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Physics
Boundary of quantum evolution under decoherence
Division of Applied Sciences, Harvard University, Cambridge, MA 02138; and ¶Institute for Organic Chemistry and Biochemistry, Technische Universität München, 85747 Garching, Germany
Edited by Alfred G. Redfield, Brandeis University, Waltham, MA and approved August 13, 2003 (received for review June 28, 2003)
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Abstract |
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In this article we derive fundamental limits on how close an ensemble of nuclear spins can be driven from its initial state to a desired target state in the presence of relaxation. In particular, we derive the maximum efficiency of polarization and coherence transfer between coupled nuclear spins. Such coherence transfer operations are important in high-resolution NMR spectroscopy (2, 3). In structural biology, NMR spectroscopy is an important technique that allows one to determine the structure of biological macromolecules, such as proteins, in aqueous solution. With increasing size of molecules or molecular complexes, the rotational tumbling of the molecules becomes slower and leads to increased relaxation losses. When these relaxation rates become comparable to the spin-spin couplings, the efficiency of coherence transfer is considerably reduced, leading to poor sensitivity and increased measurement times. Recent advances have made it possible to significantly extend the size limit of biological macromolecules amenable to study by liquid-state NMR (4-7). These techniques take advantage of the phenomenon of cross-correlated relaxation, which represents interference effects between two different relaxation mechanisms (8-13). Until now, it was not clear if further improvements can be made and what is the physical limit for the coherence transfer efficiency between coupled spins in the presence of cross-correlated relaxation.
In this article, we give analytical expressions for this maximum achievable coherence transfer efficiency for two coupled heteronuclear spins under very general decoherence mechanisms that include cross-correlated relaxation. We describe the optimal pulse sequences that achieve this efficiency and provide experimental data that support these results. In the general case of cross-correlated relaxation, we demonstrate substantial improvement over previously known sequences in NMR spectroscopy. It should be noted that the optimal transfer efficiency reported here applies to the case where the resonance frequencies of a single spin pair (I and S) are known. However, the presented approach will also make it possible to derive optimal broadband transfer schemes for a given range of resonance frequencies.
We also show theoretically that in the limit where the cross-correlated relaxation rates become identical to the autocorrelated relaxation rates, lossless transfer of coherence is possible between coupled spins. For an isolated spin pair in an isotropically tumbling molecule, this limit can be reached if the interfering interactions are axially symmetric and if the symmetry axes and the size of the interactions coincide (4, 12). Although this is not the generic case, it can be approached by many systems of practical interest (4, 6), and it may be feasible to construct molecules for quantum information processing in which a complete match is possible.
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Theory |
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denote the density operator of a quantum mechanical system coupled to a bath. Under the assumption of Markovian dynamics for the system of interest (very short correlation times with the bath) (14), the most general form for the evolution of the system density operator takes the form
 | [1] |
 | [2] |
 | [3] |
where H is the systems Hamiltonian and generates unitary evolution. All nonunitary relaxation dynamics is accounted for by LD. The Hermitian coefficient matrix A 
{ak,l} contains the information about physical relaxation parameters (lifetimes, relaxation rates) and Fk denotes operators representing various relaxation mechanisms (14).
We now consider an isolated heteronuclear spin system consisting of two coupled spins 
, denoted I (e.g., 1H) and S (e.g., 15N). To fix ideas, we first address the problem of selective population inversion of two energy levels (e.g., 
and ) as shown in Fig. 1. This is a central step in high-resolution multidimensional NMR spectroscopy and corresponds to the transfer of an initial state Iz, representing polarization on spin I, to the target state 2IzSz representing longitudinal two-spin order. We now consider the slow tumbling regime (the so-called spin diffusion limit) (2), which applies to macromolecules at high magnetic fields, where the correlation time of the molecular tumbling is much shorter than the inverse of the resonance frequencies of spins I and S. In this limit, longitudinal relaxation rates are negligible compared with transverse relaxation rates for an isolated heteronuclear spin system consisting of two coupled spins , where the two principle relaxation mechanisms are dipole-dipole (DD) relaxation and relaxation due to the chemical shift anisotropy (CSA) of spins I and S. Hence, both the initial state (Iz) and final state (2IzSz) are long-lived. However, the transfer between these two states requires the creation of coherences that in general are subject to fast transverse relaxation. In a double rotating frame chosen specifically for the pair of spins under discussion, the above set of equations (Eqs. 1-3) take the following form (Eq. 4), where F1 = 2IzSz, F2 = Iz, F3 = Sz, and we assume a23 = 0 because in the present application interference terms between the CSA of spins I and S have no effect on the involved density operator terms (see below):
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where J is the heteronuclear coupling constant. The rates kDD, 
, and 
represent autocorrelated relaxation rates due to DD relaxation, CSA relaxation of spin I, and CSA relaxation of spin S, respectively. The rates 
and 
represent cross-correlation rates of spin I and S caused by interference effects between DD and CSA relaxation. Unconventional 
factors in front of the relaxation rates results in concise expressions of optimal transfer efficiency later. The relaxation rates depend on various physical parameters, such as the gyromagnetic ratios of the spins, the internuclear distance, the CSA tensors, the strength of the magnetic field, and the correlation time of the molecular tumbling (2, 11).
