third pertains because R (0 , , γ c ) hasa simple structure and axial damping does not changethe total population in states | , n z (cid:105) . The result is thesteady-state probability for cyclotron excitation by aweak drive, P = − Ω c γ c Im (cid:34) ∞ (cid:88) n z =0 (cid:0) i R ( (cid:15) c , δ c , γ c ) − (cid:126)p ( T ) (cid:1) n z (cid:35) . (29)The vector (cid:126)p ( T ) has the Boltzmann factors p n z ( T ) as itscomponents.For the limit T = 0, the steady-state lineshape for aweak drive becomes a Lorentzian, P = (cid:18) Ω c γ c (cid:19) (cid:0) γ c (cid:1) (cid:0) (cid:15) c − δ c (cid:1) + (cid:0) γ c (cid:1) . (30)Its full width at half maximum is γ c and its maximumis shifted to (cid:15) c = δ c / c /γ c ) , given that the drive is weak. The Lorentziancomes about because only the lowest axial state is popu-lated, ¯ n z = 0, (cid:126)p ( T ) collapses to a single element p ( T ) =1, and only the reciprocal of R = − i(cid:15) c + iδ c / − γ c contributes to Eq. (29).Direct numerical integrations of the master equation(Eq. (23)) and the steady-state solution in Eq. (29) pro-vide the first fully quantum treatment of the coupled cy-clotron and axial system. (More details, including com-parisons of direct integrations and steady-state solutionsof the master equation, are in a longer work that dealswith measuring magnetic moments more generally [31].)The lineshape calculation [29, 30] previously available(and used to predict and analyze all experiments to date)assumed a classical axial oscillation undergoing Brownianmotion – producing a very different lineshape.We now investigate detector backaction and its eva-sion, with estimates first, and then with quantum line-shape calculations. The result of a thermal distributionof axial states is that a cyclotron drive will make cy-clotron transitions over a range of cyclotron drive fre-quencies, ∆ (cid:15) c > ¯ n z δ c . For the best measurement, thebath temperature was 0.3 K and above, which corre-sponds to a spread ∆ ω/ω c >
800 ppt (A part per tril-lion, ppt, is 1 part in 10 ). Line splitting made it pos-sible to obtain a 300 ppt uncertainty. Even at 0.1 Ktemperature, the backaction linewidth will still spreadthe cyclotron excitation over a broad width. Reducingthe coupling strength ( δ c in Eq. (5)) would reduce thebackaction. However, this is not an option because thissimultaneously reduces the sensitivity needed to detectthe individual states of the quantum cyclotron.The new possibility proposed here is evading backac-tion by resolving the cyclotron excitations that an elec-tron makes while it is in its axial ground state, distin-guishing these from the broad range of excitations takingplace while the system is in other axial states (estimatedabove). Resolving δ c , the cyclotron frequency shift foraxial states with n z = 0 and n z = 1, requires two condi-tions, δ c (cid:29) γ c + 2 ¯ n z γ z (31) δ c (cid:29) /t d . (32)The first (from the diagonal damping term in Eq. (24b))requires that the shift be larger than both the cyclotrondamping width, γ c , and the axial width contribution,2¯ n z γ z . The latter arises because the underlying physicsof the master equation is that probability transfers be-tween the axial oscillation and the thermal reservoir atan average rate going as ¯ n z γ z . The second requirementis a drive applied long enough that the frequency-timeuncertainty principle does not broaden the lineshape.The shift δ c / (2 π ) = 4 Hz used for measurements ismuch smaller than the extremely small cyclotron damp-ing width, γ c / (2 π ) = 0 .
