Quantum measurement corrections to CIDNP in photosynthetic reaction centers

The quantum dynamics of photosynthesis have, quite naturally, attracted a lot of attention recently. Understanding if and how Nature exploits quantum (de)coherence would have tremendous scientific and technological impact. In what follows, photosynthesis can be simply visualized as a two-step process: (i) light harvesting and (ii) photochemical processing. Light harvesting molecules absorb the incident photon and guide the subsequent excitation to the photosynthetic reaction center (RC). It is there where the photochemistry takes place, transforming the initial electronic excitation in a transmembrane proton pump that further drives the life-sustaining chemistry. Regarding step (i), quantum coherence effects in light harvesting complexes (time scales of 1–500 fs) have been addressed theoretically and experimentally [1–10]. We will focus here on step (ii), i.e. on RC quantum dynamics (time scales of 1 ps–10 μs) schematically depicted in figure 1(a).

Zoom In Zoom Out Reset image size

Photosynthetic RCs exhibit a cascade of electron transfer reactions, starting from a photo-excited donor–acceptor dyad PI (chlorophyll-type molecules) and shelving the electron further apart to quinones (Q_A, Q_B), producing a long-lived charge-separated state driving the transmembrane proton pump. In each of these steps there is a radical-ion pair (RP) formed, further perplexing the dynamics, as RPs exhibit intricate magnetic field effects [11]. RPs [12, 13] are biomolecular ions created from the photo-excited dyad ¹PI. They have two unpaired electrons (denoted by the two dots in figure 1) and any number of magnetic nuclei. The magnetic nuclei of the donor and acceptor molecular ions couple to the ion's unpaired electron via the hyperfine interaction. This coupling, along with several other intramolecule magnetic interactions, leads to singlet–triplet (S–T) mixing, i.e. a coherent oscillation of the total electronic spin, also affected by the electrons' Zeeman interaction with the external magnetic field. A spin-dependent charge recombination terminates the coherent S–T mixing and leads to the reaction products, either the original neutral dyad PI (singlet product) or a triplet intermediate (³PI). As shown in figure 1(b), singlet (triplet) radical pairs recombine at the singlet (triplet) recombination rate, denoted by k_S (k_T). Triplet products may close the reaction cycle by intersystem crossing into the singlet diamagnetic ground state at a rate k_isc.

It is apparent from figure 1 that when the lifetime of the charge-separated states (which increases with separation) is long enough that magnetic-field effects (with typical time scales of e.g. S–T mixing of the order of 0.1–1 μs) have time to build up, radical-pair spin dynamics are convolved into the RC dynamics. One aspect of this interrelation pervading a large number of photosynthetic RCs is chemically induced dynamic nuclear polarization (CIDNP) [13–20], the enhancement (by several orders of magnitude) of the ground state (PI) nuclear spin polarization resulting from the RP spin dynamics. CIDNP has emerged as a rather universal signature of spin order in photosynthetic RCs [22] and its possible operational significance for photosynthesis is still an open fundamental question. CIDNP is based on nuclear spin sorting, to be explained shortly, taking place in radical-pair reactions. Regarding solid-state CIDNP in particular, three main mechanisms have so far been known to produce CIDNP signals, the three-spin mixing (TSM) [23], differential decay (DD) and differential relaxation (DR) mechanisms [24]. The first is a coherent mechanism resulting from particular magnetic interactions within the RP, while the second is about a nuclear spin imbalance brought about by different recombination rates k_S ≠ k_T. The last mechanism (DR) will not be relevant for the following discussion.

We will demonstrate here that there is yet another, completely unanticipated mechanism at work at the fundamental quantum dynamical level of RP reactions. It is due to the inherent quantum measurement of the RP's spin state continuously going on in the RP as part of the very charge recombination process. The decoherence caused by this 'internal' quantum measurement is responsible for a new kind of nuclear spin sorting, particularly efficient at low fields, and completely unrelated to other low-field CIDNP studies [25]. This new mechanism is ubiquitous, as it is largely independent of the particular magnetic interactions in the radical pairs, and thus opens a new pathway toward exploring the biological significance of CIDNP-induced spin order in natural (Earth's field) photosynthesis. In the following, we will (i) briefly explain the basics of CIDNP signal generation for the general reader and (ii) recapitulate the recent progress in understanding the fundamental quantum dynamics of RP reactions. Combining (i) and (ii), we will then explain the quantum measurement corrections to CIDNP. Before proceeding, we mention a classical analogue of the effect to be discussed, shown in figure 2. We consider an ensemble of coupled pendulums oscillating in the anti-symmetric mode. The ensemble average of the sum of their displacements is x₁ + x₂ = 0 at all times. However, if the 'environment' perturbs the system, e.g. with random kicks, while the ensemble population decays, we will observe a non-zero total displacement rising to a maximum and decaying to zero, following suit with the population decay. This displacement is the analogue of the radical-pair nuclear spin to be discussed in the following. The non-zero value of the nuclear spin stems from the particular 'environment' pertaining to radical pairs, S–T dephasing, which produces a nuclear spin imbalance along the reaction.

