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Spin–oscillatory coupling effect in parabolic magnetic field
11 December 2000
Physics Letters A 277 (2000) 299–303 www.elsevier.nl/locate/pla
Spin–oscillatory coupling effect in parabolic magnetic field C.J. Lewa a , P. Horodecki b , R. Horodecki c,∗ , M. Horodecki c a Institute of Experimental Physics, University of Gda´ nsk, 80-952 Gda´nsk, Poland b Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk, 80-952 Gda´nsk, Poland c Institute of Theoretical Physics and Astrophysics, University of Gda´nsk, 80-952 Gda´nsk, Poland
Received 6 June 2000; received in revised form 27 October 2000; accepted 30 October 2000 Communicated by P.R. Holland
Abstract We consider eigenvalue problem of the coupled spin–oscillator system in the external parabolic magnetic field B = (B0 + G0 x + G00 x 2 )ˆz. In result we obtain the energy spectrum exhibiting highly non-standard structure, due to the second gradient term. For some values of the latter and oscillatory frequency there is large energy difference between some magnetic sublevels. The spectrum allows, in principle, to identify frequency of the oscillator via the spectrum of spin sublevels coming only from one oscillatory level. 2000 Elsevier Science B.V. All rights reserved. PACS: 76.60.-k; 76.90.+d; 87.64.Hd
In a recent decade one can observe a rapid growth of interest in the Stern–Gerlach (SG) interaction [1,2] within magnetic resonance (MR) spectroscopy [3,9]. The main reasons are the following: (i) partial saturation of the possibilities of the conventional MR methods based on Bloch–Purcell paradigm [3] (low sensitivity, especially for the nuclear MR (NMR), growing costs of apparatus, existence of cheaper, competitive methods like, e.g., ultrasonic ones), (ii) recent reports on new and promising effects (e.g., atomic force microscopy [4,5], MR microimaging [6,7], MR spectroscopy for selected Zeeman states [8,9]), (iii) the need for new solutions in the domain of NMR quantum computing (the main problem in the recent implementations of quantum computing is low signal inten* Corresponding author.
E-mail addresses: [email protected] (C.J. Lewa), [email protected] (P. Horodecki), [email protected] (R. Horodecki), [email protected] (M. Horodecki).
sity) [10–14], (iv) the recent development in the domain of ferromagnetics and superconductors allowing to produce strong magnetic fields of required shapes. In this situation one is seeking for new, both theoretical and experimental solutions for MR, in particular by exploiting the SG interaction. Recently, Sidles [5] has proposed the interesting model of spin-1/2 particle mounted to the harmonic oscillator and predicted the importance of possible oscillator-based NMR spectroscopy for molecular imaging devices. In this Letter we consider the coupled spin–oscillator system in external heterogeneous magnetic field. The model is solved exactly and provides physical spectrum in which the oscillator frequency is coupled to the field derivatives parameters. The striking feature of the latter is that for some values of the second gradient and oscillator frequency there is large energy difference between some magnetic sublevels. The spectrum can be seen as a sum of two competitive
0375-9601/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 2 7 - 1
300
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303
