Stochastic theory of optically detected single-spin coherent phenomena

PHYSICAL REVIEW B

VOLUME 56, NUMBER 1

1 JULY 1997-I

Stochastic theory of optically detected single-spin coherent phenomena: Evidence for non-Markovian dephasing of pentacene in p-terphenyl S. Ya. Kilin and A. P. Nizovtsev Institute of Physics, Belarus Academy of Science, F. Skarina Avenue, 70, 220602 Minsk, Belarus

P. R. Berman University of Michigan, Ann Arbor, Michigan 48109-1120

J. Wrachtrup and C. von Borczyskowski Institute of Physics, Technical University Chemnitz, D-09107 Chemnitz, Germany ~Received 7 January 1997! The optically detected coherent response of a single chromophore molecule in an organic host matrix to a microwave radiation resonant to a transition between triplet spin substates is analyzed using a model in which N independent random telegraph processes produce fluctuations of the spin transition frequency. Measurements of ~i! Hahn echoes, ~ii! power-broadened line shapes, ~iii! transient nutation, and ~iv! second-order correlation functions for single pentacene molecules in a p-terphenyl crystal are explained within the context of the model, assuming the fluctuations to be slow. The failure of the standard Bloch equations for this system is demonstrated and the effects of microwave-suppressed dephasing are discussed. @S0163-1829~97!03625-4#

Optically detected magnetic resonance ~ODMR! allows one to monitor the changes an individual triplet electron spin of a single chromophore molecule embedded in an organic matrix undergoes as a result of its interaction with a microwave ~MW! field.1,2 This technique has been applied successfully to study the line shapes of transitions between different triplet substates1–4 and triplet population and depopulation kinetics4 for single pentacene ~Pc! molecules in a p-terphenyl ~PT! crystal at 1.4 K. Additionally, quantum jump phenomena, resulting from shelving of the fluorescing single molecule in a metastable triplet state, have been observed and the second-order fluorescence intensity correlation function g (2) ( t ) has been measured for this guest-host pair.4,5 In Refs. 3, 4, and 6 it has been demonstrated also that, owing to the long-time scale character of ODMR experiments in which time averaging replaces the usual ensemble averaging,3 this technique may be used to observe coherent transient phenomena such as transient nutation and two-pulse Hahn echoes on a single triplet spin. Preliminary data on the rate of electron spin dephasing in the Pc1PT system have been obtained. Thus it is now possible to study single-spin coherence decay, whose behavior is very sensitive to the local environment. At 1.4 K the main source for the spin coherence losses is the mutual flip flops of the host nuclear ~proton! spins7 which cause stochastic fluctuations U t of the triplet spin resonance frequency through the hyperfine interactions. The fluctuations are expected7 to be slow: s * n , where s is the variance of the fluctuations U t and n is the average proton flip-flop rate. Under these conditions the conventional description of a dephasing process in terms of a dephasing time T 2 , adopted in the standard Markov-type Bloch equations ~BE’s!, becomes invalid and a more detailed description is needed ~see, e.g., Refs. 8–10 and references therein!. Here we extend the previous BE-based study4 of coherent transients in Pc1PT to include non-Markovian effects. 0163-1829/97/56~1!/24~4!/$10.00

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As in Ref. 4 we treat a single molecule as a five-level system in which u 0 & is the ground singlet state S 0 , u 1 & is the first excited singlet state S 1 , and u T & (T5X,Y ,Z) are three substates of the lowest triplet state T 1 . A zero-phonon optical transition 1-0 is driven by a saturating cw laser field while the transition X-Z is in near resonance with a cw or pulsed MW field. From state 1 the molecule may fluoresce back to state 0 with rate A or escape into triplet substates T with rates k 1T as a result of intersystem crossings ~ISC’s!. The ISC’s also result in relaxation of T substates to 0 state with rates k T0 . Due to the selectivity of the ISC process, the rates k 1T and k T0 are inherently different for different substates T. Typically A@k 1T , k T0 and a molecule undergoes excitation-emission cycles many times prior to a crossover to one of the triplet substates. As a result, the fluorescence photons are grouped in bunches of mean length ;2/k 1T ~Ref. 5! separated by completely dark intervals whose durations are equal to the residence times of the molecule in the triplet state. In the case of a Pc molecule in a PT crystal, the rates k 1X and k X0 exceed the others k 1X >k 1Y @ k 1Z , and k X0 >k Y0 @ k Z0 ~Ref. 4! and, as a result, the steady-state population r `XX of the X substate in the absence of the MW field and in the presence of optical saturation is significantly larger than the populations of the other states: r `XX .0.46, r `Y Y .0.2, ` r ZZ .0.03, and r `11.0.14. The effect of the MW field is to transfer the population from the state X to the longer-lived state Z, thus lengthening the lifetime of the molecule in the triplet state and decreasing the average fluorescence intensity I(t); r t11 . Moreover, the MW field mixes the states X and Z into a coherent superposition described by the density mat trix element r XZ . Fluctuations U t of the X-Z transition frequency destroy the coherence which manifests itself as a ~homogeneous! broadening of the corresponding ODMR line shape or as a decay in the transient ODMR response. These 24

