The Quantum Zeno Effect – Evolution of an Atom Impeded by Measurement
1 June 2000
Optics Communications 180Ž. 2000 115±120 www.elsevier.comrlocateroptcom
The quantum Zeno effect ± evolution of an atom impeded by measurement Chr. Balzer, R. Huesmann, W. Neuhauser, P.E. Toschek ) Institut furÈÈ Laser-Physik, UniÕersitat Hamburg, Jungiusstrasse 9, D-20355 Hamburg, Germany Received 11 April 2000; accepted 14 April 2000
Abstract
The evolution of a quantum system is supposed to be impeded by measurement of an involved observable. This effect has been proven indistinguishable from the effect of dephasing the system's wave function, except in an individual quantum system. The coherent dynamics, on an optical E2 line, of a single trapped ion driven by light of negligible phase drift has been alternated with interrogations of the internal ion state. Retardation of the ion's nutation, equivalent to the quantum Zeno effect, is demonstrated in the statistics of sequences of probe-light scattering `on' and `off' detections, the latter representing back-action-free measurement. q 2000 Published by Elsevier Science B.V. All rights reserved.
The act of measuring an observable of a system ions confined in an ion trap has resulted in complete that obeys quantum mechanics consists of recording agreement with quantum-mechanical predictionswx 6 . one of the eigenvalues and rejecting all the other However, these predictions based on the deletion, in ones. This act is accompanied by sudden transition the acts of measurement, of all superpositions of of the system's wave function into the eigenfunction eigenstates can be identified with the effect of any corresponding to the recorded eigenvalue; the re- phase perturbations by the environment upon the sponse of the system is known as the `state reduc- multi-particle wave function of the systemŽ `dephas- tion'wx 1 . It has been recognized that repeated mea- ing'. . In fact, a perturbation via the back action of surements retard, or even impede, the evolution of a the meter on the quantum system has been invoked quantum system to the extent that they may inhibit as the origin of QZEwx 7±10 . The ambiguity of the the evolutionwx 2,3 . This consequence, the `quantum initial t 2 evolution being set back by the measure- Zeno effect'wx 4 alluding to eleatic ontology, has mentswx 2,3,11 , or dephasing, ± i.e. effect of mea- aroused a great wealth of work devoted to contem- surement vs dynamical effect ± is unresolvable since plating the subjectwx 5 , and an attempt to observe it: a decision would require knowledge of the states of An experiment including the drive and probe laser all the membersŽ. the `micro-state' of any ensemble irradiation of an ensemble of some 5000 beryllium that remain unknown in a global measurement. Here, both the result of a particular measurement, and the temporal evolution of the ensemble's state, do not statistically depend on the results of previous mea- ) Corresponding author. Tel.: q49-40-42838-2381; fax: q49- surements; they are deterministic, save the `projec- 40-42838-6571; e-mail: [email protected] tion noise'wx 12,13 that affects measurements of
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S0030-4018Ž. 00 00716-1 116 Chr. Balzer et al.rOptics Communications 180() 2000 115±120 non-commuting observables and vanishes with a large light generated by frequency-doubling the light of an enough ensemble. However, with an individual sys- 822 nm diode laserŽ. Fig. 1 . The spectral width of tem, the result of a measurement as well as the the blue output did not exceed 500 Hz, with 1s of system's evolution do statistically depend on the integration. The pulses of this drive light alternated history, and the results are in general found indeter- with pulses of 369 nm probe light, generated by ministic, except after particular preparation of the frequency-doubling the output of a cw LD700 dye system, in an eigenstate of the observable to be laser. The probe light excites some 108 eventsrsof wx y detected 13 . The statistics of the results will em- resonance scattering on the S1r21P r2 line of the body the signature of the state reductions by the ion only if the ion is found in the S1r2 ground state measurements, and their effect cannot be ascribed to after the driving pulse has been appliedwx 19 . The dephasingwx 14,15 . This argument has been quanti- probe pulses were 10 ms long such that clear-cut fied, by numerical calculation, for a single spin-like presence or absence of resonance scattering represent quantum system interacting with a light mode whose measurements of the ion's internal state: Scattered photon number is measured, and for a corresponding light proves the ion to reside in the ground state ensemblewx 16 : With a single quantum system, the immediately after the detection of each photon, the evolution is not revealed by reiterated measurements. absence of light scattering instead reduces the ion to