Zoom In Zoom Out Reset image size

Consider RPs with just one nucleus coupled to the electron of e.g. the donor. At time t = 0 the RPs start out in the electronic singlet state, while the nuclear spins are thermally polarized, which for protons at the Earth's field corresponds to a polarization of about 10⁻¹⁰, practically taken to equal zero. Thus, in the ensemble of RPs half will have the nuclear spin in the | + 〉 state, while in the other half the nuclear spin will initially be in the | − 〉 state. The local magnetic field seen by the donor electron will be the external field plus the nuclear magnetic field, while the acceptor electron will only experience the external magnetic field. It is this difference in local magnetic fields experienced by the two electrons that results in a difference in their Larmor frequencies, and hence induces the coherent S–T conversion. Consequently, in the first half radical pairs (with a | + 〉 nuclear spin) the Larmor precession frequency of the donor electron, and thus the S–T mixing frequency, will be slightly higher than in the other half (due to the opposite contribution of the nuclear spin); hence the coherently formed triplet radical pairs will have their nuclear spins predominantly in the | + 〉 state, leaving the singlet radical pairs with opposite nuclear polarization. While the nuclear spins are thus sorted among singlet and triplet RPs, and unless there is a coherent mechanism such as TSM producing net nuclear polarization, the expectation value 〈I_z〉 = Tr{ρI_z} of the z-component of the single nuclear spin under consideration is still zero. Here ρ is the spin density matrix of the radical pair, describing the electron and nuclear spin degrees of freedom altogether, and z is the direction of the external magnetic field.

According to the traditional understanding of the radical-pair mechanism and the concomitant CIDNP signal generation, for this nuclear spin sorting to result in a diamagnetic ground state with net nuclear polarization, and in the absence of any coherent mechanism such as TSM, the DD mechanism should be at play in order to observe non-zero CIDNP signals. In the following, we will indeed consider a magnetic Hamiltonian that does not exhibit the TSM mechanism, and we will also take k_S = k_T, so that the DD mechanism is not operational. Hence, according to the traditional theory of RP spin dynamics, no CIDNP signal should be expected. We are then going to show that the opposite is actually true.

2.1. Nuclear spin polarization under Hamiltonian evolution

To illustrate the above considerations, we consider a radical pair with one spin-1/2 nucleus isotropically coupled to the donor electron. In figure 3(a), we display the undisturbed spin state evolution brought about just by the magnetic Hamiltonian ${\cal H}=A\mathbf {I}\cdot \mathbf {s}_{1}+\omega (s_{1z}+s_{2z})$, where ω is the electron spin Larmor precession frequency in the external magnetic field defining the z-axis and A the donor-electron hyperfine coupling with the donor nuclear spin (in units of frequency). What is plotted in figure 3(a) is the spin state's singlet character, given by the expectation value of the singlet projection operator, 〈Q_S〉 = Tr{ρQ_S}, and assuming an RP with infinite lifetime (in other words, so far we take the recombination rates to be k_S = k_T = 0). What figure 3(a) displays are the S–T oscillations brought about by ${\cal H}$. The nuclear spin, I_z, of singlet and triplet RPs is given by 〈I_z〉^S = 〈Q_SI_zQ_S〉 and 〈I_z〉^T = 〈Q_TI_zQ_T〉, respectively, and displayed in figure 3(b). The previous considerations regarding nuclear spin sorting are easily visualized by noting in figure 3(b) that although 〈I_z〉^S ≠ 0 and 〈I_z〉^T ≠ 0, the nuclear spin oscillations in the singlet and triplet subspaces are exactly out of phase, so the net RP nuclear polarization 〈I_z〉 = 〈I_z〉^S + 〈I_z〉^T = 0. In case it is not obvious, we prove that 〈I_z〉 = 〈I_z〉^S + 〈I_z〉^T in appendix A. We can also show formally that for this particular Hamiltonian ${\cal H}$ it will be 〈I_z〉 = 0, as expected since ${\cal H}$ does not qualify for the TSM mechanism [23]. The proof can be found in appendix B.