© 1997 The American Physical Society

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manifestations depend critically on whether or not the correlation time t c ;1/n of the fluctuations U t is small in comparison with the characteristic time of molecule–MW-field interactions, determined by the MW Rabi frequency 2w.8–10 In the case of fast fluctuations (2w t c !1,s t c !1) the MW field and the fluctuations act in an additive fashion and ODMR phenomena can be described in terms of the usual BE’s. In the opposite limit of slow fluctuations U t (2w t c >1,s t c *1) the MW field can modify the dephasing process. For high MW power, the fluctuation-induced decoherence ~dephasing! can be completely suppressed by the MW field interaction, an effect analogous to ‘‘spin locking’’ in the ODMR spectroscopy of an ensemble of spins ~see, e.g., Ref. 15!. As far as the ODMR technique consists of monitoring the fluorescence for a long period of time to obtain information about the MW-driven transition, the observed time-averaged fluorescence signal actually corresponds to the average over a history of stochastic process U t , which according to the ergodic principle is equivalent to conventional ensemble averaging. In transient ODMR experiments, the same sequence of MW pulses ~each constituting a ‘‘single’’ experiment! is repeated many times and the above time averaging corresponds again to the averaging of a single experiment over history of process U t . Moreover, all possible configurations of the Pc molecule’s proton spins, which are frozen during the triplet state lifetime, but can vary when the molecule returns to the singlet state, are sampled during the long observation time;1 thus, a distribution P( v XZ ) of the respective hyperfine-interaction-shifted frequencies v XZ of the resonant transition X-Z must be included in the theory of ODMR phenomena. Note that the distribution P( v XZ ) provides a single-molecule equivalent of the inhomogeneous broadening one encounters in ODMR spectroscopy for an ensemble of molecules. For Pc, P( v XZ ) is strongly asymmetric, having a width ;5 MHz.16,17 The starting point of our calculations is a matrix stochastic Liouville equation of the form r˙t 5Drt 2iFU t rt 1r0 ,

~1!

where the vector rt is composed of the density matrix elements r tnm and has the components r ti 5 r t10 , r t11 , r t00 , r t01 , t t t r tY Y , r XZ , ( r tXX 2 r ZZ )/2, r ZX (i51, . . . ,8). This equation is written in a reference frame rotating at both the optical and microwave frequencies. The matrix D, whose elements are the detunings « and d and Rabi frequencies 2 v and 2w of the optical and MW fields, respectively, and the population and depopulation rates k 1T and k T0 of the triplet sublevels T, is the same matrix which appears in the conventional BE’s in the absence of fluctuation-induced dephasing of the X-Z coherence ~see, e.g., Ref. 4!. The matrix F ~whose only nonvanishing elements are F 6651 and F 88521), multiplied by U t , accounts for the effect of fluctuations on the coherences t t r XZ and r ZX . An inhomogeneous term r0 appears in Eq. ~1! t 512( r t111 r t001 r tY Y ). since we have set r tXX 1 r ZZ The observed fluorescence intensity I(t) is proportional to the state population u 1 & ^ r t11& , averaged over realizations of the stochastic variable U t . To calculate it we need to construct the equation for the averaged vector Rt 5 ^ rt & . Supposing as in Ref. 11 that the fluctuations U t can be written as the

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sum U t 5 ( U tk of N independent random telegraph ~RT! processes U tk with the same jump rate n but different variances s k it is possible to obtain12 an exact set of 2 N equations ˙ t 5DRt 2iF R

˙ tk 5 ~ D2 n ! Rtk 2iF R

(k Rtk 1r0 ,

~2a!

( Rtk,l 2i s 2k FRt ,

~2b!

lÞk

... ˙ t1, . . . ,N 5 ~ D2N n ! Rt1, . . . ,N 2iF R

(k s 2k R1,t . . . ,k21,k11, . . . ,N , ~2c!