terms: quantum and classical oscillator, where both involve quantum field gradient parameters. Remarkably it implies that, in principle, one can recognise the frequency of oscillator basing only on signals coming from transitions between sublevels of one oscillatory level. Let us consider the particle that subjects to the simple dynamic due to the harmonic oscillator HamilΩ (x − a). Here a is the position of the minitonian Hosc mum of the oscillator potential with respect to the origin of the reference frame (we imagine that the particle is in a sample of finite dimension l situated in the origin of the reference frame, so that a can vary within the region (−l, l)). The Hamiltonian of the particle −h2
d2
−
mΩ 2x 2
¯ + (1) 2m dx 2 2 has the following spectrum and corresponding eigenvectors: 1 , En = h¯ Ω n + 2 1/4 r mΩ mΩ x Hn Ψn (Ω; x) = 22n (n!)2 hπ h¯ ¯ mΩ 2 x × exp − (2) 2h¯ Ω (x) = Hosc
where Iˆ is the identity operator. In the above formula the difference between spatial and spin degrees of freedom has been stressed. The new energy spectrum of our particle can be obtained by seeking the eigenvectors in the form |φi ⊗ |Mi (M = −S, −S + 1, . . . , S). This leads effectively to the sequence of 2S + 1 Mdependent shifted quantum oscillators. Eigenvalues of each of those oscillators can be easily solved and the final energy spectrum of the particle is given by r 2γ G00 h¯ M 1 2 EM,n = h¯ Ω − n+ m 2 0 00 2 − γ B0 + G a + G a h¯ M
with Hermite polynomials Hn . Suppose now that the particle has the spin S and recall that the quantum operator corresponding to the projection P of the spin onto the z-axis is described with the corresponding as Iˆ = SM=−S hM|MihM| ¯ spectrum and eigenvector h¯ M, |Mi. Thus effectively in the above model we have the spin object confined in spin-independent harmonic oscillator potential (1) which, in particular, can correspond to binding the spin in the larger molecule or crystal. Consider the inhomogeneous magnetic field B(x) ≡ B(x)ˆz = B0 + G0 x + G00 x 2 zˆ (3) along the zˆ -axis. Here G0 , G00 stands for the gradient and second derivative parameters. Note that the field has the value B0 = B0 zˆ for x = 0 and, in general, it depends quadratically on the spatial coordinate x. Now, if we put our particle in the field (3) then the Hamiltonian takes the form Ω ˆ (x − a) ⊗ I − γ B(x) ⊗ zˆ I, H = Hosc
(4)
[γ (G0 + 2G00 a)h¯ M]2 , 2m(Ω 2 − 2γ G00 h¯ M/m)
(5)
and the corresponding eigenvectors can be written in the form ΦM,n = |φM,n i ⊗ |Mi.
(6)
The spatial coordinate function is defined by the oscillator eigenvectors (2) in the following way: γ (G0 + 2G00 a)h¯ 2 M , φM,n (x) ≡ Ψn Ω˜ M ; x − a − 2 mΩ˜ M (7) p where Ω˜ M ≡ Ω 2 − 2γ G00 h¯ M/m. It is important to note that the squared component of the field represented by its second derivative G00 must not be too large as then it suppresses the squared term of the particle oscillator potential mΩ 2 q 2 /2 leading to Hamiltonian of the unbounded form below. This occurs for the values of G00 violating the inequality |G00 | <
mΩ 2 , 2γ h¯ Mmax
(8)
where Mmax = S. In such case the first term of spectrum (5) becomes imaginary for some M. Now, if our model is applied to the particle with spin, being a light part of molecule, it can be interpreted as dissociation of the molecule caused by the strong gradient of the magnetic field. Thus inequality (8) establishes the boundaries of our model. In this context it is convenient to introduce the new discrete parameter defined by 00 ¯ S = 2γ G hM . M 2 Ω m
(9)
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303
S inequality (8) writes as Using M M Smax < 1.
