Journal of Magnetic Resonance 239 (2014) 87–90
Contents lists available at ScienceDirect
Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Strong pulsed excitations using circularly polarized fields for ultra-low field NMR Jeong Hyun Shim ⇑, Seong-Joo Lee, Kwon-Kyu Yu, Seong-Min Hwang, Kiwoong Kim Center for Brain and Cognition Measurement, Korea Research Institute of Standards and Science, Daejeon 350-340, Republic of Korea
a r t i c l e
i n f o
Article history: Received 29 August 2013 Revised 6 December 2013 Available online 24 December 2013 Keywords: Bloch–Siegert effect Circular polarization Ultra-low field NMR Nutation SQUID detection
a b s t r a c t A pulse, which is produced by a single coil and thereby has a linear polarization, cannot coherently drive nuclear spins if the pulse is stronger than the static field B0. The inaccuracy of the pulse, which arises from the failure of the rotating wave approximation, has been an obstacle in adopting multiple pulse techniques in ultra-low field NMR where B0 is less than a few lT. Here, we show that such a limitation can be overcome by applying pulses of circular polarization using two orthogonal coils. The sinusoidal nutation of the nuclear spins was experimentally obtained, which indicates that coherent and precise controls of the nuclear spins can be achieved with circularly polarized pulses. Additional demonstration of the Carl–Purcell–Meiboom–Gill sequence verifies the feasibility of adopting multiple pulse sequences to ultra-low field NMR studies. Ó 2013 Published by Elsevier Inc.
1. Introduction The accuracy of a resonant pulse determines the quality of an NMR signal, particularly in the areas requiring multiple pulses, such as multi-dimensional NMR spectroscopy [1,2], magnetic resonance imaging (MRI) [3], and NMR quantum computing [4]. If a system’s relaxation time, T1 or T1q, is relatively short, one way to improve the accuracy is increasing the driving strength of the pulse to avoid the decay of the system’s coherence due to systemenvironment interactions while the spins are under the control of the pulse. The problem of such strong pulses having linear polarizations, unfortunately, is that the off-resonant component disturbs the coherent motion of the target spins [5]. This is known as the Bloch–Siegert (BS) effect [6]. In Ultra-Low Field (ULF) NMR experiments with a static field of a few microtesla [7–9], the strength (B1) of the pulse is comparable to or even higher than the static field strength. The relaxation times of many biological tissues are in the range of 50–200 ms at the ultra-low field condition [10]. This is a favorable condition for the notable emergence of the BS effect. A single pulse, hence, will manipulate the nuclear spins in an undesired way. Therefore, multiple pulses will perform even worse. The existing solution to improve the accuracy or driving speed of linear polarization was by applying specially shaped pulses, which are designed based on the optimal-control theory [11]. Time-optimal control schemes, for instance, were recently ⇑ Corresponding author. E-mail address: [email protected] (J.H. Shim). 1090-7807/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jmr.2013.12.007
developed, and are capable of driving the spins faster than the normal sinusoidal driving [12]. In the present study, however, we take an entirely different approach, i.e. the generation of the resonant co-rotating field alone without the counter-rotating component by using two orthogonal coils. This can never be achieved with conventional single coils. The resulting field will, therefore, have a circular polarization. Here, we show that the pulses of circular polarization are free from the BS effect regardless of the driving field strengths. As a result, the nuclear spins can be coherently driven. The demonstration of the Carl–Purcell–Meiboom–Gill (CPMG) sequence [2] with circular polarization pulses indicates that each pulse reliably controls flip angles and phases of the nuclear spins. This enables the application of multiple pulse sequences in ULFNMR experiments. 2. Background In a conventional high-field NMR apparatus, a single coil is mostly used to produce an oscillating field, which is used to excite the nuclear spins. The generated field has a linear polarization along the direction of the coil (x axis). Including the static field along z axis (B0 ) and sinusoidal field of the strength 2B1 , the Hamiltonian HðtÞ for a nuclear spin I can be written as,
HðtÞ ¼ cB0 Iz þ 2cB1 Ix cos xt;
ð1Þ
where the c is gyromagnetic ratio. The linearly polarized field along the x axis can be decomposed into two rotating fields, co-rotating and counter-rotating, as
88
2B1 Ix cosðxtÞ ¼ B1 ðeixtIz Ix eixtIz þ c:cÞ:
J.H. Shim et al. / Journal of Magnetic Resonance 239 (2014) 87–90
ð2Þ
At high magnetic fields, the counter-rotating component is mostly neglected in the Hamiltonian; this is called Rotating Wave Approximation (RWA). In fact, the RWA approach is only valid if the strength of the rotating component, B1 , is much smaller than that of the static field B0 . For this case, the BS effect will induce a minor shift in the resonance frequency. According to the average Hamiltonian theory, the B1 intensity contributes to the frequency shift in 2nd order [6]. When B1 becomes comparable to the resonance frequency, the RWA does not correctly describe the dynamics of nuclear spins. As shown in Ref. [5], the nutation of target spins does not exhibit a sinusoidal oscillation. The BS effect, therefore, must be suppressed to achieve rapid as well as accurate control of the nuclear spins. Instead of a linearly polarized field, if we apply a circularly polarized field of frequency x, the time-varying part of the Hamiltonian HðtÞ will include only the first term of Eq. (2), B1 eixtIz Ix eixtIz . Consequently, the transformation from the lab frame to the rotating frame, in which the Hamiltonian HR becomes time-independent, can be performed without approximations. The BS effect will vanish. The resulting HR can be written as,
HR ¼ ðcB0 þ xÞIz0 þ cB1 Ix0 ;
(a)
(b)
ð3Þ
where the x0 and z0 are the axes in the rotating frame. For the resonance condition x ¼ cB0 with the co-rotating field, the nuclear spins will show an ideal nutation along the x0 axis in the rotating frame. This is our main idea.