for the quantities Rt , Rtk 5 ^ U tk rt & , . . . , Rt1, . . . ,N 5 ^ U t1 •••U tN rt & . In the case of equal s k the N RT model is equivalent to the pre-Gaussian model.13 Note also that Eqs. ~2! can be written as the equations for the marginal averages Rlt , . . . ,l ~see, e.g., Ref. 14!, i.e., for the partial averages of 1 N rt over only those histories of RT processes U tk which end at time t with some specific values l k (56 s k ). In the fast fluctuations limit when n , s →` but lims 2 / n 5G ( s 5 A( s 2k ), one can neglect the high-order correlations Rtk,l , . . . ,Rt1, . . . ,N in these equations and then, after substituting the formal solution of Eq. ~2b! into Eq. ~2a!, replace Rt2 t in the time integral by Rt ~Markov approximation!. The last approximation is valid only under the conditions 2w, d ! n , s , when it is possible to neglect the density matrix evolution on the time scale t c ;1/n . As a result, Eqs. ~2! are reduced to the BE with dephasing rate g 5G1(k X0 1k Z0 )/2. In this fast fluctuation limit the role of the fluctuations U t is the same as that of a single RT having variance s . In the opposite limit of slow fluctuations s * n , the Markov approximation is no longer valid and the sum of RT processes can no longer be replaced by a single RT with some effective variance s . In this case the results of calculations based on Eqs. ~2! can be dependent on the number N of the RT processes, especially for small values of N and for low MW Rabi frequencies; for increased MW power the fluctuation-induced dephasing becomes to be dependent on 2w and d . Equations ~2! have been used to analyze the experimental data on ~i! Hahn echoes,6 ~ii! power-broadened line shapes, ~iii! transient nutation,3 and ~iv! the correlation function g (2) ( t ).4 The parameters used in the calculations were those appropriate to the Pc1PT system.4 The X-Z transition line shape, observed in ODMR as a dip in the fluorescence intensity when the MW field frequency is swept through X-Z resonance, is proportional to the steady-state population R`2 5 ^ r `11( d ) & . In the case of echo and nutation ODMR signals, one must solve Eqs. ~2! for the population ^ r T11& at time T following a single series of MW pulses, taking as initial condition the steady-state solution R` of Eqs. ~2! in the absence of the MW field. The time T coincides with the MW pulse temporal width in the case of nutation while T5 t p /21 t 1 1 t p 1 t 1 1 t p /2 in the case of an ODMR echo excited by p /2 and p pulses separated by a time delay t 1

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FIG. 1. Decay of the Hahn echo of a single Pc triplet electron spin calculated using BE’s with T 2 52 m s ~curve 1! and 3 RT model with n /2p 530 kHz and s /2p 5250 kHz ~curve 2! in comparison with the experimental data of Ref. 6; t 1 is the delay time between p /2 and p pulses.

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FIG. 3. Transient nutations of the X-Z transition calculated using BE ~curve 1! and N RT model theory ~curve 2! in comparison with the experimental data of Refs. 3. MW Rabi frequency is 4.6 MHz.

and read out by an additional p /2 pulse which follows the second pulse after a delay time t 1 .15 We calculate the t observation-time-averaged population r T115 * 0obsdt ^ r T1t 11 & / T1t t obs , where ^ r 11 & is the MW free evolution of ^ r 11& after the last MW pulse in the series and t obs is the time duration during which the fluorescence photons are counted ~typically t obs51000m s). The correlation function g (2) ( t ) was calcut lated as the ‘‘conditional’’ population ^ r (0) 11 & of the excited singlet state 1 at time t , i.e., its population at time t , provided that at t 50 the molecule was in the ground singlet state 0, normalized to the steady-state population ^ r `11& : ` 4,5 t Finally, in all cases we have integ (2) ( t )5 ^ r (0) 11 & / ^ r 11& . grated ^ r 11& over v XZ with the distribution P( v XZ ): g r 115 * d v XZ ^ r 11& P( v XZ ). It is this double-averaged quantity g r 11 which was used to fit the calculations to the experiments. It is instructive to attempt a fit to the experimental data using the conventional BE. Our calculations have shown that the observed Hahn echo decay may in principle be fitted ~Fig. 1, dotted curve 1! in terms of the BE’s using for the pure dephasing rate G of the X-Z transition the value G553105 s 21 (T 2 52 m s!. However, at high MW power,

other ODMR phenomena could not be fit using the same value for G. This is shown in Figs. 2–4, where the BE prediction ~dotted curves 1! deviate significantly from the experimental data. The BE’s predict a much broader ODMR X-Z transition line shape, faster decay of transient nutation, and faster decay of the long-time tail of g (2) ( t ). To provide consistent fits to the experimental data we have solved Eqs. ~2! using up to N54 RT processes. The value 30 kHz has been used for the jump rate n /2p .6,18 Other parameters of the model, the component variances s k and the total variance s , were the adjustable parameters. Figure 1, curve 2, shows the Hahn echo signal calculated using the 3 RT model with s /2p .250 kHz and s k 5 s / A3. At fixed s , the signal was practically independent on the specific component variances s k . The reasonable agreement of the calculated curve with the experimental data of Ref. 6 indicates that the dephasing fluctuations U t in the Pc1PT system are indeed slow, consistent with previous discussion.7 Using the same parameters we also calculated the ODMR X-Z transition line shape, the nutation signal, and the correlation function g (2) ( t ). At low MW Rabi frequencies 2w! n & s the homogeneous line width ~HWHM! H is . s while the total line shape is determined by much wider ‘‘inhomoge-

FIG. 2. Calculated and experimental ODMR line shapes of a single Pc molecule triplet spin at high MW power in the region of the X-Z transition. Curve 1 is the BE predictions; curve 2 is the calculations within N RT model theory. MW Rabi frequency is 15 MHz.