(10)
G00
that violate inequality (8) If we allow gradients the model still can be used within those values of number M of spin level for which S<1 M for G00 > 0, S < −1 for G00 < 0. M
(11)
S as the ratio of the Then we can roughly interpret M given spin level M to the one for which “dissociation” occurs. As we are interested in the effect caused by gradients, we put B0 = 0. Then the energy spectrum can be written in the following form 1 p S n + 1−M EM,n = hΩ ¯ 2 S S2 mΩ 2a 2 M mΩ 2(G0 /G00 )a M − − S S 2 2 1−M 1−M 2 0 00 2 2 S M mΩ (G /G ) . − (12) S 2 4(1 − M) Consider first the case of infinitely small sample concentrated in the origin of the reference frame (i.e., put a = 0). Then we obtain an especially appealing form of the energy spectrum p qu S EM,n (a = 0) = Eosc (Ω, n) 1 − M S2 M cl , (13) (Ω, G0 /G00 ) − Eosc S 4(1 − M) cl (Ω, G0 /G00 ) = + 1/2) and Eosc where Eosc = hΩ(n ¯ 2 0 00 2 mΩ (G /G ) /2. Here we have two competitive terms: the first one represents the energy of quantum oscillator of frequency Ω (i.e., the energy of the particle in absence of the field) while the second one is nothing but the total energy of the classical oscillator of the same frequency, of the amplitude determined by the shape of the field. Both terms have weights depending on the scaled spin number. Let us now keep the value of G00 constant (suitably chosen, in order to be not too Smax ≈ close to the dissociation regime, e.g., to have M 0.1). Now varying the linear gradient G0 we obtain smooth transition from quantum oscillator regime to the classical one. Provided G0 is sufficiently large, so that the classical part dominates, we have a kind qu
301
of amplification of the vibrations of the particle by the magnetic field. In both cases the frequency Ω influences the ratio of splitting of the spin levels. Thus by measuring the latter, we can obtain the information about the value of the frequency. Clearly, the same is impossible in homogeneous field: the possibility of monitoring vibrational frequency via magnetic resonance is exclusively due to the gradient terms. Note that the double “quantum–classical” oscillator structure of the energy levels is due to quadratic gradient term. Indeed, for G00 = 0 we cannot use the S as the latter is defined for scaled spin level number M 00 G 6= 0. In such a case we have no “dissociation” as there is no denominator singularity in formula (5). If we put, instead, G0 = 0 (with a 6= 0) we still have two oscillators, however, the latter one follows from finite dimensions of the sample (so that the centre of the oscillations may feel nonzero value of magnetic field): p qu S EM,n (G = 0) = Eosc (Ω, n) 1 − M S2 M cl . (Ω, a) − Eosc (14) S 1−M A very important feature of analysed spectrum is the presence of the “dissociation” threshold. If we fix n, S close to 1, i.e., near the threshold, the ratio then for M of the splitting between magnetic sublevels becomes larger and larger. In result, manipulating with value and sign of G00 , one can in principle, select any closer magnetic sublevel. If, e.g., G00 > 0 then the level of the largest admissible M is the most separated from other sublevels. This can be seen in Fig. 1 where we present these levels for electronic MR (EMR) versus quadratic gradient G00 for the system of spin S = 3/2 (we keep G0 being nonzero, but weak). One also can observe remarkable crossing points for sufficiently large G00 . The energy difference for n = 2, between levels with M = 1/2 and 3/2 is the largest and grows for increasing gradient G00 . An interesting feature of the obtained spectrum of the particle is that the spin quantum number M has been coupled to the characteristic frequency Ω of the particle. This means that now for fixed quantum oscillator number n the M-dependent energy sublevels form the structure which depends on the characteristic frequency Ω of the particle. Indeed, putting G0 = G00 = 0 one can immediately see that it does not hap-
302
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303
the form of classical oscillator. The effect is due to nonzero G00 . The shape of the field determines the amplitude, and the amplification is obtained for high value of the linear gradient. Note that the spin–oscillatory coupling effect can be observed if we are not too far from the “dissociation” threshold, i.e., roughly speaking, if G00 ≈
Fig. 1. Structure of the EMR energy levels (a) in parabolic field for finite dimension of the sample (S = 3/2, a = 10−4 m, Ω = 105 Hz, G0 = −0.003 T/m) and oscillatory spectrum (b).