3. Experimental To create circularly polarized fields, two coils orthogonal to each other are required. For simplicity, we assembled three square Helmholtz coils on the sides of a wooden square box as illustrated in Fig. 1(a). However, only the two that were orthogonal to the static magnetic field (Bm ), were used. Since the two coils are nearly identical, placing a phase difference in the currents was sufficient to generate circularly polarized fields. Either ±90° or 0° were used to choose the orientation of the polarization generated at the center of the coil pair. An arbitrary waveform generator (2 channels, 14 bit) produced the desired pulse shapes for the two coils. The phases and amplitudes of pulses were precisely controlled by waveform programming. The outputs were amplified using a 2Ch audio amplifier to inject sufficient current into the coils. Mechanical relays blocked the noises from the amplifier while the acquisitions were ongoing. Our ULF-NMR system uses a commercial DC-SQUID to measure the magnetic field variations produced by the precession of nuclear spins around the measurement field (Bm). The Bm is analogous to the B0 field in a high-field NMR. The SQUID sensor and a 2nd order gradiometer are inside the He dewar (Fig. 1(a)). Prior to every measurement, an 50 mT pre-polarization field (BP) was applied for two seconds to enhance the NMR signals from 1 H nuclear spins. The direction of the BP was configured to be parallel to Bm, because a resonant pulse flips the nuclear spins after the pre-polarization process, as shown in Fig. 1(b). The strength of the Bm field was 4.97 lT, which produces a resonance frequency of 212 Hz for 1 H nuclear spins. The free precession of the 1 H nuclear spins in DI water was recorded after applying the pulse. The coherence time T 2 of the 1 H spin in DI water was longer than one second. For the CPMG sequence, a constant gradient Gz , which was generated by a Maxwell coil, was applied to reduce the T 2 to produce better visibility of the spin echoes. The experiments were performed in a magnetically shielded room to suppress the environmental magnetic field noises.
Fig. 1. (a) An overview of our ULF-NMR system. The pick-up coil lies on the bottom of the He dewar. The BP coil produces the prepolarization field (BP), while the Bm coil produces the measurement field (Bm). Among the three square Helmholtz coils in the box at the center, two of them, orthogonal to the Bp and Bm fields, were chosen to generate circularly polarized fields. A gradient was applied when the echoes were recorded using the CPMG sequence. (b) The timing sequence for the measurement of the nutation. The duration of the pulse d was varied from 0 to 6 ms. The value of tm was fixed as 10 ms.