FIG. 4. Second-order correlation function of the fluorescence intensity h( t )5g ( 2 ) ( t )21 calculated using BE ~curve 1! and N RT model theory in comparison with the experimental data of Ref. 4 corrected for stray light. MW Rabi frequency is 5 MHz.

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neous’’ distribution P( v XZ ). Note that the value s /2p .250 kHz, taken for the echo experiment, is in good agreement with the homogeneous line width ;200 kHz estimated in Ref. 4 from the slope of the sharp low-frequency edge of the ODMR line shape observed at low MW power. The calculated homogeneous linewidth H increases rapidly with increasing MW Rabi frequency due to the well-known saturation-broadening effect, but the growth rate decreases when 2w* n , s . At very high MW power, when 2w is comparable with the width of distribution P( v XZ ), the calculated total line shape coincides with that of the homogeneous one and is in good agreement with experiment ~Fig. 2, curve 2!. Note that the simple analytical formula for H:H.2w @ ( g /2k Z0 (11 r 011k 1X /k X0 ) # 1/2, which can be obtained from the BE’s in this case, agrees with experiment only after substituting g .(k X0 1k Z0 )/2, i.e., replacing the total dephasing rate g by its nonadiabatic part only. This indicates clearly that the pure dephasing induced by RT processes is completely suppressed by a strong MW field. The same situation occurs in the transient nutation experiment,3 where the MW field is also sufficiently strong to suppress almost completely the dephasing induced by the RT processes. Calculations have shown ~Fig. 3, curve 2! that the experimental nutation signal decay is determined by the distribution P( v XZ ) only. Finally, our calculations of the fluorescence intensity second-

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J. Kohler, J. A. J. M. Disselhorst, M. C. J. M. Donckers, E. J. J. Groenen, J. Schmidt, and W. E. Moerner, Nature ~London! 363, 242 ~1993!. 2 J. Wrachtrup, C. von Borczyskowski, J. Bernard, M. Orrit, and R. Brown, Nature ~London! 363, 244 ~1993!. 3 J. Wrachtrup, C. von Borczyskowski, J. Bernard, M. Orrit, and R. Brown, Phys. Rev. Lett. 71, 3565 ~1993!. 4 R. Brown, J. Wrachtrup, M. Orrit, J. Bernard, and C. von Borczyskowski, J. Chem. Phys. 100, 7182 ~1994!. 5 J. Bernard, L. Fleury, H. Talon, and M. Orrit, J. Chem. Phys. 98, 850 ~1993!. 6 J. Wrachtrup, C. von Borczyskowski, J. Bernard, R. Brown, and M. Orrit, Chem. Phys. Lett. 245, 262 ~1995!. 7 C. A. Van’t Hof; and J. Schmidt, Mol. Phys. 38, 309 ~1979!. 8 A. G. Redfild, Phys. Rev. 98, 1787 ~1955!. 9 P. R. Berman and R. G. Brewer, Phys. Rev. A 32, 2784 ~1985!. 10 P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, J. Appl.

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order correlation function g (2) ( t ) ~Fig. 4, curve 2! are in qualitative agreement with observations4 which show a slowing down of the decay rate in the long-time tail of g (2) ( t ) in the presence of strong MW field. The change in the decay rate results from a MW-field-induced modification of the dephasing associated with the RT processes and a corresponding change in the saturation behavior of the ‘‘inhomogeneous’’ line P( v XZ ). In conclusion, we have developed the N-RT-model-based stochastic theory of a coherent ODMR phenomena for a single guest chromophore molecules in an organic host matrix and have demonstrated that it provides a consistent fit to a wide range of experimental data for the pentacene1 p-terphenyl pair. The conventional Bloch equations fail for this system due to the fact that the dephasing frequency fluctuations of the resonant triplet electron spin of pentacene molecule resulting from proton flip flops in the host crystal are slow in the sense of the wellknown Kubo classification. As a consequence, the MWfield-suppressed dephasing that can arise for a finite flip-flop correlation time must be taken into consideration when analyzing ODMR phenomena. This research was supported by the National Science Foundation under Grant No. PHY-9414515 and the Volkswagenstiftung under Grant No. I/72 171.

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