- Email: [email protected]
Coherent superposition of dressed spin states and pulse dressed electron spin resonance
5 March 1999
Chemical Physics Letters 301 Ž1999. 524–530
Coherent superposition of dressed spin states and pulse dressed electron spin resonance Gunnar Jeschke
)
Max-Planck-Institut fur ¨ Polymerforschung, Postfach 3148, D-55121 Mainz, Germany Received 1 December 1998
Abstract Coherent superpositions of spin eigenstates in a strong near-resonant electromagnetic field can be prepared by pulses of a secondary linearly polarized electromagnetic field, which is parallel to the static magnetic field. Transient nutations on the dressed state transition and dressed state coherence echoes are observed in a two-level system. A two-dimensional correlation experiment for dressed state transitions is introduced. It is shown that the electron Zeeman frequency can be determined with higher precision in a dressed spin resonance experiment than in a conventional electron spin resonance experiment. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The eigenstates of a quantum system change drastically if the system is brought into a strong electromagnetic field that is near-resonant with one of the quantum transitions. To treat this situation for atoms in a laser light field, Cohen–Tannoudji et al. have defined a new quantum system that includes both the original bare quantum states and the field w1,2x. The eigenstates of such a laser light dressed system are called dressed states. An analogous situation occurs in magnetic resonance experiments during strong radio frequency Žrf. or microwave Žmw. irradiation, except that spontaneous emission is no longer relevant. The energy difference between dressed states can be by orders of magnitude smaller than between the corresponding bare states, leading to a completely different regime of interactions with the environment. The most prominent example of this is probably Hartmann–Hahn cross-polarization between nuclear spins w3x, which can be considered as cross-relaxation between two pairs of dressed spin states. It may thus be of considerable interest to study the interactions of dressed states with their environment in more detail. A similar endeavor for bare spin states has greatly benefited from experiments that are based on the generation and manipulation of coherent superpositions of the states w4,5x. In this Letter we demonstrate that coherent superpositions of dressed spin states can be prepared by pulses of a linearly polarized electromagnetic field, which is parallel to the static magnetic field. We observed dressed
)
Fax: q49-6131-379100; e-mail: [email protected]
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 0 4 1 - X
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
525
spin transient nutations and dressed state coherence echoes. Dressed state coherence of electron spins in the E1X center in g-irradiated quartz glass is found to decay with a relaxation time similar to the rotating frame relaxation time T1r . For a multilevel system we study a proton spin coupled to the electron spin in an a radical derived from malonic acid. Three dressed spin transitions are observed in this case and are shown to be correlated by two-dimensional Ž2D. pulse dressed electron spin resonance. From the frequencies of these transitions, the electron Zeeman frequency can be determined with high precision.
2. Theory Consider a spin S s 1r2 in a static field B0 along the laboratory frame Z-axis and a linearly polarized electromagnetic field with frequency vem, 1 along the X-axis. The Hamiltonian in angular frequency units is given by H1 s v S SZ q 2 v 1 cos Ž vem , 1 t . S X ,
Ž 1.
where v S is the frequency of the spin transition that depends on B0 , and v 1 characterizes the strength of the electromagnetic field. Only one of the two counterrotating circularly polarized components of this field can be resonant with the spin transition, the other one is far off-resonant. When neglecting the latter component, H1 can be rendered time-independent by transformation to a frame that rotates with frequency vem, 1 about the Z-axis. In the rotating frame, the effective interaction along the Z-axis is reduced to the resonance offset V S s v S y vem , 1. Assume now that a secondary linearly polarized electromagnetic field with frequency vem, 2 is applied along the Z-axis. The rotating-frame Hamiltonian H˜2 is then given by H˜2 s V S SZ q v 1 S x q 2 v 2 cos Ž vem , 2 t . SZ .
Ž 2.
For on-resonant irradiation of the bare spin states Ž V S s 0., H˜2 has the same form as H1 , except for an interchange of coordinate axes. A parallel electromagnetic field should thus act on a dressed spin transition like a perpendicular field does on a spin transition. For off-resonant irradiation of the bare spin states Ž V S / 0., the dressed spin resonance frequency v d is given by vd s Ž v 12 q V S2 .1r2 . Only the component of the electromagnetic field perpendicular to the quantization axis drives the transition, so that the effective field is given by veff s cosŽ u . v 2 , where u s atanŽ V Srv 1 .. In fact, this consequence of the rotating frame description of spins in an electromagnetic field has been recognized immediately by Redfield w6x, who observed dressed spin resonance of nuclear spins by a continuous-wave experiment, which he termed rotary saturation. It should also be possible to prepare a coherent superposition of dressed spin states and observe its evolution. More sophisticated pulsed dressed spin resonance experiments could then be derived from the vast and well investigated body of pulsed magnetic resonance experiments w4x. Despite the similarly of the Hamiltonians, the two physical situations described by H1 and H˜2 are somewhat different. In the former case, the linearly polarized electromagnetic field could be applied along any direction in the XY-plane, while in the latter case all components of the field except for the Z-component are decoupled from the dressed spin system. This difference also prevents reiteration of the argument: transitions between the doubly-dressed states that are the eigenstates of H˜2 cannot be driven by a third electromagnetic field, whatever its direction or frequency. Such an isolation of doubly-dressed states from their environment may be of some interest, since it might lead to unusual relaxation behavior. Note also that it is possible to access the regime where the field v 2 driving the dressed state transitions is of the same order of magnitude as the transition frequency v 1. If excitation of dressed spin transitions is to be described quantitatiÕely in this case, the counterrotating component of the Z-field is no longer far off resonance and thus cannot be neglected.