4. Results and discussion To determine the coherency of the motion of nuclear spins in the presence of a pulse, we measured the nutation curve from the variation of the initial amplitude of the free precession signal as a function of the pulse duration d. The entire timing sequence is shown in Fig. 1(b). The starting time of the measurement of the FID signal (t m ) was kept constant at 10 ms from the trigger point of the pulse output. Compared to T 2 , the t m is negligible. As such, the amplitude damping during tm is not measurable. Unlike the conventional sequences used to check the z component of the nuclear magnetization remaining after a pulse [13], the sequence in Fig. 1(b) measures the in-plane components, x and y. If nuclear spins are driven coherently, the in-plane measurement should produce a sinusoidal curve that would be the same as that produced using conventional sequences, except for a different phase. The initial amplitude of the free precession signal was obtained from the integration of the real part of the spectrum, which is a Fourier transform of the time domain signal. The real spectrum was phase-corrected before the integration. When the flip angle of the pulse was over, for instance, 90°, the phase was inverted accordingly. We did not correct this phase inversion to reflect the inversion of the nuclear spins produced by the applied pulse. The left column of Fig. 2 shows the nutation curves for nuclear spins driven by the pulses of different polarizations. The blue, purple, and green curves correspond to the circular polarization of a co-rotating field in resonance with the Bm field, circular polarization of a counter-rotating field, and linear polarization, respectively. The co-rotating field corresponds to the clockwise rotation
J.H. Shim et al. / Journal of Magnetic Resonance 239 (2014) 87–90
Fig. 2. (Left column) Nutation curves measured with the sequence in Fig. 1(b). The blue, green, purple curves correspond to the circular polarization of co-rotating field, the linear polarization, and the circular polarization of counter-rotating field, respectively. (Right column) Numerical simulation results can be compared with experimental results. Inhomogeneity of 10% in the B1 field strength was assumed in the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of the B1 field in the presence of the Bm along the z axis. Likewise, the counter-rotating field corresponds the counter-clockwise rotation. Every point was measured in a single run. For comparison with the experimental results, numerical simulations were performed in the lab frame. The time was incremented with a step of 10 ns (dt). At the jth time step or at t ¼ jdt, the time-dependent Hamiltonian HðtÞ, which is defined in Eq. (4), was averaged for the e j . The timeduration dt. The averaged Hamiltonian is denoted as H evolution operator UðtÞ was obtained by sequential multiplication, Q e i.e., UðtÞ ¼ j ei H j . The magnetization along the x axis at a time t was calculated from the density matrix qðtÞ as M x ðtÞ ¼ trfqðtÞIx g. In Eq. (4), / represents the relative phase between the x and y components of the B1 field.
HðtÞ ¼ cBm Iz þ cB1 fIx cosðxtÞ þ Iy cosðxt þ /Þg:
ð4Þ
In Fig. 2, all three experimental results are in good agreement with the numerical simulations shown on the right column. However, the origin of the small difference between the experiment and simulation for the linearly polarized pulse is unknown. Only the circular polarization of the co-rotating field exhibited a sinusoidal nutation. This sinusoidal behavior indicates coherent dynamics of the nuclear spins driven by the resonant pulse. The data (blue curve) fits well with the mathematical formula, Aet=s sinðxtÞ, in which the exponential decay attributes to the inhomogeneity of the pulse strength (B1) [14]. Because the volume of the sample is finite, we can never exclude the inhomogeneity of the B1 field. Thus, the nutation frequency measured from the fitting, 510 Hz, is the spatially averaged value. In the simulation, the inhomogeneity of the B1 field over the sample volume was assumed to be 10% with a quadratic variation from the center to the edge. This reasonably explains the origin of the decay in the experiment. Because the resonance frequency is 212 Hz, the strength B1 should be sufficient to induce a strong BS effect, which, nevertheless, has vanished by virtue of the circularly polarized field. The sinusoidal curve from the circular polarization of the co-rotating field clearly contrasts with the curve from the linear polarization. It is difficult to achieve a full inversion of the nuclear spins with linearly polarized pulses, because the in-plane magnetization