Zoom In Zoom Out Reset image size

2.2. Prediction of the traditional theory

The counter-intuitive new result presented here is this: according to the new master equation describing the fundamental quantum dynamics of RP reactions [27], there actually is a non-zero RP nuclear spin polarization, depicted in figure 4. A similar result, different by just a factor of 2, follows from another variant of the quantum-measurement-based stochastic Liouville equation [31]. In this particular example, the nuclear polarization is about 10⁴ times higher than the thermal equilibrium proton polarization (left y-axis of figure 4) and roughly independent of ω as long as ω is relatively small (to be further discussed later). What is the source of this new kind of CIDNP signal? We will now go beyond the particular example considered above and provide the explanation for the general case of this new channel for photochemically enhancing nuclear polarizations at the Earth's field: the fundamental S–T decoherence underlying the radical-pair mechanism.

Zoom In Zoom Out Reset image size

To visualize this mechanism, suppose that at time t the single-nuclear-spin RP ensemble is described by the density matrix ρ. Assuming a magnetic Hamiltonian not supporting any coherent generation of net RP nuclear polarization, as in the example considered previously, it will be 〈I_z〉 = Tr{ρI_z} = 0. This, as already mentioned, implies that 〈I_z〉^S = −〈I_z〉^T. The idea behind the new kind of CIDNP signal generation is the following. The intramolecule S–T decoherence is equivalent to some RPs being projected to either the singlet or the triplet RP manifold of states, exactly due to the continuous quantum measurement of Q_S induced by the RP recombination dynamics, the measurement rate being (k_S + k_T)/2 [26]. The result of this measurement is either Q_S = 1, i.e. the RP is projected to the singlet manifold, or Q_S = 0, i.e. the RP is projected to the triplet manifold. The probability for a singlet or a triplet projection is 〈Q_S〉 or 〈Q_T〉, respectively. Hence, at time t + dt the state of the radical pairs will be (ignoring the RPs that recombine into neutral products during dt) a new mixture comprised of (a) radical pairs in the state ρ + dρ, with $\mathrm {d}\rho =-\mathrm {i}\,\mathrm {d}t[{\cal H},\rho ]$, plus (b) the singlet-projected radical pairs described by Q_SρQ_S, plus (c) triplet-projected radical pairs described by Q_TρQ_T. The nuclear polarization of states (a) will still be zero, since again, the Hamiltonian ${\cal H}$ does not generate net nuclear polarization. The combined nuclear polarization of projected states (b) and (c), denoted by 〈I_z〉^proj, will be the weighted sum of singlet RP nuclear polarization, 〈I_z〉^S, and triplet RP nuclear polarization, 〈I_z〉^T, the sum being weighted by the respective projection probabilities, i.e.

$$\begin{equation} \langle I_{z}\rangle^{\rm proj}=\langle Q_{\rm S}\rangle\langle I_{z}\rangle^{\mathrm{S}}+\langle Q_{\rm T}\rangle\langle I_{z}\rangle^{\mathrm{T}}. \end{equation} \tag{ 1 }$$

Taking into account that 〈I_z〉^T = −〈I_z〉^S and using the completeness relation 〈Q_T〉 = 1 − 〈Q_S〉, it follows that

$$\begin{equation} \langle I_{z}\rangle^{\rm proj}=\langle I_{z}\rangle^{\mathrm{S}}\Big(2\langle Q_{\rm S}\rangle-1\Big). \end{equation} \tag{ 2 }$$

Thus, if 〈Q_S〉 ≠ 〈Q_T〉 ≠ 1/2, more (when 〈Q_S〉 > 1/2) or less (when 〈Q_S〉 < 1/2) projections will take place to the singlet state than to the triplet, and 〈I_z〉^proj_t will be non-zero, if of course 〈I_z〉^S is non-zero.