pen for uniform field B(x) = B0 . It is a remarkable result for the following reasons. Imagine the molecules with spin-less heavy “core” and spin S light part coupled to it. Assume that we have a mixture of unknown molecules of that kind, each of them possessing unknown characteristic frequencies Ω1 , Ω2 , . . . . Then the fact that spectrum (5) (with nonzero G0 or G00 ) depends on the frequency will provide the possibility to identify the frequencies basing directly on the NMR techniques that generically deal with 2S + 1 spin sublevels of the given level. It is interesting that if we have G00 = 0 and nonzero G0 then n dependent term does not depend on quantum number M. Of course, the structure of spin sublevels does not depend on n. Then all of the molecules of one kind (distinguished by Ω) would give the same signal from M sublevel of all oscillatory levels. A very interesting feature of the considered interaction is the emerging amplification of the vibrational mode to
mΩ 2 . 2γ h¯ Mmax
(15)
Then the frequency leading to the effect is propor00 tional to square root √ of G with a small proportionality constant: Ω ≈ K G00 . Thus for typical molecular frequencies (≈1014 Hz) one would need extremely strong gradients to observe the effect. Fortunately, there exist molecular systems that exhibit low frequencies [14– 18] (see also [19]). Thus the effect is difficult but possible to realize experimentally. In this context it may be worth to mention that there are nanoporous materials that can be viewed as suitable model systems. Indeed in such systems the field gradients effect can come from heterogeneities in susceptibility. Finally, we believe that the quantum-mechanical spin-coupling effect in the parabolic magnetic field will provide new interesting perspectives for the MR applications. In particular, the fact that the selected quantum level is separated from the other ones may be crucial for improvement of the MR sensitivity methods, as well as, in NMR quantum computation [10–13].
Acknowledgements We are grateful also to Prof. Zdzisław Paj¸ak for helpful discussion. C.J.L. was supported by Polish Committee for Scientific Research, contract No. 2P03B 148 12. M.H., P.H. and R.H. were partially supported by Polish Committee for Scientific Research, contract No. 2P03B 103 16 and the European Science Foundation. M.H. and P.H. also acknowledge the support from the Foundation for Polish Science.
References [1] O. Stern, Z. Phys. 7 (1921) 249. [2] W. Gerlach, O. Stern, Z. Phys. 8 (1921) 110.
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303 [3] D.M. Grant, R.K. Harris (Eds.), Encyclopaedia of Nuclear Magnetic Resonance, Vol. 1, Historical Perspectives, J. Wiley & Sons, Chichester, 1995. [4] J.A. Sidles, Appl. Phys. Lett. 58 (1991) 2854. [5] J.A. Sidles, Phys. Rev. Lett. 68 (1992) 1124. [6] J. Moore, P.C. Hammel, M.L. Roukes, in: Proc. 13th Ann. Meeting SMRM, San Francisco, 1994, p. 753. [7] D. Rugar, C.S. Yannoni, J.A. Sidles, Nature 360 (1992) 563. [8] C.J. Lewa, J. Magn. Reson. Anal. 2 (1996) 110. [9] C.J. Lewa, J.D. De Certaines, Europhys. Lett. 35 (1996) 713. [10] N.A. Gershenfeld, I.L. Chuang, Science 275 (1997) 350. [11] D.G. Cory, A.F. Fahmy, T. Havel, Proc. Natl. Acad. Sci. USA 94 (1998) 1634.
303
[12] I.L. Chuang, L.M.K. Vandersypen, X. Zhou, D.W. Leung, S. Lloyd, Nature 393 (1998) 143. [13] J.A. Jones, M. Mosca, M.H. Hansen, Nature 393 (1998) 344. [14] H.N. Spiess, Ber. Bunsenges. Phys. Chem. 101 (1997) 153. [15] H.N. Spiess, Ber. Bunsenges. Phys. Chem. 97 (1997) 1294. [16] S. Hafner, H.N. Spiess, J. Magn. Reson. 121 (1996) 160. [17] S. Hafner, H.N. Spiess, Solid. States NMR 8 (1997) 17. [18] B.V.S. Murphy, K.P. Ramesh, J. Ramakrishna, Phys. Status Solidi 142 (1994) 219. [19] J.A. Tuszy´nski, E. Kimberly Strong, J. Biol. Phys. 17 (1989) 19.