89
barely becomes zero. Moreover, the phase of the in-plane magnetization evolves incoherently with the phase of the applied pulse, because the strong counter-rotating field also participates in driving the nuclear spins. The circular polarization of counter-rotating field, induces no BS effect either. The nutation, however, does not exhibit a sinusoidal behavior. According to Eq. (3), the field along the z0 axis remains because x ¼ cB0 for the counter-rotating field. Therefore, the effective field in the rotating frame is tilted from the x0 axis. This leads to the non-sinusoidal curve in the middle of Fig. 2. One of the main advantages of the coherent driving with circularly polarized pulses, as shown in Fig. 2 (blue curve), is the feasibility of adopting multiple pulse techniques that will depend on the particular purpose of the ULF-NMR/MRI studies. To substantiate this feature, we implemented the classical pulse sequences, i.e., Carr–Purcell (CP) and CPMG, with the pulses of a co-rotating field. The durations of the 90° and 180° pulses were 0.50 and 0.97 ms, respectively. Fig. 3 clearly shows the echo trains observed for both sequences. Demodulation of the NMR signal in the lab frame was performed digitally by multiplying a digital reference signal and low-pass filtering. The spin echo train of the CPMG survives longer than that of the CP, as expected from the known advantage of the CPMG sequence [2]. CPMG is more robust against pulse errors inherent in the applied pulses [15]. The fact that the CPMG sequence containing 20 refocusing pulses works well implies that the flip angle and phase of the nuclear spins can be precisely controlled by a strong pulse of the circular polarization. This implies that, all of the NMR pulse techniques will be still available even if the static magnetic field is reduced to less than one lT and even down to the regime of a near zero field [16]. The use of pulse techniques may provide a breakthrough in the improvement of the functionality of ULF-NMR/MRI. For instance, an MRI protocol using pulses may improve the quality or acquisition speed of ULF-MRI, where gradient sequences have been primarily used [8,17–19]. Furthermore, novel pulse sequences that are specific to the environment of ultra-low magnetic field can be devised. Simulation studies have revealed that the experimental factor that is the most relevant for the accuracy of the pulse, is the spatial inhomogeneity of the B1 field. As mentioned above, our current system has an inhomogeneity of approximately 10%. Nevertheless, if we take the time for the first maximum of the nutation curve (blue in Fig. 2) as the duration of the 90° pulse, the loss of the NMR signal is less than 2%. As a single pulse, this accuracy is relatively good. This is why the CP result in Fig. 3 appears to show no difference from the CPMG result until nearly the 10th refocusing pulse. Modification of the geometry of the excitation coils, such
Fig. 3. Echo trains obtained by the CP and CPMG sequences. The durations of the 90° and 180° pulses were estimated from the nutation curve (blue) in Fig. 2. From the CPMG result, T2 was measured to be 2.2 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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J.H. Shim et al. / Journal of Magnetic Resonance 239 (2014) 87–90
as the extension of the coil dimensions, will be necessary to enhance the accuracy. Finally, we believe that our scheme of using circularly polarized fields can also be applied to other systems that require strong driving because of short coherent times as well as precise operations. A good example of other systems such as this is nitrogen-vacancy centers in diamond, which can be exploited for various nano-scale sciences and applications [20]. 5. Conclusion The pulses generated by circularly polarized fields can coherently drive nuclear spins and be unhampered by the Bloch–Siegert effect, although the driving strength is higher than the static field. A coherent sinusoidal nutation curve was measured with circularly polarized pulses of a co-rotating field. With such pulses, one can rapidly and accurately manipulate nuclear spins. This will be crucial for the systems of relatively short relaxation times in the presence of a low static field. Demonstration of CP and CPMG sequences proves that such hard pulses can constitute multiple pulse sequences for ULF-NMR. We hope this work will help make a breakthrough in finding novel applications of ULF-NMR/MRI systems. Acknowledgments We appreciate the helpful comments from J.-S. Lee. This work was supported by the World Class Laboratory (WCL) Grant from Korea Research Institute of Standards and Science. References [1] R.P. Ernst, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford, 1978. [2] C. Slichter, Principles of Magnetic Resonance, Springer, Berlin, 1996. [3] D. Nishimura, Principles of Magnetic Resonance Imaging, Lulu, 2010.