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
526
Other interesting phenomena occur in two-spin systems. By dressing one of the spins with an electromagnetic field of suitable amplitude, one can create a low-field situation where level anti-crossing effects and strong mixing of the two spins can be observed. This has been shown to be the case for a system consisting of one nuclear spin I s 1r2 with anisotropic hyperfine coupling to an electron spin S s 1r2 w7x. The rotating frame Hamiltonian of this system during mw irradiation of the electron spin transitions is given by H1 S s V S SZ q v 1 Iz q AS z Iz q BS z I x q v 1 S x ,
Ž 3.
where v I is the nuclear Zeeman frequency and A and B characterize the secular and pseudosecular part of the hyperfine coupling, respectively. If mixing between the electron and nuclear spin is weak in the absence of the mw field, the Hamiltonian can be diagonalized by three consecutive unitary transformations neglecting only non-secular parts w7x. By applying the same transformations to the operator SZ that corresponds to the secondary electromagnetic field, we obtain its matrix representation in the eigenbasis of the dressed spin system. We find
U3U2 U1 SZ U1†U2†U3† s
ca
ysj sa
cj ca
0
1
ysj sa
cj2 cb y sj2 ca
sj cj Ž ca q cb .
cj cb
2
cj ca
sj cj Ž ca q cb .
sj2 cb y cj2 ca
sj sb
0
cj cb
sj sb
ycb
,
Ž 4.
where ca s cosŽ ua ., sa s sinŽ ua ., cb s cosŽ ub ., sb s sinŽ ub ., cj s cosŽ jr2., and sj s sinŽ jr2.. The angles have been defined in Ref. w7x. Obviously, the secondary field can induce any dressed state transition except for the double-quantum transition. Even this restriction vanishes, if mixing between the electron and nuclear spin is already substantial in the absence of the mw field. In any case, sufficiently strong Z-pulses can drive enough coherence transfers to obtain correlations between all dressed state transition frequencies within the same spin system. It should thus be possible to obtain and assign dressed spin spectra by means of pulse resonance experiments. Finally we wish to point out that even isotropic couplings can lead to strong mixing between dressed electron spins and nuclear spins, since the dressed spin situation corresponds to a low-field regime. Numerical simulations show that it should be possible to apply dressed electron spin resonance also to the measurement of hyperfine couplings in solution.
3. Experimental For an experimental test of the above considerations and even for performing dressed electron spin resonance routinely, commercial electron paramagnetic resonance ŽEPR. equipment is well suited. With the spectrometer ESP 380e ŽBruker., mw fields can be generated that correspond to dressed spin resonance frequencies v 1 up to 30 MHz. This is within the frequency range of electron–nuclear double resonance ŽENDOR. equipment, for instance, of the pulse ENDOR module ESP360D-P DICE for the same spectrometer and of the ENDOR probe head EN 4118X-MD-4. In our experiments, the mw field at frequency ymw is along the long axis of the probe head Ž X-axis.. The radio frequency Žrf. field would be along the Y axis for ENDOR experiments. However, by turning the probe head about its long axis, any direction of the rf field in the YZ plane can be realized. We characterize this orientation by the angle j between the rf field and the static field Ž Z . axes. For pulse excitation of dressed state transitions, the DICE output was amplified by an ENI 3200 L rf amplifier ŽENI.. In all our experiments, we use the spin-locked echo pulse sequence w3x displayed in Fig. 1a for the excitation and detection of bare electron spins. As a first model sample, we use g-irradiated Suprasil Žquartz glass., which features E1X paramagnetic centres. These centers have been investigated in some detail in crystalline quartz w8x. The overwhelming contribution to the EPR signal in this sample is due to isolated electron spins S s 1r2, since the magnetic isotopes of both
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
527
Fig. 1. Pulse sequences. Ža. Spin-locked echo sequence used for excitation of bare spin states by a transverse microwave field. Žb. Radio frequency Z-pulse. Žc. Radio frequency Z-pulse sequence for observing Hahn echoes and measuring 2D correlation spectra.
silicon and oxygen are rare, and the distance between E1X centers is large. The g anisotropy is small enough to allow for complete excitation of the EPR spectrum in a pulse experiment Ž D g s 9 = 10y4 .. Furthermore, both the longitudinal relaxation time Tl and the phase memory time Tm are conveniently long even at ambient temperature. To observe correlations between dressed state transitions, we use a g-irradiated single crystal of malonic acid that features so-called free a radicals w9x. In these radicals, one proton is adjacent to the carbon atom carrying the overwhelming part of the spin density. This sample resembles to a good approximation a spin system of one nuclear spin I s 1r2 coupled to an electron spin S s 1r2, since the hyperfine couplings of the other protons are by more than one order of magnitude smaller than the one of the a proton. Furthermore, strong mixing between electron and nuclear spins can be obtained for suitable orientations of the crystal.