To recapitulate the physics of this new effect, the S–T dephasing inherent in the radical-pair quantum dynamic evolution randomly projects radical pairs into either the singlet or the triplet subspace. The projection probability at time t depends on the singlet (or triplet) character of the electronic spin state; hence the projections will be asymmetric if 〈Q_S〉 ≠ 1/2. Nuclear spin sorting, i.e. the fact that 〈I_z〉^S = −〈I_z〉^T ≠ 0, combined with the asymmetric projections produces a non-zero nuclear spin expectation value.

We can now theoretically estimate the quantum measurement corrections to CIDNP, to be denoted by 〈I_z〉_qc, shown in figure 4. Letting k = k_S = k_T, the rate of singlet or triplet projections (the 'quantum measurement' rate) is just (k_S + k_T)/2 = k. We will focus on low magnetic fields and assume, as is the case most often, that k > A,ω. Toward the estimate, consider the following three steps. (i) As the RPs start out in the singlet state, and the S–T mixing frequency Ω ≪ k, in the short reaction time of 1/k ≪ 1/Ω the singlet character will not have enough time to change appreciably, so it will be 〈Q_S〉 ≈ 1. This means that the RPs will be projected predominantly to the singlet state. (ii) The electron spin of the donor molecule, which contains the one and only nuclear spin of the RP under consideration, will precess at frequency |ω ± A| when the nuclear spin is in the | ± 〉 state. So at time t, the number of singlet RPs with the nuclear spin in the | ± 〉 state will be N_± ≈ cos[(ω ± A)t]². The nuclear polarization of singlet RPs at time t = 1/k will thus be 〈I_z〉^S = (N₊ − N₋)/(N₊ + N₋), easily seen to be

$$\begin{equation} \langle I_{z}\rangle^{\mathrm{S}}\approx -{{\omega A}\over k^{2}}. \end{equation} \tag{ 3 }$$

(iii) S–T projections taking place at the rate k will be a source term for d〈I_z〉_qc/dt. The source term, proportional to 〈I_z〉^proj, starts at t = 0 from zero and oscillates at a rate Ω while it is drained at a rate k; thus d〈I_z〉_qc/dt = k〈I_z〉^proj e^−ktsin Ωt, yielding at time t ≈ 1/k that 〈I_z〉_qc ≈ −〈I_z〉^projΩ²/k². Putting everything together and using equation (2), we finally arrive at

$$\begin{equation} \langle I_{z}\rangle_{\rm qc}\approx {{\omega\Omega^{2} A}\over k^{4}}. \end{equation} \tag{ 4 }$$

For the particular example depicted in figure 3 it is Ω ≈ A/10; hence according to the estimate of equation (4), it is 〈I_z〉_qc ≈ 4 × 10⁻⁶, in good agreement with the exact value shown in the right y-axis of figure 4. Furthermore, the result of equation (4) can be recast in terms of the thermal polarization I_th = ℏω/4γk_BT (the expression for I_th is valid for ℏω ≪ γk_BT), where ω = γ_eB is the electron Larmor frequency in the external magnetic field B, with γ_e = 2π × 2.8 MHz G⁻¹ being the electron gyromagnetic ratio and γ = |μ_e/μ_p| = 658.5 being the ratio of electron-to-proton magnetic moment. The enhancement factor 〈I_z〉_qc/I_th is thus seen to be independent of ω and equal to

$$\begin{equation} {{\langle I_{z}\rangle_{\rm qc}}\over {I_{\rm th}}}\approx 10^{3} {{\Big(\Omega/0.01\,{\rm ns}^{-1}\Big)^2\Big(A/0.1\,{\rm ns}^{-1}\Big)}\over {\Big(k/1\,{\rm ns}^{-1}\Big)^{4}}}. \end{equation} \tag{ 5 }$$

For ease of use, in the previous expression we provided the enhancement factor in terms of the S–T mixing frequency Ω (in units of 0.01 ns⁻¹), the hyperfine coupling A (in units of 0.1 ns⁻¹) and the recombination rate k (in units of 1 ns⁻¹). For typical hyperfine couplings of about 10 G and recombination times of the order of 1 ns, it is seen that the quantum measurement corrections to chemically induced dynamic nuclear polarization amount to a significant enhancement of at least four orders of magnitude over the thermal polarization.