[4] J.A. Jones, Quantum computing with NMR, Prog. Nucl. Mag. Reson. Spectrosc. 59 (2) (2011) 91–120. [5] G.D. Fuchs, V.V. Dobrovitski, D.M. Toyli, F.J. Heremans, D.D. Awschalom, Gigahertz dynamics of a strongly driven single quantum spin, Science 326 (5959) (2009) 1520–1522. [6] F. Bloch, A. Siegert, Magnetic resonance for nonrotating fields, Phys. Rev. 57 (6) (1940) 522–527. [7] R. McDermott, A.H. Trabesinger, M. Mück, E.L. Hahn, A. Pines, J. Clarke, Liquidstate NMR and scalar couplings in microtesla magnetic fields, Science 295 (5563) (2002) 2247–2249. [8] I. Hilschenz, R. Körber, H.-J. Scheer, T. Fedele, H.-H. Albrecht, A. Mario Cassará, S. Hartwig, L. Trahms, J. Haase, M. Burghoff, Magnetic resonance imaging at frequencies below 1 kHz, Mag. Reson. Imag. 31 (2) (2013) 171–177. [9] K. Kim, S.-J. Lee, C.S. Kang, S. Min Hwang, Y.-H. Lee, K.-K. Yu, Towards a brain functional connectivity mapping modality by direct imaging of coherent brain waves: brain magnetic resonance (submitted for publication). [10] V. Zotev, A. Matlashov, I. Savukov, T. Owens, P. Volegov, J. Gomez, M. Espy, Squid-based microtesla MRI for in vivo relaxometry of the human brain, IEEE Trans. Appl. Supercond. 19 (3) (2009) 823–826. [11] S.J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N.C. Nielsen, O.W. Sørensen, C. Griesinger, Unitary control in quantum ensembles: maximizing signal intensity in coherent spectroscopy, Science 280 (5362) (1998) 421–424. [12] Z.-H. Wang, V.V. Dobrovitski, Time-optimal rotation of a spin 1/2: application to the nv center spin in diamond, Phys. Rev. B 84 (4) (2011) 045303. [13] A. Schweiger, G. Jeschke, Principles of Pulse Electron Paramagnetic Resonance, Oxford University Press, USA, 2001. [14] R. Boscaino, F.M. Gelardi, J.P. Korb, Non-bloch decay of transient nutations in s = 1/2 systems: an experimental investigation, Phys. Rev. B 48 (10) (1993) 7077–7085. [15] J.H. Shim, I. Niemeyer, J. Zhang, D. Suter, Robust dynamical decoupling for arbitrary quantum states of a single nv center in diamond, Europhys. Lett. 99 (4) (2012) 4004–4009. [16] M.P. Ledbetter, T. Theis, J.W. Blanchard, H. Ring, P. Ganssle, S. Appelt, B. Blmich, A. Pines, D. Budker, Near-zero-field nuclear magnetic resonance, Phys. Rev. Lett. 107 (10) (2011) 107601. [17] R. McDermott, S.-K. Lee, B.t. Haken, A.H. Trabesinger, A. Pines, J. Clarke, Microtesla MRI with a superconducting quantum interference device, Proc. Nat. Acad. Sci. USA 101 (21) (2004) 7857–7861. [18] V.S. Zotev, A.N. Matlashov, P.L. Volegov, A.V. Urbaitis, M.A. Espy, R.H. Kraus Jr., Squid-based instrumentation for ultralow-field MRI, Supercond. Sci. Technol. 20 (11) (2007) S367. [19] M. Espy, A. Matlashov, P. Volegov, Squid-detected ultra-low field MRI, J. Mag. Reson. 229 (0) (2013) 127–141. [20] V. Acosta, P. Hemmer, Nitrogen-vacancy centers: physics and applications, MRS Bull. 38 (02) (2013) 127–130.
Contents lists available at ScienceDirect
Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Strong pulsed excitations using circularly polarized fields for ultra-low field NMR Jeong Hyun Shim ⇑, Seong-Joo Lee, Kwon-Kyu Yu, Seong-Min Hwang, Kiwoong Kim Center for Brain and Cognition Measurement, Korea Research Institute of Standards and Science, Daejeon 350-340, Republic of Korea
a r t i c l e
i n f o
Article history: Received 29 August 2013 Revised 6 December 2013 Available online 24 December 2013 Keywords: Bloch–Siegert effect Circular polarization Ultra-low field NMR Nutation SQUID detection
a b s t r a c t A pulse, which is produced by a single coil and thereby has a linear polarization, cannot coherently drive nuclear spins if the pulse is stronger than the static field B0. The inaccuracy of the pulse, which arises from the failure of the rotating wave approximation, has been an obstacle in adopting multiple pulse techniques in ultra-low field NMR where B0 is less than a few lT. Here, we show that such a limitation can be overcome by applying pulses of circular polarization using two orthogonal coils. The sinusoidal nutation of the nuclear spins was experimentally obtained, which indicates that coherent and precise controls of the nuclear spins can be achieved with circularly polarized pulses. Additional demonstration of the Carl–Purcell–Meiboom–Gill sequence verifies the feasibility of adopting multiple pulse sequences to ultra-low field NMR studies. Ó 2013 Published by Elsevier Inc.