4. Results and discussion Experiments on the Suprasil sample where performed at a static field B0 s 347 mT corresponds to maximum intensity in the EPR spectrum at ymw s 9.7168 GHz. The first mw pulse with flip angle pr2 generates electron coherence that defocuses during the subsequent interpulse delay t s 584 ns. The magnetization component along the x axis is then spin locked by a high turning angle ŽHTA. pulse of duration t HTA . This spin-locked magnetization corresponds to non-equilibrium polarization on a dressed spin transition. As shown by Redfield w6x, such polarization features a relaxation time T lr that differs from the bare spin relaxation times T l and Tm . For our sample, we found T lr s 14 ms by varying t HTA and observing the integrated echo intensity. Such an integration over the whole echo is used in all our experiments; it ensures that only electron spins very close to resonance contribute to the signal. In the following experiments the length of the HTA pulse is fixed Ž t HTA s 7 ms.. Any change of the dressed spin polarization by resonance absorption can then be detected as a change in spin-locked echo intensity, since this echo corresponds to bare spin coherence, which in turn is generated from the dressed spin polarization by turning off the mw field. A dressed spin resonance spectrum can be obtained by sweeping the frequency of an rf Z-pulse Ž j s 0. of fixed duration t p s 5 ms Žsee Fig. 1b.. For that experiment we have directly connected the DICE module to the probe head, resulting in an rf input power of about 3.4 mW. The corresponding rf field is much weaker than the mw field, thus avoiding a significant Bloch–Siegert shift of the dressed spin resonance line w10x. Spectra were recorded for different mw field strengths of the HTA pulse. The spectra for full mw power and 6 dB attenuation are shown in Fig. 2a,b, respectively. In fair agreement with theory, the frequency of the peak maximum in the former spectrum Žy 0 dB s 20.30 MHz. is about twice as large as in the latter one Žy6 dB s 11.15 MHz.. By using
528
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
Fig. 2. Dressed spin resonance spectra obtained by sweeping the radio frequency of the Z-pulse Žsee Fig. 1.. Ža. Full microwave power during the high turning angle ŽHTA. pulse. Žb. 6 dB microwave attenuation during the HTA pulse.
a smaller sample it can be demonstrated that the asymmetric lineshape is due to inhomogeneity of the mw field in the resonator Ždata not shown.. The occurrence of transient nutations on driving a transition demonstrates generation of a coherent superposition of the two states involved. Such an experiment has been performed for the case of laser light dressed atoms before w11x, albeit on a transition, which featured only one dressed state. For the case of two dressed spin states, we observe nutations on varying t p . The result for 3 dB mw attenuation of the HTA pulse, an rf frequency yrf s 15.287 MHz, and j s 0 is displayed in Fig. 3a. We have checked that the dressed spin nutation frequency varies as expected with cos j and with the square root of rf power Ždata not shown.. From the appearance of the transient nutation signal in Fig. 3a we may conclude that the duration of an rf Z-pulse
Fig. 3. Time-domain experiments involving dressed spin coherence. Ža. Transient dressed spin nutation observed by varying the duration t p of the rf Z-pulse Žsee Fig. 1.. Žb. Dressed spin coherence echo observed by varying the interpulse delay t1 for fixed time t1 q t 2 s6.008 ms Žsee Fig. 1.. An eight-step phase cycle of the rf Z-pulses was applied. Positive intensity corresponds to inversion of the bare state spin-locked echo.
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
529
with flip angle pr2 is 32 ns. The excitation bandwidth is thus sufficient to uniformly excite the whole dressed spin resonance lines observed in the frequency sweep experiments Žsee Fig. 2.. It should then be possible to observe a dressed spin coherence echo by the rf Z-pulse sequence shown in Fig. 1c. The first two pulses generate a Hahn echo, which can be stored back to dressed state polarization by the third pulse for t 1 s t 2 . Indeed such an echo is observed on varying t 1 for fixed t 1 q t 2 s 6.008 ms, as can be seen in Fig. 3b. In this experiment, 30 dB amplification of the rf power has been used which results in lengths of the pr2 and p pulses of 48 and 96 ns at yrf s 12.281 MHz, respectively. To reject contributions from unwanted coherence transfer pathways, we have applied w0– p x phase cycles to the two pr2 pulses and a w0 q p x phase cycle to the p pulse in analogy to bare spin magnetic resonance w4x. The decay of the dressed spin coherence echo by phase relaxation has been observed by varying t 1 and keeping t 1 s t 2 . Note that such an experiment for fixed t HTA measures the difference in relaxation between dressed spin coherence and polarization. On the time scale of our experiments ŽT - 6 ms., no significant dressed spin echo decay is observed, i.e., Tm r f Tl r . From the experimental error, we can estimate a lower limit for the phase memory time Tm r of dressed spin coherence. We find Tm r ) 10 ms, which is somewhat longer than the phase memory time Tm s 6.5 ms for bare spin coherence in the same system. Finally, we have performed a 2D correlation experiment on the a radical in g-irradiated malonic acid with the pulse sequence displayed in Fig. 1c and independent incrementation of t 1 and t 2 . An orientation of the single crystal was chosen for which strong mixing between the electron and nuclear spin was observed. The measurements were performed at B0 s 348.45 mT and ymw s 9.75218 GHz corresponding to a resonance position in between the high-field pair of one allowed and one forbidden transition. By three-pulse ESEEM measurements we determined the two nuclear frequencies var2p s 7.0 and vbr2p s 33.2 MHz. From these frequencies we can calculate the spin Hamiltonian parameters except for V S . We find Ar2p s y35.4, Br2p s y12.8 MHz, using the fact that the signs and v Ir2p s y14.84 MHz are known. With an mw field strength v 1r2p s 6.8 MHz, bare spin excitation is semi-selective, i.e., only transitions within a three-level system consisting of one allowed and one forbidden transition are excited. The dressed spin transition frequencies are all expected to be larger than 3 MHz, so that we used yrf s 3 MHz as the carrier frequency of the Z-field. The Z-field pr2 pulse length was 32 ns. In the 2D spectrum displayed in Fig. 4, both the auto-correlation and cross-peaks are found in the second quadrant Žy 1 ) 0, y 2 - 0., as the Z p pulse inverts the phase of the dressed spin coherences. We observe
Fig. 4. Contour plot of a 2D dressed electron spin resonance spectrum of a g-irradiated single crystal of malonic acid. Correlations are found between two pairs of the three transitions in the selected three-level system.