The state of this system is reduced to an eigenstate, the D5r2 state. by each precise enough measurement, according to The decay of the ion's D5r2 state into the wx the result of the measurement. As long as the evolu- metastable state F7r2 of extreme lifetime 20 com- tion of the quantum system between two subsequent plicates the dynamics on the driven E2 line. How- reductions is coherent, there is no base for invoking ever, we kept the ion continuously irradiated by the dephasing. This is so because the configuration space cw output light of a diode laser at 638 nm that y2wxr of, say, an individual spin system extends only over completely saturates the F7r255 2r2 line of the the surface of the unit sphere in SU2 symmetry, and ion in order to immediately repump the ion from the the micro-state of this system is completely known F7r21level into its S r2 ground state. from the result of a measurement, in contrast with A measurement based on light scattering absent that of an ensemble of spins. Ž.`off' , i.e., when the ion is in its D5r2 state, extends In the experiment ofwx 6 , the state of the system no physical reaction on the `quantum object'Ž i.e., has been interrogated only after sequences of `mea- the spin represented by the E2 transition. , it is of the surement' pulses, such that back-and-forth transitions quantum non-demolition typewx 10 :Ž. i Both the quan- go unnoticed and falsify the probability of survival tum object and the `quantum probe'Ž the dipole on wx17 . In contrast, here the result of each measurement the resonance line. return to their initial states after is registered. We have demonstrated the retardation of the light-driven quantum evolution of an individual, lo- calized cold ion upon repeated reduction by intermit- tent probing the ion's two energy eigenstates in- volved in the driven resonance. A single ion, 172 Ybq, was localized in the node of the electric field of a 2-mm-sized electrodynamic trap in ultra-high vac- uumwx 18 . The ion was centred in the pseudo-poten- tial of the trap with less than 30 nm error, and laser-cooled to the Doppler limitŽ mean vibrational state ²:n ,10 . in order to minimize the driven micro-motion, and the free harmonic secular motion, respectively. The E2 transition S yD was co- q 1r25r2 Fig. 1. Energy levels and active transitions of 172 Yb ion, and herently driven, during time intervals ts2 ms, by s s program of alternating drive and probe light. t 2 ms, tp 10 shining upon the ion 411 nm quasi-monochromatic ms. Chr. Balzer et al.rOptics Communications 180() 2000 115±120 117 the measurement.Ž. ii The probe light does neither cause any dissipation nor back action in the combi- nation of quantum object and quantum probe. The state of this probe is indirectly measured by the null detection of scattered light, with zero recoil upon the ion. This state is correlated with the upper one of the two alternative eigenvalues of the observable to be measured, the internal energy of the quantum object. Probe-light scattering makes the ion recoil. Its net effect is spatial and velocity fluctuations, and a random phase shift upon the quantum object, the driven quadrupole. However, the ion remains cooled Fig. 2. Part of trajectory of results of measurements each of which deep inside the Lamb±Dicke regimeŽ excursion < consists of a drive and a probe pulse applied to the ion. light wavelength. , and the corresponding phase vari- ations do not exceed a small fraction of p. The condition for strong trappingwx 21 holds with the driven E2 line, such that any net recoil of the When this data acquisition is repeated either at var- drive light is indeed absorbed by the trap since the ied length t of the drive pulse, or at stepwise ion's vibrational frequency, 1.3 MHz, far exceeds progressive detuning of the drive frequency v, the the natural line width. corresponding mean rate oscillates as a result of The crucial issue is the capability, of the two sampling the effective Rabi