1. Introduction The accuracy of a resonant pulse determines the quality of an NMR signal, particularly in the areas requiring multiple pulses, such as multi-dimensional NMR spectroscopy [1,2], magnetic resonance imaging (MRI) [3], and NMR quantum computing [4]. If a system’s relaxation time, T1 or T1q, is relatively short, one way to improve the accuracy is increasing the driving strength of the pulse to avoid the decay of the system’s coherence due to systemenvironment interactions while the spins are under the control of the pulse. The problem of such strong pulses having linear polarizations, unfortunately, is that the off-resonant component disturbs the coherent motion of the target spins [5]. This is known as the Bloch–Siegert (BS) effect [6]. In Ultra-Low Field (ULF) NMR experiments with a static field of a few microtesla [7–9], the strength (B1) of the pulse is comparable to or even higher than the static field strength. The relaxation times of many biological tissues are in the range of 50–200 ms at the ultra-low field condition [10]. This is a favorable condition for the notable emergence of the BS effect. A single pulse, hence, will manipulate the nuclear spins in an undesired way. Therefore, multiple pulses will perform even worse. The existing solution to improve the accuracy or driving speed of linear polarization was by applying specially shaped pulses, which are designed based on the optimal-control theory [11]. Time-optimal control schemes, for instance, were recently ⇑ Corresponding author. E-mail address: [email protected] (J.H. Shim). 1090-7807/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jmr.2013.12.007
developed, and are capable of driving the spins faster than the normal sinusoidal driving [12]. In the present study, however, we take an entirely different approach, i.e. the generation of the resonant co-rotating field alone without the counter-rotating component by using two orthogonal coils. This can never be achieved with conventional single coils. The resulting field will, therefore, have a circular polarization. Here, we show that the pulses of circular polarization are free from the BS effect regardless of the driving field strengths. As a result, the nuclear spins can be coherently driven. The demonstration of the Carl–Purcell–Meiboom–Gill (CPMG) sequence [2] with circular polarization pulses indicates that each pulse reliably controls flip angles and phases of the nuclear spins. This enables the application of multiple pulse sequences in ULFNMR experiments. 2. Background In a conventional high-field NMR apparatus, a single coil is mostly used to produce an oscillating field, which is used to excite the nuclear spins. The generated field has a linear polarization along the direction of the coil (x axis). Including the static field along z axis (B0 ) and sinusoidal field of the strength 2B1 , the Hamiltonian HðtÞ for a nuclear spin I can be written as,
HðtÞ ¼ cB0 Iz þ 2cB1 Ix cos xt;
ð1Þ
where the c is gyromagnetic ratio. The linearly polarized field along the x axis can be decomposed into two rotating fields, co-rotating and counter-rotating, as
88
2B1 Ix cosðxtÞ ¼ B1 ðeixtIz Ix eixtIz þ c:cÞ:
J.H. Shim et al. / Journal of Magnetic Resonance 239 (2014) 87–90
ð2Þ
At high magnetic fields, the counter-rotating component is mostly neglected in the Hamiltonian; this is called Rotating Wave Approximation (RWA). In fact, the RWA approach is only valid if the strength of the rotating component, B1 , is much smaller than that of the static field B0 . For this case, the BS effect will induce a minor shift in the resonance frequency. According to the average Hamiltonian theory, the B1 intensity contributes to the frequency shift in 2nd order [6]. When B1 becomes comparable to the resonance frequency, the RWA does not correctly describe the dynamics of nuclear spins. As shown in Ref. [5], the nutation of target spins does not exhibit a sinusoidal oscillation. The BS effect, therefore, must be suppressed to achieve rapid as well as accurate control of the nuclear spins. Instead of a linearly polarized field, if we apply a circularly polarized field of frequency x, the time-varying part of the Hamiltonian HðtÞ will include only the first term of Eq. (2), B1 eixtIz Ix eixtIz . Consequently, the transformation from the lab frame to the rotating frame, in which the Hamiltonian HR becomes time-independent, can be performed without approximations. The BS effect will vanish. The resulting HR can be written as,
HR ¼ ðcB0 þ xÞIz0 þ cB1 Ix0 ;
(a)
(b)
ð3Þ
where the x0 and z0 are the axes in the rotating frame. For the resonance condition x ¼ cB0 with the co-rotating field, the nuclear spins will show an ideal nutation along the x0 axis in the rotating frame. This is our main idea.
3. Experimental To create circularly polarized fields, two coils orthogonal to each other are required. For simplicity, we assembled three square Helmholtz coils on the sides of a wooden square box as illustrated in Fig. 1(a). However, only the two that were orthogonal to the static magnetic field (Bm ), were used. Since the two coils are nearly identical, placing a phase difference in the currents was sufficient to generate circularly polarized fields. Either ±90° or 0° were used to choose the orientation of the polarization generated at the center of the coil pair. An arbitrary waveform generator (2 channels, 14 bit) produced the desired pulse shapes for the two coils. The phases and amplitudes of pulses were precisely controlled by waveform programming. The outputs were amplified using a 2Ch audio amplifier to inject sufficient current into the coils. Mechanical relays blocked the noises from the amplifier while the acquisitions were ongoing. Our ULF-NMR system uses a commercial DC-SQUID to measure the magnetic field variations produced by the precession of nuclear spins around the measurement field (Bm). The Bm is analogous to the B0 field in a high-field NMR. The SQUID sensor and a 2nd order gradiometer are inside the He dewar (Fig. 1(a)). Prior to every measurement, an 50 mT pre-polarization field (BP) was applied for two seconds to enhance the NMR signals from 1 H nuclear spins. The direction of the BP was configured to be parallel to Bm, because a resonant pulse flips the nuclear spins after the pre-polarization process, as shown in Fig. 1(b). The strength of the Bm field was 4.97 lT, which produces a resonance frequency of 212 Hz for 1 H nuclear spins. The free precession of the 1 H nuclear spins in DI water was recorded after applying the pulse. The coherence time T 2 of the 1 H spin in DI water was longer than one second. For the CPMG sequence, a constant gradient Gz , which was generated by a Maxwell coil, was applied to reduce the T 2 to produce better visibility of the spin echoes. The experiments were performed in a magnetically shielded room to suppress the environmental magnetic field noises.