530
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
auto-correlation peaks for the frequencies y 12 s 4.3, y 23 s 6.0, and y 13 s y 12 q y 23 s 10.3 MHz Žafter accounting for the carrier frequency yrf ., as well as cross-peaks Ž y 12 , y 23 . and Ž y 12 , y 13 .. From the frequencies and the Hamiltonian parameters already known, we may now calculate V Sr2p s 18.9 " 0.2 MHz with considerably better accuracy than would be possible from the electron paramagnetic resonance spectrum. There are three reasons for this improvement in precision. First, small hyperfine couplings to other protons are suppressed by the mw field w12x; second, dressed spin relaxation times are more favourable than bare spin relaxation times; and third, dressed spin nutation frequencies are quite sensitive to even small changes in V S . Since such a measurement of V S gives access to a precise value for v S , it can be used to determine g values with higher accuracy than would be possible by a conventional EPR experiment.
5. Conclusion Pulse magnetic resonance experiments on dressed spins are feasible and analogies with pulse magnetic resonance on bare spins can be exploited. Dressed spin resonance spectroscopy provides convenient access to broad distributions of bare electron spin nutation frequencies, which may be difficult to measure with transient nutation experiments. The nutation frequencies in turn have been shown earlier to be useful for disentangling overlapping EPR spectra w13–15x. However, the biggest advantages of pulsed dressed spin resonance are expected for multilevel systems. For the simple case of an S s 1r2, I s 1r2 system in an oriented sample, we have demonstrated that 2D correlation experiments can be performed on dressed spins and that dressed spin resonance allows one to determine electron Zeeman frequencies with higher precision than would be possible by conventional EPR spectroscopy. In systems with many nuclear spins andror higher spin quantum numbers, the behavior of the spin system during irradiation can become rather complex w7,12,16,17x. In such cases, we expect that 2D dressed electron spin resonance experiments will provide insights which could not be obtained otherwise.
References w1x C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10 Ž1977. 345. w2x C. Cohen-Tannoudji, in: R. Balian, S. Haroche, S. Liberman ŽEds.., Frontiers in Laser Spectroscopy, Les Houdees Session XXVII, North-Holland, Amsterdam, 1977. w3x S.R. Hartmann, E.L. Hahn, Phys. Rev. 128 Ž1962. 2042. w4x R.R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon, Oxford, 1987. w5x M. Munowitz, Coherence and NMR, John Wiley, New York, 1988. w6x A.G. Redfield, Phys. Rev. 98 Ž1955. 1787. w7x G. Jeschke, A. Schweiger, Mol. Phys. 88 Ž1996. 355. w8x R.H. Silsbee, J. Appl. Phys. 32 Ž1961. 1459. w9x H.M. McConnell, C. Heller, T. Cole, R.W. Fessenden, J. Am. Chem. Soc. 82 Ž1960. 766. w10x F. Bloch, A. Siegert, Phys. Rev. 57 Ž1940. 222. w11x C. Wei, N.B. Manson, J.P.D. Martin, Phys. Rev. Lett. 74 Ž1995. 1083. w12x G. Jeschke, A. Schweiger, J. Chem. Phys. 106 Ž1997. 9979. w13x M.K. Bowman, in: L. Kevan, M.K. Bowman ŽEds.., Modern Pulsed and Continuous-Wave Electron Spin Resonance, John Wiley, New York, 1990, p. 1. w14x A.V. Astashkin, A. Schweiger, Chem. Phys. Lett. 174 Ž1990. 595. w15x S. Stoll, G. Jeschke, M. Willer, A. Schweiger, J. Magn. Reson. 130 Ž1998. 86. w16x G. Jeschke, A. Schweiger, J. Chem. Phys. 105 Ž1996. 2199. w17x G. Jeschke, R. Rakhmatullin, A. Schweiger, J. Magn. Reson. 131 Ž1998. 261.