nutation frequency urt kinds of measurements, to distinguish retarded evolu- Ž.Fig. 3 . The observation of this modulated excita- tion from the effect of dephasing. The phase of the tion spectrum proves the coherent nature of the drive light is found to diffuse, in each 2 ms interval interaction of ion and drive light. Note the first-order of irradiation, by such a small fraction of p only, sideband, at 1.3 MHz up-tuning, generated by resid- that there is no risk of quenching the coherence. The ual vibrational phase modulation of the ionic ion's super-position state would be conserved during quadrupole. the breaks of the drive, as long as it is not measured, Let us turn to the statistics of trajectories, as in y1 for t-G s the decay time of the inversion. State Fig. 2. When starting with the ion in the ground reduction by each subsequent probing yields random state, another `on' event takes place with probability results, and the coherent dynamics is retrieved only s 2 r p0 cos Ž.Vt 2 . When starting with the ion in the when averaging over an ensemble of measurements, metastable state, an `off' result would take place after identical preparation. ± Preservation of the s with same probability, p10p , if we neglect relax- ion's vibronic state under probing the states of an ation, for the moment. In each driving pulse, the electronic resonance has permitted detection of the light-driven nutation starts anew thanks to the state corresponding nutational dynamics by a stroboscopic reduction by the previous probing, and it extends wx or `sampling' technique 22 . over the pulse duration t. The next probe pulse Fig. 2 shows a trajectory of `on' and `off' events. reduces the ion again to one of the two energy The number of onroff pairs accumulated over 500 eigenstates. Then, the probability of finding either measurements and normalized by the number of `on' `on' Ž.is0 or `off' Ž.is1 q times in a sequence, is events yields the probability of excitation, on the E2 UqŽ .sU Ž.1 Vq Žy1, . Ž. 2 line, to the metastable D5r2 level. With negligible relaxation, this transition probability is where s 2xP 2 ur q 2 q p01 cos sin Ž.2, Ž. 1 VqŽ.sp scos ŽVtr2. . Ž. 3 where tan xsDrV, us'V 22qDt, and V and In contrast, with state reduction lacking the ion's s y D v v 0 are Rabi frequency and detuning of the evolution would continue coherently over the total driving light Ž.v off its resonance v 0 , respectively. time of driving, as long as the laser-induced 118 Chr. Balzer et al.rOptics Communications 180() 2000 115±120
the drive laser. A simple model of phase diffusion wx s r 23 yields gph D 2. The diffusion constant D is related to the phase variance as ²:Ž.dw 2 sDt , such that the standard deviation of the drive's phase is dws(²:Ž.dw 2 s''22ayb . The S1r2 ground state is resolved in two Zeeman-split sublevels one of which only is excited s r by the drive light, leaving f0 1 2, whereas the 2 s less resolved upper state D5r21suggests f 1. From a fit of p01 to its contrast of modulation, i.e., to the extreme values of the data sets 7 and 9 close to s resonance in Fig. 3, one derives the values B0 0.49 and aqb,0.38. The approximate phase of nuta- tion achieved by the 2 ms-long driving pulse is derived from the contrast on the wing of the power- broadened resonanceŽ. data sets no. 15 and 16 using n yn , ( = Fig. 3. Probability of excitation, vs detuning o by 20 kHz Eq.Ž. 1 : uapp Vt 578 2p. A better value is steps of the driveŽ. top . Note the first-order vibronic sideband at revealed by the increment of nutation per each step 1.3 MHz. Within a small range close to resonance, detuning of detuning, dv s 20 kHz = 2p , that is du replaces variation of the drive-pulse length t. The spectrum of 2 absorption is superimposed by stroboscopic sampling of the ion's s(u 2 q Ž.tdv y u s 0.5p mod 2p. Compatible Rabi nutation, as demonstrated by a simulation with small steps s = s q X with uapp is only du 1.25 2p , and u 2p n u , on an expanded scaleŽ. bottom . where the integer n,640,and u X is a fraction of 2p. The numbers of sequences made up of q identical results have been evaluated from each trajectory, and quadrupole moment survives Žqt-gy1 s time of they are identified with UqŽ.. Fig. 4 shows data of residual dephasing. . This dynamics would require UqŽ.rU Ž.1 , recorded at the indicated three settings VqŽ.scos2 Ž.qVtr2. Actually the ion dynamics is modified by relax- ation: The two probabilities pi must includeŽ. i spontaneous decay, andŽ. ii phase Ž `transverse' . re- laxation of the ionic quadrupole. The Bloch equa- tions for a spin system include these processeswx 23 . From an analytic solution on resonance Ž.Ds024wx one derives s y y q 2e yŽ.aqb uye piii1 fBž/1 (1 tan ie cosŽ. i, Ž.4 s y y 2rw 2 q where tane 1 ŽŽ.a b 2 b Vt Ž.Vt x r s q r s V 2r2 8 ab . u, tane001Ž.a b u, and B V 2 q Gg , B s y s s q r s 1 B0ph,2a gt gt Ž.G 2 t ,2b Gt, u 2 s Ž.