We note for the reader who would worry about angular momentum conservation that the RP's nuclear spin polarization is exactly balanced by an opposite and equal electronic spin polarization, i.e. the sum 〈s_1z + s_2z + I_z〉 = 0 at all times, as it should be, since it is zero at time t = 0 (the initial state is ρ₀ = Q_S/Tr{Q_S}, which is a singlet state with zero nuclear polarization). The non-zero electronic spin polarization necessarily involves the admixture of the |T₊〉 = |↑↑〉 and |T₋〉 = |↓↓〉 triplet states; therefore, this new CIDNP mechanism is a low-magnetic-field effect, because at high fields the |T_±〉 states are out of reach from the singlet state $|S\rangle =(| \! \uparrow \downarrow \rangle -| \! \downarrow \uparrow \rangle )/\sqrt {2}$. How low a magnetic field depends on the recombination rate k, since the triplet RP decay broadens the triplet energy levels by about k; hence the mechanism is appreciable for magnetic fields B such that B < k/γ_e, i.e. when the Zeeman energy separation of |T_±〉 is within the broadening k. For k ≈ 1 ns⁻¹, the effect is appreciable for fields B ⩽ 50 G.

In closing, the following comments should be made.

• 1.  
  We stress that the new CIDNP signal is a fundamental phenomenon attributed to the inherent quantum dynamics of RPs.
• 2.  
  Kaptein's rules [32] provide a compact relation of the main features of the CIDNP signal to the system's parameters. For this case, the sign of the effect, i.e. whether it is emissive (+) or absorptive (−), is the same as the sign of the hyperfine coupling A.
• 3.  
  The new CIDNP effect reported here does not require any particular combination or fine-tuning of magnetic interactions, such as the coherent TSM low-field effect analyzed in [25], but is rather general and shows up whenever there is a non-zero nuclear spin polarization of singlet RPs, 〈I_z〉^S, as in figure 3(b). The rather general requirement for this to happen is that the oscillation frequencies ω_m − ω_n of the non-zero matrix elements [Q_S]_mn = 〈m|Q_S|n〉 must have terms linear in the hyperfine couplings. Here |n〉 and ω_n are the eigenvectors and eigenvalues of ${\cal H}$, with n = 1,...,4M, where M is the nuclear spin multiplicity of the radical pair.
• 4.  
  Regarding experimental detection of the new CIDNP effect, we note the following. While NMR measurements at low fields suffer loss of sensitivity, recent methods of ultra-sensitive magnetometry using superconducting quantum interference devices (SQUIDs) [33, 34] or atomic magnetometers [35, 36] have demonstrated the ability to sensitively detect NMR signals at near-zero magnetic fields. To provide a rough estimate of the magnetic field B_n produced by the polarized nuclear spins, we note that in the solid phase the concentration of RCs can be as large as 1 mM. Assuming just a single proton spin per RC (which will heavily underestimate the magnitude of the effect), choosing a magnetic field of 1 G and taking into account that the considered enhancement will boost the thermal nuclear polarization by a factor of 10⁴, we find that the magnetic field just outside a spherical volume will be B_n ≈ 10⁴P_thμ_pμ₀[RC], where [RC] is the RC concentration. At 1 G the thermal polarization is P_th ≈ 10⁻¹⁰; hence B_n ≈ 20 fT, already 200 times higher than what state-of-the-art atomic magnetometers can do [37]. In fact, this seems to be yet another exciting application of ultra-sensitive magnetometry in the biological realm [38, 39].
• 5.  
  Although we considered the special case k_S = k_T, the new kind of CIDNP signal we predict will ubiquitously affect all sorts of radical pairs, including of course the ones with asymmetric recombination rates k_S ≠ k_T, which predominantly occur in photosynthetic RCs [22]. We have chosen to deal with the special case k_S = k_T and a magnetic Hamiltonian unable to support coherent generation of CIDNP signals in order to be able to focus exclusively on this fundamentally new CIDNP process. Establishing how this mechanism attributed to S–T decoherence affects, qualitatively and quantitatively, the cases where there exists a CIDNP signal according to the mechanism's traditional understanding (e.g. TSM, DD or DR scenarios) is a more involved exercise that will be undertaken elsewhere.
• 6.  
  We have considered the CIDNP signal in the RP state, and not in the diamagnetic ground state of the molecule in a closed reaction cycle as the one shown in figure 1(b). In other words, in figure 4 we depicted the transient nuclear polarization of the RPs in between their creation at time t = 0 and the time their population has decayed to zero due to charge recombination. However, it is easily seen that the same results hold true for the diamagnetic ground state in the steady state under continuous excitation (i.e. continuous illumination) and a closed reaction cycle, as long as the intersystem crossing rate k_isc (see figure 1(b)) is fast enough.