Chemical Physics Letters 301 Ž1999. 524–530
Coherent superposition of dressed spin states and pulse dressed electron spin resonance Gunnar Jeschke
)
Max-Planck-Institut fur ¨ Polymerforschung, Postfach 3148, D-55121 Mainz, Germany Received 1 December 1998
Abstract Coherent superpositions of spin eigenstates in a strong near-resonant electromagnetic field can be prepared by pulses of a secondary linearly polarized electromagnetic field, which is parallel to the static magnetic field. Transient nutations on the dressed state transition and dressed state coherence echoes are observed in a two-level system. A two-dimensional correlation experiment for dressed state transitions is introduced. It is shown that the electron Zeeman frequency can be determined with higher precision in a dressed spin resonance experiment than in a conventional electron spin resonance experiment. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The eigenstates of a quantum system change drastically if the system is brought into a strong electromagnetic field that is near-resonant with one of the quantum transitions. To treat this situation for atoms in a laser light field, Cohen–Tannoudji et al. have defined a new quantum system that includes both the original bare quantum states and the field w1,2x. The eigenstates of such a laser light dressed system are called dressed states. An analogous situation occurs in magnetic resonance experiments during strong radio frequency Žrf. or microwave Žmw. irradiation, except that spontaneous emission is no longer relevant. The energy difference between dressed states can be by orders of magnitude smaller than between the corresponding bare states, leading to a completely different regime of interactions with the environment. The most prominent example of this is probably Hartmann–Hahn cross-polarization between nuclear spins w3x, which can be considered as cross-relaxation between two pairs of dressed spin states. It may thus be of considerable interest to study the interactions of dressed states with their environment in more detail. A similar endeavor for bare spin states has greatly benefited from experiments that are based on the generation and manipulation of coherent superpositions of the states w4,5x. In this Letter we demonstrate that coherent superpositions of dressed spin states can be prepared by pulses of a linearly polarized electromagnetic field, which is parallel to the static magnetic field. We observed dressed
)
Fax: q49-6131-379100; e-mail: [email protected]
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 0 4 1 - X
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
525
spin transient nutations and dressed state coherence echoes. Dressed state coherence of electron spins in the E1X center in g-irradiated quartz glass is found to decay with a relaxation time similar to the rotating frame relaxation time T1r . For a multilevel system we study a proton spin coupled to the electron spin in an a radical derived from malonic acid. Three dressed spin transitions are observed in this case and are shown to be correlated by two-dimensional Ž2D. pulse dressed electron spin resonance. From the frequencies of these transitions, the electron Zeeman frequency can be determined with high precision.
2. Theory Consider a spin S s 1r2 in a static field B0 along the laboratory frame Z-axis and a linearly polarized electromagnetic field with frequency vem, 1 along the X-axis. The Hamiltonian in angular frequency units is given by H1 s v S SZ q 2 v 1 cos Ž vem , 1 t . S X ,
Ž 1.
where v S is the frequency of the spin transition that depends on B0 , and v 1 characterizes the strength of the electromagnetic field. Only one of the two counterrotating circularly polarized components of this field can be resonant with the spin transition, the other one is far off-resonant. When neglecting the latter component, H1 can be rendered time-independent by transformation to a frame that rotates with frequency vem, 1 about the Z-axis. In the rotating frame, the effective interaction along the Z-axis is reduced to the resonance offset V S s v S y vem , 1. Assume now that a secondary linearly polarized electromagnetic field with frequency vem, 2 is applied along the Z-axis. The rotating-frame Hamiltonian H˜2 is then given by H˜2 s V S SZ q v 1 S x q 2 v 2 cos Ž vem , 2 t . SZ .
Ž 2.
For on-resonant irradiation of the bare spin states Ž V S s 0., H˜2 has the same form as H1 , except for an interchange of coordinate axes. A parallel electromagnetic field should thus act on a dressed spin transition like a perpendicular field does on a spin transition. For off-resonant irradiation of the bare spin states Ž V S / 0., the dressed spin resonance frequency v d is given by vd s Ž v 12 q V S2 .1r2 . Only the component of the electromagnetic field perpendicular to the quantization axis drives the transition, so that the effective field is given by veff s cosŽ u . v 2 , where u s atanŽ V Srv 1 .. In fact, this consequence of the rotating frame description of spins in an electromagnetic field has been recognized immediately by Redfield w6x, who observed dressed spin resonance of nuclear spins by a continuous-wave experiment, which he termed rotary saturation. It should also be possible to prepare a coherent superposition of dressed spin states and observe its evolution. More sophisticated pulsed dressed spin resonance experiments could then be derived from the vast and well investigated body of pulsed magnetic resonance experiments w4x. Despite the similarly of the Hamiltonians, the two physical situations described by H1 and H˜2 are somewhat different. In the former case, the linearly polarized electromagnetic field could be applied along any direction in the XY-plane, while in the latter case all components of the field except for the Z-component are decoupled from the dressed spin system. This difference also prevents reiteration of the argument: transitions between the doubly-dressed states that are the eigenstates of H˜2 cannot be driven by a third electromagnetic field, whatever its direction or frequency. Such an isolation of doubly-dressed states from their environment may be of some interest, since it might lead to unusual relaxation behavior. Note also that it is possible to access the regime where the field v 2 driving the dressed state transitions is of the same order of magnitude as the transition frequency v 1. If excitation of dressed spin transitions is to be described quantitatiÕely in this case, the counterrotating component of the Z-field is no longer far off resonance and thus cannot be neglected.