Physics Letters A 277 (2000) 299–303 www.elsevier.nl/locate/pla
Spin–oscillatory coupling effect in parabolic magnetic field C.J. Lewa a , P. Horodecki b , R. Horodecki c,∗ , M. Horodecki c a Institute of Experimental Physics, University of Gda´ nsk, 80-952 Gda´nsk, Poland b Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk, 80-952 Gda´nsk, Poland c Institute of Theoretical Physics and Astrophysics, University of Gda´nsk, 80-952 Gda´nsk, Poland
Received 6 June 2000; received in revised form 27 October 2000; accepted 30 October 2000 Communicated by P.R. Holland
Abstract We consider eigenvalue problem of the coupled spin–oscillator system in the external parabolic magnetic field B = (B0 + G0 x + G00 x 2 )ˆz. In result we obtain the energy spectrum exhibiting highly non-standard structure, due to the second gradient term. For some values of the latter and oscillatory frequency there is large energy difference between some magnetic sublevels. The spectrum allows, in principle, to identify frequency of the oscillator via the spectrum of spin sublevels coming only from one oscillatory level. 2000 Elsevier Science B.V. All rights reserved. PACS: 76.60.-k; 76.90.+d; 87.64.Hd
In a recent decade one can observe a rapid growth of interest in the Stern–Gerlach (SG) interaction [1,2] within magnetic resonance (MR) spectroscopy [3,9]. The main reasons are the following: (i) partial saturation of the possibilities of the conventional MR methods based on Bloch–Purcell paradigm [3] (low sensitivity, especially for the nuclear MR (NMR), growing costs of apparatus, existence of cheaper, competitive methods like, e.g., ultrasonic ones), (ii) recent reports on new and promising effects (e.g., atomic force microscopy [4,5], MR microimaging [6,7], MR spectroscopy for selected Zeeman states [8,9]), (iii) the need for new solutions in the domain of NMR quantum computing (the main problem in the recent implementations of quantum computing is low signal inten* Corresponding author.
E-mail addresses: [email protected] (C.J. Lewa), [email protected] (P. Horodecki), [email protected] (R. Horodecki), [email protected] (M. Horodecki).
sity) [10–14], (iv) the recent development in the domain of ferromagnetics and superconductors allowing to produce strong magnetic fields of required shapes. In this situation one is seeking for new, both theoretical and experimental solutions for MR, in particular by exploiting the SG interaction. Recently, Sidles [5] has proposed the interesting model of spin-1/2 particle mounted to the harmonic oscillator and predicted the importance of possible oscillator-based NMR spectroscopy for molecular imaging devices. In this Letter we consider the coupled spin–oscillator system in external heterogeneous magnetic field. The model is solved exactly and provides physical spectrum in which the oscillator frequency is coupled to the field derivatives parameters. The striking feature of the latter is that for some values of the second gradient and oscillator frequency there is large energy difference between some magnetic sublevels. The spectrum can be seen as a sum of two competitive
0375-9601/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 2 7 - 1
300
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303