03 Hz, realized using a microwavecavity to inhibit spontaneous emission [41]. At the am-bient temperature of experiments, T = 0 . γ z / (2 π ) (cid:28) . γ z / (2 π ) = 1 Hz of the bestmeasurement. Resolving axial quantum structure thusrequires reducing γ z by about two orders magnitude. Thesecond condition (Eq. (32)) is met by simply applying thecyclotron drive for much longer than 40 ms.The axial damping rate cannot be reduced by this largefactor during the time that the frequency of the axial fre-quency is measured to determine the quantum state ofthe cyclotron motion (Eq. (11)). The detected signal isproportional to this damping rate, and a large value isrequired to detect the signal from a single particle. How-ever, it seems possible to electronically switch the axialdamping rate between a small value during the time thatexcitations are driven, and a large value for subsequentdetection of the axial frequency [31].Quantum calculations of the cyclotron lineshape c d / c e frequency detuning r e l a t i v e p r obab ili t y )=0.3 Hz p /(2 z g )=0.03 Hz p /(2 z g )=0.003 Hz p /(2 z g (a) (b) z g z n+2 c g c g =0 z n c g =0.1 c W c g = c W FIG. 2. (a) Quantum cyclotron lineshape for a weak drive(Ω c = 0 . γ c ) resolves the axial states as the axial dampingrates γ z is reduced. (b) The n z = 0 peak for γ z = 0 . T = 0 K Lorentzian lineshape (dotted). demonstrate the backaction evasion. Figure 2(a) showssteady-state lineshapes (Eq. (29)) for three values of theaxial damping rate, γ z , at a temperature T = 0 . δ c / (2 π ) = 4 Hz. For the dashed lineshape,2¯ n z γ z / (2 π ) = 6 Hz does not satisfy Eq. (31) and the ax-ial quantum states is not resolved. For a ten times lower γ z / (2 π ) = 0 .
03 Hz, the quantum structure of the axialmotion manifests itself in the dotted lineshape. For an-other 10-fold reduction in γ z , the solid lineshape showscompletely resolved peaks.The extremely narrow left peak for n z = 0 is good newsfor measurement. Its width, γ c + 2¯ n z γ z , is only about3 times the cyclotron decay width γ c , and much smallerthan the total cyclotron linewidth (Fig. 2(b)). More goodnews is that this n z = 0 peak is very symmetric aboutits center frequency – a big help in precisely identifyingthe center frequency of the resonance. The next peak tothe right is for n z = 1, and so on. There are many peaksbecause ¯ n z = 10 for T = 0 . c =0 . γ c , is only 3 . × − . However, increasing the cy-clotron drive strength to Ω c = γ c (dashed curve inFig. 2(b)) increases the excitation probability to 2 . × − while power broadening the full linewidth from 3to only 3.6 cyclotron decay widths. (The 300 differentialequations for the vector master equation were integrateddirectly to time 10 /γ c for Ω c = γ c because the steadystate solution applies only for Ω c (cid:28) γ c .) Stronger drivesmay be useful for tracking slow magnetic field drifts [2].The offset of the n z = 0 resonance from (cid:15) c = 0 to (cid:15) c = δ c / ω z to ω z + δ c can be measured.In summary, detector backaction that was thought tolimit the accuracy of electron magnetic moment measure-ments can be evaded. This is demonstrated using the firstquantum mechanical treatment of a quantum cyclotronthat has a QND coupling to a harmonic detection oscil-lation. The cyclotron resonance lineshapes that are pre-dicted contain extremely narrow peaks that correspond to resolved quantum states of the detection oscillation.This work was supported by the NSF, with partial sup-port of X. Fan from the Masason Foundation. B. D’Ursomade early contributions. B. D’Urso, S. E. Fayer, T.G. Myers, B. A. D. Sukra and G. Nahal provided usefulcomments. [1] D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev.Lett. , 120801 (2008).