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
526
Other interesting phenomena occur in two-spin systems. By dressing one of the spins with an electromagnetic field of suitable amplitude, one can create a low-field situation where level anti-crossing effects and strong mixing of the two spins can be observed. This has been shown to be the case for a system consisting of one nuclear spin I s 1r2 with anisotropic hyperfine coupling to an electron spin S s 1r2 w7x. The rotating frame Hamiltonian of this system during mw irradiation of the electron spin transitions is given by H1 S s V S SZ q v 1 Iz q AS z Iz q BS z I x q v 1 S x ,
Ž 3.
where v I is the nuclear Zeeman frequency and A and B characterize the secular and pseudosecular part of the hyperfine coupling, respectively. If mixing between the electron and nuclear spin is weak in the absence of the mw field, the Hamiltonian can be diagonalized by three consecutive unitary transformations neglecting only non-secular parts w7x. By applying the same transformations to the operator SZ that corresponds to the secondary electromagnetic field, we obtain its matrix representation in the eigenbasis of the dressed spin system. We find
U3U2 U1 SZ U1†U2†U3† s
ca
ysj sa
cj ca
0
1
ysj sa
cj2 cb y sj2 ca
sj cj Ž ca q cb .
cj cb
2
cj ca
sj cj Ž ca q cb .
sj2 cb y cj2 ca
sj sb
0
cj cb
sj sb
ycb
,
Ž 4.
where ca s cosŽ ua ., sa s sinŽ ua ., cb s cosŽ ub ., sb s sinŽ ub ., cj s cosŽ jr2., and sj s sinŽ jr2.. The angles have been defined in Ref. w7x. Obviously, the secondary field can induce any dressed state transition except for the double-quantum transition. Even this restriction vanishes, if mixing between the electron and nuclear spin is already substantial in the absence of the mw field. In any case, sufficiently strong Z-pulses can drive enough coherence transfers to obtain correlations between all dressed state transition frequencies within the same spin system. It should thus be possible to obtain and assign dressed spin spectra by means of pulse resonance experiments. Finally we wish to point out that even isotropic couplings can lead to strong mixing between dressed electron spins and nuclear spins, since the dressed spin situation corresponds to a low-field regime. Numerical simulations show that it should be possible to apply dressed electron spin resonance also to the measurement of hyperfine couplings in solution.
3. Experimental For an experimental test of the above considerations and even for performing dressed electron spin resonance routinely, commercial electron paramagnetic resonance ŽEPR. equipment is well suited. With the spectrometer ESP 380e ŽBruker., mw fields can be generated that correspond to dressed spin resonance frequencies v 1 up to 30 MHz. This is within the frequency range of electron–nuclear double resonance ŽENDOR. equipment, for instance, of the pulse ENDOR module ESP360D-P DICE for the same spectrometer and of the ENDOR probe head EN 4118X-MD-4. In our experiments, the mw field at frequency ymw is along the long axis of the probe head Ž X-axis.. The radio frequency Žrf. field would be along the Y axis for ENDOR experiments. However, by turning the probe head about its long axis, any direction of the rf field in the YZ plane can be realized. We characterize this orientation by the angle j between the rf field and the static field Ž Z . axes. For pulse excitation of dressed state transitions, the DICE output was amplified by an ENI 3200 L rf amplifier ŽENI.. In all our experiments, we use the spin-locked echo pulse sequence w3x displayed in Fig. 1a for the excitation and detection of bare electron spins. As a first model sample, we use g-irradiated Suprasil Žquartz glass., which features E1X paramagnetic centres. These centers have been investigated in some detail in crystalline quartz w8x. The overwhelming contribution to the EPR signal in this sample is due to isolated electron spins S s 1r2, since the magnetic isotopes of both
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
527
Fig. 1. Pulse sequences. Ža. Spin-locked echo sequence used for excitation of bare spin states by a transverse microwave field. Žb. Radio frequency Z-pulse. Žc. Radio frequency Z-pulse sequence for observing Hahn echoes and measuring 2D correlation spectra.
silicon and oxygen are rare, and the distance between E1X centers is large. The g anisotropy is small enough to allow for complete excitation of the EPR spectrum in a pulse experiment Ž D g s 9 = 10y4 .. Furthermore, both the longitudinal relaxation time Tl and the phase memory time Tm are conveniently long even at ambient temperature. To observe correlations between dressed state transitions, we use a g-irradiated single crystal of malonic acid that features so-called free a radicals w9x. In these radicals, one proton is adjacent to the carbon atom carrying the overwhelming part of the spin density. This sample resembles to a good approximation a spin system of one nuclear spin I s 1r2 coupled to an electron spin S s 1r2, since the hyperfine couplings of the other protons are by more than one order of magnitude smaller than the one of the a proton. Furthermore, strong mixing between electron and nuclear spins can be obtained for suitable orientations of the crystal.