terms: quantum and classical oscillator, where both involve quantum field gradient parameters. Remarkably it implies that, in principle, one can recognise the frequency of oscillator basing only on signals coming from transitions between sublevels of one oscillatory level. Let us consider the particle that subjects to the simple dynamic due to the harmonic oscillator HamilΩ (x − a). Here a is the position of the minitonian Hosc mum of the oscillator potential with respect to the origin of the reference frame (we imagine that the particle is in a sample of finite dimension l situated in the origin of the reference frame, so that a can vary within the region (−l, l)). The Hamiltonian of the particle −h2
d2
−
mΩ 2x 2
¯ + (1) 2m dx 2 2 has the following spectrum and corresponding eigenvectors: 1 , En = h¯ Ω n + 2 1/4 r mΩ mΩ x Hn Ψn (Ω; x) = 22n (n!)2 hπ h¯ ¯ mΩ 2 x × exp − (2) 2h¯ Ω (x) = Hosc
where Iˆ is the identity operator. In the above formula the difference between spatial and spin degrees of freedom has been stressed. The new energy spectrum of our particle can be obtained by seeking the eigenvectors in the form |φi ⊗ |Mi (M = −S, −S + 1, . . . , S). This leads effectively to the sequence of 2S + 1 Mdependent shifted quantum oscillators. Eigenvalues of each of those oscillators can be easily solved and the final energy spectrum of the particle is given by r 2γ G00 h¯ M 1 2 EM,n = h¯ Ω − n+ m 2 0 00 2 − γ B0 + G a + G a h¯ M
with Hermite polynomials Hn . Suppose now that the particle has the spin S and recall that the quantum operator corresponding to the projection P of the spin onto the z-axis is described with the corresponding as Iˆ = SM=−S hM|MihM| ¯ spectrum and eigenvector h¯ M, |Mi. Thus effectively in the above model we have the spin object confined in spin-independent harmonic oscillator potential (1) which, in particular, can correspond to binding the spin in the larger molecule or crystal. Consider the inhomogeneous magnetic field B(x) ≡ B(x)ˆz = B0 + G0 x + G00 x 2 zˆ (3) along the zˆ -axis. Here G0 , G00 stands for the gradient and second derivative parameters. Note that the field has the value B0 = B0 zˆ for x = 0 and, in general, it depends quadratically on the spatial coordinate x. Now, if we put our particle in the field (3) then the Hamiltonian takes the form Ω ˆ (x − a) ⊗ I − γ B(x) ⊗ zˆ I, H = Hosc
(4)
[γ (G0 + 2G00 a)h¯ M]2 , 2m(Ω 2 − 2γ G00 h¯ M/m)
(5)
and the corresponding eigenvectors can be written in the form ΦM,n = |φM,n i ⊗ |Mi.
(6)
The spatial coordinate function is defined by the oscillator eigenvectors (2) in the following way: γ (G0 + 2G00 a)h¯ 2 M , φM,n (x) ≡ Ψn Ω˜ M ; x − a − 2 mΩ˜ M (7) p where Ω˜ M ≡ Ω 2 − 2γ G00 h¯ M/m. It is important to note that the squared component of the field represented by its second derivative G00 must not be too large as then it suppresses the squared term of the particle oscillator potential mΩ 2 q 2 /2 leading to Hamiltonian of the unbounded form below. This occurs for the values of G00 violating the inequality |G00 | <
mΩ 2 , 2γ h¯ Mmax
(8)
where Mmax = S. In such case the first term of spectrum (5) becomes imaginary for some M. Now, if our model is applied to the particle with spin, being a light part of molecule, it can be interpreted as dissociation of the molecule caused by the strong gradient of the magnetic field. Thus inequality (8) establishes the boundaries of our model. In this context it is convenient to introduce the new discrete parameter defined by 00 ¯ S = 2γ G hM . M 2 Ω m
(9)
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303
S inequality (8) writes as Using M M Smax < 1.