Fig. 1. (a) An overview of our ULF-NMR system. The pick-up coil lies on the bottom of the He dewar. The BP coil produces the prepolarization field (BP), while the Bm coil produces the measurement field (Bm). Among the three square Helmholtz coils in the box at the center, two of them, orthogonal to the Bp and Bm fields, were chosen to generate circularly polarized fields. A gradient was applied when the echoes were recorded using the CPMG sequence. (b) The timing sequence for the measurement of the nutation. The duration of the pulse d was varied from 0 to 6 ms. The value of tm was fixed as 10 ms.
4. Results and discussion To determine the coherency of the motion of nuclear spins in the presence of a pulse, we measured the nutation curve from the variation of the initial amplitude of the free precession signal as a function of the pulse duration d. The entire timing sequence is shown in Fig. 1(b). The starting time of the measurement of the FID signal (t m ) was kept constant at 10 ms from the trigger point of the pulse output. Compared to T 2 , the t m is negligible. As such, the amplitude damping during tm is not measurable. Unlike the conventional sequences used to check the z component of the nuclear magnetization remaining after a pulse [13], the sequence in Fig. 1(b) measures the in-plane components, x and y. If nuclear spins are driven coherently, the in-plane measurement should produce a sinusoidal curve that would be the same as that produced using conventional sequences, except for a different phase. The initial amplitude of the free precession signal was obtained from the integration of the real part of the spectrum, which is a Fourier transform of the time domain signal. The real spectrum was phase-corrected before the integration. When the flip angle of the pulse was over, for instance, 90°, the phase was inverted accordingly. We did not correct this phase inversion to reflect the inversion of the nuclear spins produced by the applied pulse. The left column of Fig. 2 shows the nutation curves for nuclear spins driven by the pulses of different polarizations. The blue, purple, and green curves correspond to the circular polarization of a co-rotating field in resonance with the Bm field, circular polarization of a counter-rotating field, and linear polarization, respectively. The co-rotating field corresponds to the clockwise rotation
J.H. Shim et al. / Journal of Magnetic Resonance 239 (2014) 87–90
Fig. 2. (Left column) Nutation curves measured with the sequence in Fig. 1(b). The blue, green, purple curves correspond to the circular polarization of co-rotating field, the linear polarization, and the circular polarization of counter-rotating field, respectively. (Right column) Numerical simulation results can be compared with experimental results. Inhomogeneity of 10% in the B1 field strength was assumed in the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of the B1 field in the presence of the Bm along the z axis. Likewise, the counter-rotating field corresponds the counter-clockwise rotation. Every point was measured in a single run. For comparison with the experimental results, numerical simulations were performed in the lab frame. The time was incremented with a step of 10 ns (dt). At the jth time step or at t ¼ jdt, the time-dependent Hamiltonian HðtÞ, which is defined in Eq. (4), was averaged for the e j . The timeduration dt. The averaged Hamiltonian is denoted as H evolution operator UðtÞ was obtained by sequential multiplication, Q e i.e., UðtÞ ¼ j ei H j . The magnetization along the x axis at a time t was calculated from the density matrix qðtÞ as M x ðtÞ ¼ trfqðtÞIx g. In Eq. (4), / represents the relative phase between the x and y components of the B1 field.