4. Results and discussion Experiments on the Suprasil sample where performed at a static field B0 s 347 mT corresponds to maximum intensity in the EPR spectrum at ymw s 9.7168 GHz. The first mw pulse with flip angle pr2 generates electron coherence that defocuses during the subsequent interpulse delay t s 584 ns. The magnetization component along the x axis is then spin locked by a high turning angle ŽHTA. pulse of duration t HTA . This spin-locked magnetization corresponds to non-equilibrium polarization on a dressed spin transition. As shown by Redfield w6x, such polarization features a relaxation time T lr that differs from the bare spin relaxation times T l and Tm . For our sample, we found T lr s 14 ms by varying t HTA and observing the integrated echo intensity. Such an integration over the whole echo is used in all our experiments; it ensures that only electron spins very close to resonance contribute to the signal. In the following experiments the length of the HTA pulse is fixed Ž t HTA s 7 ms.. Any change of the dressed spin polarization by resonance absorption can then be detected as a change in spin-locked echo intensity, since this echo corresponds to bare spin coherence, which in turn is generated from the dressed spin polarization by turning off the mw field. A dressed spin resonance spectrum can be obtained by sweeping the frequency of an rf Z-pulse Ž j s 0. of fixed duration t p s 5 ms Žsee Fig. 1b.. For that experiment we have directly connected the DICE module to the probe head, resulting in an rf input power of about 3.4 mW. The corresponding rf field is much weaker than the mw field, thus avoiding a significant Bloch–Siegert shift of the dressed spin resonance line w10x. Spectra were recorded for different mw field strengths of the HTA pulse. The spectra for full mw power and 6 dB attenuation are shown in Fig. 2a,b, respectively. In fair agreement with theory, the frequency of the peak maximum in the former spectrum Žy 0 dB s 20.30 MHz. is about twice as large as in the latter one Žy6 dB s 11.15 MHz.. By using
528
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
Fig. 2. Dressed spin resonance spectra obtained by sweeping the radio frequency of the Z-pulse Žsee Fig. 1.. Ža. Full microwave power during the high turning angle ŽHTA. pulse. Žb. 6 dB microwave attenuation during the HTA pulse.
a smaller sample it can be demonstrated that the asymmetric lineshape is due to inhomogeneity of the mw field in the resonator Ždata not shown.. The occurrence of transient nutations on driving a transition demonstrates generation of a coherent superposition of the two states involved. Such an experiment has been performed for the case of laser light dressed atoms before w11x, albeit on a transition, which featured only one dressed state. For the case of two dressed spin states, we observe nutations on varying t p . The result for 3 dB mw attenuation of the HTA pulse, an rf frequency yrf s 15.287 MHz, and j s 0 is displayed in Fig. 3a. We have checked that the dressed spin nutation frequency varies as expected with cos j and with the square root of rf power Ždata not shown.. From the appearance of the transient nutation signal in Fig. 3a we may conclude that the duration of an rf Z-pulse
Fig. 3. Time-domain experiments involving dressed spin coherence. Ža. Transient dressed spin nutation observed by varying the duration t p of the rf Z-pulse Žsee Fig. 1.. Žb. Dressed spin coherence echo observed by varying the interpulse delay t1 for fixed time t1 q t 2 s6.008 ms Žsee Fig. 1.. An eight-step phase cycle of the rf Z-pulses was applied. Positive intensity corresponds to inversion of the bare state spin-locked echo.
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
529
with flip angle pr2 is 32 ns. The excitation bandwidth is thus sufficient to uniformly excite the whole dressed spin resonance lines observed in the frequency sweep experiments Žsee Fig. 2.. It should then be possible to observe a dressed spin coherence echo by the rf Z-pulse sequence shown in Fig. 1c. The first two pulses generate a Hahn echo, which can be stored back to dressed state polarization by the third pulse for t 1 s t 2 . Indeed such an echo is observed on varying t 1 for fixed t 1 q t 2 s 6.008 ms, as can be seen in Fig. 3b. In this experiment, 30 dB amplification of the rf power has been used which results in lengths of the pr2 and p pulses of 48 and 96 ns at yrf s 12.281 MHz, respectively. To reject contributions from unwanted coherence transfer pathways, we have applied w0– p x phase cycles to the two pr2 pulses and a w0 q p x phase cycle to the p pulse in analogy to bare spin magnetic resonance w4x. The decay of the dressed spin coherence echo by phase relaxation has been observed by varying t 1 and keeping t 1 s t 2 . Note that such an experiment for fixed t HTA measures the difference in relaxation between dressed spin coherence and polarization. On the time scale of our experiments ŽT - 6 ms., no significant dressed spin echo decay is observed, i.e., Tm r f Tl r . From the experimental error, we can estimate a lower limit for the phase memory time Tm r of dressed spin coherence. We find Tm r ) 10 ms, which is somewhat longer than the phase memory time Tm s 6.5 ms for bare spin coherence in the same system. Finally, we have performed a 2D correlation experiment on the a radical in g-irradiated malonic acid with the pulse sequence displayed in Fig. 1c and independent incrementation of t 1 and t 2 . An orientation of the single crystal was chosen for which strong mixing between the electron and nuclear spin was observed. The measurements were performed at B0 s 348.45 mT and ymw s 9.75218 GHz corresponding to a resonance position in between the high-field pair of one allowed and one forbidden transition. By three-pulse ESEEM measurements we determined the two nuclear frequencies var2p s 7.0 and vbr2p s 33.2 MHz. From these frequencies we can calculate the spin Hamiltonian parameters except for V S . We find Ar2p s y35.4, Br2p s y12.8 MHz, using the fact that the signs and v Ir2p s y14.84 MHz are known. With an mw field strength v 1r2p s 6.8 MHz, bare spin excitation is semi-selective, i.e., only transitions within a three-level system consisting of one allowed and one forbidden transition are excited. The dressed spin transition frequencies are all expected to be larger than 3 MHz, so that we used yrf s 3 MHz as the carrier frequency of the Z-field. The Z-field pr2 pulse length was 32 ns. In the 2D spectrum displayed in Fig. 4, both the auto-correlation and cross-peaks are found in the second quadrant Žy 1 ) 0, y 2 - 0., as the Z p pulse inverts the phase of the dressed spin coherences. We observe
Fig. 4. Contour plot of a 2D dressed electron spin resonance spectrum of a g-irradiated single crystal of malonic acid. Correlations are found between two pairs of the three transitions in the selected three-level system.