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(0) = A and denote the density operator at time t by (t). The maximum efficiency of transfer between A and target operator C is defined as the largest possible value of trace (C(t)) for any time t (3) (by convention operators A and C are normalized).
The main result of this article is as follows. The maximal efficiency of transfer between the operators Iz and 2IzSz depends only on the scalar coupling constant J and the net autocorrelated and cross-correlated relaxation rates of spin I, given by 
and 
, respectively. Here the rates ka and kc are a factor of smaller than in conventional definitions of the rates, e.g., ka = 1/(T2) if T2 is the transverse relaxation rate in the absence of cross-correlation effects (15). The physical limit 
of the maximal transfer efficiency is given by
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where
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The derivation of the maximal efficiency rests on the basic principles of optimal control theory (16, 17) (for details, see Supporting Methods, which is published as supporting information on the PNAS web site). The optimal transfer scheme (CROP, cross-correlated relaxation optimized pulse) has two constants of motion. If l1(t) and l2(t) denote the two-dimensional vectors (
) and (
), respectively, then throughout the transfer process the ratio of the magnitudes of the vectors l2 and l1 is maintained constant at . Furthermore, the angle 
* between l1 and l2 is constant throughout. The two constants of motion of the optimal transfer scheme determine the amplitude and phase of the rf field at each point in time and explicit expressions for the optimal pulse sequence can be derived (see Supporting Methods).
We now consider two important limiting cases of this problem: (i) In the case when ka > 0 and kc = 0 (no cross-correlated relaxation), the optimal efficiency is equal to 
(see curve for kc/ka = 0 in Fig. 2A) and the optimal angle * is /2 (15).
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approaches 1 (see curve for kc/ka = 1 in Fig. 2), and * approaches . Surprisingly, in this case using optimal control it is possible to transfer coherence without any loss in the presence of relaxation. In the limit of large relaxation rates ka, this relaxation-optimized transfer mechanism gains up to 100% compared with state-of-the-art transfer schemes.
The optimal transfer scheme is best illustrated by decomposing the initial operator Iz as a sum of the two single-transition operators IzS = Iz/2 + IzSz and IzS = Iz/2 - IzSz (2). The transverse components IxS, IyS and IxS, IyS relax with rates ka + kc and ka - kc, respectively. When kc/ka approaches 1, the transverse single-transition operators IxS and IyS do not relax. The optimal pulse sequence in this case reduces to selectively inverting IzS to -IzS by weak rf irradiation at the frequency (-J/2) of the slowly relaxing multiplet component. Such selective inversions have been performed in the past for various applications, including the selective measurement of relaxation rates (11, 20-23). Because the component IzS, which we do not want to invert, has a large transverse relaxation rate given by ka + kc, it is possible to carry out the selective inversion process much more rapidly than in the absence of relaxation. In Fig. 3, optimal trajectories of the two multiplet components are shown for several cross-correlation coefficients kc/ka and ka = J.
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is shown in Fig. 2 A for several ratios kc/ka as a function of the autocorrelated relaxation rate ka/J. For the case kc/ka = 0.75, the physical limit of the transfer efficiency is compared in Fig. 2B to the transfer efficiency of conventional transfer schemes INEPT (18), CRIPT (19), and CRINEPT (7).