(10)
G00
that violate inequality (8) If we allow gradients the model still can be used within those values of number M of spin level for which S<1 M for G00 > 0, S < −1 for G00 < 0. M
(11)
S as the ratio of the Then we can roughly interpret M given spin level M to the one for which “dissociation” occurs. As we are interested in the effect caused by gradients, we put B0 = 0. Then the energy spectrum can be written in the following form 1 p S n + 1−M EM,n = hΩ ¯ 2 S S2 mΩ 2a 2 M mΩ 2(G0 /G00 )a M − − S S 2 2 1−M 1−M 2 0 00 2 2 S M mΩ (G /G ) . − (12) S 2 4(1 − M) Consider first the case of infinitely small sample concentrated in the origin of the reference frame (i.e., put a = 0). Then we obtain an especially appealing form of the energy spectrum p qu S EM,n (a = 0) = Eosc (Ω, n) 1 − M S2 M cl , (13) (Ω, G0 /G00 ) − Eosc S 4(1 − M) cl (Ω, G0 /G00 ) = + 1/2) and Eosc where Eosc = hΩ(n ¯ 2 0 00 2 mΩ (G /G ) /2. Here we have two competitive terms: the first one represents the energy of quantum oscillator of frequency Ω (i.e., the energy of the particle in absence of the field) while the second one is nothing but the total energy of the classical oscillator of the same frequency, of the amplitude determined by the shape of the field. Both terms have weights depending on the scaled spin number. Let us now keep the value of G00 constant (suitably chosen, in order to be not too Smax ≈ close to the dissociation regime, e.g., to have M 0.1). Now varying the linear gradient G0 we obtain smooth transition from quantum oscillator regime to the classical one. Provided G0 is sufficiently large, so that the classical part dominates, we have a kind qu
301
of amplification of the vibrations of the particle by the magnetic field. In both cases the frequency Ω influences the ratio of splitting of the spin levels. Thus by measuring the latter, we can obtain the information about the value of the frequency. Clearly, the same is impossible in homogeneous field: the possibility of monitoring vibrational frequency via magnetic resonance is exclusively due to the gradient terms. Note that the double “quantum–classical” oscillator structure of the energy levels is due to quadratic gradient term. Indeed, for G00 = 0 we cannot use the S as the latter is defined for scaled spin level number M 00 G 6= 0. In such a case we have no “dissociation” as there is no denominator singularity in formula (5). If we put, instead, G0 = 0 (with a 6= 0) we still have two oscillators, however, the latter one follows from finite dimensions of the sample (so that the centre of the oscillations may feel nonzero value of magnetic field): p qu S EM,n (G = 0) = Eosc (Ω, n) 1 − M S2 M cl . (Ω, a) − Eosc (14) S 1−M A very important feature of analysed spectrum is the presence of the “dissociation” threshold. If we fix n, S close to 1, i.e., near the threshold, the ratio then for M of the splitting between magnetic sublevels becomes larger and larger. In result, manipulating with value and sign of G00 , one can in principle, select any closer magnetic sublevel. If, e.g., G00 > 0 then the level of the largest admissible M is the most separated from other sublevels. This can be seen in Fig. 1 where we present these levels for electronic MR (EMR) versus quadratic gradient G00 for the system of spin S = 3/2 (we keep G0 being nonzero, but weak). One also can observe remarkable crossing points for sufficiently large G00 . The energy difference for n = 2, between levels with M = 1/2 and 3/2 is the largest and grows for increasing gradient G00 . An interesting feature of the obtained spectrum of the particle is that the spin quantum number M has been coupled to the characteristic frequency Ω of the particle. This means that now for fixed quantum oscillator number n the M-dependent energy sublevels form the structure which depends on the characteristic frequency Ω of the particle. Indeed, putting G0 = G00 = 0 one can immediately see that it does not hap-
302
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303
the form of classical oscillator. The effect is due to nonzero G00 . The shape of the field determines the amplitude, and the amplification is obtained for high value of the linear gradient. Note that the spin–oscillatory coupling effect can be observed if we are not too far from the “dissociation” threshold, i.e., roughly speaking, if G00 ≈
Fig. 1. Structure of the EMR energy levels (a) in parabolic field for finite dimension of the sample (S = 3/2, a = 10−4 m, Ω = 105 Hz, G0 = −0.003 T/m) and oscillatory spectrum (b).