ŽVt 2 y ayb .2. The supplementary factor fi is unity for non-degenerate levels, the net proba- s y Fig. 4. Probability UqŽ.rU Ž.1 of uninterrupted sequences of q bility of excitation is p011 p 0, and g ph is the rate of phase diffusion, and Gr2 is the decay rate of `on' resultsŽ. white dots and `off' results Ž. black dots . The lines show the distributions of probabilities VqŽ.y1 for the ion's state D . X 5r2 evolution on its drive transition, according to Eqs.Ž. 2 and Ž. 3 . u The probabilities p and p no longer agree. The - 01 and f11from fit; values f 1 indicate redistribution, over sub- rate gph results from residual phase fluctuations of levels, by cycles of spontaneous decay and reexcitation. Chr. Balzer et al.rOptics Communications 180() 2000 115±120 119 of the frequency detuning close to the principal peated measurements on the system, i.e., the quan- resonance of Fig. 3. It shows also values of VqŽ.y1 tum Zeno effect, by alternately driving and probing 172 q calculated from the solutions of p01and p , and an individual Yb ion on a weak Ž.E2 and a multiplied by the factor Ž.Nyqq1 rN that takes strongŽ. resonance transition, respectively. We have care of the finite length of the data sets. From the fit revealed the statistics of uninterrupted sequences of to the `on' sequences of data set 7, aqbs0.395 observations which find the ion either in the ground and u X sŽ1q10y4 .p are derived. The constant of state subjected to resonance scattering of the probe total relaxation is inserted when fitting the `on' light, or in the D5r2 state that does not show reso- statistics of the neighbouring data sets for their frac- nance scattering since it is decoupled from the probe tional phases u X. The same phase values u X at- resonance. The evolution derived from this statistics tributed to the `off' statistics require f1 decreasing agrees with the evolution of the ion's wave function from unity with increasing transition probability p10 : assumed interrupted by state reductions, and the Cycles of spontaneous decay and reexcitation in- retardation cannot be attributed to dephasing. The creasingly contribute to the `off' sequences and re- frustrated attempts of detecting fluorescence qualify distribute the ion over the D5r2 sublevels such that as quantum non-demolition measurements of the in- potential deexcitation becomes selective, and f1 is ternal ion energy. The statistical distribution of se- expected to approach 1r6. quences of such measurements identifies the degree The agreement of UqŽ.rU Ž.1 and Vq Žy1 . con- to which the measurements, intertwined among the firms state reduction by each measurement that goes pulses of driving, have impeded the quantum evolu- along with probing the ion's resonance scattering, tion of the ion. and it proves the coherent evolution of the ion to set out again after each of these state reductions. The statistics of `no count' sequences provides such a Acknowledgements proof even under QND conditions. The overall effect of the reiterated resettings is the ion's quantum dynamics being impeded. This work was supported by the Korber-Stiftung,È Since u42p , the resonance of Fig. 3 is substan- Hamburg, by the Hamburgische Wissenschaftliche tially power-broadened, and the other natural mode Stiftung, and by the ZEIT-Stiftung, Hamburg. of relaxation given by ayb cannot be determined. However, aqb is compatible with the lifetime of wx wx the D5r2 levelŽ 5,7 ms 20 , 7,2 ms 25. . It serves References also as an upper limit of a: For the standard devia- tion of the driving phase, the limit yields dw-1.2 wx1 J. von Neumann, Mathematische Grundlagen der Quanten- <2p. This phase fluctuation corresponds to the mechanik, Springer, Berlin, 1932. wx2 L.A. Khalfin, Pis'ma Zh. Eksp. Teor. Fiz. 8Ž. 1968 106, bandwidth less than 100 Hz, which is compatible w x dns JETP Lett. 8Ž. 1968 65 . with the controlled 1-s frequency bandwidth, wx3 L. Fonda, G.C. Ghirardi, A. Rimini, T. Weber, Nuovo dwrtQ500 Hz. 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