The optimal control methods for the transfer from Iz to 2IzSz in the presence of cross-correlated relaxation immediately extend to other routinely used transfers, such as in-phase to in-phase transfer (Ix 
Sx) (24) and single-transition to single-transition transfer (e.g., 2IxS 2ISx) (4). Because the operators Iz, Sz, and 2IzSz do not decay, the optimal efficiency for the transfer Ix to Sx is achieved by first rotating Ix to Iz (which can be done rapidly with negligible loss). Then Iz is transferred optimally to 2IzSz with efficiency (Eq. 5), followed by the optimal transfer of 2IzSz to Sz, which is finally rotated rapidly to Sx. The optimal transfer 2IzSz Sz is analogous to the optimal transfer Iz 2IzSz. The efficiency ' for this transfer is also given by Eq. 5, where the rates ka and kc are replaced by the corresponding rates 
and 
for spin S and 
is replaced by the corresponding '. The maximal efficiency for the transfer Ix Sx is the product of the efficiencies of the individual steps (see Table 1).
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2ISx is important in NMR applications to structural biology. It is of both theoretical and practical interest to establish the physical limits for this transfer. This transfer can be achieved optimally as a sequence of the following steps. First, the term 2IxS is rapidly rotated to 2IzS = Iz - 2IzSz. In a second step, -2IzSz is transferred by means of CROP to Sz, followed by the CROP transfer of Iz to -2IzSz. This completes the transfer from 2IxS to 2ISz, which is finally rapidly rotated to 2ISx. The maximal overall transfer efficiency is given by 
(cf. Table 1). |
Experimental Results |
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1.1 (Fig. 5A), and 260 K, where ka/J 0.6 (Fig. 5B). At a magnetic field of 17.6 T, the experimentally determined ratio of cross- to autocorrelation rate was kc/ka 0.75. In the preparation phase of the experiments, the thermal equilibrium 1H magnetization was dephased by applying a 90° proton pulse followed by a pulsed magnetic field gradient. The transfer efficiency of 13C polarization Iz to 2IzSz was measured for the novel CROP sequence, as well as for INEPT (18), CRIPT (19), and CRINEPT (7) sequences. Finally, a hard 90°y proton pulse was applied to transform 2IzSz to 2IzSx, and the amplitude of the resulting proton antiphase signal was measured. The resulting experimental transfer amplitudes are shown in Fig. 5 as a function of the transfer time. CROP sequences were truncated symmetrically to acquire transfer amplitudes also for finite mixing times. Experimentally, the optimal transfer time of the CROP sequence was found to be 7.5 ms and 15 ms, respectively. This is a compromise between losses due to the truncation of the (very long) CROP sequence and losses due to the nonzero relaxation rates of the terms IzSz. The experimentally determined relaxation time of these terms was about 45 ms and 80 ms, respectively. Despite these nonidealities of the model system, the CROP sequences are substantially more efficient than the conventional sequences. In Fig. 5, the experimental gains compared with CRINEPT are 34% and 22%, respectively. Although the optimal pulse sequences were designed for specific rates ka and kc, they were found to be robust to variations in these parameters.
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Conclusion |
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The most surprising aspect of the presented results is that despite large relaxation rates, it is possible to exploit the structure of relaxation and have decoherence-free evolution by steering the system through a decoherence-free subspace (when kc = ka, the operators IxS, IyS, and IzS span a decoherence-free subspace). Decoherence-free subspaces (DFS) have generated considerable interest in the area of quantum information processing recently. It has been shown that by encoding qubits within the subspaces of the Hilbert space that do not decohere, it is possible to perform error-free quantum computations (27). Interference effects among various decoherence mechanisms (13, 28) provide a way for creating DFS. It is possible that in some future implementations of quantum computing devices, by suitably engineering interference between various decoherence mechanisms, a DFS can be synthesized for error-free computation. The methods presented here can be extended to find optimal pulse sequences that in the presence of relaxation will produce a Liouville evolution that is closest to a desired unitary evolution. Such relaxation-optimized implementations of unitary propagators can then be used to minimize decoherence losses in quantum information processing.
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Acknowledgements |
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Footnotes |
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Abbreviations: CSA, chemical shift anisotropy; DD, dipole-dipole; CROP, cross-correlated relaxation optimized pulse.
To whom correspondence should be addressed. E-mail: navin{at}hrl.harvard.edu.
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  N. Khaneja, J.-S. Li, C. Kehlet, B. Luy, and S. J. Glaser Broadband relaxation-optimized polarization transfer in magnetic resonance PNAS, October 12, 2004; 101(41): 14742 - 14747. [Abstract] [Full Text] [PDF]
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and 
for ka = J and kc/ka = 0 (green curves), kc/ka = 0.75 (red curves), kc/ka = 0.95 (blue curves), and kc/ka = 0.999 (black curves).
/J. The dotted line corresponds to the frequency of the narrow multiplet component that is inverted by the CROP sequence.