[2] D. Hanneke, S. Fogwell Hoogerheide, and G. Gabrielse,Phys. Rev. A , 073002 (2011).[3] G. Gabrielse, S. Fayer, T. Myers, and X. Fan, Atoms ,45 (2019).[4] T. Aoyama, T. Kinoshita, and M. Nio, Atoms , (2019).[5] P. A. M. Dirac and R. H. Fowler, Proceedings of theRoyal Society of London. Series A, Containing Papers ofa Mathematical and Physical Character , 610 (1928).[6] S. Laporta, Physics Letters B , 232 (2017).[7] T. Aoyama, T. Kinoshita, and M. Nio, Phys. Rev. D ,036001 (2018).[8] D. Nomura and T. Teubner, Nucl. Phys. B , 236(2013).[9] A. Kurz, T. Liu, P. Marquard, and M. Steinhauser,Physics Letters B , 144 (2014).[10] Jegerlehner, Fred, EPJ Web Conf. , 01003 (2019).[11] K. Fujikawa, B. W. Lee, and A. I. Sanda, Phys. Rev. D , 2923 (1972).[12] A. Czarnecki, B. Krause, and W. J. Marciano, Phys. Rev.Lett. , 3267 (1996).[13] M. Knecht, M. Perrottet, E. de Rafael, and S. Peris, Jour-nal of High Energy Physics , 003 (2002).[14] A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys.Rev. D , 073006 (2003).[15] C. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A.Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl,T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D.Rovera, C. Salomon, P. Laurent, and T. W. H¨ansch,Phys. Rev. Lett. , 203001 (2011).[16] H. Fleurbaey, S. Galtier, S. Thomas, M. Bonnaud, L.Julien, F. m. c. Biraben, F. m. c. Nez, M. Abgrall, andJ. Gu´ena, Phys. Rev. Lett. , 183001 (2018).[17] A. Beyer, L. Maisenbacher, A. Matveev, R. Pohl, K.Khabarova, A. Grinin, T. Lamour, D. C. Yost, T. W.H¨ansch, N. Kolachevsky, and T. Udem, Science , 79(2017).[18] M. P. Bradley, J. V. Porto, S. Rainville, J. K. Thompson,and D. E. Pritchard, Phys. Rev. Lett. , 4510 (1999).[19] S. Sturm, F. K¨ohler, J. Zatorski, A. Wagner, Z. Harman,G. Werth, W. Quint, C. H. Keitel, and K. Blaum, Nature , 467 (2014). [20] E. Myers, Atoms , 37 (2019).[21] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. M¨uller,Science , 191 (2018).[22] R. Bouchendira, P. Clad´e, S. Guellati-Kh´elifa, F. m. c.Nez, and F. m. c. Biraben, Phys. Rev. Lett. , 080801(2011).[23] S. Gardner and X. Yan, Light scalars with lepton numberto solve the ( g − e anomaly, 2019.[24] J. Liu, C. E. M. Wagner, and X.-P. Wang, Journal ofHigh Energy Physics , 8 (2019).[25] H. Davoudiasl and W. J. Marciano, Phys. Rev. D ,075011 (2018).[26] A. Crivellin, M. Hoferichter, and P. Schmidt-Wellenburg,Phys. Rev. D , 113002 (2018).[27] X.-F. Han, T. Li, L. Wang, and Y. Zhang, Phys. Rev. D , 095034 (2019).[28] S. Peil and G. Gabrielse, Phys. Rev. Lett. , 1287(1999).[29] L. S. Brown, Phys. Rev. Lett. , 2013 (1984).[30] L. S. Brown, Ann. Phys. (N.Y.) , 62 (1985).[31] X. Fan and G. Gabrielse, (Manuscript in preparation.)(2020).[32] L. Landau, Zeitschrift f¨ur Physik , 629 (1930).[33] L. S. Brown and G. Gabrielse, Rev. Mod. Phys. , 233(1986).[34] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Sci-ence , 547 (1980).[35] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sand-berg, and M. Zimmermann, Rev. Mod. Phys. , 341(1980).[36] V. B. Braginsky and F. Y. Khalili, Rev. Mod. Phys. ,1 (1996).[37] R. Van Dyck, Jr., P. Ekstrom, and H. Dehmelt, Nature , 776 (1976).[38] G. Gabrielse, H. Dehmelt, and W. Kells, Phys. Rev. Lett. , 537 (1985).[39] G. Lindblad, Comm. Math. Phys. , 119 (1976).[40] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,Journal of Mathematical Physics , 821 (1976).[41] G. Gabrielse and H. Dehmelt, Phys. Rev. Lett.55