530
G. Jeschker Chemical Physics Letters 301 (1999) 524–530
auto-correlation peaks for the frequencies y 12 s 4.3, y 23 s 6.0, and y 13 s y 12 q y 23 s 10.3 MHz Žafter accounting for the carrier frequency yrf ., as well as cross-peaks Ž y 12 , y 23 . and Ž y 12 , y 13 .. From the frequencies and the Hamiltonian parameters already known, we may now calculate V Sr2p s 18.9 " 0.2 MHz with considerably better accuracy than would be possible from the electron paramagnetic resonance spectrum. There are three reasons for this improvement in precision. First, small hyperfine couplings to other protons are suppressed by the mw field w12x; second, dressed spin relaxation times are more favourable than bare spin relaxation times; and third, dressed spin nutation frequencies are quite sensitive to even small changes in V S . Since such a measurement of V S gives access to a precise value for v S , it can be used to determine g values with higher accuracy than would be possible by a conventional EPR experiment.
5. Conclusion Pulse magnetic resonance experiments on dressed spins are feasible and analogies with pulse magnetic resonance on bare spins can be exploited. Dressed spin resonance spectroscopy provides convenient access to broad distributions of bare electron spin nutation frequencies, which may be difficult to measure with transient nutation experiments. The nutation frequencies in turn have been shown earlier to be useful for disentangling overlapping EPR spectra w13–15x. However, the biggest advantages of pulsed dressed spin resonance are expected for multilevel systems. For the simple case of an S s 1r2, I s 1r2 system in an oriented sample, we have demonstrated that 2D correlation experiments can be performed on dressed spins and that dressed spin resonance allows one to determine electron Zeeman frequencies with higher precision than would be possible by conventional EPR spectroscopy. In systems with many nuclear spins andror higher spin quantum numbers, the behavior of the spin system during irradiation can become rather complex w7,12,16,17x. In such cases, we expect that 2D dressed electron spin resonance experiments will provide insights which could not be obtained otherwise.
References w1x C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10 Ž1977. 345. w2x C. Cohen-Tannoudji, in: R. Balian, S. Haroche, S. Liberman ŽEds.., Frontiers in Laser Spectroscopy, Les Houdees Session XXVII, North-Holland, Amsterdam, 1977. w3x S.R. Hartmann, E.L. Hahn, Phys. Rev. 128 Ž1962. 2042. w4x R.R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon, Oxford, 1987. w5x M. Munowitz, Coherence and NMR, John Wiley, New York, 1988. w6x A.G. Redfield, Phys. Rev. 98 Ž1955. 1787. w7x G. Jeschke, A. Schweiger, Mol. Phys. 88 Ž1996. 355. w8x R.H. Silsbee, J. Appl. Phys. 32 Ž1961. 1459. w9x H.M. McConnell, C. Heller, T. Cole, R.W. Fessenden, J. Am. Chem. Soc. 82 Ž1960. 766. w10x F. Bloch, A. Siegert, Phys. Rev. 57 Ž1940. 222. w11x C. Wei, N.B. Manson, J.P.D. Martin, Phys. Rev. Lett. 74 Ž1995. 1083. w12x G. Jeschke, A. Schweiger, J. Chem. Phys. 106 Ž1997. 9979. w13x M.K. Bowman, in: L. Kevan, M.K. Bowman ŽEds.., Modern Pulsed and Continuous-Wave Electron Spin Resonance, John Wiley, New York, 1990, p. 1. w14x A.V. Astashkin, A. Schweiger, Chem. Phys. Lett. 174 Ž1990. 595. w15x S. Stoll, G. Jeschke, M. Willer, A. Schweiger, J. Magn. Reson. 130 Ž1998. 86. w16x G. Jeschke, A. Schweiger, J. Chem. Phys. 105 Ž1996. 2199. w17x G. Jeschke, R. Rakhmatullin, A. Schweiger, J. Magn. Reson. 131 Ž1998. 261.