HðtÞ ¼ cBm Iz þ cB1 fIx cosðxtÞ þ Iy cosðxt þ /Þg:
ð4Þ
In Fig. 2, all three experimental results are in good agreement with the numerical simulations shown on the right column. However, the origin of the small difference between the experiment and simulation for the linearly polarized pulse is unknown. Only the circular polarization of the co-rotating field exhibited a sinusoidal nutation. This sinusoidal behavior indicates coherent dynamics of the nuclear spins driven by the resonant pulse. The data (blue curve) fits well with the mathematical formula, Aet=s sinðxtÞ, in which the exponential decay attributes to the inhomogeneity of the pulse strength (B1) [14]. Because the volume of the sample is finite, we can never exclude the inhomogeneity of the B1 field. Thus, the nutation frequency measured from the fitting, 510 Hz, is the spatially averaged value. In the simulation, the inhomogeneity of the B1 field over the sample volume was assumed to be 10% with a quadratic variation from the center to the edge. This reasonably explains the origin of the decay in the experiment. Because the resonance frequency is 212 Hz, the strength B1 should be sufficient to induce a strong BS effect, which, nevertheless, has vanished by virtue of the circularly polarized field. The sinusoidal curve from the circular polarization of the co-rotating field clearly contrasts with the curve from the linear polarization. It is difficult to achieve a full inversion of the nuclear spins with linearly polarized pulses, because the in-plane magnetization
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barely becomes zero. Moreover, the phase of the in-plane magnetization evolves incoherently with the phase of the applied pulse, because the strong counter-rotating field also participates in driving the nuclear spins. The circular polarization of counter-rotating field, induces no BS effect either. The nutation, however, does not exhibit a sinusoidal behavior. According to Eq. (3), the field along the z0 axis remains because x ¼ cB0 for the counter-rotating field. Therefore, the effective field in the rotating frame is tilted from the x0 axis. This leads to the non-sinusoidal curve in the middle of Fig. 2. One of the main advantages of the coherent driving with circularly polarized pulses, as shown in Fig. 2 (blue curve), is the feasibility of adopting multiple pulse techniques that will depend on the particular purpose of the ULF-NMR/MRI studies. To substantiate this feature, we implemented the classical pulse sequences, i.e., Carr–Purcell (CP) and CPMG, with the pulses of a co-rotating field. The durations of the 90° and 180° pulses were 0.50 and 0.97 ms, respectively. Fig. 3 clearly shows the echo trains observed for both sequences. Demodulation of the NMR signal in the lab frame was performed digitally by multiplying a digital reference signal and low-pass filtering. The spin echo train of the CPMG survives longer than that of the CP, as expected from the known advantage of the CPMG sequence [2]. CPMG is more robust against pulse errors inherent in the applied pulses [15]. The fact that the CPMG sequence containing 20 refocusing pulses works well implies that the flip angle and phase of the nuclear spins can be precisely controlled by a strong pulse of the circular polarization. This implies that, all of the NMR pulse techniques will be still available even if the static magnetic field is reduced to less than one lT and even down to the regime of a near zero field [16]. The use of pulse techniques may provide a breakthrough in the improvement of the functionality of ULF-NMR/MRI. For instance, an MRI protocol using pulses may improve the quality or acquisition speed of ULF-MRI, where gradient sequences have been primarily used [8,17–19]. Furthermore, novel pulse sequences that are specific to the environment of ultra-low magnetic field can be devised. Simulation studies have revealed that the experimental factor that is the most relevant for the accuracy of the pulse, is the spatial inhomogeneity of the B1 field. As mentioned above, our current system has an inhomogeneity of approximately 10%. Nevertheless, if we take the time for the first maximum of the nutation curve (blue in Fig. 2) as the duration of the 90° pulse, the loss of the NMR signal is less than 2%. As a single pulse, this accuracy is relatively good. This is why the CP result in Fig. 3 appears to show no difference from the CPMG result until nearly the 10th refocusing pulse. Modification of the geometry of the excitation coils, such
Fig. 3. Echo trains obtained by the CP and CPMG sequences. The durations of the 90° and 180° pulses were estimated from the nutation curve (blue) in Fig. 2. From the CPMG result, T2 was measured to be 2.2 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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as the extension of the coil dimensions, will be necessary to enhance the accuracy. Finally, we believe that our scheme of using circularly polarized fields can also be applied to other systems that require strong driving because of short coherent times as well as precise operations. A good example of other systems such as this is nitrogen-vacancy centers in diamond, which can be exploited for various nano-scale sciences and applications [20]. 5. Conclusion The pulses generated by circularly polarized fields can coherently drive nuclear spins and be unhampered by the Bloch–Siegert effect, although the driving strength is higher than the static field. A coherent sinusoidal nutation curve was measured with circularly polarized pulses of a co-rotating field. With such pulses, one can rapidly and accurately manipulate nuclear spins. This will be crucial for the systems of relatively short relaxation times in the presence of a low static field. Demonstration of CP and CPMG sequences proves that such hard pulses can constitute multiple pulse sequences for ULF-NMR. We hope this work will help make a breakthrough in finding novel applications of ULF-NMR/MRI systems. Acknowledgments We appreciate the helpful comments from J.-S. Lee. This work was supported by the World Class Laboratory (WCL) Grant from Korea Research Institute of Standards and Science. References [1] R.P. Ernst, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford, 1978. [2] C. Slichter, Principles of Magnetic Resonance, Springer, Berlin, 1996. [3] D. Nishimura, Principles of Magnetic Resonance Imaging, Lulu, 2010.
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