In conclusion, this is the first step of a very promising research program, namely the particular repercussions of the recently unraveled fundamental quantum dynamics of the radical-pair mechanism for understanding the dynamics of photosynthetic RCs. We have extended the fundamental understanding of CIDNP by pointing to a new and qualitatively different way of obtaining enhanced nuclear spin polarization. Interestingly, this polarization naturally turns out to be significant at low magnetic fields pertinent to natural photosynthesis. It stems from the random singlet or triplet projections naturally taking place during the quantum evolution of radical pairs and leading to spin decoherence. Thus, an intriguing point raised is if and how the quantum mechanical process of spin decoherence affects the biological workings of photosynthetic RCs. In other words, what is the biological significance of Earth-field CIDNP produced by S–T decoherence?

I warmly acknowledge helpful comments from Professors Gunnar Jeschke and Ulrich Steiner, and several comments clarifying CIDNP-related issues from Professor Jörg Matysik.

Since I_z commutes with the electron spins s₁ and s₂, it also commutes with Q_S = 1/4 − s₁·s₂ and with Q_T = 1 − Q_S. Thus, since the two projection operators are orthogonal, i.e. Q_SQ_T = Q_TQ_S = 0, it will be Q_SI_zQ_T = Q_TI_zQ_S = 0. Furthermore, since Q_S + Q_T = 1, it will be Tr{ρI_z} = Tr{ρ(Q_S + Q_T )I_z(Q_S + Q_T )} and by using the previous relations it follows that indeed 〈I_z〉 = 〈I_z〉^S + 〈I_z〉^T.

Since I_z commutes with s₁ and [I_z,I_x] = iI_y,[I_z,I_y] = −iI_x, it follows that $[I_{z},{\cal H}]=\mathrm {i}A(s_{1x}I_{y}-s_{1y}I_{x})$. Then, Heisenberg's equation of motion $\mathrm {d}I_{z}/\mathrm {d}t=\mathrm {i}[I_{z},{\cal H}]$ leads to

$$\begin{equation} \mathrm{d}\langle I_{z}\rangle/\mathrm{d}t=-A\langle s_{1x}I_{y}-s_{1y}I_{x}\rangle. \end{equation} \tag{ B.1 }$$

The most general pure spin state of the RP can be written as $|\psi \rangle =\sum _{i=S,T_{0},T_{++},T_{--}}|i\rangle |\chi _{i}\rangle$, where |i〉 are the four electron-spin S–T basis states and |χ_i〉 is a nuclear spin state. It is straightforward to show that 〈ψ|s_1xI_y − s_1yI_x|ψ〉 = 0; therefore d〈I_z〉/dt = 0 at all times. The same holds true for any mixture of pure states. Since at time t = 0 we have an RP mixture with zero nuclear polarization, at all times it will be $\langle I_{z}\rangle =\langle [{\cal H},I_{z}]\rangle =0$.

The traditional master equation is $\mathrm {d}\rho /\mathrm {d}t=-\mathrm {i}[{\cal H},\rho ]-k_{\mathrm { S}}(Q_{\mathrm { S}}\rho +\rho Q_{\mathrm { S}})/2-k_{\mathrm { T}}(Q_{\mathrm { T}}\rho +\rho Q_{\mathrm { T}})/2$. For the case under study k_S = k_T = k, and taking into account that Q_S + Q_T = 1, the traditional master equation becomes $\mathrm {d}\rho /\mathrm {d}t=-\mathrm {i}[{\cal H},\rho ]-k\rho$. After multiplying by I_z and taking the trace, it follows that according to the traditional theory it will be $\mathrm {d}\langle I_{z}\rangle /\mathrm {d}t=-\mathrm {i}\,{\mathrm { Tr}}\{[{\cal H},\rho ] I_{z}\}-k\langle I_{z}\rangle$. But ${\mathrm { Tr}}\{[{\cal H},\rho ] I_{z}\}=-{\mathrm { Tr}}\{[{\cal H}, I_{z}]\rho \}=-\langle [{\cal H},I_{z}]\rangle$, which was shown to equal zero in appendix B. Therefore, d〈I_z〉/dt = −k〈I_z〉. Since at t = 0 it is 〈I_z〉 = 0, the same is true at all times.