pen for uniform field B(x) = B0 . It is a remarkable result for the following reasons. Imagine the molecules with spin-less heavy “core” and spin S light part coupled to it. Assume that we have a mixture of unknown molecules of that kind, each of them possessing unknown characteristic frequencies Ω1 , Ω2 , . . . . Then the fact that spectrum (5) (with nonzero G0 or G00 ) depends on the frequency will provide the possibility to identify the frequencies basing directly on the NMR techniques that generically deal with 2S + 1 spin sublevels of the given level. It is interesting that if we have G00 = 0 and nonzero G0 then n dependent term does not depend on quantum number M. Of course, the structure of spin sublevels does not depend on n. Then all of the molecules of one kind (distinguished by Ω) would give the same signal from M sublevel of all oscillatory levels. A very interesting feature of the considered interaction is the emerging amplification of the vibrational mode to
mΩ 2 . 2γ h¯ Mmax
(15)
Then the frequency leading to the effect is propor00 tional to square root √ of G with a small proportionality constant: Ω ≈ K G00 . Thus for typical molecular frequencies (≈1014 Hz) one would need extremely strong gradients to observe the effect. Fortunately, there exist molecular systems that exhibit low frequencies [14– 18] (see also [19]). Thus the effect is difficult but possible to realize experimentally. In this context it may be worth to mention that there are nanoporous materials that can be viewed as suitable model systems. Indeed in such systems the field gradients effect can come from heterogeneities in susceptibility. Finally, we believe that the quantum-mechanical spin-coupling effect in the parabolic magnetic field will provide new interesting perspectives for the MR applications. In particular, the fact that the selected quantum level is separated from the other ones may be crucial for improvement of the MR sensitivity methods, as well as, in NMR quantum computation [10–13].
Acknowledgements We are grateful also to Prof. Zdzisław Paj¸ak for helpful discussion. C.J.L. was supported by Polish Committee for Scientific Research, contract No. 2P03B 148 12. M.H., P.H. and R.H. were partially supported by Polish Committee for Scientific Research, contract No. 2P03B 103 16 and the European Science Foundation. M.H. and P.H. also acknowledge the support from the Foundation for Polish Science.
References [1] O. Stern, Z. Phys. 7 (1921) 249. [2] W. Gerlach, O. Stern, Z. Phys. 8 (1921) 110.
C.J. Lewa et al. / Physics Letters A 277 (2000) 299–303 [3] D.M. Grant, R.K. Harris (Eds.), Encyclopaedia of Nuclear Magnetic Resonance, Vol. 1, Historical Perspectives, J. Wiley & Sons, Chichester, 1995. [4] J.A. Sidles, Appl. Phys. Lett. 58 (1991) 2854. [5] J.A. Sidles, Phys. Rev. Lett. 68 (1992) 1124. [6] J. Moore, P.C. Hammel, M.L. Roukes, in: Proc. 13th Ann. Meeting SMRM, San Francisco, 1994, p. 753. [7] D. Rugar, C.S. Yannoni, J.A. Sidles, Nature 360 (1992) 563. [8] C.J. Lewa, J. Magn. Reson. Anal. 2 (1996) 110. [9] C.J. Lewa, J.D. De Certaines, Europhys. Lett. 35 (1996) 713. [10] N.A. Gershenfeld, I.L. Chuang, Science 275 (1997) 350. [11] D.G. Cory, A.F. Fahmy, T. Havel, Proc. Natl. Acad. Sci. USA 94 (1998) 1634.
303
[12] I.L. Chuang, L.M.K. Vandersypen, X. Zhou, D.W. Leung, S. Lloyd, Nature 393 (1998) 143. [13] J.A. Jones, M. Mosca, M.H. Hansen, Nature 393 (1998) 344. [14] H.N. Spiess, Ber. Bunsenges. Phys. Chem. 101 (1997) 153. [15] H.N. Spiess, Ber. Bunsenges. Phys. Chem. 97 (1997) 1294. [16] S. Hafner, H.N. Spiess, J. Magn. Reson. 121 (1996) 160. [17] S. Hafner, H.N. Spiess, Solid. States NMR 8 (1997) 17. [18] B.V.S. Murphy, K.P. Ramesh, J. Ramakrishna, Phys. Status Solidi 142 (1994) 219. [19] J.A. Tuszy´nski, E. Kimberly Strong, J. Biol. Phys. 17 (1989) 19.